The Project Gutenberg eBook #32625: A treatise on probability
Author: John Maynard Keynes
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Title: A Treatise of Probability

Author: John Maynard Keynes

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BY THE SAME AUTHOR

INDIAN CURRENCY AND FINANCE

8vo. Pp. viii + 263. 1913.

7s. 6d. net.

THE ECONOMIC CONSEQUENCES

OF THE PEACE

8vo. Pp. vii + 279. 1919.

8s. 6d. net.

A TREATISE ON PROBABILITY

MACMILLAN AND CO., Limited

LONDON · BOMBAY · CALCUTTA · MADRAS

MELBOURNE

THE MACMILLAN COMPANY

NEW YORK · BOSTON · CHICAGO

DALLAS · SAN FRANCISCO

THE MACMILLAN CO. OF CANADA, Ltd.

TORONTO

A TREATISE

ON PROBABILITY

BY

JOHN MAYNARD KEYNES

FELLOW OF KING’S COLLEGE, CAMBRIDGE

MACMILLAN AND CO., LIMITED

ST. MARTIN’S STREET, LONDON



PREFACE

The subject matter of this book was first broached in the brain
of Leibniz, who, in the dissertation, written in his twenty-third year,
on the mode of electing the kings of Poland, conceived of Probability
as a branch of Logic. A few years before, “un problème,” in the
words of Poisson, “proposé à un austère janséniste par un homme du
monde, a été l’origine du calcul des probabilitiés.” In the intervening
centuries the algebraical exercises, in which the Chevalier de la Méré
interested Pascal, have so far predominated in the learned world over
the profounder enquiries of the philosopher into those processes of
human faculty which, by determining reasonable preference, guide our
choice, that Probability is oftener reckoned with Mathematics than
with Logic. There is much here, therefore, which is novel and, being
novel, unsifted, inaccurate, or deficient. I propound my systematic
conception of this subject for criticism and enlargement at the hand
of others, doubtful whether I myself am likely to get much further,
by waiting longer, with a work, which, beginning as a Fellowship

Dissertation, and interrupted by the war, has already extended over
many years.

It may be perceived that I have been much influenced by W. E.

Johnson, G. E. Moore, and Bertrand Russell, that is to say by

Cambridge, which, with great debts to the writers of Continental

Europe, yet continues in direct succession the English tradition of

Locke and Berkeley and Hume, of Mill and Sidgwick, who, in spite of
their divergences of doctrine, are united in a preference for what is
matter of fact, and have conceived their subject as a branch rather of
science than of the creative imagination, prose writers, hoping to be
understood.

J. M. KEYNES.

King’s College, Cambridge,

May 1, 1920.

v

CONTENTS

PART I
fundamental ideas

CHAPTER I
page

The Meaning of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

CHAPTER II

Probability in Relation to the Theory of Knowledge

9

CHAPTER III

The Measurement of Probabilities . . . . . . . . . . . . . . . . . . . . . .

20

CHAPTER IV

The Principle of Indifference . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

CHAPTER V

Other Methods of Determining Probabilities . . . . . . . . .

71
vi

CONTENTS
vii

CHAPTER VI
page

The Weight of Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

CHAPTER VII

Historical Retrospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

CHAPTER VIII

The Frequency Theory of Probability . . . . . . . . . . . . . . . . .

102

CHAPTER IX

The Constructive Theory of Part I. Summarized . . . . . . 123

PART II
fundamental theorems

CHAPTER X

Introductory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

CHAPTER XI

The Theory of Groups, with special reference to

Logical Consistence, Inference, and Logical Pri-
ority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

CHAPTER XII

The Definitions and Axioms of Inference and Proba-
bility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

A TREATISE ON PROBABILITY
viii

CHAPTER XIII
page

The Fundamental Theorems of Necessary Inference . 153

CHAPTER XIV

The Fundamental Theorems of Probable Inference . . 161

CHAPTER XV

Numerical Measurement and Approximation of Prob-
abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

CHAPTER XVI

Observations on the Theorems of Chapter XIV. and
their Developments, including Testimony . . . . . . . . . 188

CHAPTER XVII

Some Problems in Inverse Probability,
including

Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

PART III
induction and analogy

CHAPTER XVIII

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

CHAPTER XIX

The Nature of Argument by Analogy . . . . . . . . . . . . . . . . . .

256

CONTENTS
ix

CHAPTER XX
page

The Value of Multiplication of Instances, or Pure

Induction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

CHAPTER XXI

The Nature of Inductive Argument Continued . . . . . . . . 279

CHAPTER XXII

The Justification of these Methods . . . . . . . . . . . . . . . . . . . . . 289

CHAPTER XXIII

Some Historical Notes on Induction . . . . . . . . . . . . . . . . . . . . 305

Notes on Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

PART IV
some philosophical applications of probability

CHAPTER XXIV

The Meanings of Objective Chance, and of Random-
ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

CHAPTER XXV

Some Problems arising out of the Discussion of

Chance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335

A TREATISE ON PROBABILITY
x

CHAPTER XXVI
page

The Application of Probability to Conduct . . . . . . . . . . . 351

PART V
the foundations of statistical inference

CHAPTER XXVII

The Nature of Statistical Inference . . . . . . . . . . . . . . . . . . . . 371

CHAPTER XXVIII

The Law of Great Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

CHAPTER XXIX

The Use of à priori Probabilities for the Predic-
tion of Statistical Frequency—the Theorems of

Bernoulli, Poisson, and Tchebycheff . . . . . . . . . . . . . . . 382

CHAPTER XXX

The Mathematical use of Statistical Frequencies for
the Determination of Probability à posteriori —the

Methods of Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

CHAPTER XXXI

The Inversion of Bernoulli’s Theorem . . . . . . . . . . . . . . . . . . 438

CONTENTS
xi

CHAPTER XXXII
page

The Inductive use of Statistical Frequencies for
the Determination of Probability à posteriori —the

Methods of Lexis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

CHAPTER XXXIII

Outline of a Constructive Theory . . . . . . . . . . . . . . . . . . . . . . 464

PART I
fundamental ideas

CHAPTER I
the meaning of probability

“J’ai dit plus d’une fois qu’il faudrait une nouvelle espèce de logique, qui
traiteroit des degrés de Probabilité.”—Leibniz.

1.

Part of our knowledge we obtain direct; and part by
argument. The Theory of Probability is concerned with that part
which we obtain by argument, and it treats of the different degrees in
which the results so obtained are conclusive or inconclusive.

In most branches of academic logic, such as the theory of the
syllogism or the geometry of ideal space, all the arguments aim at
demonstrative certainty. They claim to be conclusive. But many other
arguments are rational and claim some weight without pretending to
be certain. In Metaphysics, in Science, and in Conduct, most of the
arguments, upon which we habitually base our rational beliefs, are
admitted to be inconclusive in a greater or less degree. Thus for a
philosophical treatment of these branches of knowledge, the study of
probability is required.

The course which the history of thought has led Logic to follow
has encouraged the view that doubtful arguments are not within
its scope. But in the actual exercise of reason we do not wait on
certainty, or doom it irrational to depend on a doubtful argument. If
logic investigates the general principles of valid thought, the study of
arguments, to which it is rational to attach some weight, is as much
a part of it as the study of those which are demonstrative.

2.

The terms certain and probable describe the various degrees
of rational belief about a proposition which different amounts of
knowledge authorise us to entertain. All propositions are true or false,
but the knowledge we have of them depends on our circumstances;
and while it is often convenient to speak of propositions as certain or
probable, this expresses strictly a relationship in which they stand to
a corpus of knowledge, actual or hypothetical, and not a characteristic

2
ch. i

FUNDAMENTAL IDEAS

3
of the propositions in themselves. A proposition is capable at the
same time of varying degrees of this relationship, depending upon the
knowledge to which it is related, so that it is without significance to
call a proposition probable unless we specify the knowledge to which
we are relating it.

To this extent, therefore, probability may be called subjective.

But in the sense important to logic, probability is not subjective. It
is not, that is to say, subject to human caprice. A proposition is not
probable because we think it so. When once the facts are given which
determine our knowledge, what is probable or improbable in these
circumstances has been fixed objectively, and is independent of our
opinion. The Theory of Probability is logical, therefore, because it is
concerned with the degree of belief which it is rational to entertain in
given conditions, and not merely with the actual beliefs of particular
individuals, which may or may not be rational.

Given the body of direct knowledge which constitutes our ultimate
premisses, this theory tells us what further rational beliefs, certain or
probable, can be derived by valid argument from our direct knowledge.

This involves purely logical relations between the propositions which
embody our direct knowledge and the propositions about which we
seek indirect knowledge. What particular propositions we select as
the premisses of our argument naturally depends on subjective factors
peculiar to ourselves; but the relations, in which other propositions
stand to these, and which entitle us to probable beliefs, are objective
and logical.

3.

Let our premisses consist of any set of propositions h, and our
conclusion consist of any set of propositions a, then, if a knowledge
of h justifies a rational belief in a of degree α, we say that there is a
probability-relation of degree α between a and h.∗

In ordinary speech we often describe the conclusion as being
doubtful, uncertain, or only probable. But, strictly, these terms

∗This will be written a/h = α.

pt. i

A TREATISE ON PROBABILITY

4
ought to be applied, either to the degree of our rational belief in
the conclusion, or to the relation or argument between two sets
of propositions, knowledge of which would afford grounds for a
corresponding degree of rational belief.∗

4.

With the term “event,” which has taken hitherto so important
a place in the phraseology of the subject, I shall dispense altogether.†

Writers on Probability have generally dealt with what they term the

“happening” of “events.” In the problems which they first studied
this did not involve much departure from common usage. But these
expressions are now used in a way which is vague and ambiguous;
and it will be more than a verbal improvement to discuss the truth
and the probability of propositions instead of the occurrence and the
probability of events.‡

5.

These general ideas are not likely to provoke much criticism.

In the ordinary course of thought and argument, we are constantly
assuming that knowledge of one statement, while not proving the
truth of a second, yields nevertheless some ground for believing it.

We assert that we ought on the evidence to prefer such and such
a belief. We claim rational grounds for assertions which are not
conclusively demonstrated. We allow, in fact, that statements may
be unproved, without, for that reason, being unfounded. And it
does not seem on reflection that the information we convey by these
expressions is wholly subjective. When we argue that Darwin gives
valid grounds for our accepting his theory of natural selection, we do
not simply mean that we are psychologically inclined to agree with

∗See also Chapter II. § 5.

†Except in those chapters (Chap. XVII., for example) where I am dealing
chiefly with the work of others.

‡The first writer I know of to notice this was Ancillon in Doutes sur les
bases du calcul des probabilités (1794):

“Dire qu’un fait passé, présent ou à
venir est probable, c’est dire qu’une proposition est probable.” The point was
emphasised by Boole, Laws of Thought, pp. 7 and 167.

See also Czuber,

Wahrscheinlichkeitsrechnung, vol. i. p. 5, and Stumpf, Über den Begriff der
mathematischen Wahrscheinlichkeit.

ch. i

FUNDAMENTAL IDEAS

5
him; it is certain that we also intend to convey our belief that we are
acting rationally in regarding his theory as probable. We believe that
there is some real objective relation between Darwin’s evidence and
his conclusions, which is independent of the mere fact of our belief,
and which is just as real and objective, though of a different degree,
as that which would exist if the argument were as demonstrative as
a syllogism. We are claiming, in fact, to cognise correctly a logical
connection between one set of propositions which we call our evidence
and which we suppose ourselves to know, and another set which we
call our conclusions, and to which we attach more or less weight
according to the grounds supplied by the first. It is this type of
objective relation between sets of propositions—the type which we
claim to be correctly perceiving when we make such assertions as
these—to which the reader’s attention must be directed.

6.

It is not straining the use of words to speak of this as the
relation of probability. It is true that mathematicians have employed
the term in a narrower sense; for they have often confined it to
the limited class of instances in which the relation is adapted to an
algebraical treatment. But in common usage the word has never
received this limitation.

Students of probability in the sense which is meant by the authors
of typical treatises on Wahrscheinlichkeitsrechnung or Calcul des
probabilités, will find that I do eventually reach topics with which
they are familiar. But in making a serious attempt to deal with
the fundamental difficulties with which all students of mathematical
probabilities have met and which are notoriously unsolved, we must
begin at the beginning (or almost at the beginning) and treat our
subject widely. As soon as mathematical probability ceases to be
the merest algebra or pretends to guide our decisions, it immediately
meets with problems against which its own weapons are quite
powerless. And even if we wish later on to use probability in a narrow
sense, it will be well to know first what it means in the widest.

7.

Between two sets of propositions, therefore, there exists
pt. i

A TREATISE ON PROBABILITY

6
a relation, in virtue of which, if we know the first, we can attach
to the latter some degree of rational belief. This relation is the
subject-matter of the logic of probability.

A great deal of confusion and error has arisen out of a failure to
take due account of this relational aspect of probability. From the
premisses “a implies b” and “a is true,” we can conclude something
about b—namely that b is true—which does not involve a. But, if a
is so related to b, that a knowledge of it renders a probable belief in b
rational, we cannot conclude anything whatever about b which has
not reference to a; and it is not true that every set of self-consistent
premisses which includes a has this same relation to b. It is as useless,
therefore, to say “b is probable” as it would be to say “b is equal,”

or “b is greater than,” and as unwarranted to conclude that, because
a makes b probable, therefore a and c together make b probable, as
to argue that because a is less than b, therefore a and c together are
less than b.

Thus, when in ordinary speech we name some opinion as probable
without further qualification, the phrase is generally elliptical. We
mean that it is probable when certain considerations, implicitly or
explicitly present to our minds at the moment, are taken into account.

We use the word for the sake of shortness, just as we speak of a place
as being three miles distant, when we mean three miles distant from
where we are then situated, or from some starting-point to which we
tacitly refer. No proposition is in itself either probable or improbable,
just as no place can be intrinsically distant; and the probability of
the same statement varies with the evidence presented, which is, as
it were, its origin of reference. We may fix our attention on our own
knowledge and, treating this as our origin, consider the probabilities
of all other suppositions,—according to the usual practice which leads
to the elliptical form of common speech; or we may, equally well, fix it
on a proposed conclusion and consider what degree of probability this
would derive from various sets of assumptions, which might constitute
the corpus of knowledge of ourselves or others, or which are merely
ch. i

FUNDAMENTAL IDEAS

7
hypotheses.

Reflection will show that this account harmonises with familiar
experience.

There is nothing novel in the supposition that the
probability of a theory turns upon the evidence by which it is
supported; and it is common to assert that an opinion was probable
on the evidence at first to hand, but on further information was
untenable.

As our knowledge or our hypothesis changes, our
conclusions have new probabilities, not in themselves, but relatively
to these new premisses. New logical relations have now become
important, namely those between the conclusions which we are
investigating and our new assumptions; but the old relations between
the conclusions and the former assumptions still exist and are just
as real as these new ones. It would be as absurd to deny that an
opinion was probable, when at a later stage certain objections have
come to light, as to deny, when we have reached our destination, that
it was ever three miles distant; and the opinion still is probable in
relation to the old hypotheses, just as the destination is still three
miles distant from our starting-point.

8.

A definition of probability is not possible, unless it contents us
to define degrees of the probability-relation by reference to degrees of
rational belief. We cannot analyse the probability-relation in terms of
simpler ideas. As soon as we have passed from the logic of implication
and the categories of truth and falsehood to the logic of probability
and the categories of knowledge, ignorance, and rational belief, we
are paying attention to a new logical relation in which, although it
is logical, we were not previously interested, and which cannot be
explained or defined in terms of our previous notions.

This opinion is, from the nature of the case, incapable of positive
proof. The presumption in its favour must arise partly out of our
failure to find a definition, and partly because the notion presents
itself to the mind as something new and independent. If the statement
that an opinion was probable on the evidence at first to hand, but
became untenable on further information, is not solely concerned with
pt. i

A TREATISE ON PROBABILITY

8
psychological belief, I do not know how the element of logical doubt
is to be defined, or how its substance is to be stated, in terms of the
other indefinables of formal logic. The attempts at definition, which
have been made hitherto, will be criticised in later chapters. I do not
believe that any of them accurately represent that particular logical
relation which we have in our minds when we speak of the probability
of an argument.

In the great majority of cases the term “probable” seems to be
used consistently by different persons to describe the same concept.

Differences of opinion have not been due, I think, to a radical
ambiguity of language. In any case a desire to reduce the indefinables
of logic can easily be carried too far.

Even if a definition is
discoverable in the end, there is no harm in postponing it until our
enquiry into the object of definition is far advanced. In the case of

“probability” the object before the mind is so familiar that the danger
of misdescribing its qualities through lack of a definition is less than if
it were a highly abstract entity far removed from the normal channels
of thought.

9.

This chapter has served briefly to indicate, though not
to define, the subject matter of the book. Its object has been
to emphasise the existence of a logical relation between two sets of
propositions in cases where it is not possible to argue demonstratively
from one to the other. This is a contention of a most fundamental
character. It is not entirely novel, but has seldom received due
emphasis, is often overlooked, and sometimes denied. The view, that
probability arises out of the existence of a specific relation between
premiss and conclusion, depends for its acceptance upon a reflective
judgment on the true character of the concept. It will be our object
to discuss, under the title of Probability, the principal properties of
this relation. First, however, we must digress in order to consider
briefly what we mean by knowledge, rational belief, and argument.

CHAPTER II
probability in relation to the theory of
knowledge

1.

I do not wish to become involved in questions of epistemology
to which I do not know the answer; and I am anxious to reach as
soon as possible the particular part of philosophy or logic which is the
subject of this book. But some explanation is necessary if the reader
is to be put in a position to understand the point of view from which
the author sets out; I will, therefore, expand some part of what has
been outlined or assumed in the first chapter.

2.

There is, first of all, the distinction between that part of
our belief which is rational and that part which is not. If a man
believes something for a reason which is preposterous or for no reason
at all, and what he believes turns out to be true for some reason not
known to him, he cannot be said to believe it rationally, although he
believes it and it is in fact true. On the other hand, a man may
rationally believe a proposition to be probable, when it is in fact false.

The distinction between rational belief and mere belief, therefore, is
not the same as the distinction between true beliefs and false beliefs.

The highest degree of rational belief, which is termed certain rational
belief, corresponds to knowledge. We may be said to know a thing
when we have a certain rational belief in it, and vice versa. For
reasons which will appear from our account of probable degrees of
rational belief in the following paragraph, it is preferable to regard
knowledge as fundamental and to define rational belief by reference
to it.

3.

We come next to the distinction between that part of our
rational belief which is certain and that part which is only probable.

Belief, whether rational or not, is capable of degree. The highest
degree of rational belief, or rational certainty of belief, and its relation
to knowledge have been introduced above. What, however, is the

9
pt. i

A TREATISE ON PROBABILITY

10
relation to knowledge of probable degrees of rational belief?

The proposition (say, q) that we know in this case is not the
same as the proposition (say, p) in which we have a probable degree

(say, α) of rational belief. If the evidence upon which we base our
belief is h, then what we know, namely q, is that the proposition p
bears the probability-relation of degree α to the set of propositions h;
and this knowledge of ours justifies us in a rational belief of degree α
in the proposition p. It will be convenient to call propositions such
as p, which do not contain assertions about probability-relations,

“primary propositions”; and propositions such as q, which assert the
existence of a probability-relation, “secondary propositions.”∗

4.

Thus knowledge of a proposition always corresponds to
certainty of rational belief in it and at the same time to actual truth
in the proposition itself. We cannot know a proposition unless it is
in fact true. A probable degree of rational belief in a proposition,
on the other hand, arises out of knowledge of some corresponding
secondary proposition. A man may rationally believe a proposition to
be probable when it is in fact false, if the secondary proposition on
which he depends is true and certain; while a man cannot rationally
believe a proposition to be probable even when it is in fact true, if
the secondary proposition on which he depends is not true. Thus
rational belief of whatever degree can only arise out of knowledge,
although the knowledge may be of a proposition secondary, in the
above sense, to the proposition in which the rational degree of belief
is entertained.

5.

At this point it is desirable to colligate the three senses in
which the term probability has been so far employed. In its most
fundamental sense, I think, it refers to the logical relation between
two sets of propositions, which in § 4 of Chapter I. I have termed the
probability-relation. It is with this that I shall be mainly concerned
in the greater part of this Treatise. Derivative from this sense, we

∗This classification of “primary” and “secondary” propositions was suggested
to me by Mr. W. E. Johnson.

ch. ii

FUNDAMENTAL IDEAS

11
have the sense in which, as above, the term probable is applied to
the degrees of rational belief arising out of knowledge of secondary
propositions which assert the existence of probability-relations in the
fundamental logical sense. Further it is often convenient, and not
necessarily misleading, to apply the term probable to the proposition
which is the object of the probable degree of rational belief, and
which bears the probability-relation in question to the propositions
comprising the evidence.

6.

I turn now to the distinction between direct and indirect
knowledge—between that part of our rational belief which we know
directly and that part which we know by argument.

We start from things, of various classes, with which we have,
what I choose to call without reference to other uses of this term,
direct acquaintance. Acquaintance with such things does not in itself
constitute knowledge, although knowledge arises out of acquaintance
with them. The most important classes of things with which we
have direct acquaintance are our own sensations, which we may be
said to experience, the ideas or meanings, about which we have
thoughts and which we may be said to understand, and facts or
characteristics or relations of sense-data or meanings, which we may
be said to perceive;—experience, understanding, and perception being
three forms of direct acquaintance.

The objects of knowledge and belief—as opposed to the objects
of direct acquaintance which I term sensations, meanings, and
perceptions—I shall term propositions.

Now our knowledge of propositions seems to be obtained in
two ways: directly, as the result of contemplating the objects
of acquaintance; and indirectly, by argument, through perceiving
the probability-relation of the proposition, about which we seek
knowledge, to other propositions. In the second case, at any rate
at first, what we know is not the proposition itself but a secondary
proposition involving it. When we know a secondary proposition
involving the proposition p as subject, we may be said to have indirect
pt. i

A TREATISE ON PROBABILITY

12
knowledge about p.

Indirect knowledge about p may in suitable conditions lead to
rational belief in p of an appropriate degree. If this degree is that of
certainty, then we have not merely indirect knowledge about p, but
indirect knowledge of p.

7.

Let us take examples of direct knowledge. From acquaintance
with a sensation of yellow I can pass directly to a knowledge of the
proposition “I have a sensation of yellow.” From acquaintance with
a sensation of yellow and with the meanings of “yellow,” “colour,”

“existence,” I may be able to pass to a direct knowledge of the
propositions “I understand the meaning of yellow,” “my sensation of
yellow exists,” “yellow is a colour.” Thus, by some mental process
of which it is difficult to give an account, we are able to pass from
direct acquaintance with things to a knowledge of propositions about
the things of which we have sensations or understand the meaning.

Next, by the contemplation of propositions of which we have direct
knowledge, we are able to pass indirectly to knowledge of or about
other propositions. The mental process by which we pass from direct
knowledge to indirect knowledge is in some cases and in some degree
capable of analysis. We pass from a knowledge of the proposition a to
a knowledge about the proposition b by perceiving a logical relation
between them. With this logical relation we have direct acquaintance.

The logic of knowledge is mainly occupied with a study of the logical
relations, direct acquaintance with which permits direct knowledge
of the secondary proposition asserting the probability-relation, and
so to indirect knowledge about, and in some cases of, the primary
proposition.

It is not always possible, however, to analyse the mental process
in the case of indirect knowledge, or to say by the perception of what
logical relation we have passed from the knowledge of one proposition
to knowledge about another. But although in some cases we seem to
pass directly from one proposition to another, I am inclined to believe
that in all legitimate transitions of this kind some logical relation
ch. ii

FUNDAMENTAL IDEAS

13
of the proper kind must exist between the propositions, even when
we are not explicitly aware of it. In any case, whenever we pass to
knowledge about one proposition by the contemplation of it in relation
to another proposition of which we have knowledge—even when the
process is unanalysable—I call it an argument. The knowledge, such
as we have in ordinary thought by passing from one proposition to
another without being able to say what logical relations, if any, we
have perceived between them, may be termed uncompleted knowledge.

And knowledge, which results from a distinct apprehension of the
relevant logical relations, may be termed knowledge proper.

8.

In this way, therefore, I distinguish between direct and
indirect knowledge, between that part of our rational belief which is
based on direct knowledge and that part which is based on argument.

About what kinds of things we are capable of knowing propositions
directly, it is not easy to say. About our own existence, our own
sense-data, some logical ideas, and some logical relations, it is usually
agreed that we have direct knowledge. Of the law of gravity, of the
appearance of the other side of the moon, of the cure for phthisis,
of the contents of Bradshaw, it is usually agreed that we do not
have direct knowledge. But many questions are in doubt. Of which
logical ideas and relations we have direct acquaintance, as to whether
we can ever know directly the existence of other people, and as
to when we are knowing propositions about sense-data directly and
when we are interpreting them—it is not possible to give a clear
answer. Moreover, there is another and peculiar kind of derivative
knowledge—by memory.

At a given moment there is a great deal of our knowledge which
we know neither directly nor by argument—we remember it. We
may remember it as knowledge, but forget how we originally knew
it. What we once knew and now consciously remember, can fairly
be called knowledge. But it is not easy to draw the line between
conscious memory, unconscious memory or habit, and pure instinct
or irrational associations of ideas (acquired or inherited)—the last of
pt. i

A TREATISE ON PROBABILITY

14
which cannot fairly be called knowledge, for unlike the first two it did
not even arise (in us at least) out of knowledge. Especially in such
a case as that of what our eyes tell us, it is difficult to distinguish
between the different ways in which our beliefs have arisen. We
cannot always tell, therefore, what is remembered knowledge and
what is not knowledge at all; and when knowledge is remembered, we
do not always remember at the same time whether, originally, it was
direct or indirect.

Although it is with knowledge by argument that I shall be mainly
concerned in this book there is one kind of direct knowledge, namely
of secondary propositions, with which I cannot help but be involved.

In the case of every argument, it is only directly that we can know
the secondary proposition which makes the argument itself valid and
rational. When we know something by argument this must be through
direct acquaintance with some logical relation between the conclusion
and the premiss. In all knowledge, therefore, there is some direct
element; and logic can never be made purely mechanical. All it can
do is so to arrange the reasoning that the logical relations, which have
to be perceived directly, are made explicit and are of a simple kind.

9.

It must be added that the term certainty is sometimes used
in a merely psychological sense to describe a state of mind without
reference to the logical grounds of the belief. With this sense I am not
concerned. It is also used to describe the highest degree of rational
belief; and this is the sense relevant to our present purpose. The
peculiarity of certainty is that knowledge of a secondary proposition
involving certainty, together with knowledge of what stands in this
secondary proposition in the position of evidence, leads to knowledge
of, and not merely about, the corresponding primary proposition.

Knowledge, on the other hand, of a secondary proposition involving
a degree of probability lower than certainty, together with knowledge
of the premiss of the secondary proposition, leads only to a rational
belief of the appropriate degree in the primary proposition.

The
knowledge present in this latter case I have called knowledge about
ch. ii

FUNDAMENTAL IDEAS

15
the primary proposition or conclusion of the argument, as distinct
from knowledge of it.

Of probability we can say no more than that it is a lower degree
of rational belief than certainty; and we may say, if we like, that
it deals with degrees of certainty.∗ Or we may make probability
the more fundamental of the two and regard certainty as a special
case of probability, as being, in fact, the maximum probability.

Speaking somewhat loosely we may say that, if our premisses make
the conclusion certain, then it follows from the premisses; and if they
make it very probable, then it very nearly follows from them.

It is sometimes useful to use the term “impossibility” as the
negative correlative of “certainty,” although the former sometimes has
a different set of associations. If a is certain, then the contradictory
of a is impossible. If a knowledge of a makes b certain, then a
knowledge of a makes the contradictory of b impossible. Thus a
proposition is impossible with respect to a given premiss, if it is
disproved by the premiss; and the relation of impossibility is the
relation of minimum probability.†

10.

We have distinguished between rational belief and irrational
belief and also between rational beliefs which are certain in degree
and those which are only probable. Knowledge has been distinguished
according as it is direct or indirect, according as it is of primary or

∗This view has often been taken, e.g., by Bernoulli and, incidentally, by

Laplace; also by Fries (see Czuber, Entwicklung, p. 12). The view, occasionally
held, that probability is concerned with degrees of truth, arises out of a
confusion between certainty and truth. Perhaps the Aristotelian doctrine that
future events are neither true nor false arose in this way.

†Necessity and Impossibility, in the senses in which these terms are used in
the theory of Modality, seem to correspond to the relations of Certainty and

Impossibility in the theory of probability, the other modals, which comprise the
intermediate degrees of possibility, corresponding to the intermediate degrees of
probability.

Almost up to the end of the seventeenth century the traditional
treatment of modals is, in fact, a primitive attempt to bring the relations of
probability within the scope of formal logic.

pt. i

A TREATISE ON PROBABILITY

16
secondary propositions, and according as it is of or merely about its
object.

In order that we may have a rational belief in a proposition p
of the degree of certainty, it is necessary that one of two conditions
should be fulfilled—(i.) that we know p directly; or (ii.) that we know
a set of propositions h, and also know some secondary proposition q
asserting a certainty-relation between p and h. In the latter case
h may include secondary as well as primary propositions, but it is
a necessary condition that all the propositions h should be known.

In order that we may have rational belief in p of a lower degree
of probability than certainty, it is necessary that we know a set of
propositions h, and also know some secondary proposition q asserting
a probability-relation between p and h.

In the above account one possibility has been ruled out. It is
assumed that we cannot have a rational belief in p of a degree less
than certainty except through knowing a secondary proposition of the
prescribed type. Such belief can only arise, that is to say, by means of
the perception of some probability-relation. To employ a common use
of terms (though one inconsistent with the use adopted above), I have
assumed that all direct knowledge is certain. All knowledge, that is
to say, which is obtained in a manner strictly direct by contemplation
of the objects of acquaintance and without any admixture whatever
of argument and the contemplation of the logical bearing of any other
knowledge on this, corresponds to certain rational belief and not to
a merely probable degree of rational belief. It is true that there do
seem to be degrees of knowledge and rational belief, when the source
of the belief is solely in acquaintance, as there are when its source is
in argument. But I think that this appearance arises partly out of the
difficulty of distinguishing direct from indirect knowledge, and partly
out of a confusion between probable knowledge and vague knowledge.

I cannot attempt here to analyse the meaning of vague knowledge. It
is certainly not the same thing as knowledge proper, whether certain
or probable, and it does not seem likely that it is susceptible of strict
ch. ii

FUNDAMENTAL IDEAS

17
logical treatment. At any rate I do not know how to deal with it, and
in spite of its importance I will not complicate a difficult subject by
endeavouring to treat adequately the theory of vague knowledge.

I assume then that only true propositions can be known, that the
term “probable knowledge” ought to be replaced by the term “probable
degree of rational belief,” and that a probable degree of rational belief
cannot arise directly but only as the result of an argument, out of
the knowledge, that is to say, of a secondary proposition asserting
some logical probability-relation in which the object of the belief
stands to some known proposition. With arguments, if they exist, the
ultimate premisses of which are known in some other manner than
that described above, such as might be called “probable knowledge,”

my theory is not adequate to deal without modification.∗

For the objects of certain belief which is based on direct
knowledge, as opposed to certain belief arising indirectly, there is a
well-established expression; propositions, in which our rational belief
is both certain and direct, are said to be self-evident.

11.

In conclusion, the relativity of knowledge to the individual
may be briefly touched on. Some part of knowledge—knowledge of
our own existence or of our own sensations—is clearly relative to
individual experience. We cannot speak of knowledge absolutely—
only of the knowledge of a particular person.

Other parts of
knowledge—knowledge of the axioms of logic, for example—may seem
more objective. But we must admit, I think, that this too is relative
to the constitution of the human mind, and that the constitution
of the human mind may vary in some degree from man to man.

What is self-evident to me and what I really know, may be only a
probable belief to you, or may form no part of your rational beliefs
at all. And this may be true not only of such things as my existence,
but of some logical axioms also. Some men—indeed it is obviously
the case—may have a greater power of logical intuition than others.

∗I do not mean to imply, however, at any rate at present, that the ultimate
premisses of an argument need always be primary propositions.

pt. i

A TREATISE ON PROBABILITY

18

Further, the difference between some kinds of propositions over which
human intuition seems to have power, and some over which it has
none, may depend wholly upon the constitution of our minds and
have no significance for a perfectly objective logic. We can no
more assume that all true secondary propositions are or ought to be
universally known than that all true primary propositions are known.

The perceptions of some relations of probability may be outside the
powers of some or all of us.

What we know and what probability we can attribute to our
rational beliefs is, therefore, subjective in the sense of being relative to
the individual. But given the body of premisses which our subjective
powers and circumstances supply to us, and given the kinds of logical
relations, upon which arguments can be based and which we have
the capacity to perceive, the conclusions, which it is rational for us
to draw, stand to these premisses in an objective and wholly logical
relation. Our logic is concerned with drawing conclusions by a series
of steps of certain specified kinds from a limited body of premisses.

With these brief indications as to the relation of Probability, as I
understand it, to the Theory of Knowledge, I pass from problems of
ultimate analysis and definition, which are not the primary subject
matter of this book, to the logical theory and superstructure, which
occupies an intermediate position between the ultimate problems
and the applications of the theory, whether such applications take a
generalised mathematical form or a concrete and particular one. For
this purpose it would only encumber the exposition, without adding
to its clearness or its accuracy, if I were to employ the perfectly
exact terminology and minute refinements of language, which are
necessary for the avoidance of error in very fundamental enquiries.

While taking pains, therefore, to avoid any divergence between the
substance of this chapter and of those which succeed it, and to employ
only such periphrases as could be translated, if desired, into perfectly
exact language, I shall not cut myself off from the convenient, but
looser, expressions, which have been habitually employed by previous
ch. ii

FUNDAMENTAL IDEAS

19
writers and have the advantage of being, in a general way at least,
immediately intelligible to the reader.∗

∗This question, which faces all contemporary writers on logical and
philosophical subjects, is in my opinion much more a question of style—and
therefore to be settled on the same sort of considerations as other such
questions—than is generally supposed.

There are occasions for very exact
methods of statement,
such as are employed in Mr. Russell’s Principia

Mathematica. But there are advantages also in writing the English of Hume.

Mr. Moore has developed in Principia Ethica an intermediate style which in his
hands has force and beauty.

But those writers, who strain after exaggerated
precision without going the whole hog with Mr. Russell, are sometimes merely
pedantic.

They lose the reader’s attention, and the repetitious complication
of their phrases eludes his comprehension, without their really attaining, to
compensate, a complete precision.

Confusion of thought is not always best
avoided by technical and unaccustomed expressions, to which the mind has no
immediate reaction of understanding; it is possible, under cover of a careful
formalism, to make statements, which, if expressed in plain language, the mind
would immediately repudiate. There is much to be said, therefore, in favour of
understanding the substance of what you are saying all the time, and of never
reducing the substantives of your argument to the mental status of an x or y.

CHAPTER III
the measurement of probabilities

1.

I have spoken of probability as being concerned with
degrees of rational belief.

This phrase implies that it is in some
sense quantitative and perhaps capable of measurement. The theory
of probable arguments must be much occupied, therefore, with
comparisons of the respective weights which attach to different
arguments. With this question we will now concern ourselves.

It has been assumed hitherto as a matter of course that probability
is, in the full and literal sense of the word, measurable. I shall have
to limit, not extend, the popular doctrine. But, keeping my own
theories in the background for the moment, I will begin by discussing
some existing opinions on the subject.

2.

It has been sometimes supposed that a numerical comparison
between the degrees of any pair of probabilities is not only conceivable
but is actually within our power. Bentham, for instance, in his

Rationale of Judicial Evidence,∗ proposed a scale on which witnesses
might mark the degree of their certainty; and others have suggested
seriously a ‘barometer of probability.’†

That such comparison is theoretically possible, whether or not we
are actually competent in every case to make the comparison, has
been the generally accepted opinion. The following quotation‡ puts
this point of view very well:

∗Book i. chap vi. (referred to by Venn).

†The reader may be reminded of Gibbon’s proposal that:—“A Theological

Barometer might be formed,
of which the Cardinal (Baronius) and our
countryman,

Dr.

Middleton,
should
constitute
the
opposite
and
remote
extremities, as the former sunk to the lowest degree of credulity, which was
compatible with learning, and the latter rose to the highest pitch of scepticism,
in any wise consistent with Religion.”

‡W. F. Donkin, Phil. Mag., 1851. He is replying to an article by J. D.

Forbes (Phil. Mag., Aug. 1849) which had cast doubt upon this opinion.

20
ch. iii

FUNDAMENTAL IDEAS

21

“I do not see on what ground it can be doubted that every definite
state of belief concerning a proposed hypothesis is in itself capable
of being represented by a numerical expression, however difficult or
impracticable it may be to ascertain its actual value. It would be
very difficult to estimate in numbers the vis viva of all of the particles
of a human body at any instant; but no one doubts that it is
capable of numerical expression. I mention this because I am not sure
that Professor Forbes has distinguished the difficulty of ascertaining
numbers in certain cases from a supposed difficulty of expression by
means of numbers. The former difficulty is real, but merely relative
to our knowledge and skill; the latter, if real, would be absolute and
inherent in the subject-matter, which I conceive is not the case.”

De Morgan held the same opinion on the ground that, wherever we
have differences of degree, numerical comparison must be theoretically
possible.∗ He assumes, that is to say, that all probabilities can be
placed in an order of magnitude, and argues from this that they
must be measurable. Philosophers, however, who are mathematicians,
would no longer agree that, even if the premiss is sound, the
conclusion follows from it. Objects can be arranged in an order,
which we can reasonably call one of degree or magnitude, without its
being possible to conceive a system of measurement of the differences
between the individuals.

This opinion may also have been held by others, if not by

De Morgan, in part because of the narrow associations which

Probability has had for them. The Calculus of Probability has
received far more attention than its logic, and mathematicians, under
no compulsion to deal with the whole of the subject, have naturally
confined their attention to those special cases, the existence of which

∗“Whenever the terms greater and less can be applied, there twice, thrice,
etc., can be conceived, though not perhaps measured by us.”—“Theory of

Probabilities,” Encyclopaedia Metropolitana, p. 395. He is a little more guarded
in his Formal Logic, pp. 174, 175; but arrives at the same conclusion so far as
probability is concerned.

pt. i

A TREATISE ON PROBABILITY

22
will be demonstrated at a later stage, where algebraical representation
is possible. Probability has become associated, therefore, in the minds
of theorists with those problems in which we are presented with a
number of exclusive and exhaustive alternatives of equal probability;
and the principles, which are readily applicable in such circumstances,
have been supposed, without much further enquiry, to possess general
validity.

3.

It is also the case that theories of probability have been
propounded and widely accepted, according to which its numerical
character is necessarily involved in the definition. It is often said, for
instance, that probability is the ratio of the number of “favourable
cases” to the total number of “cases.” If this definition is accurate,
it follows that every probability can be properly represented by a
number and in fact is a number; for a ratio is not a quantity at
all. In the case also of definitions based upon statistical frequency,
there must be by definition a numerical ratio corresponding to every
probability. These definitions and the theories based on them will
be discussed in Chapter VIII.; they are connected with fundamental
differences of opinion with which it is not necessary to burden the
present argument.

4.

If we pass from the opinions of theorists to the experience
of practical men, it might perhaps be held that a presumption in
favour of the numerical valuation of all probabilities can be based on
the practice of underwriters and the willingness of Lloyd’s to insure
against practically any risk. Underwriters are actually willing, it
might be urged, to name a numerical measure in every case, and
to back their opinion with money. But this practice shows no more
than that many probabilities are greater or less than some numerical
measure, not that they themselves are numerically definite. It is
sufficient for the underwriter if the premium he names exceeds the
probable risk. But, apart from this, I doubt whether in extreme
cases the process of thought, through which he goes before naming
a premium, is wholly rational and determinate; or that two equally
ch. iii

FUNDAMENTAL IDEAS

23
intelligent brokers acting on the same evidence would always arrive
at the same result. In the case, for instance, of insurances effected
before a Budget, the figures quoted must be partly arbitrary. There
is in them an element of caprice, and the broker’s state of mind,
when he quotes a figure, is like a bookmaker’s when he names odds.

Whilst he may be able to make sure of a profit, on the principles of
the bookmaker, yet the individual figures that make up the book are,
within certain limits, arbitrary. He may be almost certain, that is to
say, that there will not be new taxes on more than one of the articles
tea, sugar, and whisky; there may be an opinion abroad, reasonable
or unreasonable, that the likelihood is in the order—whisky, tea,
sugar; and he may, therefore be able to effect insurances for equal
amounts in each at 30 per cent, 40 per cent, and 45 per cent. He
has thus made sure of a profit of 15 per cent, however absurd and
arbitrary his quotations may be. It is not necessary for the success of
underwriting on these lines that the probabilities of these new taxes
are really measurable by the figures 3 , 4 , and 45 ; it is sufficient

10

10

100
that there should be merchants willing to insure at these rates. These
merchants, moreover, may be wise to insure even if the quotations
are partly arbitrary; for they may run the risk of insolvency unless
their possible loss is thus limited. That the transaction is in principle
one of bookmaking is shown by the fact that, if there is a specially
large demand for insurance against one of the possibilities, the rate
rises;—the probability has not changed, but the “book” is in danger
of being upset. A Presidential election in the United States supplies
a more precise example. On August 23, 1912, 60 per cent was quoted
at Lloyd’s to pay a total loss should Dr. Woodrow Wilson be elected,

30 per cent should Mr. Taft be elected, and 20 per cent should

Mr. Roosevelt be elected. A broker, who could effect insurances in
equal amounts against the election of each candidate, would be certain
at these rates of a profit of 10 per cent. Subsequent modifications of
these terms would largely depend upon the number of applicants for
each kind of policy. Is it possible to maintain that these figures in
pt. i

A TREATISE ON PROBABILITY

24
any way represent reasoned numerical estimates of probability?

In some insurances the arbitrary element seems even greater.

Consider, for instance, the reinsurance rates for the Waratah, a vessel
which disappeared in South African waters. The lapse of time made
rates rise; the departure of ships in search of her made them fall;
some nameless wreckage is found and they rise; it is remembered that
in similar circumstances thirty years ago a vessel floated, helpless but
not seriously damaged, for two months, and they fall. Can it be
pretended that the figures which were quoted from day to day—75 per
cent, 83 per cent, 78 per cent—were rationally determinate, or
that the actual figure was not within wide limits arbitrary and
due to the caprice of individuals? In fact underwriters themselves
distinguish between risks which are properly insurable, either because
their probability can be estimated between comparatively narrow
numerical limits or because it is possible to make a “book” which
covers all possibilities, and other risks which cannot be dealt with
in this way and which cannot form the basis of a regular business
of insurance,—although an occasional gamble may be indulged in. I
believe, therefore, that the practice of underwriters weakens rather
than supports the contention that all probabilities can be measured
and estimated numerically.

5.

Another set of practical men, the lawyers, have been
more subtle in this matter than the philosophers.∗ A distinction,
interesting for our present purpose, between probabilities, which can
be estimated within somewhat narrow limits, and those which cannot,
has arisen in a series of judicial decisions respecting damages. The
following extract† from the Times Law Reports seems to me to deal

∗Leibniz notes the subtle distinctions made by Jurisconsults between degrees
of probability; and in the preface to a work, projected but unfinished, which was
to have been entitled Ad stateram juris de gradibus probationum et probabilitatum
he recommends them as models of logic in contingent questions (Couturat,

Logique de Leibniz, p. 240).

†I have considerably compressed the original report (Sapwell v. Bass).

ch. iii

FUNDAMENTAL IDEAS

25
very clearly in a mixture of popular and legal phraseology, with the
logical point at issue:

This was an action brought by a breeder of racehorses to recover
damages for breach of a contract. The contract was that Cyllene,
a racehorse owned by the defendant, should in the season of the
year 1909 serve one of the plaintiff’s brood mares. In the summer
of 1908 the defendant, without the consent of the plaintiff, sold

Cyllene for £30,000 to go to South America. The plaintiff claimed a
sum equal to the average profit he had made through having a mare
served by Cyllene during the past four years. During those four years
he had had four colts which had sold at £3300. Upon that basis his
loss came to 700 guineas.

Mr. Justice Jelf said that he was desirous, if he properly could,
to find some mode of legally making the defendant compensate the
plaintiff; but the question of damages presented formidable and, to his
mind, insuperable difficulties. The damages, if any, recoverable here
must be either the estimated loss of profit or else nominal damages.

The estimate could only be based on a succession of contingencies.

Thus it was assumed that (inter alia) Cyllene would be alive and
well at the time of the intended service; that the mare sent would
be well bred and not barren; that she would not slip her foal; and
that the foal would be born alive and healthy. In a case of this kind
he could only rely on the weighing of chances; and the law generally
regarded damages which depended on the weighing of chances as too
remote, and therefore irrecoverable. It was drawing the line between
an estimate of damage based on probabilities, as in “Simpson v.

L. and N.W. Railway Co.” (1, Q.B.D., 274), where Cockburn, C.J.,
said: “To some extent, no doubt, the damage must be a matter of
speculation, but that is no reason for not awarding any damages at
all,” and a claim for damages of a totally problematical character.

He (Mr. Justice Jelf) thought the present case was well over the
line. Having referred to “Mayne on Damages” (8th ed., p. 70), he
pointed out that in “Watson v. Ambergah Railway Co.” (15, Jur., 448)

pt. i

A TREATISE ON PROBABILITY

26

Patteson, J., seemed to think that the chance of a prize might be
taken into account in estimating the damages for breach of a contract
to send a machine for loading barges by railway too late for a show;
but Erle, J., appeared to think such damage was too remote. In his

Lordship’s view the chance of winning a prize was not of sufficiently
ascertainable value at the time the contract was made to be within
the contemplation of the parties. Further, in the present case, the
contingencies were far more numerous and uncertain. He would enter
judgment for the plaintiff for nominal damages, which were all he was
entitled to. They would be assessed at 1s.

One other similar case may be quoted in further elucidation of
the same point, and because it also illustrates another point—the
importance of making clear the assumptions relative to which the
probability is calculated. This case∗ arose out of an offer of a Beauty

Prize† by the Daily Express. Out of 6000 photographs submitted, a
number were to be selected and published in the newspaper in the
following manner:

The United Kingdom was to be divided into districts and the
photographs of the selected candidates living in each district were to
be submitted to the readers of the paper in the district, who were to
select by their votes those whom they considered the most beautiful,
and a Mr. Seymour Hicks was then to make an appointment with
the 50 ladies obtaining the greatest number of votes and himself
select 12 of them. The plaintiff, who came out head of one of
the districts, submitted that she had not been given a reasonable
opportunity of keeping an appointment, that she had thereby lost the
value of her chance of one of the 12 prizes, and claimed damages
accordingly.

The jury found that the defendant had not taken
reasonable means to give the plaintiff an opportunity of presenting
herself for selection, and assessed the damages, provided they were

∗Chaplin v. Hicks (1911).

†The prize was to be a theatrical engagement and, according to the article,
the probability of subsequent marriage into the peerage.

ch. iii

FUNDAMENTAL IDEAS

27
capable of assessment, at £100, the question of the possibility of
assessment being postponed. This was argued before Mr. Justice

Pickford, and subsequently in the Court of Appeal before Lord

Justices Vaughan Williams, Fletcher Moulton, and Harwell. Two
questions arose—relative to what evidence ought the probability to
be calculated, and was it numerically measurable? Counsel for the
defendant contended that, “if the value of the plaintiff’s chance was
to be considered, it must be the value as it stood at the beginning of
the competition, not as it stood after she had been selected as one of
the 50. As 6000 photographs had been sent in, and there was also
the personal taste of the defendant as final arbiter to be considered,
the value of the chance of success was really incalculable.” The first
contention that she ought to be considered as one of 6000 not as
one of 50 was plainly preposterous and did not hoodwink the court.

But the other point, the personal taste of the arbiter, presented more
difficulty. In estimating the chance, ought the Court to receive and
take account of evidence respecting the arbiter’s preferences in types
of beauty? Mr. Justice Pickford, without illuminating the question,
held that the damages were capable of estimation. Lord Justice

Vaughan Williams in giving judgment in the Court of Appeal argued
as follows:

As he understood it, there were some 50 competitors, and there
were 12 prizes of equal value, so that the average chance of success
was about one in four. It was then said that the questions which
might arise in the minds of the persons who had to give the decisions
were so numerous that it was impossible to apply the doctrine of
averages. He did not agree. Then it was said that if precision and
certainty were impossible in any case it would be right to describe
the damages as unassessable. He agreed that there might be damages
so unassessable that the doctrine of averages was not possible of
application because the figures necessary to be applied were not
forthcoming. Several cases were to be found in the reports where it
had been so held, but he denied the proposition that because precision
pt. i

A TREATISE ON PROBABILITY

28
and certainty had not been arrived at, the jury had no function or
duty to determine the damages. . . . He (the Lord Justice) denied that
the mere fact that you could not assess with precision and certainty
relieved a wrongdoer from paying damages for his breach of duty.

He would not lay down that in every case it could be left to the
jury to assess the damages; there were cases where the loss was so
dependent on the mere unrestricted volition of another person that it
was impossible to arrive at any assessable loss from the breach. It was
true that there was no market here; the right to compete was personal
and could not be transferred. He could not admit that a competitor
who found herself one of 50 could have gone into the market and sold
her right to compete. At the same time the jury might reasonably
have asked themselves the question whether, if there was a right to
compete, it could have been transferred, and at what price. Under
these circumstances he thought the matter was one for the jury.

The attitude of the Lord Justice is clear. The plaintiff had
evidently suffered damage, and justice required that she should
be compensated.

But it was equally evident, that, relative to
the completest information available and account being taken of
the arbiter’s personal taste, the probability could be by no means
estimated with numerical precision. Further, it was impossible to say
how much weight ought to be attached to the fact that the plaintiff
had been head of her district (there were fewer than 50 districts); yet
it was plain that it made her chance better than the chances of those
of the 50 left in, who were not head of their districts. Let rough
justice be done, therefore. Let the case be simplified by ignoring some
part of the evidence. The “doctrine of averages” is then applicable, or,
in other words, the plaintiff’s loss may be assessed at twelve-fiftieths
of the value of the prize.∗

6.

How does the matter stand, then?

Whether or not

∗The jury in assessing the damages at £100, however, cannot have argued
so subtly as this; for the average value of a prize (I have omitted the details
bearing on their value) could not have been fairly estimated so high as £400.

ch. iii

FUNDAMENTAL IDEAS

29
such a thing is theoretically conceivable, no exercise of the practical
judgment is possible, by which a numerical value can actually be
given to the probability of every argument. So far from our being
able to measure them, it is not even clear that we are always able to
place them in an order of magnitude. Nor has any theoretical rule for
their evaluation ever been suggested.

The doubt, in view of these facts, whether any two probabilities
are in every case even theoretically capable of comparison in terms
of numbers, has not, however, received serious consideration. There
seems to me to be exceedingly strong reasons for entertaining the
doubt. Let us examine a few more instances.

7.

Consider an induction or a generalisation. It is usually held
that each additional instance increases the generalisation’s probability.

A conclusion, which is based on three experiments in which the
unessential conditions are varied, is more trustworthy than if it were
based on two. But what reason or principle can be adduced for
attributing a numerical measure to the increase?∗

Or, to take another class of instances, we may sometimes have
some reason for supposing that one object belongs to a certain
category if it has points of similarity to other known members of the
category (e.g. if we are considering whether a certain picture should
be ascribed to a certain painter), and the greater the similarity the
greater the probability of our conclusion. But we cannot in these
cases measure the increase; we can say that the presence of certain

∗It is true that Laplace and others (even amongst contemporary writers)

have believed that the probability of an induction is measurable by means of a
formula known as the rule of succession, according to which the probability of
n + 1
an induction based on n instances is

. Those who have been convinced
n + 2
by the reasoning employed to establish this rule must be asked to postpone
judgment until it has been examined in Chapter XXX.

But we may point
out here the absurdity of supposing that the odds are 2 to 1 in favour of a
generalisation based on a single instance—a conclusion which this formula would
seem to justify.

pt. i

A TREATISE ON PROBABILITY

30
peculiar marks in a picture increases the probability that the artist
of whom those marks are known to be characteristic painted it, but
we cannot say that the presence of these marks makes it two or three
or any other number of times more probable than it would have been
without them. We can say that one thing is more like a second object
than it is like a third; but there will very seldom be any meaning in
saying that it is twice as like. Probability is, so far as measurement
is concerned, closely analogous to similarity.∗

Or consider the ordinary circumstances of life. We are out for a
walk—what is the probability that we shall reach home alive? Has
this always a numerical measure? If a thunderstorm bursts upon us,
the probability is less than it was before; but is it changed by some
definite numerical amount? There might, of course, be data which
would make these probabilities numerically comparable; it might be
argued that a knowledge of the statistics of death by lightning would
make such a comparison possible. But if such information is not
included within the knowledge to which the probability is referred, this
fact is not relevant to the probability actually in question and cannot
affect its value. In some cases, moreover, where general statistics
are available, the numerical probability which might be derived from
them is inapplicable because of the presence of additional knowledge
with regard to the particular case. Gibbon calculated his prospects

∗There are very few writers on probability who have explicitly admitted that
probabilities, though in some sense quantitative, may be incapable of numerical
comparison. Edgeworth, “Philosophy of Chance” (Mind, 1884, p. 225), admitted
that “there may well be important quantitative,
although not numerical,
estimates” of probabilities.

Goldschmidt (Wahrscheinlichkeitsrechnung, p. 43)

may also be cited as holding a somewhat similar opinion. He maintains that a
lack of comparability in the grounds often stands in the way of the measurability
of the probable in ordinary usage, and that there are not necessarily good
reasons for measuring the value of one argument against that of another. On
the other hand, a numerical statement for the degree of the probable, although
generally impossible, is not in itself contradictory to the notion; and of three
statements, relating to the same circumstances, we can well say that one is
more probable than another, and that one is the most probable of the three.

ch. iii

FUNDAMENTAL IDEAS

31
of life from the volumes of vital statistics and the calculations of
actuaries. But if a doctor had been called to his assistance the nice
precision of these calculations would have become useless; Gibbon’s
prospects would have been better or worse than before, but he would
no longer have been able to calculate to within a day or week the
period for which he then possessed an even chance of survival.

In these instances we can, perhaps, arrange the probabilities in
an order of magnitude and assert that the new datum strengthens
or weakens the argument, although there is no basis for an estimate
how much stronger or weaker the new argument is than the old.

But in another class of instances is it even possible to arrange the
probabilities in an order of magnitude, or to say that one is the
greater and the other less?

8.

Consider three sets of experiments, each directed towards
establishing a generalisation. The first set is more numerous; in the
second set the irrelevant conditions have been more carefully varied;
in the third case the generalisation in view is wider in scope than in
the others. Which of these generalisations is on such evidence the
most probable? There is, surely, no answer; there is neither equality
nor inequality between them. We cannot always weigh the analogy
against the induction, or the scope of the generalisation against the
bulk of the evidence in support of it. If we have more grounds than
before, comparison is possible; but, if the grounds in the two cases
are quite different, even a comparison of more and less, let alone
numerical measurement, may be impossible.

This leads up to a contention, which I have heard supported, that,
although not all measurements and not all comparisons of probability
are within our power, yet we can say in the case of every argument
whether it is more or less likely than not. Is our expectation of rain,
when we start out for a walk, always more likely than not, or less
likely than not, or as likely as not? I am prepared to argue that
on some occasions none of these alternatives hold, and that it will
be an arbitrary matter to decide for or against the umbrella. If the
pt. i

A TREATISE ON PROBABILITY

32
barometer is high, but the clouds are black, it is not always rational
that one should prevail over the other in our minds, or even that we
should balance them,—though it will be rational to allow caprice to
determine us and to waste no time on the debate.

9.

Some cases, therefore, there certainly are in which no rational
basis has been discovered for numerical comparison. It is not the case
here that the method of calculation, prescribed by theory, is beyond
our powers or too laborious for actual application. No method of
calculation, however impracticable, has been suggested. Nor have we
any prima facie indications of the existence of a common unit to
which the magnitudes of all probabilities are naturally referrible. A
degree of probability is not composed of some homogeneous material,
and is not apparently divisible into parts of like character with one
another. An assertion, that the magnitude of a given probability is in
a numerical ratio to the magnitude of every other, seems, therefore,
unless it is based on one of the current definitions of probability, with
which I shall deal separately in later chapters, to be altogether devoid
of the kind of support, which can usually be supplied in the case of
quantities of which the mensurability is not open to denial. It will be
worth while, however, to pursue the argument a little further.

10.

There appear to be four alternatives. Either in some cases
there is no probability at all; or probabilities do not all belong to a
single set of magnitudes measurable in terms of a common unit; or
these measures always exist, but in many cases are, and must remain,
unknown; or probabilities do belong to such a set and their measures
are capable of being determined by us, although we are not always
able so to determine them in practice.

11.

Laplace and his followers excluded the first two alternatives.

They argued that every conclusion has its place in the numerical
range of probabilities from 0 to 1, if only we knew it, and they
developed their theory of unknown probabilities.

In dealing with this contention, we must be clear as to what
we mean by saying that a probability is unknown. Do we mean
ch. iii

FUNDAMENTAL IDEAS

33
unknown through lack of skill in arguing from given evidence, or
unknown through lack of evidence? The first is alone admissible,
for new evidence would give us a new probability, not a fuller
knowledge of the old one; we have not discovered the probability
of a statement on given evidence, by determining its probability in
relation to quite different evidence. We must not allow the theory
of unknown probabilities to gain plausibility from the second sense.

A relation of probability does not yield us, as a rule, information of
much value, unless it invests the conclusion with a probability which
lies between narrow numerical limits. In ordinary practice, therefore,
we do not always regard ourselves as knowing the probability of a
conclusion, unless we can estimate it numerically. We are apt, that
is to say, to restrict the use of the expression probable to these
numerical examples, and to allege in other cases that the probability
is unknown. We might say, for example, that we do not know, when
we go on a railway journey, the probability of death in a railway
accident, unless we are told the statistics of accidents in former years;
or that we do not know our chances in a lottery, unless we are told the
number of the tickets. But it must be clear upon reflection that if we
use the term in this sense,—which is no doubt a perfectly legitimate
sense,—we ought to say that in the case of some arguments a relation
of probability does not exist, and not that it is unknown. For it is
not this probability that we have discovered, when the accession of
new evidence makes it possible to frame a numerical estimate.

Possibly this theory of unknown probabilities may also gain
strength from our practice of estimating arguments, which, as I
maintain, have no numerical value, by reference to those that have.

We frame two ideal arguments, that is to say, in which the general
character of the evidence largely resembles what is actually within our
knowledge, but which is so constituted as to yield a numerical value,
and we judge that the probability of the actual argument lies between
these two. Since our standards, therefore, are referred to numerical
measures in many cases where actual measurement is impossible, and
pt. i

A TREATISE ON PROBABILITY

34
since the probability lies between two numerical measures, we come to
believe that it must also, if only we knew it, possess such a measure
itself.

12.

To say, then, that a probability is unknown ought to mean
that it is unknown to us through our lack of skill in arguing from
given evidence. The evidence justifies a certain degree of knowledge,
but the weakness of our reasoning power prevents our knowing what
this degree is. At the best, in such cases, we only know vaguely
with what degree of probability the premisses invest the conclusion.

That probabilities can be unknown in this sense or known with less
distinctness than the argument justifies, is clearly the case. We
can through stupidity fail to make any estimate of a probability at
all, just as we may through the same cause estimate a probability
wrongly. As soon as we distinguish between the degree of belief which
it is rational to entertain and the degree of belief actually entertained,
we have in effect admitted that the true probability is not known to
everybody.

But this admission must not be allowed to carry us too far.

Probability is, vide Chapter II. (§ 12), relative in a sense to the
principles of human reason. The degree of probability, which it is
rational for us to entertain, does not presume perfect logical insight,
and is relative in part to the secondary propositions which we in fact
know; and it is not dependent upon whether more perfect logical
insight is or is not conceivable. It is the degree of probability to which
those logical processes lead, of which our minds are capable; or, in the
language of Chapter II., which those secondary propositions justify,
which we in fact know. If we do not take this view of probability, if
we do not limit it in this way and make it, to this extent, relative
to human powers, we are altogether adrift in the unknown; for we
cannot ever know what degree of probability would be justified by
the perception of logical relations which we are, and must always be,
incapable of comprehending.

13.

Those who have maintained that, where we cannot assign
ch. iii

FUNDAMENTAL IDEAS

35
a numerical probability, this is not because there is none, but simply
because we do not know it, have really meant, I feel sure, that with
some addition to our knowledge a numerical value would be assignable,
that is to say that our conclusions would have a numerical probability
relative to slightly different premisses. Unless, therefore, the reader
clings to the opinion that, in every one of the instances I have cited
in the earlier paragraphs of this chapter, it is theoretically possible on
that evidence to assign a numerical value to the probability, we are
left with the first two of the alternatives of § 10, which were as follows:

either in some cases there is no probability at all; or probabilities do
not all belong to a single set of magnitudes measurable in terms of a
common unit. It would be difficult to maintain that there is no logical
relation whatever between our premiss and our conclusion in those
cases where we cannot assign a numerical value to the probability;
and if this is so, it is really a question of whether the logical relation
has characteristics, other than mensurability, of a kind to justify us
in calling it a probability-relation. Which of the two we favour is,
therefore, partly a matter of definition. We might, that is to say,
pick out from probabilities (in the widest sense) a set, if there is
one, all of which are measurable in terms of a common unit, and call
the members of this set, and them only, probabilities (in the narrow
sense). To restrict the term ‘probability’ in this way would be, I
think, very inconvenient. For it is possible, as I shall show, to find
several sets, the members of each of which are measurable in terms
of a unit common to all the members of that set; so that it would be
in some degree arbitrary∗ which we chose. Further, the distinction
between probabilities, which would be thus measurable and those
which would not, is not fundamental.

At any rate I aim here at dealing with probability in its widest
sense, and am averse to confining its scope to a limited type of
argument. If the opinion that not all probabilities can be measured

∗Not altogether; for it would be natural to select the set to which the
relation of certainty belongs.

pt. i

A TREATISE ON PROBABILITY

36
seems paradoxical, it may be due to this divergence from a usage which
the reader may expect. Common usage, even if it involves, as a rule, a
flavour of numerical measurement, does not consistently exclude those
probabilities which are incapable of it. The confused attempts, which
have been made, to deal with numerically indeterminate probabilities
under the title of unknown probabilities, show how difficult it is
to confine the discussion within the intended limits, if the original
definition is too narrow.

14.

I maintain, then, in what follows, that there are some
pairs of probabilities between the members of which no comparison
of magnitude is possible; that we can say, nevertheless, of some pairs
of relations of probability that the one is greater and the other less,
although it is not possible to measure the difference between them;
and that in a very special type of case, to be dealt with later, a
meaning can be given to a numerical comparison of magnitude. I
think that the results of observation, of which examples have been
given earlier in this chapter, are consistent with this account.

By saying that not all probabilities are measurable, I mean that
it is not possible to say of every pair of conclusions, about which we
have some knowledge, that the degree of our rational belief in one
bears any numerical relation to the degree of our rational belief in
the other; and by saying that not all probabilities are comparable in
respect of more and less, I mean that it is not always possible to say
that the degree of our rational belief in one conclusion is either equal
to, greater than, or less than the degree of our belief in another.

We must now examine a philosophical theory of the quantitative
properties of probability, which would explain and justify the
conclusions, which reflection discovers, if the preceding discussion is
correct, in the practice of ordinary argument. We must bear in mind
that our theory must apply to all probabilities and not to a limited
class only, and that, as we do not adopt a definition of probability
which presupposes its numerical mensurability, we cannot directly
argue from differences in degree to a numerical measurement of these
ch. iii

FUNDAMENTAL IDEAS

37
differences. The problem is subtle and difficult, and the following
solution is, therefore, proposed with hesitation; but I am strongly
convinced that something resembling the conclusion here set forth is
true.

15.

The so-called magnitudes or degrees of knowledge or
probability, in virtue of which one is greater and another less, really
arise out of an order in which it is possible to place them. Certainty,
impossibility, and a probability, which has an intermediate value, for
example, constitute an ordered series in which the probability lies
between certainty and impossibility.

In the same way there may
exist a second probability which lies between certainty and the first
probability. When, therefore, we say that one probability is greater
than another, this precisely means that the degree of our rational
belief in the first case lies between certainty and the degree of our
rational belief in the second case.

On this theory it is easy to see why comparisons of more and less
are not always possible. They exist between two probabilities, only
when they and certainty all lie on the same ordered series. But if
more than one distinct series of probabilities exist, then it is clear
that only those, which belong to the same series, can be compared.

If the attribute ‘greater’ as applied to one of two terms arises solely
out of the relative order of the terms in a series, then comparisons
of greater and less must always be possible between terms which are
members of the same series, and can never be possible between two
terms which are not members of the same series. Some probabilities
are not comparable in respect of more and less, because there exists
more than one path, so to speak, between proof and disproof, between
certainty and impossibility; and neither of two probabilities, which lie
on independent paths, bears to the other and to certainty the relation
of ‘between’ which is necessary for quantitative comparison.

If we are comparing the probabilities of two arguments, where the
conclusion is the same in both and the evidence of one exceeds the
evidence of the other by the inclusion of some fact which is favourably
pt. i

A TREATISE ON PROBABILITY

38
relevant, in such a case a relation seems clearly to exist between the
two in virtue of which one lies nearer to certainty than the other.

Several types of argument can be instanced in which the existence
of such a relation is equally apparent. But we cannot assume its
presence in every case or in comparing in respect of more and less the
probabilities of every pair of arguments.

16.

Analogous instances are by no means rare, in which, by a
convenient looseness, the phraseology of quantity is misapplied in the
same manner as in the case of probability. The simplest example is
that of colour. When we describe the colour of one object as bluer
than that of another, or say that it has more green in it, we do not
mean that there are quantities blue and green of which the object’s
colour possesses more or less; we mean that the colour has a certain
position in an order of colours and that it is nearer some standard
colour than is the colour with which we compare it.

Another example is afforded by the cardinal numbers. We say
that the number three is greater than the number two, but we do
not mean that these numbers are quantities one of which possesses
a greater magnitude than the other. The one is greater than the
other by reason of its position in the order of numbers; it is further
distant from the origin zero. One number is greater than another if
the second number lies between zero and the first.

But the closest analogy is that of similarity. When we say of
three objects A, B, and C that B is more like A than C is, we mean,
not that there is any respect in which B is in itself quantitatively
greater than C, but that, if the three objects are placed in an order
of similarity, B is nearer to A than C is. There are also, as in the
case of probability, different orders of similarity. For instance, a book
bound in blue morocco is more like a book bound in red morocco
than if it were bound in blue calf; and a book bound in red calf is
more like the book in red morocco than if it were in blue calf. But
there may be no comparison between the degree of similarity which
exists between books bound in red morocco and blue morocco, and
ch. iii

FUNDAMENTAL IDEAS

39
that which exists between books bound in red morocco and red calf.

This illustration deserves special attention, as the analogy between
orders of similarity and probability is so great that its apprehension
will greatly assist that of the ideas I wish to convey. We say that one
argument is more probable than another (i.e. nearer to certainty) in
the same kind of way as we can describe one object as more like than
another to a standard object of comparison.

17.

Nothing has been said up to this point which bears
on the question whether probabilities are ever capable of numerical
comparison. It is true of some types of ordered series that there are
measurable relations of distance between their members as well as
order, and that the relation of one of its members to an ‘origin’ can
be numerically compared with the relation of another member to the
same origin. But the legitimacy of such comparisons must be matter
for special enquiry in each case.

It will not be possible to explain in detail how and in what sense
a meaning can sometimes be given to the numerical measurement
of probabilities until Part II. is reached. But this chapter will be more complete if I indicate briefly the conclusions at which we
shall arrive later. It will be shown that a process of compounding
probabilities can be defined with such properties that it can be
conveniently called a process of addition. It will sometimes be the
case, therefore, that we can say that one probability C is equal to
the sum of two other probabilities A and B, i.e. C = A + B. If in
such a case A and B are equal, then we may write this C = 2A
and say that C is double A. Similarly if D = C + A, we may write

D = 3A, and so on. We can attach a meaning, therefore, to the
equation P = n A, where P and A are relations of probability, and
n is a number. The relation of certainty has been commonly taken
as the unit of such conventional measurements. Hence if P represents
certainty, we should say, in ordinary language, that the magnitude
of the probability A is 1 .

It will be shown also that we can
n
define a process, applicable to probabilities, which has the properties
pt. i

A TREATISE ON PROBABILITY

40
of arithmetical multiplication.

Where numerical measurement is
possible, we can in consequence perform algebraical operations of
considerable complexity. The attention, out of proportion to their real
importance, which has been paid, on account of the opportunities of
mathematical manipulation which they afford, to the limited class of
numerical probabilities, seems to be a part explanation of the belief,
which it is the principal object of this chapter to prove erroneous,
that all probabilities must belong to it.

18.

We must look, then, at the quantitative characteristics of
probability in the following way. Some sets of probabilities we can
place in an ordered series, in which we can say of any pair that one is
nearer than the other to certainty,—that the argument in one case is
nearer proof than in the other, and that there is more reason for one
conclusion than for the other. But we can only build up these ordered
series in special cases. If we are given two distinct arguments, there
is no general presumption that their two probabilities and certainty
can be placed in an order. The burden of establishing the existence
of such an order lies on us in each separate case. An endeavour
will be made later to explain in a systematic way how and in what
circumstances such orders can be established. The argument for the
theory here proposed will then be strengthened. For the present it
has been shown to be agreeable to common sense to suppose that an
order exists in some cases and not in others.

19.

Some of the principal properties of ordered series of
probabilities are as follows:

(i.) Every probability lies on a path between impossibility
and certainty; it is always true to say of a degree of
probability, which is not identical either with impossibility
or with certainty, that it lies between them. Thus certainty,
impossibility and any other degree of probability form an
ordered series. This is the same thing as to say that every
argument amounts to proof, or disproof, or occupies an
intermediate position.

ch. iii

FUNDAMENTAL IDEAS

41

(ii.) A path or series, composed of degrees of probability, is
not in general compact. It is not necessarily true, that is to
say, that any pair of probabilities in the same series have a
probability between them.

(iii.) The same degree of probability can lie on more than one
path (i.e. can belong to more than one series). Hence, if

B lies between A and C, and also lies between A and C
it does not follow that of A and A either lies between the
other and certainty. The fact, that the same probability can
belong to more than one distinct series, has its analogy in
the case of similarity.

(iv.) If ABC forms an ordered series, B lying between A and C,
and BCD forms an ordered series, C lying between B and D,
then ABCD forms an ordered series, B lying between

A and D.

20.

The different series of probabilities and their mutual relations
can be most easily pictured by means of a diagram. Let us represent
an ordered series by points lying upon a path, all the points on a
given path belonging to the same series. It follows from (i.) that
the points O and I, representing the relations of impossibility and
certainty, lie on every path, and that all paths lie wholly between
these points. It follows from (iii.) that the same point can lie on more
than one path. It is possible, therefore, for paths to intersect and
cross. It follows from (iv.) that the probability represented by a given
point is greater than that represented by any other point which can
be reached by passing along a path with a motion constantly towards
the point of impossibility, and less than that represented by any point
which can be reached by moving along a path towards the point of
certainty. As there are independent paths there will be some pairs
of points representing relations of probability such that we cannot
reach one by moving from the other along a path always in the same
direction.

These properties are illustrated in the annexed diagram, O rep-
pt. i

A TREATISE ON PROBABILITY

42
resents impossibility, I certainty, and A a numerically measurable
probability intermediate between O and I; U, V, W, X, Y, Z are non-
numerical probabilities, of which, however, V is less than the numerical
probability A, and is also less
than W, X, and Y. X and Y
are both greater than W, and
greater than V, but are not
comparable with one another, or
with A.

V and Z are both
less than W, X, and Y, but
are not comparable with one
another; U is not quantitatively
comparable with any of the probabilities V, W, X, Y, Z. Probabilities
which are numerically comparable will all belong to one series, and
the path of this series, which we may call the numerical path or
strand, will be represented by OAI.

21.

The chief results which have been reached so far are collected
together below, and expressed with precision:—

(i.) There are amongst degrees of probability or rational belief
various sets, each set composing an ordered series. These
series are ordered by virtue of a relation of ‘between.’ If B is

‘between’ A and C, ABC form a series.

(ii.) There are two degrees of probability O and I between
which all other probabilities lie. If, that is to say, A is a
probability, OAI form a series. O represents impossibility
and I certainty.

(iii.) If A lies between O and B, we may write this AB, so that

OA and AI are true for all probabilities.

(iv.) If AB, the probability B is said to be greater than the
probability A, and this can be expressed by B > A.

(v.) If the conclusion a bears the relation of probability P
to the premiss h, or if, in other words, the hypothesis h
invests the conclusion a with probability P, this may be
ch. iii

FUNDAMENTAL IDEAS

43
written aPh. It may also be written a/h = P.

This latter expression, which proves to be the more useful of
the two for most purposes, is of fundamental importance. If aPh
and a Ph , i.e. if the probability of a relative to h is the same as the
probability of a relative to h , this may be written a/h = a /h . The
value of the symbol a/h, which represents what is called by other
writers ‘the probability of a,’ lies in the fact that it contains explicit
reference to the data to which the probability relates the conclusion,
and avoids the numerous errors which have arisen out of the omission
of this reference.

CHAPTER IV
the principle of indifference

Absolute. ‘Sure, Sir, this is not very reasonable, to summon my affection
for a lady I know nothing of.’

Sir Anthony. ‘I am sure, Sir, ’tis more unreasonable in you to object to a
lady you know nothing of.’∗

1.

In the last chapter it was assumed that in some cases the
probabilities of two arguments may be equal. It was also argued
that there are other cases in which one probability is, in some sense,
greater than another. But so far there has been nothing to show
how we are to know when two probabilities are equal or unequal.

The recognition of equality, when it exists, will be dealt with in this
chapter, and the recognition of inequality in the next. An historical
account of the various theories about this problem, which have been
held from time to time, will be given in Chapter VII.

2.

The determination of equality between probabilities has
received hitherto much more attention than the determination of
inequality. This has been due to the stress which has been laid
on the mathematical side of the subject. In order that numerical
measurement may be possible, we must be given a number of
equally probable alternatives.

The discovery of a rule, by which
equiprobability could be established, was, therefore, essential. A
rule, adequate to the purpose, introduced by James Bernoulli, who
was the real founder of mathematical probability,† has been widely
adopted, generally under the title of The Principle of Non-Sufficient

Reason, down to the present time. This description is clumsy and
unsatisfactory, and, if it is justifiable to break away from tradition, I
prefer to call it The Principle of Indifference.

∗Quoted by Mr. Bosanquet with reference to the Principle of Non-Sufficient

Reason.

†See also Chap. VII.

44
ch. iv

FUNDAMENTAL IDEAS

45

The Principle of Indifference asserts that if there is no known
reason for predicating of our subject one rather than another of
several alternatives, then relatively to such knowledge the assertions
of each of these alternatives have an equal probability. Thus equal
probabilities must be assigned to each of several arguments, if there
is an absence of positive ground for assigning unequal ones.

This rule, as it stands, may lead to paradoxical and even
contradictory conclusions. I propose to criticise it in detail, and then
to consider whether any valid modification of it is discoverable. For
several of the criticisms which follow I am much indebted to Von

Kries’s Die Principien der Wahrscheinlichkeit.∗

3.

If every probability was necessarily either greater than,
equal to, or less than any other, the Principle of Indifference would
be plausible. For if the evidence affords no ground for attributing
unequal probabilities to the alternative predications, it seems to follow
that they must be equal. If, on the other hand, there need be
neither equality nor inequality between probabilities, this method of
reasoning fails. Apart, however, from this objection, which is based
on the arguments of Chapter III., the plausibility of the principle will
be most easily shaken by an exhibition of the contradictions which it
involves. These fall under three or four distinct heads. In §§ 4–9 my
criticism will be purely destructive, and I shall not attempt in these
paragraphs to indicate my own way out of the difficulties.

4.

Consider a proposition, about the subject of which we know
only the meaning, and about the truth of which, as applied to this
subject, we possess no external relevant evidence. It has been held
that there are here two exhaustive and exclusive alternatives—the
truth of the proposition and the truth of its contradictory—while our
knowledge of the subject affords no ground for preferring one to the
other. Thus if a and ¯a are contradictories, about the subject of which

∗Published in 1886. A brief account of Von Kries’s principal conclusions will
be given on p. 96. A useful summary of his book will be found in a review by Meinong, published in the Göttingische gelehrte Anzeigen for 1890 (pp. 56–75).

pt. i

A TREATISE ON PROBABILITY

46
we have no outside knowledge, it is inferred that the probability of
each is 1.∗ In the same way the probabilities of two other propositions,

2
b and c, having the same subject as a, may be each 1. But without

2
having any evidence bearing on the subject of these propositions we
may know that the predicates are contraries amongst themselves, and,
therefore, exclusive alternatives—a supposition which leads by means
of the same principle to values inconsistent with those just obtained.

If, for instance, having no evidence relevant to the colour of this book,
we could conclude that 1 is the probability of ‘This book is red,’ we

2
could conclude equally that the probability of each of the propositions

‘This book is black’ and ‘This book is blue’ is also 1. So that we are

2
faced with the impossible case of three exclusive alternatives all as
likely as not. A defender of the Principle of Indifference might rejoin
that we are assuming knowledge of the proposition: ‘Two different
colours cannot be predicated of the same subject at the same time’;
and that, if we know this, it constitutes relevant outside evidence.

But such evidence is about the predicate, not about the subject.

∗Cf. (e.g.) the well-known passage in Jevons’s Principles of Science, vol. i.

p. 243, in which he assigns the probability 1 to the proposition “A Platythliptic

2

Coefficient is positive.”

Jevons points out, by way of proof, that no other
probability could reasonably be given. This, of course, involves the assumption
that every proposition must have some numerical probability. Such a contention
was first criticised, so far as I am aware, by Bishop Terrot in the Edin. Phil.

Trans. for 1856. It was deliberately rejected by Boole in his last published work
on probability: “It is a plain consequence,” he says (Edin. Phil. Trans. vol. xxi.

p. 624), “of the logical theory of probabilities, that the state of expectation
which accompanies entire ignorance of an event is properly represented, not by
the fraction 1 , but by the indefinite form 0 .”

Jevons’s particular example,

2

0
however, is also open to the objection that we do not even know the meaning
of the subject of the proposition. Would he maintain that there is any sense in
saying that for those who know no Arabic the probability of every statement
expressed in Arabic is even?

How far has he been influenced in the choice
of his example by known characteristics of the predicate ‘positive’ ? Would he
have assigned the probability 1 to the proposition ‘A Platythliptic Coefficient

2
is a perfect cube’ ? What about the proposition ‘A Platythliptic Coefficient is
allogeneous’ ?

ch. iv

FUNDAMENTAL IDEAS

47

Thus the defender of the Principle will be driven on, either to confine
it to cases where we know nothing about either the subject or the
predicate, which would be to emasculate it for all practical purposes,
or else to revise and amplify it, which is what we propose to do
ourselves.

The difficulty cannot be met by saying that we must know and
take account of the number of possible contraries. For the number of
contraries to any proposition on any evidence is always infinite; ¯ab is
contrary to a for all values of b. The same point can be put in a form
which does not involve contraries or contradictories. For example,
a/h = 1 and ab/h = 1, if h is irrelevant both to a and to b, in the

2

2
sense required by the crude Principle of Indifference.∗ It follows from
this that, if a is true, b must be true also. If it follows from the
absence of positive data that ‘A is a red book’ has a probability of 1,

2
and that the probability of ‘A is red’ is also 1, then we may deduce

2
that, if A is red, it must certainly be a book.

We may take it, then, that the probability of a proposition,
about the subject of which we have no extraneous evidence, is not
necessarily 1. Whether or not this conclusion discredits the Principle

2
of Indifference, it is important on its own account, and will help later
on to confute some famous conclusions of Laplace’s school.

5.

Objection can now be made in a somewhat different shape.

Let us suppose as before that there is no positive evidence relating to
the subjects of the propositions under examination which would lead
us to discriminate in any way between certain alternative predicates.

If, to take an example, we have no information whatever as to the
area or population of the countries of the world, a man is as likely
to be an inhabitant of Great Britain as of France, there being no
reason to prefer one alternative to the other.† He is also as likely to

∗a/h stands for ‘the probability of a on hypothesis h.’

†This example raises a difficulty similar to that raised by Von Kries’s
example of the meteor.

Stumpf has propounded an invalid solution of Von

Kries’s difficulty. Against the example proposed here, Stumpf’s solution has less
pt. i

A TREATISE ON PROBABILITY

48
be an inhabitant of Ireland as of France. And on the same principle
he is as likely to be an inhabitant of the British Isles as of France.

And yet these conclusions are plainly inconsistent. For our first two
propositions together yield the conclusion that he is twice as likely to
be an inhabitant of the British Isles as of France.

Unless we argue, as I do not think we can, that the knowledge
that the British Isles are composed of Great Britain and Ireland is a
ground for supposing that a man is more likely to inhabit them than

France, there is no way out of the contradiction. It is not plausible to
maintain, when we are considering the relative populations of different
areas, that the number of names of subdivisions which are within our
knowledge, is, in the absence of any evidence as to their size, a piece
of relevant evidence.

At any rate, many other similar examples could be invented, which
would require a special explanation in each case; for the above is an
instance of a perfectly general difficulty. The possible alternatives
may be a, b, c, and d, and there may be no means of discriminating
between them; but equally there may be no means of discriminating
between (a or b), c, and d. This difficulty could be made striking in a
variety of ways, but it will be better to criticise the principle further
from a somewhat different side.

6.

Consider the specific volume of a given substance.∗ Let us
suppose that we know the specific volume to lie between 1 and 3, but
that we have no information as to whereabouts in this interval its
exact value is to be found. The Principle of Indifference would allow
us to assume that it is as likely to lie between 1 and 2 as between

2 and 3; for there is no reason for supposing that it lies in one interval
rather than in the other. But now consider the specific density. The
specific density is the reciprocal of the specific volume, so that if the
latter is v the former is 1. Our data remaining as before, we know
v
plausibility than against Von Kries’s.

∗This example is taken from Von Kries, op. cit. p. 24. Von Kries does not
seem to me to explain correctly how the contradiction arises.

ch. iv

FUNDAMENTAL IDEAS

49
that the specific density must lie between 1 and 1, and, by the same

3
use of the Principle of Indifference as before, that it is as likely to
be between 1 and 2 as between 2 and 1. But the specific volume

3

3

3
being a determinate function of the specific density, if the latter lies
between 1 and 2, the former lies between 1 and 11, and if the latter

3

2
lies between 2 and 1, the former lies between 11 and 3. It follows,

3

3

2
therefore, that the specific volume is as likely to lie between 1 and 112
as between 11 and 3; whereas we have already proved, relatively to

2
precisely the same data, that it is as likely to lie between 1 and 2
as between 2 and 3. Moreover, any other function of the specific
volume would have suited our purpose equally well, and by a suitable
choice of this function we might have proved in a similar manner
that any division whatever of the interval 1 to 3 yields sub-intervals
of equal probability. Specific volume and specific density are simply
alternative methods of measuring the same objective quantity; and
there are many methods which might be adopted, each yielding on
the application of the Principle of Indifference a different probability
for a given objective variation in the quantity.∗

∗A. Nitsche (“Die Dimensionen der Wahrscheinlichkeit und die Evidenz der

Ungewissheit,” Vierteljahrsschr. f. wissensch. Philos. vol. xvi. p. 29, 1892), in
criticising Von Kries, argues that the alternatives to which the principle must
be applied are the smallest physically distinguishable intervals, and that the
probability of the specific volume’s lying within a certain range of values turns
on the number of such distinguishable intervals in the range. This procedure
might conceivably provide the correct method of computation, but it does not
therefore restore the credit of the Principle of Indifference.

For it is argued,
not that the results of applying the principle are always wrong, but that it
does not lead unambiguously to the correct procedure. If we do not know the
number of distinguishable intervals we have no reason for supposing that the
specific volume lies between 1 and 2 rather than 2 and 3, and the principle can
therefore be applied as it has been applied above.

And even if we do know
the number and reckon intervals as equal which contain an equal number of

‘physically distinguishable’ parts, is it certain that this does not simply provide
us with a new system of measurement, which has the same conventional basis
as the methods of specific volume and specific density, and is no more the one
pt. i

A TREATISE ON PROBABILITY

50

The arbitrary nature of particular methods of measurement of
this and of many other physical quantities is easily explained.

The objective quality measured may not, strictly speaking, possess
numerical quantitativeness, although it has the properties necessary
for measurement by means of correlation with numbers. The values
which it can assume may be capable of being ranged in an order,
and it will sometimes happen that the series which is thus formed
is continuous, so that a value can always be found whose order in
the series is between any two selected values; but it does not follow
from this that there is any meaning in the assertion that one value
is twice another value. The relations of continuous order can exist
between the terms of a series of values, without the relations of
numerical quantitativeness necessarily existing also, and in such cases
we can adopt a largely arbitrary measure of the successive terms,
which yields results which may be satisfactory for many purposes,
those, for instance, of mathematical physics, though not for those of
probability. This method is to select some other series of quantities
or numbers, each of the terms of which corresponds in order to one
and only one of the terms of the series which we wish to measure.

For instance, the series of characteristics, differing in degree, which
are measured by specific volume, have this relation to the series
of numerical ratios between the volumes of equal masses of the
substances, the specific volumes of which are in question, and of
water. They have it also to the corresponding ratios which give rise
to the measure of specific density. But these only yield conventional
measurements, and the numbers with which we correlate the terms
which we wish to measure can be selected in a variety of ways. It
follows that equal intervals between the numbers which represent the
ratios do not necessarily correspond to equal intervals between the
qualities under measurement; for these numerical differences depend
correct measure than these are?

ch. iv

FUNDAMENTAL IDEAS

51
upon which convention of measurement we have selected.

7.

A somewhat analogous difficulty arises in connection with
the problems of what is known as ‘geometrical’ or ‘local’ probability.∗

In these problems we are concerned with the position of a point or
infinitesimal area or volume within a continuum.† The number of
cases here is indefinite, but the Principle of Indifference has been
held to justify the supposition that equal lengths or areas or volumes
of the continuum are, in the absence of discriminating evidence,
equally likely to contain the point. It has long been known that
this assumption leads in numerous cases to contradictory conclusions.

If, for instance, two points A and A are taken at random on the
surface of a sphere, and we seek the probability that the lesser of the
two arcs of the great circle AA is less than a, we get one result by
assuming that the probability of a point’s lying on a given portion of
the sphere’s surface is proportional to the area of that portion, and
another result by assuming that, if a point lies on a given great circle,
the probability of its lying on a given arc of that circle is proportional
to the length of the arc, each of these assumptions being equally
justified by the Principle of Indifference.

Or consider the following problem: if a chord in a circle is drawn
at random, what is the probability that it will be less than the side
of the inscribed equilateral triangle. One can argue:—

∗The best accounts of this subject are to be found in Czuber, Geometrische

Wahrscheinlichkeiten
und

Mittelwerte;

Czuber,

Wahrscheinlichkeitsrechnung,
vol. i. pp. 75–109; Crofton, Encycl. Brit. (9th edit.), article ‘Probability’; Borel,

Eléments de la théorie des probabilités, chaps. vi.–viii.; a few other references
are given in the following pages, and a number of discussions of individual
problems will be found in the mathematical volumes of the Educational Times.

The interest of the subject is primarily mathematical, and no discussion of its
principal problems will be attempted here.

†As Czuber points out (Wahrscheinlichkeiterechnung, vol. i. p. 84), all
problems, whether geometrical or arithmetical, which deal with a continuum
and with non-enumerable aggregates, are commonly discussed under the name
of ‘geometrical probability.’ See also Lämmel, Untersuchungen.

pt. i

A TREATISE ON PROBABILITY

52

(a) It is indifferent at what point one end of the chord lies. If
we suppose this end fixed, the direction is then chosen at
random. In this case the answer is easily shown to be 2.

3

(b) It is indifferent in what direction we suppose the chord
to lie. Beginning with this apparently not less justifiable
assumption, we find that the answer is 1.

2

(c) To choose a chord at random, one must choose its middle
point at random. If the chord is to be less than the side of
the inscribed equilateral triangle, the middle point must be
at a greater distance from the centre than half the radius.

But the area at a greater distance than this is 3 of the

4
whole. Hence our answer is 3.∗

4

In general, if x and f(x) are both continuous variables, varying
always in the same or in the opposite sense, and x must lie between
a and b, then the probability that x lies between c and d, where
a < c < d < b, seems to be d − c, and the probability that f (x) lies
b − a
between f(c) and f(d) to be f(d) − f(c). These expressions, which
f (b) − f (a)

represent the probabilities of necessarily concordant conclusions, are
not, as they ought to be, equal.†

8.

More than one attempt has been made to separate the cases
in which the Principle of Indifference can be legitimately applied to
examples of geometrical probability from those in which it cannot.

M. Borel argues that the mathematician can define the geometrical
probability that a point M lies on a certain segment PQ of AD as
proportional to the length of the segment, but that this definition is
conventional until its consequences have been confirmed à posteriori
by their conformity with the results of empirical observation. He
points out that in actual cases there are generally some considerations
present which lead us to prefer one of the possible assumptions to the

∗Bertrand, Calcul des probabilités, p. 5.

†See (e.g.) Borel, Éléments de la théorie des probabilités, p. 85.

ch. iv

FUNDAMENTAL IDEAS

53
others. Whether or not this is so, the proposed procedure amounts to
an abandonment of the Principle of Indifference as a valid criterion,
and leaves our choice undetermined when further evidence is not
forthcoming.

M. Poincaré, who also held that judgments of equiprobability in
such cases depend upon a ‘convention,’ endeavoured to minimise the
importance of the arbitrary element by showing that, under certain
conditions, the result is independent of the particular convention
which is chosen. Instead of assuming that the point is equally likely to
lie in every infinitesimal interval dx we may represent the probability
of its lying in this interval by the function φ(x) dx. M. Poincaré
showed that, in the game of rouge et noir, for instance, where
we have a number of compartments arranged in a circle coloured
alternately black and white, if we can assume that φ(x) is a regular
function, continuous and with continuous differential coefficients, then,
whatever the particular form of the function, the probability of black
is approximately equal to that of white.∗

Whether or not investigations on these lines prove to have a
practical value, they have not, I think, any theoretical importance.

If, as I maintain, the probability φ(x) is not necessarily numerical,
it is not a generally justifiable assumption to take its continuity
for granted. We have, in the particular example quoted, a number
of alternatives, half of which lead to black and half to white; the
assumption of continuity amounts to the assumption that for every
white alternative there is a black alternative whose probability is very
nearly equal to that of the white. Naturally in such a case we can
get an approximately equal probability for the whites as a whole and
for the blacks as a whole, without assuming equal probability for
each alternative individually. But this fact has no bearing on the
theoretical difficulties which we are discussing.

M. Bertrand is so much impressed by the contradictions of

∗Poincaré, Calcul des probabilités, pp. 126 et seq.

pt. i

A TREATISE ON PROBABILITY

54
geometrical probability that he wishes to exclude all examples in
which the number of alternatives is infinite.∗ It will be argued in the
sequel that something resembling this is true. The discussion of this
question will be resumed in §§ 21–25.

9.

There is yet another group of cases, distinct in character
from those considered so far, in which the principle does not seem to
provide us with unambiguous guidance. The typical example is that
of an urn containing black and white balls in an unknown proportion.†

The Principle of Indifference can be claimed to support the most
usual hypothesis, namely, that all possible numerical ratios of black
and white are equally probable. But we might equally well assume
that all possible constitutions‡ of the system of balls are equally
probable, so that each individual ball is assumed equally likely to
be black or white. It would follow from this that an approximately
equal number of black and white balls is more probable than a large
excess of one colour. On this hypothesis, moreover, the drawing of
one ball and the resulting knowledge of its colour leaves unaltered the
probabilities of the various possible constitutions of the rest of the
bag; whereas on the first hypothesis knowledge of the colour of one
ball, drawn and not replaced, manifestly alters the probability of the
colour of the next ball to be drawn. Either of these hypotheses seems

∗Bertrand, Calcul des probabilités, p. 4: “L’infini n’est pas un nombre; on
ne doit pas, sans explication, l’introduire dans les raisonnements. La précision
illusoire des mots pourrait faire naitre des contradictions.

Choisir au hasard,
entre un nombre infini de cas possibles, n’est pas une indication suffisante.”

†The difficulty in question was first pointed out by Boole, Laws of Thought,
pp. 369–370. After discussing the Law of Succession, Boole proceeds to show
that “there are other hypotheses, as strictly involving the principle of the ‘equal
distribution of knowledge or ignorance’ which would also conduct to conflicting
results.” See also Von Kries, op. cit. pp. 31–34, 59, and Stumpf, Über den Begriff
der mathematischen Wahrscheinlichkeit, Bavarian Academy, 1892, pp. 64–68.

‡If A and B are two balls, A white, B black, and A black, B white, are
different ‘constitutions.’ But if we consider different numerical ratios, these two
cases are indistinguishable, and count as one only.

ch. iv

FUNDAMENTAL IDEAS

55
to satisfy the Principle of Indifference, and a believer in the absolute
validity of the principle will doubtless adopt that one which enters
his mind first.∗

The same point is very clearly illustrated by an example which

I take from Von Kries. Two cards, chosen from different packs, are
placed face downwards on the table; one is taken up and found to be
of a black suit: what is the chance that the other is black also? One
would naturally reply that the chance is even. But this is based on
the supposition, relatively unpopular with writers on the subject, that
every ‘constitution’ is equally probable, i.e. that each individual card
is as likely to be black as red. If we prefer this assumption, we must
relinquish the text-book theory that the drawing of a black ball from
an urn, containing black and white balls in unknown proportions,
affects our knowledge as to the proportion of black and white amongst
the remaining balls.

The alternative—or text-book—theory assumes that there are
three equal possibilities—one of each colour, both black, both red.

If both cards are black, we are twice as likely to turn up a black
card than if only one is black. After we have turned up a black, the
probability that the other is black is, therefore, twice as great as the
probability that it is red. The chance of the second’s being black is
therefore 2.† The Principle of Indifference has nothing to say against

3
either solution. Until some further criterion has been proposed we
seem compelled to agree with Poincaré that a preference for either

∗C. S. Peirce in his Theory of Probable Inference (Johns Hopkins Studies in

Logic), pp. 172, 173, argues that the ‘constitution’ hypothesis is alone valid,
on the ground that, of the two hypotheses, only this one is consistent with
itself. I agree with his conclusion, and shall give at the close of the chapter the
fundamental considerations which lead to the rejection of the ‘ratio’ hypothesis.

Stumpf points out that the probability of drawing a white ball is, in any case, 1 .

2

This is true; but the probability of a second white clearly depends upon which
of the two hypotheses has been preferred. Nitsche (loc. cit. p. 31) seems to miss
the point of the difficulty in the same way.

†This is Poisson’s solution, Recherches, p. 96.

pt. i

A TREATISE ON PROBABILITY

56
hypothesis is wholly arbitrary.

10.

Such, then, are the kinds of result to which an unguarded
use of the Principle of Indifference may lead us. The difficulties, to
which attention has been drawn, have been noticed before; but the
discredit has not been emphatically thrown on the original source
of error. Yet the principle certainly remains as a negative criterion;
two propositions cannot be equally probable, so long as there is any
ground for discriminating between them. The principle is a necessary,
but not, as it seems, a sufficient condition.

The enunciation of some sufficient rule is certainly essential if
we are to make any progress in the subject. But the difficulty
of discovering a correct principle is considerable. This difficulty is
partly responsible, I think, for the doubts which philosophers and
many others have often felt regarding any practical application of the

Calculus. Many candid persons, when confronted with the results of

Probability, feel a strong sense of the uncertainty of the logical basis
upon which it seems to rest. It is difficult to find an intelligible
account of the meaning of ‘probability,’ or of how we are ever to
determine the probability of any particular proposition; and yet
treatises on the subject profess to arrive at complicated results of the
greatest precision and the most profound practical importance.

The incautious methods and exaggerated claims of the school of

Laplace have undoubtedly contributed towards the existence of these
sentiments. But the general scepticism, which I believe to be much
more widely spread than the literature of the subject admits, is more
fundamental. In this matter Hume need not have felt “affrighted
and confounded with that forelorn solitude, in which I am placed
in my philosophy,” or have fancied himself “some strange uncouth
monster, who not being able to mingle and unite in society, has
been expell’d all human commerce, and left utterly abandon’d and
disconsolate.” In his views on probability, he stands for the plain man
against the sophisms and ingenuities of “metaphysicians, logicians,
mathematicians, and even theologians.”

ch. iv

FUNDAMENTAL IDEAS

57

Yet such scepticism goes too far. The judgments of probability,
upon which we depend for almost all our beliefs in matters of
experience, undoubtedly depend on a strong psychological propensity
in us to consider objects in a particular light. But this is no ground
for supposing that they are nothing more than “lively imaginations.”

The same is true of the judgments in virtue of which we assent to
other logical arguments; and yet in such cases we believe that there
may be present some element of objective validity, transcending the
psychological impulsion, with which primarily we are presented. So
also in the case of probability, we may believe that our judgments
can penetrate into the real world, even though their credentials are
subjective.

11.

We must now inquire how far it is possible to rehabilitate
the Principle of Indifference or find a substitute for it. There are
several distinct difficulties which need attention in a discussion of the
problems raised in the preceding paragraphs. Our first object must
be to make the Principle itself more precise by disclosing how far its
application is mechanical and how far it involves an appeal to logical
intuition.

12.

Without compromising the objective character of relations of
probability, we must nevertheless admit that there is little likelihood
of our discovering a method of recognising particular probabilities,
without any assistance whatever from intuition or direct judgment.

Inasmuch as it is always assumed that we can sometimes judge directly
that a conclusion follows from a premiss, it is no great extension
of this assumption to suppose that we can sometimes recognise
that a conclusion partially follows from, or stands in a relation of
probability to, a premiss. Moreover, the failure to explain or define

‘probability’ in terms of other logical notions, creates a presumption
that particular relations of probability must be, in the first instance,
directly recognised as such, and cannot be evolved by rule out of data
which themselves contain no statements of probability.

On the other hand, although we cannot exclude every element
pt. i

A TREATISE ON PROBABILITY

58
of direct judgment, these judgments may be limited and controlled,
perhaps, by logical rules and principles which possess a general
application. While we may possess a faculty of direct recognition of
many relations of probability, as in the case of many other logical
relations, yet some may be much more easily recognisable than others.

The object of a logical system of probability is to enable us to know
the relations, which cannot be easily perceived, by means of other
relations which we can recognise more distinctly—to convert, in fact,
vague knowledge into more distinct knowledge.∗

13.

Let us seek to distinguish between the element of direct
judgment and the element of mechanical rule in the Principle of

Indifference. The enunciation of this principle, as it is ordinarily
expressed, cloaks, but does not avoid, the former element. It is in
part a formula and in part an appeal to direct inspection; but in
addition to the obscurity and ambiguity of the formula, the appeal
to intuition is not as explicit as it should be. The principle states
that ‘there must be no known reason for preferring one of a set of
alternatives to any other.’ What does this mean? What are ‘reasons,’

and how are we to know whether they do or do not justify us in
preferring one alternative to another? I do not know any discussion
of Probability in which this question has been so much as asked. If,
for example, we are considering the probability of drawing a black

∗As it is the aim of trigonometry to determine the position of an object,
which is in a sense visible, not by a direct observation of it, but by observing
some other object together with certain relations, so an indirect method of this
kind is the aim of all logical system. If the truth of some propositions, and the
validity of some arguments, could not be recognised directly, we could make no
progress. We may have, moreover, some power of direct recognition where it is
not necessary in our logical system that we should make use of it.

In these
cases the method of logical proof increases the certainty of knowledge, which we
might be able to possess in a more doubtful manner without it. In other cases,
that, for instance, of a complicated mathematical theorem, it enables us to know
propositions to be true, which are altogether beyond the reach of our direct
insight; just as we can often obtain knowledge about the position of a partially
visible or even invisible object by starting with observations of other objects.

ch. iv

FUNDAMENTAL IDEAS

59
ball from an urn containing balls which are black and white, we
assume that the difference of colour between the balls is not a reason
for preferring either alternative. But how do we know this, unless
by a judgment that, on the evidence in hand, our knowledge of the
colours of the balls is irrelevant to the probability in question? We
know of some respects in which the alternatives differ; but we judge
that a knowledge of these differences is not relevant. If, on the other
hand, we were taking the balls out of the urn with a magnet, and
knew that the black balls were of iron and the white of tin, we might
regard the fact, that a ball was iron and not tin, as very important
in determining the probability of its being drawn. Before, then, we
can begin to apply the Principle of Indifference, we must have made a
number of direct judgments to the effect that the probabilities under
consideration are unaffected by the inclusion in the evidence of certain
particular details. We have no right to say of any known difference
between the two alternatives that it is ‘no reason’ for preferring one
of them, unless we have judged that a knowledge of this difference is
irrelevant to the probability in question.

14.

A brief digression is now necessary, in order to introduce some
new terms. There are in general two principal types of probabilities,
the magnitudes of which we seek to compare,—those in which the
evidence is the same and the conclusions different, and those in
which the evidence is different but the conclusion the same. Other
types of comparison may be required, but these two are by far the
commonest. In the first we compare the likelihood of two conclusions
on given evidence; in the second we consider what difference a change
of evidence makes to the likelihood of a given conclusion. In symbolic
language we may wish to compare x/h with y/h, or x/h with x/h1h.

We may call the first type judgments of preference, or, when there is
equality between x/h and y/h, of indifference; and the second type we
may call judgments of relevance, or, when there is equality between
x/h and x/h1h, of irrelevance. In the first we consider whether or
not x is to be preferred to y on evidence h; in the second we consider
pt. i

A TREATISE ON PROBABILITY

60
whether the addition of h1 to evidence h is relevant to x.

The Principle of Indifference endeavours to formulate a rule which
will justify judgments of indifference. But the rule that there must
be no ground for preferring one alternative to another, involves, if it
is to be a guiding rule at all, and not a petitio principii, an appeal to
judgments of irrelevance.

The simplest definition of Irrelevance is as follows: h1 is irrelevant
to x on evidence h, if the probability of x on evidence hh1 is the
same as its probability on evidence h.∗ But for a reason which will
appear in Chapter VI., a stricter and more complicated definition, as
follows, is theoretically preferable: h1 is irrelevant to x on evidence h,
if there is no proposition, inferrible from h1h but not from h, such
that its addition to evidence h affects the probability of x.† Any
proposition which is irrelevant in the strict sense is, of course, also
irrelevant in the simpler sense; but if we were to adopt the simpler
definition, it would sometimes occur that a part of evidence would be
relevant, which taken as a whole was irrelevant. The more elaborate
definition by avoiding this proves in the sequel more convenient. If
the condition x/h1h = x/h alone is satisfied, we may say that the
evidence h1 is ‘irrelevant as a whole.’‡

It will be convenient to define also two other phrases. h1 and h2
are independent and complementary parts of the evidence, if between
them they make up h and neither can be inferred from the other. If
x is the conclusion, and h1 and h2 are independent and complementary
parts of the evidence, then h1 is relevant if the addition of it to h2
affects the probability of x.§

∗That is to say, h1 is irrelevant to x/h if x/h1h = x/h.

†That is to say, h1 is irrelevant to x/h, if there is no propsition h such

1
that h /h

/h =

| 1, and x/h h =

| x/h.

1

1h = 1, h1

1

‡Where no misunderstanding can arise, the qualification ‘as a whole’ will be
sometimes omitted.

§I.e. (in symbolism) h1 and h2 are independent and complementary parts
of h if h1h2 = h, h1/h2 =

| 1, and h2/h1 =

| 1. Also h1 is relevant if x/h =

| x/h2.

ch. iv

FUNDAMENTAL IDEAS

61

Some propositions regarding irrelevance will be proved in Part II.

If ¯h1 is the contradictory of h1 and x/h1h = x/h, then x/¯h1h = x/h.

Thus the contradictory of irrelevant evidence is also irrelevant. Also,
if x/yh = x/h, it follows that y/xh = y/h. Hence if, on initial
evidence h, y is irrelevant to x, then, on the same initial evidence,
x is irrelevant to y, i.e. if in a given state of knowledge one occurrence
has no bearing on another then equally the second has no bearing on
the first.

15.

This distinction enables us to formulate the Principle of

Indifference at any rate more precisely. There must be no relevant
evidence relating to one alternative, unless there is corresponding
evidence relating to the other; our relevant evidence, that is to say,
must be symmetrical with regard to the alternatives, and must be
applicable to each in the same manner. This is the rule at which
the Principle of Indifference somewhat obscurely aims. We must first
determine what parts of our evidence are relevant on the whole by
a series of judgments of relevance, not easily reduced to rule, of the
type described above. If this relevant evidence is of the same form
for both alternatives, then the Principle authorises a judgment of
indifference.

16.

This rule can be expressed more precisely in symbolic
language.

Let us assume, to begin with, that the alternative
conclusions are expressible in the forms φ(a) and φ(b), where φ(x) is
a propositional function.∗ The difference between them, that is to
say, can be represented in terms of a single variable.

The Principle of Indifference is applicable to the alternatives
φ(a) and φ(b), when the evidence h is so constituted that, if f (a) is
an independent part of h (see § 14) which is relevant to φ(a), and
does not contain any independent parts which are irrelevant to φ(a),
then h includes f(b) also.

The rule can be extended by successive steps to cases in which

∗If φ(a), φ(b), etc., are propositions, and x is a variable, capable of taking
the values a, b, etc, then φ(x) is a propositional function.

pt. i

A TREATISE ON PROBABILITY

62
we have more than one variable.

We can, if the necessary
conditions are fulfilled, successively compare the probabilities of
φ(a1a2) and φ(b1a2), and of φ(b1a2) and φ(b1b2) and establish equality
between φ(a1a2) and φ(b1b2).

This elucidation is suited to most of the cases to which the

Principle of Indifference is ordinarily applied. Thus in the favourite
examples in which balls are drawn from urns, we can infer from
our evidence no relevant proposition about white balls, such that we
cannot infer a corresponding proposition about black balls. Most of
the examples, to which the mathematical theory of chances has been
applied, and which depend upon the Principle of Indifference, can be
arranged, I think, in the forms which the rule requires as formulated
above.

17.

We can now clear up the difficulties which arose over
the group of cases dealt with in § 9, the typical example of which
was the problem of the urn containing black and white balls in an
unknown proportion. This more precise enunciation of the Principle
enables us to show that of the two solutions the equiprobability
of each ‘constitution’ is alone legitimate, and the equiprobability of
each numerical ratio erroneous. Let us write the alternative ‘The
proportion of black balls is x’ ≡ φ(x), and the datum ‘There are
n balls in the bag, with regard to none of which it is known whether
they are black or white’ ≡ h. On the ‘ratio’ hypothesis it is argued
that the Principle of Indifference justifies the judgment of indifference,
φ(x)/h = φ(y)/h. In order that this may be valid, it must be possible
to state the relevant evidence in the form f(x) f(y). But this is not
the case. If x = 1 and y = 1, we have relevant knowledge about the

2

4
way in which a proportion of black balls of one half can arise, which
is not identical with our knowledge of the way in which a proportion
of one quarter can arise. If there are four balls, A, B, C, D, one half
are black, if A, B or A, C or A, D or B, C or B, D or C, D are black;
and one quarter are black, if A or B or C or D are black. These
propositions are not identical in form, and only by a false judgment
ch. iv

FUNDAMENTAL IDEAS

63
of irrelevance can we ignore them. On the ‘constitution’ hypothesis,
however, where A, B black and A, C black are treated as distinct
alternatives, this want of symmetry in our relevant evidence cannot
arise.

18.

We can also deal with the point which was illustrated by
the difficulty raised in § 4. We considered there the probabilities of a
and its contradictory ¯a when there is no external evidence relevant
to either. What exactly do we mean by saying that there is no
relevant evidence? Is the addition of the word external significant? If
a represents a particular proposition, we must know something about
it, namely, its meaning. May not the apprehension of its meaning
afford us some relevant evidence? If so, such evidence must not be
excluded. If, then, we say that there is no relevant evidence, we must
mean no evidence beyond what arises from the mere apprehension of
the meaning of the symbol a. If we attach no meaning to the symbol,
it is useless to discuss the value of the probability; for the probability,
which belongs to a proposition as an object of knowledge, not as a
form of words, cannot in such a case exist.

What exactly does the symbol a stand for in the above? Does it
stand for any proposition of which we know no more than that it is
a proposition? Or does it stand for a particular proposition which
we understand but of which we know no more than is involved in
understanding it? In the former case we cannot extend our result
to a proposition of which we know even the meaning; for we should
then know more than that it is a proposition; and in the latter case
we cannot say what the probability of a is as compared with that of
its contradictory, until we know what particular proposition it stands
for; for, as we have seen, the proposition itself may supply relevant
evidence.

This suggests that a source of much confusion may lie in the use
of symbols and the notion of variables in probability. In the logic of
implication, which deals not with probability but with truth, what is
true of a variable must be equally true of all instances of the variable.

pt. i

A TREATISE ON PROBABILITY

64

In Probability, on the other hand, we must be on our guard wherever
a variable occurs. In Implication we may conclude that ψ is true of
anything of which φ is true. In Probability we may conclude no more
than that ψ is probable of anything of which we only know that φ is
true of it. If x stands for anything of which φ(x) is true, as soon
as we substitute in probability any particular value, whose meaning
we know, for x, the value of the probability may be affected; for
knowledge, which was irrelevant before, may now become relevant.

Take the following example: Does φ(a)/ψ(a) = φ(b)/ψ(b)? That is to
say, is the probability of φ’s being true of a, given only that ψ is true
of a, equal to the probability of φ’s being true of b, given only that
ψ is true of b? If this simply means that the probability of an object’s
satisfying φ about which nothing is known except that it satisfies ψ
is equal to ditto ditto, the equation is an identity. For in this case
φ(a)/ψ(a) means the same as φ(b)/ψ(b), i.e. we know nothing about
x and y except that they satisfy ψ, and there is nothing whatever
by which we can distinguish a from b. But if a and b represent
specific entities, which we can distinguish, then the equality does not
necessarily hold. If, for instance, φ(x) stands for ‘x is Socrates,’ then
it is plainly false that φ(a)/ψ(a) = φ(b)/ψ(b), where a stands for

Socrates and b does not.

19.

Bearing this danger in mind, we can now give further
precision to the enunciation of the Principle of Indifference given in

§ 16. Our knowledge of the meaning of a must be taken account
of so far as it is relevant; and the Principle is only satisfied if
we have corresponding knowledge about the meaning of b. Thus
φ(a)/h = φ(b)/h may be true for one pair of values a, b, and not true
for another pair of values a , b .

This makes it possible to explain in part the contradiction
discussed in § 4. Even if it were true that the probability of a is 1,

2
when we know nothing except that a is a proposition, it does not
follow that the probability of ‘This book is red’ is 1, when we know

2
the meanings of ‘book’ and ‘red,’ even if we know no more than this.

ch. iv

FUNDAMENTAL IDEAS

65

Knowledge arising directly out of acquaintance with the meaning of

‘red’ may be sufficient to enable us to infer that ‘red’ and ‘not-red’

are not satisfactory alternatives to which to apply the Principle of

Indifference. How this may come about will be discussed in §§ 20, 21.

But the contradictions are not yet really solved; for some of the
difficulties discussed in § 4 can arise even when we know no more of a
and b than that they are different propositions. In fact, although we
have now stated more clearly than before how the Principle should
be enunciated, it is not yet possible to explain or to avoid all the
contradictions to which it led us in §§ 4 to 7. For this purpose we
must proceed to a further qualification.

20.

The examples, in which the Principle of Indifference broke
down, had a great deal in common. We broke up the field of
possibility, as we may term it, into a number of areas by a series of
disjunctive judgments. But the alternative areas were not ultimate.

They were capable of further subdivision into other areas similar in
kind to the former. The paradoxes and contradictions arose, in each
case, when the alternatives, which the Principle of Indifference treated
as equivalent, actually contained or might contain a different or an
indefinite number of more elementary units.

In the type of cases in which the Principle of Indifference seemed
to permit the assertion that, in the absence of relevant evidence, a
proposition is as likely as its contradictory, its contradictory is not
an ultimate and indivisible alternative (in the sense to be explained
in § 21 below), even if the proposition itself satisfies this condition.

For its contradictory can be disjunctively resolved into an indefinite
number of sets of contraries to the proposition. It was out of this that
our difficulties first arose. ‘This book is not red’ includes amongst
others the alternatives ‘This book is black’ and ‘This book is blue.’

It is not, therefore, an ultimate alternative.

In the same way the contradiction of § 5 arose out of the possibility
of splitting the alternatives ‘He inhabits the British Isles’ into the
sub-alternatives ‘He inhabits Ireland or he inhabits Great Britain.’

pt. i

A TREATISE ON PROBABILITY

66

And in the third type of case, to which the example of specific volume
and density belongs, the alternative ‘v lies in the interval 1 to 2’ can
be broken up into the sub-alternatives ‘v lies in the interval 1 to 112
or 11 to 2.’

2

21.

This, then, seems to point the way to the qualification of
which we are in search. We must enunciate some formal rule which
will exclude those cases, in which one of the alternatives involved is
itself a disjunction of sub-alternatives of the same form. For this
purpose the following condition is proposed.

Let the alternatives, the equiprobability of which we seek to
establish by means of the Principle of Indifference, be φ(a1),
φ(a2) . . . φ(ar),∗ and let the evidence be h. Then it is a necessary
condition for the application of the principle, that these should be,
relatively to the evidence, indivisible alternatives of the form φ(x).

We may define a divisible alternative in the following manner:

An alternative φ(ar) is divisible if

(i.) [φ(ar) ≡ φ(ar ) + φ(ar )]/h = 1,

(ii.) φ(ar ) φ(ar )/h = 0,

(iii.) φ(ar )/h =| 0 and φ(ar )/h =| 0

The condition that the sub-alternatives must be of the same form
as the original alternatives, i.e. expressible by means of the same
propositional function φ(x), deserves attention. It might be the case
that the original alternatives had nothing substantial in common; i.e.

φ(x) ≡ (x = x) is the only propositional function common to all of
them, the alternatives being a1, a2, . . . , ar. In these circumstances
the condition in question cannot be satisfied. For the proposition ar
can always be resolved into the disjunction a

¯
r b + ar b, where b is
any proposition and ¯b its contradictory. If, on the other hand, the

∗The more complicated cases in which the propositional function, of which
the alternatives are instances, involves more than one variable (see § 16), can be
dealt with in a similar manner mutatis mutandis.

ch. iv

FUNDAMENTAL IDEAS

67
alternatives which we are comparing can be expressed in the forms
φ(a1) and φ(a2), where the function φ(x) is distinct from x, it is
not necessarily the case that either of these can be resolved into a
disjunctive combination of terms which can be expressed in their turn
in the same form.

Dispensing with symbolism, we can express these conditions
as follows: Our knowledge must not enable us to split up the
alternative φ(ar) into a disjunction of two sub-alternatives, (i.) which
are themselves expressible in the same form φ, (ii.) which are mutually
exclusive, and (iii.) which, on the evidence, are possible.

In short, the Principle of Indifference is not applicable to a pair
of alternatives, if we know that either of them is capable of being
further split up into a pair of possible but incompatible alternatives
of the same form as the original pair.

22.

This rule commends itself to common sense. If we know
that the two alternatives are compounded of a different number or of
an indefinite number of sub-alternatives which are in other respects
similar, so far as our evidence goes to the original alternatives, then
this is a relevant fact of which we must take account. And as it
affects the two alternatives in differing and unsymmetrical ways, it
breaks down the fundamental condition for the valid application of
the Principle of Indifference.

Neither this consideration nor that discussed in §§ 18 and 19
substantially modify the Principle of Indifference as enunciated in

§ 16. They have only served to make explicit what was always
implicit in the Principle, by explaining the manner in which our
knowledge of the form and meaning of the alternatives may be a
relevant part of the evidence. The apparent contradictions arose from
paying attention to what we may term the extraneous evidence only,
to the neglect of such part of the evidence as bore upon the form and
meaning of the alternatives.

23.

The application of this result to the examples cited in § 18
is not difficult. It excludes the class of cases in which a proposition
pt. i

A TREATISE ON PROBABILITY

68
and its contradictory constitute the alternatives. For if b is the
proposition and ¯b its contradictory, we cannot find a propositional
function φ(x) which will satisfy the necessary conditions. It deals
also with the type of contradiction which arose in considering the
probability that an individual taken at random was an inhabitant of
a given region. If, on the other hand, the term ‘country’ is so defined
that one country cannot include two countries, then an individual is,
relatively to suitable hypotheses, as likely to be an inhabitant of one
as of another. For the function φ(x), where φ(x) ≡ ‘the individual
is an inhabitant of country x,’ satisfies the conditions. And it deals
with the example of ranges of specific volume and specific density,
because there is no range which does not contain within itself two
similar ranges. As there are in this case no definite units by which we
can define equal ranges, the device, which will be referred to in § 25
for dealing with geometrical probabilities, is not available.

24.

It is worth while to add that the qualification of § 21 is fatal
to the practical utility of the Principle of Indifference in those cases
only in which it is possible to find no ultimate alternatives which
satisfy the conditions. For if the original alternatives each comprise
a definite number of indivisible and indifferent sub-alternatives, we
can compute their probabilities. It is often the case, however, that
we cannot by any process of finite subdivision arrive at indivisible
sub-alternatives, or that, if we can, they are not on the evidence
indifferent. In the examples given above, for instance, where φ(x) ≡ x,
or where x is a part of unspecified magnitude in a continuum, there
are no indivisible sub-alternatives. The first type comprises all cases,
amongst others, in which we weigh the probabilities of a proposition
and its contradictory; and the second includes a great number of
cases in which physical or geometrical quantities are involved.

25.

We can now return to the numerous paradoxes which arise
in the study of geometrical probability (see §§ 7, 8). The qualification
of § 21 enables us, I think, to discover the source of the confusion.

Our alternatives in these problems relate to certain areas or segments
ch. iv

FUNDAMENTAL IDEAS

69
or arcs, and however small the elements are which we adopt as our
alternatives, they are made up of yet smaller elements which would
also serve as alternatives. Our rule, therefore, is not satisfied, and,
as long as we enunciate them in this shape, we cannot employ the

Principle of Indifference. But it is easy in most cases to discover
another set of alternatives which do satisfy the condition, and which
will often serve our purpose equally well. Suppose, for instance, that
a point lies on a line of length m.l., we may write the alternative

‘the interval of length l on which the point lies is the xth interval
of that length as we move along the line from left to right’ ≡ φ(x);
and the Principle of Indifference can then be applied safely to the
m alternatives φ(1), φ(2) . . . φ(m), the number m increasing as the
length l of the intervals is diminished. There is no reason why
l should not be of any definite length however small.

If we deal with the problems of geometrical probability in this
way, we shall avoid the contradictory conclusions, which arise from
confusing together distinct elementary areas. In the problem, for
instance, of the chord drawn at random in a circle, which is discussed
in § 7, the chord is regarded, not as a one-dimensional line, but as the
limit of an area, the shape of which is different in each of the variant
solutions. In the first solution it is the limit of a triangle, the length
of the base of which tends to zero; in the second solution it is the
limit of a quadrilateral, two of the sides of which are parallel and at a
distance apart which tends to zero; and in the third solution the area
is defined by the limiting position of a central section of undefined
shape. These distinct hypotheses lead inevitably to different results.

If we were dealing with a strictly linear chord, the Principle of

Indifference would yield us no result, as we could not enunciate the
alternatives in the required form; and if the chord is an elementary
area, we must know the shape of the area of which it is the limit.

So long as we are careful to enunciate the alternatives in a form
to which the Principle of Indifference can be applied unambiguously,
we shall be prevented from confusing together distinct problems, and
pt. i

A TREATISE ON PROBABILITY

70
shall be able to reach conclusions in geometrical probability which are
unambiguously valid.

The substance of this explanation can be put in a slightly different
way by saying that it is not a matter of indifference in these
cases in what manner we proceed to the limit. We must assign
the probabilities before proceeding to the limit, which we can do
unambiguously. But if the problem in hand does not stop at small
finite lengths, areas, or volumes, and we have to proceed to the
limit, then the final result depends upon the shape in which the
body approaches the limit. Mathematicians will recognise an analogy
between this case and the determination of potential at points within
a conductor. Its value depends upon the shape of the area which in
the limit represents the point.

26.

The positive contributions of this chapter to the
determination of valid judgments of equiprobability are two. In the
first place we have stated the Principle of Indifference in a more
accurate form, by displaying its necessary dependence upon judgments
of relevance and so bringing out the hidden element of direct judgment
or intuition, which it has always involved. It has been shown that the

Principle lays down a rule by which direct judgments of relevance and
irrelevance can lead on to judgments of preference and indifference.

In the second place, some types of consideration, which are in fact
relevant, but which are in danger of being overlooked, have been
brought into prominence. By this means it has been possible to avoid
the various types of doubtful and contradictory conclusions to which
the Principle seemed to lead, so long as we applied it without due
qualification.

CHAPTER V
other methods of determining probabilities

1.

The recognition of the fact, that not all probabilities
are numerical, limits the scope of the Principle of Indifference. It
has always been agreed that a numerical measure can actually be
obtained in those cases only in which a reduction to a set of exclusive
and exhaustive equiprobable alternatives is practicable. Our previous
conclusion that numerical measurement is often impossible agrees very
well, therefore, with the argument of the preceding chapter that the
rules, in virtue of which we can assert equiprobability, are somewhat
limited in their field of application.

But the recognition of this same fact makes it more necessary to
discuss the principles which will justify comparisons of more and less
between probabilities, where numerical measurement is theoretically,
as well as practically, impossible. We must, for the reasons given in
the preceding chapter, rely in the last resort on direct judgment. The
object of the following rules and principles is to reduce the judgments
of preference and relevance, which we are compelled to make, to a
few relatively simple types.∗

2.

We will enquire first in what circumstances we can expect a
comparison of more and less to be theoretically possible. I am inclined
to think that this is a matter about which, rather unexpectedly
perhaps, we are able to lay down definite rules. We are able, I think,
always to compare a pair of probabilities which are

(i.) of the type ab/h and a/h,
or

(ii.) of the type a/hh1 and a/h,
provided the additional evidence h1 contains only one independent
piece of relevant information.

∗Parts of Chap. XV. are closely connected with the topics of the following
paragraphs, and the discussion which is commenced here is concluded there.

71
pt. i

A TREATISE ON PROBABILITY

72

(i.) The propositions of Part II. will enable us to prove that ab/h < a/h unless b/ah = 1;
that is to say, the probability of our conclusion is diminished by the
addition to it of something, which on the hypothesis of our argument
cannot be inferred from it. This proposition will be self-evident to
the reader. The rule, that the probability of two propositions jointly
is, in general, less than that of either of them separately, includes the
rule that the attribution of a more specialised concept is less probable
than the attribution of a less specialised concept.

(ii.)

This condition requires a little more explanation.

It
states that the probability a/hh1 is always greater than, equal
to, or less than the probability a/h, if h1 contains no pair of
complementary and independent parts∗ both relevant to a/h. If h1 is
favourable, a/hh1 > a/h. Similarly, if h2 is favourable to a/hh1,
a/hh1h2 > a/hh1. The reverse holds if h1 and h2 are unfavourable.

Thus we can compare a/hh and a/h, in every case in which the
relevant independent parts of the additional evidence h are either
all favourable, or all unfavourable. In cases in which our additional
evidence is equivocal, part taken by itself being favourable and part
unfavourable, comparison is not necessarily possible. In ordinary
language we may assert that, according to our rule, the addition to
our evidence of a single fact always has a definite bearing on our
conclusion. It either leaves its probability unaffected and is irrelevant,
or it has a definitely favourable or unfavourable bearing, being
favourably or unfavourably relevant. It cannot affect the conclusion
in an indefinite way, which allows no comparison between the two
probabilities. But if the addition of one fact is favourable, and the
addition of a second is unfavourable, it is not necessarily possible to
compare the probability of our original argument with its probability
when it has been modified by the addition of both the new facts.

∗See Chap. IV. § 14 for the meaning of these terms.

ch. v

FUNDAMENTAL IDEAS

73

Other comparisons are possible by a combination of these two
principles with the Principle of Indifference.

We may find, for
instance, that a/hh1 > a/h, that a/h = b/h, that b/h > b/hh2, and
that, therefore, a/hh1 > b/hh2. We have thus obtained a comparison
between a pair of probabilities, which are not of the types discussed
above, but without the introduction of any fresh principle. We may
denote comparisons of this type by (iii.).

3.

Whether any comparisons are possible which do not fall
within any of the categories (i.), (ii.), or (iii.), I do not feel certain.

We undoubtedly make a number of direct comparisons which do not
seem to be covered by them. We judge it more probable, for instance,
that Caesar invaded Britain than that Romulus founded Rome. But
even in such cases as this, where a reduction into the regular form
is not obvious, it might prove possible if we could clearly analyse
the real grounds of our judgment. We might argue in this instance
that, whereas Romulus’s founding of Rome rests solely on tradition,
we have in addition evidence of another kind for Caesar’s invasion of

Britain, and that, in so far as our belief in Caesar’s invasion rests on
tradition, we have reasons of a precisely similar kind as for our belief
in Romulus without the additional doubt involved in the maintenance
of a tradition between the times of Romulus and Caesar. By some
such analysis as this our judgment of comparison might be brought
within the above categories.

The process of reaching a judgment of comparison in this way
may be called ‘schematisation.’∗ We take initially an ideal scheme
which falls within the categories of comparison. Let us represent ‘the
historical tradition x has been handed down from a date many years
previous to the time of Caesar’ by ψ1(x); ‘the historical tradition x
has been handed down from the time of Caesar’ by ψ2(x); ‘the
historical tradition x has extra-traditional support’ by ψ3(x); and
the two traditions, the Romulus tradition and the Caesar tradition

∗This phrase is used by Von Kries, op. cit. p. 179, in a somewhat similar
connection.

pt. i

A TREATISE ON PROBABILITY

74
respectively, by a and b. Then if our relevant evidence h were of the
form ψ1(a)ψ2(b)ψ3(b), it is easily seen that the comparison a/h < b/h
could be justified on the lines laid down above.∗ A further judgment,
that our actual evidence presented no relevant divergence from this
schematic form, would then establish the practical conclusion. As

I am not aware of any plausible judgment of comparison which we
make in common practice, but which is clearly incapable of reduction
to some schematic form, and as I see no logical basis for such a
comparison, I feel justified in doubting the possibility of comparing
the probabilities of arguments dissimilar in form and incapable of
schematic reduction. But the point must remain very doubtful until
this part of the subject has received a more prolonged consideration.

4.

Category (ii.) is very wide, and evidently covers a great
variety of cases. If we are to establish general principles of argument
and so avoid excessive dependence on direct individual judgments of
relevance, we must discover some new and more particular principles
included within it. Two of these—those of Analogy and of Induction—
are excessively important, and will be the subject of Part III. of this book. In addition to these a few criteria will be examined and
established in Chapter XIV., §§ 4 and 8 (49.1). We must be content
here (pending the symbolic developments of Part II.) with the two observations following:

(1) The addition of new† evidence h1 to a doubtful‡ argument a/h
is favourably relevant, if either of the following conditions is fulfilled:—

(a) if a/hh1 = 0; (b) if a/hh1 = 1. Divested of symbolism, this merely
amounts to a statement that a piece of evidence is favourable if, in
conjunction with the previous evidence, it is either a necessary or a
sufficient condition for the truth of our conclusion.

(2) It might plausibly be supposed that evidence would be

∗For
a/ψ2(a) = b/ψ2(b);
a/ψ1(a) < a/ψ2(a);
b/ψ2(b) < b/ψ2(b)ψ3(b);
a/ψ1(a) = a/h; and b/ψ2(b)ψ3(b) = b/h.

†h1 is new evidence so long as h1/h =| 1.

‡The argument is doubtful so long as a/h is neither certain nor impossible.

ch. v

FUNDAMENTAL IDEAS

75
favourable to our conclusion which is favourable to favourable
evidence—i.e. that, if h1 is favourable to x/h and x is favourable
to a/h, h1 is favourable to a/h. Whilst, however, this argument
is frequently employed under conditions, which, if explicitly stated,
would justify it, there are also conditions in which this is not so, so
that it is not necessarily valid. For the very deceptive fallacy involved
in the above supposition, Mr. Johnson has suggested to me the name
of the Fallacy of the Middle Term. The general question—If h1 is
favourable to x/h and x is favourable to a/h, in what conditions is
h1 favourable to a/h?—will be examined in Chapter XIV. §§ 4 and

8 (49.1). In the meantime, the intuition of the reader towards the
fallacy may be assisted by the following observations, which are due
to Mr. Johnson:

Let x, x , x . . . be exclusive and exhaustive alternatives under
datum h. Let h1 and a be concordant in regard to each of these
alternatives: i.e. any hypothesis which is strengthened by h1 will
strengthen a, and any hypothesis which is weakened by h1 will
weaken a. It is obvious that, if h1 strengthens some of the hypotheses
x, x , x . . ., it will weaken others. This fact helps us to see why we
cannot consider the concordance of h1 and a in regard to one single
alternative, but must be able to assert their concordance with regard
to every one of the exclusive and exhaustive alternatives, including
the particular one taken. But a further condition is needed, which (as
we shall show) is obviously satisfied in two typical problems at least.

This further condition is that, for each hypothesis x, x , x . . ., it
shall hold that, were this hypothesis known to be true, the knowledge
of h1 would not weaken the probability of a.

These two conditions are sufficient to ensure that h1 shall
strengthen a (independently of knowledge of x, x , x . . .); and, in a
sense, they appear to be necessary; for, unless they are satisfied, the
dependence of h1 upon a would be (so to speak) accidental as regards
the ‘middle terms,’ (x, x , x . . .).

The necessity for reference to all the alternatives x, x , x . . . is
pt. i

A TREATISE ON PROBABILITY

76
analogous to the requirement of distribution of the middle term in
ordinary syllogism. Thus, from premises “All P is x, all S is x,” the
conclusion that “S’s are P” does not formally follow; but given “all P
is x and all S is x ” it does follow that “no S are P”, where x is any
contrary to x. The two conditions taken together would be analogous
to the argument: all x S is P; all x S is P; all x S is P; . . . therefore
all S is P.

First Typical Problem.—An urn contains an unknown proportion
of differently coloured balls. A ball is drawn and replaced. Then
x, x , x . . . stand for the various possible proportions. Let h1 mean

“a white ball has been drawn”; and let a mean “a white ball will
be again drawn.” Then any hypothesis which is strengthened by h1
will strengthen a; and any hypothesis which is weakened by h1 will
weaken a. Moreover, were any one of these hypotheses known to be
true, the knowledge of h1 would not weaken the probability of a.

Hence, in the absence of definite knowledge as regards x, x , x . . .,
the knowledge of h1 would strengthen the probability of a.

Second Typical Problem.—Let a certain event have taken place;
which may have been x, x , x or . . . . Let h1 mean that A reports
so and so; and let a mean that B reports similarly or identically.

The phrase similarly merely indicates that any hypothesis as to the
actual fact, which would be strengthened by A’s report, would be
strengthened by B’s report. Of course, even if the reports were
verbally identical, A’s evidence would not necessarily strengthen the
hypothesis in an equal degree with B’s; because A and B may be
unequally expert or intelligent. Now, in such cases, we may further
affirm (in general), that, were the actual nature of the event known,
the knowledge of A’s report on it would not weaken (though it also
need not strengthen) the probability that B would give a similar
report. Hence, in the absence of such knowledge, the knowledge of h1
would strengthen the probability of a.

5.

Before leaving this part of the argument we must emphasise
the part played by direct judgment in the theory here presented.

ch. v

FUNDAMENTAL IDEAS

77

The rules for the determination of equality and inequality between
probabilities all depend upon it at some point. This seems to me
quite unavoidable. But I do not feel that we should regard it as a
weakness. For we have seen that most, and perhaps all, cases can
be determined by the application of general principles to one simple
type of direct judgment. No more is asked of the intuitive power
applied to particular cases than to determine whether a new piece of
evidence tells, on the whole, for or against a given conclusion. The
application of the rules involves no wider assumptions than those of
other branches of logic.

While it is important, in establishing a control of direct judgment
by general principles, not to conceal its presence, yet the fact that
we ultimately depend upon an intuition need not lead us to suppose
that our conclusions have, therefore, no basis in reason, or that they
are as subjective in validity as they are in origin. It is reasonable
to maintain with the logicians of the Port Royal that we may draw
a conclusion which is truly probable by paying attention to all the
circumstances which accompany the case, and we must admit with
as little concern as possible Hume’s taunt that “when we give the
preference to one set of arguments above another, we do nothing but
decide from our feeling concerning the superiority of their influence.”

CHAPTER VI
the weight of arguments

1.

The question to be raised in this chapter is somewhat
novel; after much consideration I remain uncertain as to how much
importance to attach to it.

The magnitude of the probability
of an argument, in the sense discussed in Chapter III., depends
upon a balance between what may be termed the favourable and
the unfavourable evidence; a new piece of evidence which leaves
this balance unchanged, also leaves the probability of the argument
unchanged. But it seems that there may be another respect in which
some kind of quantitative comparison between arguments is possible.

This comparison turns upon a balance, not between the favourable
and the unfavourable evidence, but between the absolute amounts of
relevant knowledge and of relevant ignorance respectively.

As the relevant evidence at our disposal increases, the magnitude
of the probability of the argument may either decrease or increase,
according as the new knowledge strengthens the unfavourable or the
favourable evidence; but something seems to have increased in either
case,—we have a more substantial basis upon which to rest our
conclusion. I express this by saying that an accession of new evidence
increases the weight of an argument. New evidence will sometimes
decrease the probability of an argument, but it will always increase
its ‘weight.’

2.

The measurement of evidential weight presents similar
difficulties to those with which we met in the measurement of
probability. Only in a restricted class of cases can we compare the
weights of two arguments in respect of more and less. But this must
always be possible where the conclusion of the two arguments is the
same, and the relevant evidence in the one includes and exceeds the
evidence in the other. If the new evidence is ‘irrelevant,’ in the more
precise of the two senses defined in § 14 of Chapter IV., the weight is

78
ch. vi

FUNDAMENTAL IDEAS

79
left unchanged. If any part of the new evidence is relevant, then the
value is increased.

The reason for our stricter definition of ‘relevance’ is now apparent.

If we are to be able to treat ‘weight’ and ‘relevance’ as correlative
terms, we must regard evidence as relevant, part of which is
favourable and part unfavourable, even if, taken as a whole, it leaves
the probability unchanged. With this definition, to say that a new
piece of evidence is ‘relevant’ is the same thing as to say that it
increases the ‘weight’ of the argument.

A proposition cannot be the subject of an argument, unless we
at least attach some meaning to it, and this meaning, even if it
only relates to the form of the proposition, may be relevant in
some arguments relating to it. But there may be no other relevant
evidence; and it is sometimes convenient to term the probability of
such an argument an à priori probability. In this case the weight
of the argument is at its lowest. Starting, therefore, with minimum
weight, corresponding to à priori probability, the evidential weight of
an argument rises, though its probability may either rise or fall, with
every accession of relevant evidence.

3.

Where the conclusions of two arguments are different, or
where the evidence for the one does not overlap the evidence for the
other, it will often be impossible to compare their weights, just as
it may be impossible to compare their probabilities. Some rules of
comparison, however, exist, and there seems to be a close, though
not a complete, correspondence between the conditions under which
pairs of arguments are comparable in respect of probability and of
weight respectively. We found that there were three principal types
in which comparison of probability was possible, other comparisons
being based on a combination of these:—

(i.) Those based on the Principle of Indifference, subject to certain
conditions, and of the form φa/ψa h1 = φb/ψb h2, where h1 and h2
are irrelevant to the arguments.

(ii.) a/hh1 ≷ a/h, where h1 is a single unit of information,
pt. i

A TREATISE ON PROBABILITY

80
containing no independent parts which are relevant.

(iii.) ab/h ≤ a/h

Let us represent the evidential weight of the argument, whose
probability is a/h, by V(a/h). Then, corresponding to the above, we
find that the following comparisons of weight are possible:—

(i.) V(φa/ψa h1) = V(φb/ψb h2), where h1 and h2 are irrelevant
in the strict sense. Arguments, that is to say, to which the Principle
of Indifference is applicable, have equal evidential weights.

(ii.) V(a/hh1) > V(a/h), unless h1 is irrelevant, in which case

V(a/hh1) = V(a/h). The restriction on the composition of h1, which
is necessary in the case of comparisons of magnitude, is not necessary
in the case of weight.

There is, however, no rule for comparisons of weight corresponding
to (iii.) above. It might be thought that V(ab/h) < V(a/h), on the
ground that the more complicated an argument is, relative to given
premisses, the less is its evidential weight. But this is invalid. The
argument ab/h is further off proof than was the argument a/h; but
it is nearer disproof. For example, if ab/h = 0 and a/h > 0, then

V(ab/h) > V(A/h). In fact it would seem to be the case that the
weight of the argument a/h is always equal to that of ¯a/h, where ¯a
is the contradictory of a; i.e., V(a/h) = V(¯a/h). For an argument
is always as near proving or disproving a proposition, as it is to
disproving or proving its contradictory.

4.

It may be pointed out that if a/h = b/h, it does not
necessarily follow that V(a/h) = V(b/h). It has been asserted already
that if the first equality follows directly from a single application of
the Principle of Indifference, the second equality also holds. But the
first equality can exist in other cases also. If, for instance, a and b
are members respectively of different sets of three equally probable
exclusive and exhaustive alternatives, then a/h = b/h; but these
arguments may have very different weights. If, however, a and b can
each, relatively to h, be inferred from the other, i.e. if a/bh = 1 and
b/ah = 1, then V(a/h) = V(b/h). For in proving or disproving one,
ch. vi

FUNDAMENTAL IDEAS

81
we are necessarily proving or disproving the other.

Further principles could, no doubt, be arrived at. The above
can be combined to reach results in cases upon which unaided
common sense might feel itself unable to pronounce with confidence.

Suppose, for instance, that we have three exclusive and exhaustive
alternatives, a, b, and c, and that a/h = b/h in virtue of the

Principle of Indifference, then we have V(a/h) = V(b/h) and

V(a/h) = V(¯
a/h), so that V(b/h) = V(¯a/h). It is also true, since

¯
a/(b + c)h = 1 and (b + c)/¯ah = 1, that V(¯a/h) = V((b + c)/h). Hence

V(b/h) = V((b + c)/h).

5.

The preceding paragraphs will have made it clear that the
weighing of the amount of evidence is quite a separate process from
the balancing of the evidence for and against. In so far, however,
as the question of weight has been discussed at all, attempts have
been made, as a rule, to explain the former in terms of the latter.

If x/h1h2 = 2 and x/h

, it has sometimes been supposed that

3

1 = 3

4
it is more probable that x/h1h2 really is 2 than that x/h

3

1
really
is 3. According to this view, an increase in the amount of evidence

4
strengthens the probability of the probability, or, as De Morgan
would say, the presumption of the probability. A little reflection will
show that such a theory is untenable. For the probability of x on
hypothesis h1 is independent of whether as a matter of fact x is or is
not true, and if we find out subsequently that x is true, this does not
make it false to say that on hypothesis h1 the probability of x is 3.

4

Similarly the fact that x/h1h2 is 2 does not impugn the conclusion

3
that x/h1 is 3, and unless we have made a mistake in our judgment

4
or our calculation on the evidence, the two probabilities are 2 and 3

3

4
respectively.

6.

A second method, by which it might be thought, perhaps,
that the question of weight has been treated, is the method of
probable error.

But while probable error is sometimes connected
with weight, it is primarily concerned with quite a different question.

‘Probable error,’ it should be explained, is the name given, rather
pt. i

A TREATISE ON PROBABILITY

82
inconveniently perhaps, to an expression which arises when we
consider the probability that a given quantity is measured by one of
a number of different magnitudes. Our data may tell us that one of
these magnitudes is the most probable measure of the quantity; but in
some cases it will also tell us how probable each of the other possible
magnitudes of the quantity is. In such cases we can determine the
probability that the quantity will have a magnitude which does not
differ from the most probable by more than a specified amount. The
amount, which the difference between the actual value of the quantity
and its most probable value is as likely as not to exceed, is the

‘probable error.’ In many practical questions the existence of a small
probable error is of the greatest importance, if our conclusions are to
prove valuable. The probability that the quantity has any particular
magnitude may be very small; but this may matter very little, if
there is a high probability that it lies within a certain range.

Now it is obvious that the determination of probable error is
intrinsically a different problem from the determination of weight.

The method of probable error is simply a summation of a number
of alternative and exclusive probabilities. If we say that the most
probable magnitude is x and the probable error y, this is a way,
convenient for many purposes, of summing up a number of probable
conclusions regarding a variety of magnitudes other than x which,
on the evidence, the quantity may possess. The connection between
probable error and weight, such as it is, is due to the fact that in
scientific problems a large probable error is not uncommonly due to
a great lack of evidence, and that as the available evidence increases
there is a tendency for the probable error to diminish. In these cases
the probable error may conceivably be a good practical measure of
the weight.

It is necessary, however, in a theoretical discussion, to point out
that the connection is casual, and only exists in a limited class of
cases. This is easily shown by an example. We may have data
on which the probability of x = 5 is 1, of x = 6 is 1, of x = 7

3

4
ch. vi

FUNDAMENTAL IDEAS

83
is 1, of x = 8 is 1, and of x = 9 is 1 . Additional evidence might

5

6

20
show that x must either be 5 or 8 or 9, the probabilities of each
of these conclusions being 7 , 5 , 4 . The evidential weight of the

16

16

16
latter argument is greater than that of the former, but the probable
error, so far from being diminished, has been increased. There is, in
fact, no reason whatever for supposing that the probable error must
necessarily diminish, as the weight of the argument is increased.

The typical case, in which there may be a practical connection
between weight and probable error, may be illustrated by the two
cases following of balls drawn from an urn. In each case we require
the probability of drawing a white ball; in the first case we know
that the urn contains black and white in equal proportions; in the
second case the proportion of each colour is unknown, and each ball
is as likely to be black as white. It is evident that in either case
the probability of drawing a white ball is 1 but that the weight

2
of the argument in favour of this conclusion is greater in the first
case. When we consider the most probable proportion in which balls
will be drawn in the long run, if after each withdrawal they are
replaced, the question of probable error enters in, and we find that
the greater evidential weight of the argument on the first hypothesis
is accompanied by the smaller probable error.

This conventionalised example is typical of many scientific
problems. The more we know about any phenomenon, the less likely,
as a rule, is our opinion to be modified by each additional item of
experience. In such problems, therefore, an argument of high weight
concerning some phenomenon is likely to be accompanied by a low
probable error, when the character of a series of similar phenomena
is under consideration.

7.

Weight cannot, then, be explained in terms of probability.

An argument of high weight is not ‘more likely to be right’ than
one of low weight; for the probabilities of these arguments only state
relations between premiss and conclusion, and these relations are
stated with equal accuracy in either case. Nor is an argument of
pt. i

A TREATISE ON PROBABILITY

84
high weight one in which the probable error is small; for a small
probable error only means that magnitudes in the neighbourhood of
the most probable magnitude have a relatively high probability, and
an increase of evidence does not necessarily involve an increase in
these probabilities.

The conclusion, that the ‘weight’ and the ‘probability’ of an
argument are independent properties, may possibly introduce a
difficulty into the discussion of the application of probability to
practice.∗ For in deciding on a course of action, it seems plausible
to suppose that we ought to take account of the weight as well as
the probability of different expectations. But it is difficult to think of
any clear example of this, and I do not feel sure that the theory of

‘evidential weight’ has much practical significance.

Bernoulli’s second maxim, that we must take into account all
the information we have, amounts to an injunction that we should
be guided by the probability of that argument, amongst those of
which we know the premisses, of which the evidential weight is the
greatest. But should not this be re-enforced by a further maxim,
that we ought to make the weight of our arguments as great as
possible by getting all the information we can?† It is difficult to see,
however, to what point the strengthening of an argument’s weight
by increasing the evidence ought to be pushed. We may argue that,
when our knowledge is slight but capable of increase, the course of
action, which will, relative to such knowledge, probably produce the
greatest amount of good, will often consist in the acquisition of more
knowledge. But there clearly comes a point when it is no longer
worth while to spend trouble, before acting, in the acquisition of
further information, and there is no evident principle by which to
determine how far we ought to carry our maxim of strengthening the

∗See also Chapter XXVI. § 7.

†Cf. Locke, Essay concerning Human Understanding, book ii. chap. xxi. § 67:

“He that judges without informing himself to the utmost that he is capable,
cannot acquit himself of judging amiss.”

ch. vi

FUNDAMENTAL IDEAS

85
weight of our argument. A little reflection will probably convince the
reader that this is a very confusing problem.

8.

The fundamental distinction of this chapter may be briefly
repeated. One argument has more weight than another if it is based
upon a greater amount of relevant evidence; but it is not always, or
even generally, possible to say of two sets of propositions that one set
embodies more evidence than the other. It has a greater probability
than another if the balance in its favour, of what evidence there is,
is greater than the balance in favour of the argument with which we
compare it; but it is not always, or even generally, possible to say
that the balance in the one case is greater than the balance in the
other. The weight, to speak metaphorically, measures the sum of the
favourable and unfavourable evidence, the probability measures the
difference.

9.

The phenomenon of ‘weight’ can be described from the
point of view of other theories of probability than that which is
adopted here. If we follow certain German logicians in regarding
probability as being based on the disjunctive judgment, we may
say that the weight is increased when the number of alternatives
is reduced, although the ratio of the number of favourable to the
number of unfavourable alternatives may not have been disturbed; or,
to adopt the phraseology of another German school, we may say that
the weight of the probability is increased, as the field of possibility is
contracted.

The same distinction may be explained in the language of the
frequency theory.∗ We should then say that the weight is increased
if we are able to employ as the class of reference a class which is
contained in the original class of reference.

10.

The subject of this chapter has not usually been discussed by
writers on probability, and I know of only two by whom the question
has been explicitly raised:† Meinong, who threw out a suggestion at

∗See Chap. VIII.

†There are also some remarks by Czuber (Wahrscheinlichkeitsrechnung, vol. i.

pt. i

A TREATISE ON PROBABILITY

86
the conclusion of his review of Von Kries’ “Principien,” published in
the Göttingische gelehrte Anzeigen for 1890 (see especially pp. 70–74),
and A. Nitsche, who took up Meinong’s suggestion in an article in the

Vierteljahrsschrift für wissenschaftliche Philosophie, 1892, vol. xvi.

pp. 20–35, entitled “Die Dimensionen der Wahrscheinlichkeit und die

Evidenz der Ungewissheit.”

Meinong, who does not develop the point in any detail, distin-
guishes probability and weight as ‘Intensität’ and ‘Qualität,’ and is
inclined to regard them as two independent dimensions in which
the judgment is free to move—they are the two dimensions of the

‘Urteils-Continuum.’ Nitsche regards the weight as being the measure
of the reliability (Sicherheit) of the probability, and holds that the
probability continually approximates to its true magnitude (reale

Geltung) as the weight increases. His treatment is too brief for it to
be possible to understand very clearly what he means, but his view
seems to resemble the theory already discussed that an argument of
high weight is ‘more likely to be right’ than one of low weight.

p. 202) on the Erkenninisswert of probabilities obtained by different methods,
which may have been intended to have some bearing on it.

CHAPTER VII
historical retrospect

1.

The characteristic features of our Philosophy of Probability
must be determined by the solutions which we offer to the problems
attacked in Chapters III and IV. Whilst a great part of the logical
calculus, which will be developed in Part II., would be applicable with slight modification to several distinct theories of the subject, the
ultimate problems of establishing the premisses of the calculus bring
into the light every fundamental difference of opinion.

These problems are often, for this reason perhaps, left on one side
by writers whose interest chiefly lies in the more formal parts of the
subject. But Probability is not yet on so sound a basis that the formal
or mathematical side of it can be safely developed in isolation, and
some attempts have naturally been made to solve the problem which

Bishop Butler sets to the logician in the concluding words of the brief
discussion on probability with which he prefaces the Analogy.∗

In this chapter, therefore, we will review in their historical order
the answers of Philosophy to the questions, how we know relations
of probability, what ground we have for our judgments, and by what
method we can advance our knowledge.

2.

The natural man is disposed to the opinion that probability
is essentially connected with the inductions of experience and, if he
is a little more sophisticated, with the Laws of Causation and of the

Uniformity of Nature. As Aristotle says, “the probable is that which
usually happens.” Events do not always occur in accordance with

∗“It is not my design to inquire further into the nature, the foundation and
measure of probability; or whence it proceeds that likeness should beget that
presumption, opinion and full conviction, which the human mind is formed to
receive from it, and which it does necessarily produce in every one; or to guard
against the errors to which reasoning from analogy is liable. This belongs to the
subject of logic, and is a part of that subject which has not yet been thoroughly
considered.”

87
pt. i

A TREATISE ON PROBABILITY

88
the expectations of experience; but the laws of experience afford us
a good ground for supposing that they usually will. The occasional
disappointment of these expectations prevents our predictions from
being more than probable; but the ground of their probability must
be sought in this experience, and in this experience only.

This is, in substance, the argument of the authors of the Port

Royal Logic (1662), who were the first to deal with the logic of
probability in the modern manner: “In order for me to judge of the
truth of an event, and to be determined to believe it or not believe
it, it is not necessary to consider it abstractly, and in itself, as we
should consider a proposition in geometry; but it is necessary to pay
attention to all the circumstances which accompany it, internal as
well as external. I call internal circumstances those which belong to
the fact itself, and external those which belong to the persons by
whose testimony we are led to believe it. This being done, if all
the circumstances are such that it never or rarely happens that the
like circumstances are the concomitants of falsehood, our mind is led,
naturally, to believe that it is true.”∗ Locke follows the Port Royal

Logicians very closely: “Probability is likeliness to be true. . . . The
grounds of it are, in short, these two following. First, the conformity
of anything with our own knowledge, observation, and experience.

Secondly, the testimony of others, vouching their observation and
experience”;† and essentially the same opinion is maintained by Bishop

Butler: “When we determine a thing to be probably true, suppose
that an event has or will come to pass, it is from the mind’s remarking
in it a likeness to some other event, which we have observed has
come to pass. And this observation forms, in numberless instances, a
presumption, opinion, or full conviction that such event has or will
come to pass.”‡

∗Eng. Trans., p. 353.

†An Essay concerning Human Understanding, book iv. “Of Knowledge and

Opinion.”

‡Introduction to the Analogy.

ch. vii

FUNDAMENTAL IDEAS

89

Against this view of the subject the criticisms of Hume were
directed: “The idea of cause and effect is derived from experience,
which informs us, that such particular objects, in all past instances,
have been constantly conjoined with each other. . . . According to
this account of things . . . probability is founded on the presumption
of a resemblance betwixt those objects, of which we have had
experience, and those, of which we have had none; and therefore

’tis impossible this presumption can arise from probability.”∗ “When
we are accustomed to see two impressions conjoined together, the
appearance or idea of the one immediately carries us to the idea of
the other. . . . Thus all probable reasoning is nothing but a species of
sensation. ’Tis not solely in poetry and music, we must follow our
taste and sentiment, but likewise in philosophy. When I am convinced
of any principle, ’tis only an idea, which strikes more strongly upon
me. When I give the preference to one set of arguments above
another, I do nothing but decide from my feeling concerning the
superiority of their influence.”† Hume, in fact, points out that, while
it is true that past experience gives rise to a psychological anticipation
of some events rather than of others, no ground has been given for
the validity of this superior anticipation.

3.

But in the meantime the subject had fallen into the hands
of the mathematicians, and an entirely new method of approach was
in course of development. It had become obvious that many of the
judgments of probability which we in fact make do not depend upon
past experience in a way which satisfies the canons laid down by
the Port Royal Logicians or by Locke. In particular, alternatives are
judged equally probable, without there being necessarily any actual
experience of their approximately equal frequency of occurrence in
the past. And, apart from this, it is evident that judgments based
on a somewhat indefinite experience of the past do not easily lend
themselves to precise numerical appraisement. Accordingly James

∗Treatise of Human Nature, p. 391 (Green’s edition).

†Op. cit. p. 403.

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90

Bernoulli,∗ the real founder of the classical school of mathematical
probability, while not repudiating the old test of experience, had
based many of his conclusions on a quite different criterion—the rule
which I have named the Principle of Indifference. The traditional
method of the mathematical school essentially depends upon reducing
all the possible conclusions to a number of ‘equiprobable cases.’ And,
according to the Principle of Indifference, ‘cases’ are held to be
equiprobable when there is no reason for preferring any one to any
other, when there is nothing, as with Buridan’s ass, to determine the
mind in any one of the several possible directions. To take Czuber’s
example of dice,† this principle permits us to assume that each face
is equally likely to fall, if there is no reason to suppose any particular
irregularity, and it does not require that we should know that the
construction is regular, or that each face has, as a matter of fact,
fallen equally often in the past.

On this Principle, extended by Bernoulli beyond those problems
of gaming in which by its tacit assumption Pascal and Huyghens had
worked out a few simple exercises, the whole fabric of mathematical
probability was soon allowed to rest. The older criterion of experience,
never repudiated, was soon subsumed under the new doctrine. First,
in virtue of Bernoulli’s famous Law of Great Numbers, the fractions
representing the probabilities of events were thought to represent
also the actual proportion of their occurrences, so that experience,
if it were considerable, could be translated into the cyphers of
arithmetic. And next, by the aid of the Principle of Indifference,

Laplace established his Law of Succession by which the influence of
any experience, however limited, could be numerically measured, and
which purported to prove that, if B has been seen to accompany A
twice, it is two to one that B will again accompany A on A’s next
appearance. No other formula in the alchemy of logic has exerted

∗See especially Ars Conjectandi, p. 224. Cf. Laplace, Théorie analytique,
p. 178.

†Wahrscheinlichkeitsrechnung, p. 9.

ch. vii

FUNDAMENTAL IDEAS

91
more astonishing powers, For it has established the existence of

God from the premiss of total ignorance; and it has measured with
numerical precision the probability that the sun will rise to-morrow.

Yet the new principles did not win acceptance without opposition.

D’Alembert,∗ Hume, and Ancillon† stand out as the sceptical critics
of probability, against the credulity of eighteenth-century philosophers
who were ready to swallow without too many questions the conclusions
of a science which claimed and seemed to bring an entire new field
within the dominion of Reason.‡

The first effective criticism came from Hume, who was also the
first to distinguish the method of Locke and the philosophers from
the method of Bernoulli and the mathematicians. “Probability,” he
says, “or reasoning from conjecture, may be divided into two kinds,
viz. that which is founded on chance and that which arises from
causes.”§ By these two kinds he evidently means the mathematical
method of counting the equal chances based on Indifference, and the

∗D’Alembert’s scepticism was directed towards the current mathematical
theory
only,
and
was
not,
like

Hume’s,
fundamental
and
far-reaching.

His opposition to the received opinions was, perhaps, more splendid than
discriminating.

†Ancillon’s communication to the Berlin Academy in 1794, entitled Doutes
sur les bases du calcul des probabilités, is not as well known as it deserves to be.

He writes as a follower of Hume, but adds much that is original and interesting.

An historian, who also wrote on a variety of philosophical subjects, Ancillon
was, at one time, the Prussian Minister of Foreign Affairs.

‡French philosophy of the latter half of the eighteenth century was profoundly
affected by the supposed conquests of the Calculus of Probability in all fields
of thought.

Nothing seemed beyond its powers of prediction, and it almost
succeeded in men’s minds to the place previously occupied by Revelation. It was
under these influences that Condorcet evolved his doctrine of the perfectibility of
the human race. The continuity and oneness of modern European thought may
be illustrated, if such things amuse the reader, by the reflection that Condorcet
derived from Bernoulli, that Godwin was inspired by Condorcet, that Malthus
was stimulated by Godwin’s folly into stating his famous doctrine, and that
from the reading of Malthus on Population Darwin received his earliest impulse.

§Treatise of Human Nature, p. 424 (Green’s edition).

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A TREATISE ON PROBABILITY

92
inductive method based on the experience of uniformity. He argues
that ‘chance’ alone can be the foundation of nothing, and “that there
must always be a mixture of causes among the chances, in order to
be the foundation of any reasoning.”∗ His previous argument against
probabilities, which were based on an assumption of cause, is thus
extended to the mathematical method also.

But the great prestige of Laplace and the ‘verifications’ of his
principles which his more famous results were supposed to supply had,
by the beginning of the nineteenth century, established the science
on the Principle of Indifference in an almost unquestioned position.

It may be noted, however, that De Morgan, the principal student of
the subject in England, seems to have regarded the method of actual
experiment and the method of counting cases, which were equally
probable on grounds of Indifference, as alternative methods of equal
validity.

4.

The reaction against the traditional teaching during the
past hundred years has not possessed sufficient force to displace the
established doctrine, and the Principle of Indifference is still very
widely accepted in an unqualified form. Criticism has proceeded along
two distinct lines; the one, originated by Leslie Ellis, and developed
by Dr. Venn, Professor Edgeworth, and Professor Karl Pearson, has
been almost entirely confined in its influence to England; the other,
of which the beginnings are to be seen in Boole’s Laws of Thought,
has been developed in Germany, where its ablest exponent has been

Von Kries. France has remained uninfluenced by either, and faithful,
on the whole, to the tradition of Laplace. Even Henri Poincaré, who
had his doubts, and described the Principle of Indifference as “very
vague and very elastic,” regarded it as our only guide in the choice
of that convention, “which has always something arbitrary about it,”

∗Op. cit. p. 425.

ch. vii

FUNDAMENTAL IDEAS

93
but upon which calculation in probability invariably rests.∗

5.

Before following up in detail these two lines of development,

I will summarise again the earlier doctrine with which the leaders of
the new schools found themselves confronted.

The earlier philosophers had in mind in dealing with probability
the application to the future of the inductions of experience, to
the almost complete exclusion of other problems. For the data of
probability, therefore, they looked only to their own experience and
to the recorded experiences of others; their principal refinement was
to distinguish these two grounds, and they did not attempt to make
a numerical estimate of the chances. The mathematicians, on the
other hand, setting out from the simple problems presented by dice
and playing cards, and requiring for the application of their methods
a basis of numerical measurement, dwelt on the negative rather than
the positive side of their evidence, and found it easier to measure
equal degrees of ignorance than equivalent quantities of experience.

This led to the explicit introduction of the Principle of Indifference,

∗Poincaré’s opinions on Probability are to be found in his Calcul des

Probabilités and in his Science et Hypothèse.

Neither of these books appears
to me to be in all respects a considered work, but his view is sufficiently
novel to be worth a reference. Briefly, he shows that the current mathematical
definition is circular, and argues from this that the choice of the particular
probabilities, which we are to regard as initially equal before the application
of our mathematics, is entirely a matter of ‘convention.’

Much epigram is,
therefore, expended in pointing out that the study of probability is no more
than a polite exercise, and he concludes:

“Le calcul des probabilités offre
une contradiction dans les termes mêmes qui servent à le désigner, et, si je
ne craignais de rappeler ici un mot trop souvent répété, je dirais qu’il nous
enseigne surtout une chose; c’est de savoir que nous ne savons rien.”

On the
other hand, the greater part of his book is devoted to working out instances
of practical application, and he speaks of ‘metaphysics’ legitimising particular
conventions. How this comes about is not explained. He seems to endeavour to
save his reputation as a philosopher by the surrender of probability as a valid
conception, without at the same time forfeiting his claim as a mathematician to
work out probable formulae of practical importance.

pt. i

A TREATISE ON PROBABILITY

94
or, as it was then termed, the Principle of Non-Sufficient Reason.

The great achievement of the eighteenth century was, in the eyes of
the early nineteenth, the reconciliation of the two points of view and
the measurement of probabilities, which were grounded on experience,
by a method whose logical basis was the Principle of Non-Sufficient

Reason. This would indeed have been a very astonishing discovery,
and would, as its authors declared, have gradually brought almost
every phase of human activity within the power of the most refined
mathematical analysis.

But it was not long before more sceptical persons began to suspect
that this theory proved too much. Its calculations, it is true, were
constructed from the data of experience, but the more simple and the
less complex the experience the better satisfied was the theory. What
was required was not a wide experience or detailed information, but
a completeness of symmetry in the little information there might be.

It seemed to follow from the Laplacian doctrine that the primary
qualification for one who would be well informed was an equally
balanced ignorance.

6.

The obvious reaction from a teaching, which seemed to derive
from abstractions results relevant to experience, was into the arms
of empiricism; and in the state of philosophy at that time England
was the natural home of this reaction. The first protest, of which

I am aware, came from Leslie Ellis in 1842.∗ At the conclusion
of his Remarks on an alleged proof of the Method of least squares,†

“Mere ignorance,” he says, “is no ground for any inference whatever.

Ex nihilo nihil.” In Venn’s Logic of Chance Ellis’s suggestions are
developed into a complete theory:‡ “Experience is our sole guide.

If we want to discover what is in reality a series of things, not
a series of our own conceptions, we must appeal to the things
themselves to obtain it, for we cannot find much help elsewhere.”

∗On the Foundations of the Theory of Probabilities.

†Republished in Miscellaneous Writings.

‡Logic of Chance, p. 74.

ch. vii

FUNDAMENTAL IDEAS

95

Professor Edgeworth∗ was an early disciple of the same school: “The
probability,” he says, “of head occurring n times if the coin is of
the ordinary make is approximately at least (1)n. This value is

2
rigidly deducible from positive experience, the observations made by
gamesters, the experiments recorded by Jevons and De Morgan.”

The doctrines of the empirical school will be examined in Chapter

VIII., and I postpone my detailed criticism to that chapter. Venn
rejects the applications of Bernoulli’s theorem, which he describes as

“one of the last remaining relics of Realism,” as well as the later

Laplacian Law of Succession, thus destroying the link between the
empirical and the à priori methods. But, apart from this, his
view that statements of probability are simply a particular class of
statements about the actual world of phenomena, would have led
him to a closer dependence on actual experience. He holds that
the probability of an event’s having a certain attribute is simply the
fraction expressing the proportion of cases in which, as a matter of
actual fact, this attribute is present. Our knowledge, however, of this
proportion is often reached inductively, and shares the uncertainty to
which all inductions are liable. And, besides, in referring an event to
a series we do not postulate that all the members of the series should
be identical, but only that they should not be known to differ in a
relevant manner. Even on this theory, therefore, we are not solely
determined by positive knowledge and the direct data of experience.

7.

The Empirical School in their reaction against the pretentious
results, which the Laplacian theory affected to develop out of nothing,
have gone too far in the opposite direction. If our experience and
our knowledge were complete, we should be beyond the need of the

Calculus of Probability. And where our experience is incomplete, we
cannot hope to derive from it judgments of probability without the aid
either of intuition or of some further à priori principle. Experience, as
opposed to intuition, cannot possibly afford us a criterion by which to

∗Metretike, p. 4.

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A TREATISE ON PROBABILITY

96
judge whether on given evidence the probabilities of two propositions
are or are not equal.

However essential the data of experience may be, they cannot by
themselves, it seems, supply us with what we want. Czuber,∗ who
prefers what he calls the Principle of Compelling Reason (das Prinzip
des zwingenden Grundes), and holds that Probability has an objective
and not merely formal interpretation only when it is grounded on
definite knowledge, is rightly compelled to admit that we cannot get
on altogether without the Principle of Non-Sufficient Reason. On the
grounds both of its own intuitive plausibility and of that of some of
the conclusions for which it is necessary, we are inevitably led towards
this principle as a necessary basis for judgments of probability. In
some sense, judgments of probability do seem to be based on equally
balanced degrees of ignorance.

8.

It is from this starting-point that the German logicians
have set out. They have perceived that there are few judgments
of probability which are altogether independent of some principle
resembling that of Non-Sufficient Reason. But they also apprehend,
with Boole, that this may be a very arbitrary method of procedure.

It was pointed out in § 18 of Chapter IV. that the cases,
in which the Principle of Indifference (or Non-Sufficient Reason)

breaks down, have a great deal in common, and that we break up
the field of possibility into a number of areas, actually unequal,
but indistinguishable on the evidence. Several German logicians,
therefore, have endeavoured to determine some rule by which it might
be possible to postulate actual equality of area for the fields of the
various possibilities.

By far the most complete and closely reasoned solution on these
lines is that of Von Kries.† He is primarily anxious to discover a
proper basis for the numerical measurement of probabilities, and he

∗Wahrscheinlichkeitsrechnung, p. 11.

†Die Principien der Wahrscheinlichkeitsrechnung. Eine logische Untersuchung.

Freiburg, 1886.

ch. vii

FUNDAMENTAL IDEAS

97
is thus led to examine with care the grounds of valid judgments
of equiprobability. His criticisms of the Principle of Non-Sufficient

Reason are searching, and, to meet them, he elaborates a number of
qualifying conditions which are, he argues, necessary and sufficient.

The value of his book, however, lies, in the opinion of the present
writer, in the critical rather than in the constructive parts. The
manner in which his qualifying conditions are expressed is often,
to an English reader at any rate, somewhat obscure, and he seems
sometimes to cover up difficulties, rather than solve them, by the
invention of new technical terms. These characteristics render it
difficult to expound him adequately in a summary, and the reader
must be referred to the original for a proper exposition of the

Doctrine of Spielräume. Briefly, but not very intelligibly perhaps,
he may be said to hold that the hypotheses for the probabilities of
which we wish to obtain a numerical comparison, must refer to ‘fields’

(Spielräume) which are ‘indifferent,’ ‘comparable’ in magnitude, and

‘original’ (ursprünglich). Two fields are ‘indifferent’ if they are equal
before the Principle of Non-Sufficient Reason; they are ‘comparable’

if it is true that the fields are actually of equal extent; and they are

‘original’ or ultimate if they are not derived from some other field.

The last condition is exceedingly obscure, but it seems to mean that
the objects with which we are ultimately dealing must be directly
represented by the ‘fields’ of our hypotheses, and there must not be
merely correlation between these objects and the objects of the fields.

The qualification of comparability is intended to deal with difficulties
such as that connected with the population of different areas of
unknown extent; and the qualification of originality with those arising
from indirect measurement, as in the case of specific density.

Von Kries’s solution is highly suggestive, but it does not seem, so
far as I understand it, to supply an unambiguous criterion for all cases.

His discussion of the philosophical character of probability is brief and
inadequate, and the fundamental error in his treatment of the subject
is the physical, rather than logical, bias which seems to direct the
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A TREATISE ON PROBABILITY

98
formulation of his conditions. The condition of Ursprünglichkeit, for
instance, seems to depend upon physical rather than logical criteria,
and is, as a result, much more restricted in its applicability than a
condition, which will really meet the difficulties of the case, ought to
be. But, although I differ from him in his philosophical conception of
probability, the treatment of the Principle of Indifference, which fills
the greater part of his book, is, I think, along fruitful lines, and I
have been deeply indebted to it in formulating my own conditions in

Chapter IV.

Of less closely reasoned and less detailed treatments, which aim
at the same kind of result, those of Sigwart and Lotze are worth
noticing. Sigwart’s∗ position is sufficiently explained by the following
extract: “The possibility of a mathematical treatment lies primarily
in the fact that in the disjunctive judgment the number of terms
in the disjunction plays a decisive part. Inasmuch as a limited
number of mutually exclusive possibilities is presented, of which one
alone is actual, the element of number forms an essential part of our
knowledge. . . . Our knowledge must enable us to assume that the
particular terms of the disjunction are so far equivalent that they
express an equal degree of specialisation of a general concept, or that
they cover equal parts of the whole extension of the concept. . . . This
equivalence is most intuitable where we are dealing with equal parts
of a spatial area, or equal parts of a period of time. . . . But even
where this obvious quality is not forthcoming, we may ground our
expectations upon a hypothetical equivalence, where we see no reason
for considering the extent of one possibility to be greater than that
of the others. . . .”

In the beginning of this passage Sigwart seems to be aware of
the fundamental difficulty, although exception may be taken to the
vagueness of the phrase “equal degree of specialisation of a general
concept.” But in the last sentence quoted he surrenders the advantages

∗Sigwart Logic (Eng. edition), vol. ii. p. 220.

ch. vii

FUNDAMENTAL IDEAS

99
he has gained in the earlier part of his explanation, and, instead of
insisting on a knowledge of an equal degree of specialisation, he is
satisfied with an absence of any knowledge to the contrary. Hence, in
spite of his initial qualifications, he ends unrestrainedly in the arms
of Non-Sufficient Reason.∗

Lotze,† in a brief discussion of the subject, throws out some
remarks well worth quoting: “We disclaim all knowledge of the
circumstances which condition the real issue, so that when we talk
of equally possible cases we can only mean coördinated as equivalent
species in the compass of an universal case; that is to say, if we
enumerate the special forms, which the genus can assume, we get a
disjunctive judgment of the form: if the condition B is fulfilled, one
of the kinds f1f2f3 . . . of the universal consequent F will occur to the
exclusion of the rest. Which of all those different consequents will,
in fact, occur, depends in all cases on the special form b1b2b3 . . . in
which that universal condition is fulfilled. . . . A coördinated case is
a case which answers to one and only one of the mutually exclusive
values b1b2 . . . of the condition B, and these rival values may occur in
reality; it does not answer to a more general form B, of this condition,
which can never exist in reality, because it embraces several of the
particular values b1b2. . . .”

This certainly meets some of the difficulties, and its resemblance
to the conditions formulated in Chapter IV. will be evident to the
careful reader. But it is not very precise, and not easily applicable
to all cases, to those, for instance, of the measurement of continuous
quantity. By combining the suggestions of Von Kries, Sigwart, and

Lotze, we might, perhaps, patch up a fairly comprehensive rule. We
might say, for instance, that if b1 and b2 are classes, their members
must be finite in number and enumerable or they must compose

∗Sigwart’s treatment of the subject of probability is curiously inaccurate. Of
his four fundamental rules of probability, for instance, three are, as he states
them, certainly false.

†Lotze, Logic (Eng. edition), pp. 364, 365.

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A TREATISE ON PROBABILITY

100
stretches; that, if they are finite in number, they must be equal in
number; and that, if their members compose stretches, the stretches
must be equal stretches; and that if b1 and b2 are concepts, they
must represent concepts of an equal degree of specialisation. But
qualifications so worded would raise almost as many difficulties as
they solved. How, for instance, are we to know when concepts are of
an equal degree of specialisation?

9.

That probability is a relation has often received incidental
recognition from logicians, in spite of the general failure to place
proper emphasis on it.

The earliest writer, with whom I am
acquainted, explicitly to notice this, is Kahle in his Elementa logicae

Probabilium methodo mathematica in usum Scientiarum et Vitae
adornata published at Halle in 1735.∗ Amongst more recent writers
casual statements are common to the effect that the probability of a
conclusion is relative to the grounds upon which it is based. Take

Boole† for instance: “It is implied in the definition that probability
is always relative to our actual state of information and varies with

∗This work, which seems to have soon fallen into complete neglect and
is now extremely rare, is full of interest and original thought.

The following
quotations will show the fundamental position taken up: “Est cognitio probabilis,
si desunt quaedam requisita ad veritatem demonstrativam (p. 15). Propositio
probabilis esse potest falsa, et improbabilis esse potest vera; ergo cognitio
hodie possibilis, crastina luce mutari potest improbabilem, si accedunt reliqua
requisita omnia, in certitudinem (p. 26). . . .

Certitudo est terminus relativus:

considerare potest ratione representationum in intellectu nostro. . . . Incerta nobis
dependent a defectu cognitionis (p. 35). . . .

Actionem imprudenter et contra
regulas probabilitatis susceptam eventus felix sequi potest.

Ergo prudentia
actionum ex successu solo non est aestimanda (p. 62). . . . Logica probabilium
est scientia dijudicandi gradum certitudinis eorum, quibus desunt requisita ad
veritatem demonstrativam (p. 94).”

†“On a General Method in the Theory of Probabilities,” Phil. Mag.,

4th Series, viii., 1854.

See also, “On the Application of the Theory of

Probabilities to the Question of the Combination of Testimonies or Judgments”

(Edin. Phil. Trans. xxi. p. 600): “Our estimate of the probability of an event
varies not absolutely with the circumstances which actually affect its occurrence,
but with our knowledge of those circumstances.”

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FUNDAMENTAL IDEAS

101
that state of information.” Or Bradley:∗ “Probability tells us what
we ought to believe, what we ought to believe on certain data . . . .

Probability is no more ‘relative’ and ‘subjective’ than is any other
act of logical inference from hypothetical premises. It is relative
to the data with which it has to deal, and is not relative in any
other sense.” Or even Laplace, when he is explaining the diversity
of human opinions: “Dans les choses qui ne sont que vraisemblables,
la différence des données que chaque homme a sur elles, est une des
causes principales de la diversité des opinions que l’on voit régner
sur les mêmes objets . . . c’est ainsi que le même fait, récité devant
une nombreuse assemblée, obtient divers degrés de croyance, suivant
l’étendue des connaissances des auditeurs.”†

10.

Here we may leave this account of the various directions in
which progress has seemed possible, with the hope that it may assist
the reader, who is dissatisfied with the solution proposed in Chapter

IV., to determine the line of argument along which he is likeliest to
discover the solution of a difficult problem.

∗The Principles of Logic, p. 208.

†Essai philosophique, p. 7.

CHAPTER VIII
the frequency theory of probability

1.

The theory of probability, outlined in the preceding chapters,
has serious difficulties to overcome. There is a theoretical, as well as a
practical, difficulty in measuring or comparing degrees of probability,
and a further difficulty in determining them à priori. We must
now examine an alternative theory which is much freer from these
troubles, and is widely held at the present time.

2.

The theory is in its essence a very old one, Aristotle
foreshadowed it when he held that “the probable is that which for
the most part happens”;∗ and, as we have seen in Chapter VII., an
opinion not unlike this was entertained by those philosophers of the
seventeenth and eighteenth centuries who approached the problems
of probability uninfluenced by the work of mathematicians. But the
underlying conception of earlier writers received at the hands of some

English logicians during the latter half of the nineteenth century a
new and much more complicated form.

The theory in question, which I shall call the Frequency Theory
of Probability, first appears† as the basis of a proposed logical
scheme in a brief essay by Leslie Ellis On the Foundations of the

Theory of Probabilities, and is somewhat further developed in his

Remarks on the Fundamental Principles of the Theory of Probabilities.‡

∗Rhet. i. 2, 1357 a 34.

†I give Ellis the priority because his paper, published in 1843, was read
on Feb. 14, 1842. The same conception, however, is to be found in Cournot’s

Exposition, also published in 1843: “La théorie des probabilités a pour objet
certains rapports numériques qui prendraient des valeurs fixes et complétement
déterminées, si l’on pouvait répéter à l’infini les épreuves des mêmes hasards, et
qui, pour un nombre fini d’épreuves, oscillent entre des limites d’autant plus
resserées, d’autant plus voisines des valeurs finales, que le nombre des épreuves
est plus grand.”

‡These essays were published in the Transactions of the Camb. Phil. Soc.,

102
ch. viii

FUNDAMENTAL IDEAS

103

“If the probability of a given event be correctly determined,” he
says, “the event will on a long run of trials tend to recur with
frequency proportional to their probability. This is generally proved
mathematically. It seems to me to be true à priori. . . . I have been
unable to sever the judgment that one event is more likely to happen
than another from the belief that in the long run it will occur more
frequently.” Ellis explicitly introduces the conception that probability
is essentially concerned with a group or series.

Although the priority of invention must be allowed to Leslie

Ellis, the theory is commonly associated with the name of Venn.

In his Logic of Chance∗ it first received elaborate and systematic
treatment, and, in spite of his having attracted a number of followers,
there has been no other comprehensive attempt to meet the theory’s
special difficulties or the criticisms directed against it. I shall begin,
therefore, by examining it in the form in which Venn has expounded
it. Venn’s exposition is much coloured by an empirical view of logic,
which is not perhaps as necessary to the essential part of his doctrine
as he himself implies, and is not shared by all of those who must
be classed as in general agreement with him about probability. It
will be necessary, therefore, to supplement a criticism of Venn by an
account of a more general frequency theory of probability, divested of
the empiricism with which he has clothed it.

3.

The following quotations from Venn’s Logic of Chance
will show the general drift of his argument: The fundamental
conception is that of a series (p. 4). The series is of events which
have a certain number of features or attributes in common (p. 10).

The characteristic distinctive of probability is this,—the occasional
the first in 1843 (vol. viii.), and the second in 1854 (vol. ix.).

Both were
reprinted in Mathematical and other Writings (1863), together with three other
brief papers on Probability and the Method of Least Squares. All five are full
of spirit and originality, and are not now so well known as they deserve to be.

∗The first edition appeared in 1866. Revised editions were issued in 1876
and 1888. References are given to the third edition of 1888.

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A TREATISE ON PROBABILITY

104
attributes, as distinguished from the permanent, are found on an
examination to tend to exist in a certain definite proportion of the
whole number of cases (p. 11). We require that there should be in
nature large classes of objects, throughout all the individual members
of which a general resemblance extends. For this purpose the existence
of natural kinds or groups is necessary (p. 55). The distinctive
characteristics of probability prevail principally in the properties of
natural kinds, both in the ultimate and in the derivative or accidental
properties (p. 63). The same peculiarity prevails again in the force
and frequency of most natural agencies (p. 64). There seems reason
to believe that it is in such things only, as distinguished from things
artificial, that the property in question is to be found (p. 65). How, in
any particular case, are we to establish the existence of a probability
series? Experience is our sole guide. If we want to discover what
is in reality a series of things, not a series of our own conceptions,
we must appeal to the things themselves to obtain it, for we cannot
find much help elsewhere (p. 174). When probability is divorced from
direct reference to objects, as it substantially is by not being founded
upon experience, it simply resolves itself into the common algebraical
doctrine of Permutations and Combinations (p. 87). By assigning
an expectation in reference to the individual, we mean nothing more
than to make a statement about the average of his class (p. 151).

When we say of a conclusion within the strict province of probability,
that it is not certain, all that we mean is that in some proportion of
cases only will such conclusion be right, in the other cases it will be
wrong (p. 210).

The essence of this theory can be expressed in a few words. To
say, that the probability of an event’s having a certain characteristic
is x, is to mean that the event is one of a number of events, a
y
proportion x of which have the characteristic in question; and the
y
fact, that there is such a series of events possessing this frequency in
respect of the characteristic, is purely a matter of experience to be
determined in the same manner as any other question of fact. That
ch. viii

FUNDAMENTAL IDEAS

105
such series do exist happens to be a characteristic of the real world as
we know it, and from this the practical importance of the calculation
of probabilities is derived.

Such a theory possesses manifest advantages. There is no mystery
about it—no new indefinables, no appeals to intuition. Measurement
leads to no difficulties; our probabilities or frequencies are ordinary
numbers, upon which the arithmetical apparatus can be safely brought
to bear. And at the same time it seems to crystallise in a clear,
explicit shape the floating opinion of common sense that an event is
or is not probable in certain supposed circumstances according as it
is or is not usual as a matter of fact and experience.

The two principal tenets, then, of Venn’s system are these,—that
probability is concerned with series or groups of events, and that
all the requisite facts must be determined empirically, a statement
in probability merely summing up in a convenient way a group of
experiences. Aggregate regularity combined with individual difference
happens, he says, to be characteristic of many events in the real world.

It will often be the case, therefore, that we can make statements
regarding the average of a certain class, or regarding its characteristics
in the long run, which we cannot make about any of its individual
members without great risk of error. As our knowledge regarding the
class as a whole may give us valuable guidance in dealing with an
individual instance, we require a convenient way of saying that an
individual belongs to a class in which certain characteristics appear
on the average with a known frequency; and this the conventional
language of probability gives us. The importance of probability
depends solely upon the actual existence of such groups or real kinds
in the world of experience, and a judgment of probability must
necessarily depend for its validity upon our empirical knowledge of
them.

4.

It is the obvious, as well as the correct, criticism of such a
theory, that the identification of probability with statistical frequency
is a very grave departure from the established use of words; for it
pt. i

A TREATISE ON PROBABILITY

106
clearly excludes a great number of judgments which are generally
believed to deal with probability. Venn himself was well aware of
this, and cannot be accused of supposing that all beliefs, which
are commonly called probable, are really concerned with statistical
frequency. But some of his followers, to judge from their published
work, have not always seen, so clearly as he did, that his theory is
not concerned with the same subject as that with which other writers
have dealt under the same title. Venn justifies his procedure by
arguing that no other meaning, of which it is possible to take strict
logical cognisance, can reasonably be given to the term, and that the
other meanings, with which it has been used, have not enough in
common to permit their reduction to a single logical scheme. It is
useless, therefore, for a critic of Venn to point out that many supposed
judgments of probability are not concerned with statistical frequency;
for, as I understand the Logic of Chance, he admits it; and the critic
must show that the sense different from Venn’s in which the term
probability is often employed has an important logical interpretation
about which we can generalise. This position I seek to establish. It is,
in my opinion, this other sense alone which has importance; Venn’s
theory by itself has few practical applications, and if we allow it to
hold the field, we must admit that probability is not the guide of life,
and that in following it we are not acting according to reason.

5.

Part of the plausibility of Venn’s theory is derived, I think,
from a failure to recognise the narrow limits of its applicability, or
to notice his own admissions regarding this. “In every case,” he says

(p. 124), “in which we extend our inferences by Induction or Analogy,
or depend upon the witness of others, or trust to our own memory
of the past, or come to a conclusion through conflicting arguments,
or even make a long and complicated deduction by mathematics or
logic, we have a result of which we can scarcely feel as certain as of
the premisses from which it was obtained. In all these cases, then,
we are conscious of varying quantities of belief, but are the laws
according to which the belief is produced and varied the same? If
ch. viii

FUNDAMENTAL IDEAS

107
they cannot be reduced to one harmonious scheme, if, in fact, they
can at best be brought to nothing but a number of different schemes,
each with its own body of laws and rules, then it is vain to endeavour
to force them into one science.” All these cases, therefore, in which
we are ‘not certain,’ Venn explicitly excludes from what he chooses
to call the science of probability, and he pays no further attention to
them. The science of probability is, according to him, no more than
a method which enables us to express in a convenient form statistical
statements of frequency. “The province of probability,” he says again
on page 160, “is not so extensive as that over which variation of belief
might be observed. Probability only considers the case in which this
variation is brought about in a certain definite statistical way.”∗ He
points out on p. 194 that for the purposes of probability we must take
the statistical frequency from which we start ready made and ask no
questions about the process or completeness of its manufacture: “It
may be obtained by any of the numerous rules furnished by Induction,
or it may be inferred deductively, or given by our own observation;
its value may be diminished by its depending upon the testimony of
witnesses, or its being recalled by our own memory. Its real value may
be influenced by these causes or any combinations of them; but all
these are preliminary questions with which we have nothing directly
to do. We assume our statistical proposition to be true, neglecting

∗Edgeworth uses the term ‘probability’ widely, as I do; but he makes
a distinction corresponding to Venn’s by limiting the subject-matter of the

Calculus of Probabilities.

He writes (‘Philosophy of Chance,’ Mind, 1884,
p. 223):

“The Calculus of Probabilities is concerned with the estimation of
degrees of probability; not every species of estimate, but that which is founded
on a particular standard.

That standard is the phenomenon of statistical
uniformity: the fact that a genus can very frequently be subdivided into species
such that the number of individuals in each species bears an approximately
constant ratio to the number of individuals in the genus.” This use of terms is
legitimate, though it is not easy to follow it consistently. But, like Venn’s, it
leaves aside the most important questions. The Calculus of Probabilities, thus
interpreted, is no guide by itself as to which opinion we ought to follow, and is
not a measure of the weight we should attach to conflicting arguments.

pt. i

A TREATISE ON PROBABILITY

108
the diminution of its value by the processes of attainment.”

It must be recognised, therefore, that Venn has deliberately
excluded from his survey almost all the cases in which we regard our
judgments as ‘only probable’; and, whatever the value or consistency
of his own scheme, he has left untouched a wide field of study for
others.

6.

The main grounds, which have induced Venn to regard
judgments based on statistical frequency as the only cases of
probability which possess logical importance, seem to be two: (i.) that
other cases are mainly subjective, and (ii.) that they are incapable of
accurate measurement.

With regard to the first it must be admitted that there are
many instances in which variation of belief is occasioned by purely
psychological causes, and that his argument is valid against those
who have defined probability as measuring the degree of subjective
belief. But this has not been the usual way of looking at the subject.

Probability is the study of the grounds which lead us to entertain a
rational preference for one belief over another. That there are rational
grounds other than statistical frequency, for such preferences, Venn
does not deny; he admits in the quotation given above that the ‘real
value’ of our conclusion is influenced by many other considerations
than that of statistical frequency. Venn’s theory, therefore, cannot be
fairly propounded by his disciples as alternative to such a theory as
is propounded here. For my Treatise is concerned with the general
theory of arguments from premisses leading to conclusions which are
reasonable but not certain; and this is a subject which Venn has,
deliberately, not treated in the Logic of Chance.

7.

Apart from two circumstances, it would scarcely be necessary
to say anything further; but in the first place some writers have
believed that Venn has propounded a complete theory of probability,
failing to realise that he is not at all concerned with the sense in
which we may say that one induction or analogy, or testimony, or
memory, or train of argument is more probable than another; and in
ch. viii

FUNDAMENTAL IDEAS

109
the second place he himself has not always kept within the narrow
limits, which he has himself laid down as proper to his theory.

For he has not remained content with defining a probability as
identical with a statistical frequency, but has often spoken as if his
theory told us which alternatives it is reasonable to prefer. When he
states, for instance, that modality ought to be banished from Logic
and relegated to Probability (p. 296), he forgets his own dictum
that of premisses, the distinctive characteristic of which is their lack
of certainty, Probability takes account of one class only, Induction
concerning itself with another class, and so forth (p. 321). He forgets
also that, when he comes to consider the practical use of statistical
frequencies, he has to admit that an event may possess more than
one frequency, and that we must decide which of these to prefer on
extraneous grounds (p. 213). The device, he says, must be to a great
extent arbitrary, and there are no logical grounds of decision; but
would he deny that it is often reasonable to found our probability on
one statistical frequency rather than on another? And if our grounds
are reasonable, are they not in an important sense logical?

Even in those cases, therefore, in which we derive our preference
for one alternative over another from a knowledge of statistical
frequencies, a statistical frequency by itself is insufficient to determine
us. We may call a statistical frequency a probability, if we choose; but
the fundamental problem of determining which of several alternatives
is logically preferable still awaits solution. We cannot be content with
the only counsel Venn can offer, that we should choose a frequency
which is derived from a series neither too large nor too small.

The same difficulty, that a probability in Venn’s sense is insufficient
to determine which alternative is logically preferable, arises in another
connection. In most cases the statistical frequency is not given in
experience for certain, but is arrived at by a process of induction, and
inductions, he admits, are not certain. If, in the past, three infants
out of every ten have died in their first four years, induction may
base on this the doubtful assertion, All infants die in that proportion.

pt. i

A TREATISE ON PROBABILITY

110

But we cannot assert on this ground, as Venn wishes to do, that the
probability of the death of an infant in its first four years is 3 ths.

10

We can say no more than that it is probable (in my sense) that there
is such a probability (in his sense). For the purpose of coming to a
decision we cannot compare the value of this conclusion with that of
others until we know the probability (in my sense) that the statistical
frequency really is 3 ths. The cases in which we can determine

10
the logical value of a conclusion entirely on grounds of statistical
frequency would seem to be extremely few in number.

8.

The second main reason which led Venn to develop his
theory is to be found in his belief that probabilities which are based
on statistical frequencies are alone capable of accurate measurement.

The term ‘probabilities,’ he argues, is properly confined to the case
of chances which can be calculated, and all calculable chances can be
made to depend upon statistical frequency. In attempting to establish
this latter contention he is involved in some paradoxical opinions.

“In many cases,” he admits, “it is undoubtedly true that we do not
resort to direct experience at all. If I want to know what is my
chance of holding ten trumps in a game of whist, I do not enquire
how often such a thing has occurred before. . . . In practice, à priori
determination is often easy, whilst à posteriori appeal to experience
would be not merely tedious but utterly impracticable.” But these
cases which are usually based on the Principle of Indifference can,
he maintains, be justified on statistical grounds. In the case of coin
tossing there is a considerable experience of the equally frequent
occurrence of heads and tails; the experience gained in this simple case
is to be extended to the complex cases by “Induction and Analogy.” In
one simple case the result to which the Principle of Indifference would
lead is that which experience recommends. Therefore in complex
cases, where there is no basis of experiment at all, we may assume
that Experience, if experience there was, would speak with the same
voice as Indifference. This is to assert that, because in one case,
where there is no known reason to the contrary, there actually is
ch. viii

FUNDAMENTAL IDEAS

111
none, therefore in other cases incapable of verification the absence of
known reason to the contrary proves that actually there is none.

The attempt to justify the rules of inverse probability on statistical
grounds I have failed to understand; and after a careful reading, I am
unable to produce an intelligible account of the argument involved in
the latter part of chapter vii. of the Logic of Chance.∗ I am doubtful
whether Venn should not have excluded à posteriori arguments in
probability from his scheme as well as inductive arguments. The
attempt to include them may have been induced by a desire to deal
with all cases in which numerical calculation has been commonly
thought possible.

9.

The argument so far has been solely concerned with the
case for the frequency theory developed in the Logic of Chance. The
criticisms which follow will be directed against a more general form
of the same theory which may conceivably have recommended itself
to some readers. It is unfortunate that no adherent of the doctrine,
with the exception of Venn, has attempted to present the theory of it
in detail. Professor Karl Pearson, for instance, probably agrees with

Venn in a general way only, and it is very likely that many of the
foregoing remarks do not apply to his view of probability; but while

I generally disagree with the fundamental premisses upon which his
work in probability and statistics seems to rest, I am not clearly aware
of the nature of the philosophical theory from which he thinks that he
derives them and which makes them appear to him to be satisfactory.

A careful exposition of his logical presuppositions would greatly add
to the completeness of his work. In the meantime it is only possible
to raise general objections to any theory of probability which seeks to
found itself upon the conception of statistical frequency.

∗Let the reader, who is acquainted with this chapter, consider what precise
assumption Venn’s reasoning requires on p. 187 in the example which seeks to
show the efficacy of Lord Lister’s antiseptic treatment à posteriori. What is the

‘inevitable assumption about the bags’ when it is translated into the language
of this example?

pt. i

A TREATISE ON PROBABILITY

112

The generalised frequency theory which I propose to put forward,
as perhaps representative of what adherents of this doctrine have
in mind, differs from Venn’s in several important respects.∗ In the
first place, it does not regard probability as being identical with
statistical frequency, although it holds that all probabilities must be
based on statements of frequency, and can be defined in terms of
them. It accepts the theory that propositions rather than events
should be taken as the subject-matter of probability; and it adopts
the comprehensive view of the subject according to which it includes
induction and all other cases in which we believe that there are
logical grounds for preferring one alternative out of a set none of
which are certain. Nor does it follow Venn in supposing any special
connection to exist between a frequency theory of probability and
logical empiricism.

10.

A proposition can be a member of many distinct classes of
propositions, the classes being merely constituted by the existence of
particular resemblances between their members or in some such way.

We may know of a given proposition that it is one of a particular class
of propositions, and we may also know, precisely or within defined
limits, what proportion of this class are true, without our being aware
whether or not the given proposition is true. Let us, therefore, call the
actual proportion of true propositions in a class the truth-frequency†
of the class, and define the measure of the probability of a proposition
relative to a class, of which it is a member, as being equal to the
truth-frequency of the class.

The fundamental tenet of a frequency theory of probability is,
then, that the probability of a proposition always depends upon
referring it to some class whose truth-frequency is known within wide
or narrow limits.

∗In what follows I am much indebted for some suggestions in favour of the
frequency theory communicated to me by Dr. Whitehead; but it is not to be
supposed that the exposition which follows represents his own opinion.

†This is Dr. Whitehead’s phrase.

ch. viii

FUNDAMENTAL IDEAS

113

Such a theory possesses most of the advantages of Venn’s, but
escapes his narrowness. There is nothing in it so far which could
not be easily expressed with complete precision in the terms of
ordinary logic. Nor is it necessarily confined to probabilities which
are numerical. In some cases we may know the exact number which
expresses the truth-frequency of our class; but a less precise knowledge
is not without value, and we may say that one probability is greater
than another, without knowing how much greater, and that it is
large or small or negligible, if we have knowledge of corresponding
accuracy about the truth-frequencies of the classes to which the
probabilities refer. The magnitudes of some pairs of probabilities we
shall be able to compare numerically, others in respect of more and
less only, and others not at all. A great deal, therefore, of what has
been said in Chapter III. would apply equally to the present theory,
with this difference that the probabilities would, as a matter of fact,
have numerical values in all cases, and the less complete comparisons
would only hold the field in cases where the real probabilities were
partially unknown. On the frequency theory, therefore, there is an
important sense in which probabilities can be unknown, and the
relative vagueness of the probabilities employed in ordinary reasoning
is explained as belonging not to the probabilities themselves but only
to our knowledge of them. For the probabilities are relative, not to
our knowledge, but to some objective class, possessing a perfectly
definite truth-frequency, to which we have chosen to refer them.

The frequency theory expounded in this manner cannot easily
avoid mention of the relativity of probabilities which is implicit here,
as it is in Venn’s. Whether or not the probability of a proposition
is relative to given data, it is clearly relative to the particular class
or series to which we choose to refer it. A given proposition has
a great variety of different probabilities corresponding to each of
the various distinct classes of which it is a member; and before an
intelligible meaning can be given to a statement that the probability
of a proposition is so-and-so, the class must be specified to which the
pt. i

A TREATISE ON PROBABILITY

114
proposition is being referred. Most adherents of the frequency theory
would probably go further, and agree that the class of reference must
be determined in any particular case by the data at our disposal.

Here, then, is another point on which it is not necessary for the
frequency theory to diverge from the theory of this Treatise. It should,

I think, be generally agreed by every school of thought that the
probability of a conclusion is in an important sense relative to given
premisses. On this issue and also on the point that our knowledge of
many probabilities is not numerically definite, there might well be for
the future an end of disagreement, and disputation might be reserved
for the philosophical interpretation of these settled facts, which it is
unreasonable to deny, however we may explain them.

11.

I now proceed to those contentions upon which my
fundamental criticism of the frequency theory is founded. The first
of these relates to the method by which the class of reference is
to be determined. The magnitude of a probability is always to be
measured by the truth-frequency of some class; and this class, it is
allowed, must be determined by reference to the premisses, on which
the probability of the conclusion is to be determined. But, as a
given proposition belongs to innumerable different classes, how are
we to know which class the premisses indicate as appropriate? What
substitute has the frequency theory to offer for judgments of relevance
and indifference? And without something of this kind, what principle
is there for uniquely determining the class, the truth-frequency of
which is to measure the probability of the argument? Indeed the
difficulties of showing how given premisses determine the class of
reference, by means of rules expressed in terms of previous ideas, and
without the introduction of any notion, which is new and peculiar to
probability, appear to me insurmountable.

Whilst no general criterion of choice seems to exist, where of two
alternative classes neither includes the other, it might be thought
that where one does include the other, the obvious course would be
to take the narrowest and most specialised class. This procedure was
ch. viii

FUNDAMENTAL IDEAS

115
examined and rejected by Venn; though the objection to it is due,
not, as he supposed, to the lack of sufficient statistics in such cases
upon which to found a generalisation, but to the inclusion in the
class-concept of marks characteristic of the proposition in question,
but nevertheless not relevant to the matter in hand. If the process of
narrowing the class were to be carried to its furthest point, we should
generally be left with a class whose only member is the proposition in
question, for we generally know something about it which is true of no
other proposition. We cannot, therefore, define the class of reference
as being the class of propositions of which everything is true which
is known to be true of the proposition whose probability we seek to
determine. And, indeed, in those examples for which the frequency
theory possesses the greatest prima facie plausibility, the class of
reference is selected by taking account of some only of the known
characteristics of the quaesitum, those characteristics, namely, which
are relevant in the circumstances. In those cases in which one can
admit that the probability can be measured by reference to a known
truth-frequency, the class of reference is formed of propositions about
which our relevant knowledge is the same as about the proposition
under consideration. In these special cases we get the same result
from the frequency theory as from the Principle of Indifference. But
this does not serve to rehabilitate the frequency theory as a general
explanation of probability, and goes rather to show that the theory of
this Treatise is the generalised theory, comprehending within it such
applications of the idea of statistical truth-frequency as have validity.

‘Relevance’ is an important term in probability, of which the
meaning is readily intelligible. I have given my own definition of it
already. But I do not know how it is to be explained in terms of
the frequency theory. Whether supporters of this theory have fully
appreciated the difficulty I much doubt. It is a fundamental issue
involving the essence of the peculiarity of probability, which prevents
its being explained away in terms of statistical frequency or anything
pt. i

A TREATISE ON PROBABILITY

116
else.

12.

Yet perhaps a modified view of the frequency theory could be
evolved which would avoid this difficulty, and I proceed, therefore, to
some further criticisms. It might be agreed that a novel element must
be admitted at this point, and that relevancy must be determined in
some such manner as has been explained in earlier chapters. With
this admission, it might be argued, the theory would still stand,
divested, it is true, of some of its original simplicity, but nevertheless
a substantial theory differing in important respects, although not
quite so fundamentally as before, from alternative schemes.

The next important objection, then, is concerned with the manner
in which the principal theorems of probability are to be established
on a theory of frequency. This will involve an anticipation in some
part of later arguments; and the reader may be well advised to return
to the following paragraph after he has finished Part II.

13.

Let us begin by a consideration of the ‘Addition Theorem.’

If a/h denotes the probability of a on hypothesis h, this theorem
may be written (a + b)/h = a/h + b/h − ab/h, and may be read ‘On
hypothesis h the probability of “a or b” is equal to the probability
of a + the probability of b − the probability of “both a and b.” ’ This
theorem, interpreted in some way or other, is universally assumed;
and we must, therefore, inquire what proof of it the frequency
theory can afford. A little symbolism will assist the argument: Let
αf represent the truth-frequency of any class α, and let aα/h stand
for ‘the probability of a on hypothesis h, α being the class of reference
determined by this hypothesis.’∗ We then have aα/h = αf, and
we require to prove a proposition, for values of γ and δ not yet
determined, which will be of the form:

(a + b)δ/h = aα/h + bβ/h − abγ/h.

Now if δ is the class of propositions (a + b) such that a is an α

∗The question, previously at issue, as to how the class of reference is
determined by the hypothesis, is now ignored.

ch. viii

FUNDAMENTAL IDEAS

117
and b a β, it is easily shown by the ordinary arithmetic of classes
that δ = α
f
f + βf − αβf where αβ is the class of propositions which
are members of both α and β. In the case, therefore, where δ = δ
and γ = αβ, an addition theorem of the required kind has been
established.

But it does not follow by any reasonable rule that, if h determines
α and β as the appropriate classes of reference for a and b, h must
necessarily determine δ and αβ as the appropriate classes of reference
for (a + b) and ab; it may, for instance, be the case that h, while
it renders α and β determinate, yields no information whatever
regarding αβ, and points to some quite different class µ as the
suitable class of reference for ab. On the frequency theory, therefore,
we cannot maintain that the addition theorem is true in general, but
only in those special cases where it happens that δ = δ and γ = αβ.

The following is a good example: We are given that the proportion
of black-haired men in the population is p1 and the proportion of
q
colour-blind men p2 , and there is no known connection between
q
black-hair and colour-blindness: what is the probability that a man,
about whom nothing special is known, is∗ either black-haired or
colour-blind? If we represent the hypotheses by h and the alternatives
by a and b, it would usually be held that, colour-blindness and
black hair being independent for knowledge† relative to the given
data,
p
ab/h =

1p2 , and that, therefore, by the addition theorem,
q2
p
p
p

(a + b)/h = 1 + 2 − 1p2 . But, on the frequency theory, this result
q
q
q2
might be invalid; for
p
αβ

1p2
f =

, only if this is the actual proportion
q2
in fact of persons who are both colour-blind and black-haired, and
that this is the actual proportion cannot possibly be inferred from

∗In the course of the present discussion the disjunctive a + b is never
interpreted so as to exclude the conjunctive ab.

†For a discussion of this term see Chapter XVI. § 2.

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A TREATISE ON PROBABILITY

118
the independence for knowledge of the characters in question.∗

Precisely the same difficulty arises in connection with the
multiplication theorem ab/h = a/bh b/h.† In the frequency notation,
which is proposed above, the corresponding theorem will be of the
form abδ/h = aγ/bh bβ/h. For this equation to be satisfied it is easily
seen that δ must be the class of propositions xy such that x is a
member of α and y of β, and γ the class of propositions xb such that
x is a member of α; and, as in the case of the addition theorem, we
have no guarantee that these classes γ and δ will be those which the
hypotheses bh and h will respectively determine as the appropriate
classes of reference for a and ab.

In the case of the theorem of inverse probability‡
b/ah
a/bh
b/h

=
c/ah
a/ch
c/h
the same difficulty again arises, with an additional one when
practical applications are considered. For the relative probabilities
of our à priori hypotheses, b and c, will scarcely ever be capable
of determination by means of known frequencies, and in the
most legitimate instances of the inverse principle’s operation we
depend either upon an inductive argument or upon the Principle of

Indifference. It is hard to think of an example in which the frequency
conditions are even approximately satisfied.

Thus an important class of case, in which arguments in probability,
generally accepted as satisfactory, do not satisfy the frequency
conditions given above, are those in which the notion is introduced of
two propositions being, on certain data, independent for knowledge.

∗Venn argues (Logic of Chance, pp. 173, 174) that there is an inductive
ground for making this inference. The question of extending the fundamental
theorems of a frequency theory of probability by means of induction is discussed
in § 14 below.

†Vide Chapter XII. § 6, and Chapter XIV. § 4.

‡Vide Chapter XIV. § 5.

ch. viii

FUNDAMENTAL IDEAS

119

The meaning and definition of this expression is discussed more fully
in Part II.; but I do not see what interpretation the frequency theory can put upon it. Yet if the conception of ‘independence for
knowledge’ is discarded, we shall be brought to a standstill in the vast
majority of problems, which are ordinarily considered to be problems
in probability, simply from the lack of sufficiently detailed data. Thus
the frequency theory is not adequate to explain the processes of
reasoning which it sets out to explain. If the theory restricts its
operation, as would seem necessary, to those cases in which we know
precisely how far the true members of α and β overlap, the vast
majority of arguments in which probability has been employed must
be rejected.

14.

An appeal to some further principle is, therefore, required
before the ordinary apparatus of probable inference can be established
on considerations of statistical frequency; and it may have occurred
to some readers that assistance may be obtained from the principles
of induction. Here also it will be necessary to anticipate a subsequent
discussion. If the argument of Part III. is correct, nothing is more fatal than Induction to the theory now under criticism. For, so
far from Induction’s lending support to the fundamental rules of
probability, it is itself dependent on them. In any case, it is generally
agreed that an inductive conclusion is only probable, and that its
probability increases with the number of instances upon which it
is founded. According to the frequency theory, this belief is only
justified if the majority of inductive conclusions actually are true, and
it will be false, even on our existing data, that any of them are even
probable, if the acknowledged possibility that a majority are false
is an actuality. Yet what possible reason can the frequency theory
offer, which does not beg the question, for supposing that a majority
are true? And failing this, what ground have we for believing the
inductive process to be reasonable? Yet we invariably assume that
with our existing knowledge it is logically reasonable to attach some
weight to the inductive method, even if future experience shows that
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A TREATISE ON PROBABILITY

120
not one of its conclusions is verified in fact. The frequency theory,
therefore, in its present form at any rate, entirely fails to explain or
justify the most important source of the most usual arguments in the
field of probable inference.

15.

The failure of the frequency theory to explain or justify
arguments from induction or analogy suggests some remarks of a
more general kind. While it is undoubtedly the case that many
valuable judgments in probability are partly based on a knowledge of
statistical frequencies, and that many more can be held, with some
plausibility, to be indirectly derived from them, there remains a great
mass of probable argument which it would be paradoxical to justify in
the same manner. It is not sufficient, therefore, even if it is possible,
to show that the theory can be developed in a self-consistent manner;
it must also be shown how the body of probable argument, upon
which the greater part of our generally accepted knowledge seems to
rest, can be explained in terms of it; for it is certain that much of it
does not appear to be derived from premisses of statistical frequency.

Take, for instance, the intricate network of arguments upon which
the conclusions of The Origin of Species are founded: how impossible
it would be to transform them into a shape in which they would be
seen to rest upon statistical frequency! Many individual arguments, of
course, are explicitly founded upon such considerations; but this only
serves to differentiate them more clearly from those which are not.

Darwin’s own account of the nature of the argument may be quoted:

“The belief in Natural Selection must at present be grounded entirely
on general considerations: (1) on its being a vera causa, from the
struggle for existence and the certain geological fact that species do
somehow change; (2) from the analogy of change under domestication
by man’s selection; (3) and chiefly from this view connecting under
an intelligible point of view a host of facts. When we descend to
details . . . we cannot prove that a single species has changed; nor
can we prove that the supposed changes are beneficial, which is the
groundwork of the theory; nor can we explain why some species have
ch. viii

FUNDAMENTAL IDEAS

121
changed and others have not.”∗ Not only in the main argument, but
in many of the subsidiary discussions,† an elaborate combination of
induction and analogy is superimposed upon a narrow and limited
knowledge of statistical frequency. And this is equally the case in
almost all everyday arguments of any degree of complexity. The class
of judgments, which a theory of statistical frequency can comprehend,
is too narrow to justify its claim to present a complete theory of
probability.

16.

Before concluding this chapter, we should not overlook
the element of truth which the frequency theory embodies and which
provides its plausibility. In the first place, it gives a true account, so
long as it does not argue that probability and frequency are identical,
of a large number of the most precise arguments in probability, and of
those to which mathematical treatment is easily applicable. It is this
characteristic which has recommended it to statisticians, and explains
the large measure of its acceptance in England at the present time;
for the popularity in this country of an opinion, which has, so far as I
know, no thorough supporters abroad, may reasonably be attributed
to the chance which has led most of the English writers, who have
paid much attention to probability in recent years, to approach the
subject from the statistical side.

In the second place, the statement that the probability of an
event is measured by its actual frequency of occurrence ‘in the long
run’ has a very close connection with a valid conclusion which can
be derived, in certain cases, from Bernoulli’s theorem. This theorem
and its connection with the theory of frequency will be the subject of

Chapter XXIX.

17.

The absence of a recent exposition of the logical basis
of the frequency theory by any of its adherents has been a great
disadvantage to me in criticising it. It is possible that some of the

∗Letter to G. Bentham, Life and Letters, vol. iii. p. 25.

†E.g. in the discussion on the relative effect of disuse and selection in
reducing unnecessary organs to a rudimentary condition.

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A TREATISE ON PROBABILITY

122
opinions, which I have examined at length, are now held by no one;
nor am I absolutely certain, at the present stage of the inquiry, that
a partial rehabilitation of the theory may not be possible. But I am
sure that the objections which I have raised cannot be met without
a great complication of the theory, and without robbing it of the
simplicity which is its greatest preliminary recommendation. Until
the theory has been given new foundations, its logical basis is not
so secure as to permit controversial applications of it in practice. A
good deal of modern statistical work may be based, I think, upon an
inconsistent logical scheme, which, avowedly founded upon a theory
of frequency, introduces principles which this theory has no power to
justify.

CHAPTER IX
the constructive theory of part i. summarized

1.

That part of our knowledge which we obtain directly, supplies
the premisses of that part which we obtain by argument. From these
premisses we seek to justify some degree of rational belief about all
sorts of conclusions. We do this by perceiving certain logical relations
between the premisses and the conclusions. The kind of rational
belief which we infer in this manner is termed probable (or in the
limit certain), and the logical relations, by the perception of which it
is obtained, we term relations of probability.

The probability of a conclusion a derived from premisses h we
write a/h; and this symbol is of fundamental importance.

2.

The object of the Theory or Logic of Probability is to
systematise such processes of inference. In particular it aims at
elucidating rules by means of which the probabilities of different
arguments can be compared. It is of great practical importance
to determine which of two conclusions is on the evidence the more
probable.

The most important of these rules is the Principle of Indifference.

According to this Principle we must rely upon direct judgment
for discriminating between the relevant and the irrelevant parts of
the evidence. We can only discard those parts of the evidence
which are irrelevant by seeing that they have no logical bearing on
the conclusion. The irrelevant evidence being thus discarded, the

Principle lays it down that if the evidence for either conclusion is the
same (i.e. symmetrical), then their probabilities also are the same

(i.e. equal).

If, on the other hand, there is additional evidence (i.e. in addition
to the symmetrical evidence) for one of the conclusions, and this
evidence is favourably relevant, then that conclusion is the more
probable. Certain rules have been given by which to judge whether

123
pt. i

A TREATISE ON PROBABILITY

124
or not evidence is favourably relevant. And by combinations of these
judgments of preference with the judgments of indifference warranted
by the Principle of Indifference more complicated comparisons are
possible.

3.

There are, however, many cases in which these rules furnish no
means of comparison; and in which it is certain that it is not actually
within our power to make the comparison. It has been argued that
in these cases the probabilities are, in fact, not comparable. As in the
example of similarity, where there are different orders of increasing
and diminishing similarity, but where it is not possible to say of every
pair of objects which of them is on the whole the more like a third
object, so there are different orders of probability, and probabilities,
which are not of the same order, cannot be compared.

4.

It is sometimes of practical importance, when, for example,
we wish to evaluate a chance or to determine the amount of our
expectation, to say not only that one probability is greater than
another, but by how much it is greater. We wish, that is to say, to
have a numerical measure of degrees of probability.

This is only occasionally possible.

A rule can be given for
numerical measurement when the conclusion is one of a number
of equiprobable, exclusive, and exhaustive alternatives, but not
otherwise.

5.

In Part II. I proceed to a symbolic treatment of the subject, and to the greater systematisation, by symbolic methods on the basis
of certain axioms, of the rules of probable argument.

In Parts III., IV., and V. the nature of certain very important types of probable argument of a complex kind will be treated in
detail; in Part III. the methods of Induction and Analogy, in Part

IV. certain semi-philosophical problems, and in Part V. the logical foundations of the methods of inference now commonly known as
statistical.

PART II
fundamental theorems

CHAPTER X
introductory

1.

In Part I. we have been occupied with the epistemology of our subject, that is to say, with what we know about the characteristics
and the justification of probable Knowledge. In Part II. I pass to its Formal Logic. I am not certain of how much positive value this Part
will prove to the reader. My object in it is to show that, starting from
the philosophical ideas of Part I., we can deduce by rigorous methods out of simple and precise definitions the usually accepted results, such
as the theorems of the addition and multiplication of probabilities
and of inverse probability. The reader will readily perceive that
this Part would never have been written except under the influence
of Mr. Russell’s Principia Mathematica. But I am sensible that it
may suffer from the over-elaboration and artificiality of this method
without the justification which its grandeur of scale affords to that
great work. In common, however, with other examples of formal
method, this attempt has had the negative advantage of compelling
the author to make his ideas precise and of discovering fallacies
and mistakes. It is a part of the spade-work which a conscientious
author has to undertake; though the process of doing it may be of
greater value to him than the results can be to the reader, who is
concerned to know, as a safeguard of the reliability of the rest of the
construction, that the thing can be done, rather than to examine the
architectural plans in detail. In the development of my own thought,
the following chapters have been of great importance. For it was
through trying to prove the fundamental theorems of the subject on
the hypothesis that Probability was a relation that I first worked my
way into the subject; and the rest of this Treatise has arisen out of
attempts to solve the successive questions to which the ambition to
treat Probability as a branch of Formal Logic first gave rise.

A further occasion of diffidence and apology in introducing this

126
ch. x

FUNDAMENTAL THEOREMS

127

Part of my Treatise arises out of the extent of my debt to Mr. W. E.

Johnson. I worked out the first scheme in complete independence
of his work and ignorant of the fact that he had thought, more
profoundly than I had, along the same lines; I have also given the
exposition its final shape with my own hands. But there was an
intermediate stage, at which I submitted what I had done for his
criticism, and received the benefit not only of criticism but of his own
constructive exercises. The result is that in its final form it is difficult
to indicate the exact extent of my indebtedness to him. When the
following pages were first in proof, there seemed little likelihood of the
appearance of any work on Probability from his own pen, and I do
not now proceed to publication with so good a conscience, when he
is announcing the approaching completion of a work on Logic which
will include “Problematic Inference.”

I propose to give here a brief summary of the five chapters
following, without attempting to be rigorous or precise. I shall then
be free to write technically in Chapters XI.–XV., inviting the reader,
who is not specially interested in the details of this sort of technique,
to pass them by.

2.

Probability is concerned with arguments, that is to say,
with the “bearing” of one set of propositions upon another set. If
we are to deal formally with a generalised treatment of this subject,
we must be prepared to consider relations of probability between
any pair of sets of propositions, and not only between sets which
are actually the subject of knowledge. But we soon find that some
limitation must be put on the character of sets of propositions which
we can consider as the hypothetical subject of an argument, namely,
that they must be possible subjects of knowledge. We cannot, that
is to say, conveniently apply our theorems to premisses which are
self-contradictory and formally inconsistent with themselves.

For the purpose of this limitation we have to make a distinction
between a set of propositions which is merely false in fact and a
pt. ii

A TREATISE ON PROBABILITY

128
set which is formally inconsistent with itself.∗ This leads us to the
conception of a group of propositions, which is defined as a set of
propositions such that—(i.) if a logical principle belongs to it, all
propositions which are instances of that logical principle also belong
to it; (ii.) if the proposition p and the proposition ‘not-p or q’ both
belong to it, then the proposition q also belongs to it; (iii.) if any
proposition p belongs to it, then the contradictory of p is excluded
from it. If the group defined by one part of a set of propositions
excludes a proposition which belongs to a group defined by another
part of the set, then the set taken as a whole is inconsistent with
itself and is incapable of forming the premiss of an argument.

The conception of a group leads on to a precise definition of
one proposition requiring another (which in the realm of assertion
corresponds to relevance in the realm of probability), and of logical
priority as being an order of propositions arising out of their relation
to those special groups, or real groups, which are in fact the subject
of knowledge. Logical priority has no absolute signification, but is

∗Spinoza had in mind, I think, the distinction between Truth and Probability
in his treatment of Necessity, Contingence, and Possibility.

Res enim omnes
ex data Dei natura necessario sequutae sunt, et ex necessitate naturae Dei
determinatae sunt ad certo modo existendum et operandum (Ethices i. 33). That
is to say, everything is, without qualification, true or false.

At res aliqua
nulla alia de causa contingens dicitur, nisi respectu defectus nostrae cognitionis

(Ethices i. 33, scholium).

That is to say, Contingence, or, as I term it,

Probability, solely arises out of the limitations of our knowledge. Contingence
in this wide sense, which includes every proposition which, in relation to our
knowledge, is only probable (this term covering all intermediate degrees of
probability), may be further divided into Contingence in the strict sense, which
corresponds to an à priori or formal probability exceeding zero, and Possibility;
that is to say, into formal possibility and empirical possibility. Res singulares
voco contingentes, quatenus, dum ad earum solam essentiam attendimus, nihil
invenimus, quod earum existentiam necessario ponat, vel quod ipsam necessario
secludat.

Easdem res singulares voco possibiles, quatenus dum ad causas, ex
quibus produci debent, attendimus, nescimus, an ipsae determinatae sint ad
easdem producendum (Ethices iv. Def 3, 4).

ch. x

FUNDAMENTAL THEOREMS

129
relative to a specific body of knowledge, or, as it has been termed in
the traditional logic, to the Universe of Reference.

It also enables us to reach a definition of inference distinct from
implication, as defined by Mr. Russell. This is a matter of very great
importance. Readers who are acquainted with the work of Mr. Russell
and his followers will probably have noticed that the contrast between
his work and that of the traditional logic is by no means wholly
due to the greater precision and more mathematical character of his
technique. There is a difference also in the design. His object is
to discover what assumptions are required in order that the formal
propositions generally accepted by mathematicians and logicians may
be obtainable as the result of successive steps or substitutions of a few
very simple types, and to lay bare by this means any inconsistencies
which may exist in received results. But beyond the fact that the
conclusions to which he seeks to lead up are those of common sense,
and that the uniform type of argument, upon the validity of which
each step of his system depends, is of a specially obvious kind, he is
not concerned with analysing the methods of valid reasoning which
we actually employ. He concludes with familiar results, but he reaches
them from premisses, which have never occurred to us before, and by
an argument so elaborate that our minds have difficulty in following
it. As a method of setting forth the system of formal truth, which
shall possess beauty, interdependence, and completeness, his is vastly
superior to any which has preceded it. But it gives rise to questions
about the relation in which ordinary reasoning stands to this ordered
system, and, in particular, as to the precise connection between the
process of inference, in which the older logicians were principally
interested but which he ignores, and the relation of implication on
which his scheme depends.

‘p implies q’ is, according to his definition, exactly equivalent to
the disjunction ‘q is true or p is false.’ If q is true, ‘p implies q’

holds for all values of p; and similarly if p is false, the implication
holds for all values of q. This is not what we mean when we say
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A TREATISE ON PROBABILITY

130
that q can be inferred or follows from p. For whatever the exact
meaning of inference may be, it certainly does not hold between all
pairs of true propositions, and is not of such a character that every
proposition follows from a false one. It is not true that ‘A male
now rules over England’ follows or can be inferred from ‘A male now
rules over France’; or ‘A female now rules over England’ from ‘A
female now rules over France’; whereas, on Mr. Russell’s definition,
the corresponding implications hold simply in virtue of the facts that

‘A male now rules over England’ is true and ‘A female now rules over

France’ is false.

The distinction between the Relatival Logic of Inference and

Probability, and Mr. Russell’s Universal Logic of Implication, seems
to be that the former is concerned with the relations of propositions
in general to a particular limited group. Inference and Probability
depend for their importance upon the fact that in actual reasoning
the limitation of our knowledge presents us with a particular set of
propositions, to which we must relate any other proposition about
which we seek knowledge. The course of an argument and the results
of reasoning depend, not simply on what is true, but on the particular
body of knowledge from which we have set out. Ultimately, indeed,

Mr. Russell cannot avoid concerning himself with groups. For his
aim is to discover the smallest set of propositions which specify our
formal knowledge, and then to show that they do in fact specify it.

In this enterprise, being human, he must confine himself to that part
of formal truth which we know, and the question, how far his axioms
comprehend all formal truth, must remain insoluble. But his object,
nevertheless, is to establish a train of implications between formal
truths; and the character and the justification of rational argument
as such is not his subject.

3.

Passing on from these preliminary reflections, our first
task is to establish the axioms and definitions which are to make
operative our symbolical processes.

These processes are almost
entirely a development of the idea of representing a probability by
ch. x

FUNDAMENTAL THEOREMS

131
the symbol a/h, where h is the premiss of an argument and a its
conclusion. It might have been a notation more in accordance with
our fundamental ideas, to have employed the symbol a/h to designate
the argument from h to a, and to have represented the probability of
the argument, or rather the degree of rational belief about a which the
argument authorises, by the symbol P(a/h). This would correspond
to the symbol V(a/h) which has been employed in Chapter VI. for the
evidential value of the argument as distinct from its probability. But
in a section where we are only concerned with probabilities, the use of

P(a/h) would have been unnecessarily cumbrous, and it is, therefore,
convenient to drop the prefix P and to denote the probability itself
by a/h.

The discovery of a convenient symbol, like that of an essential
word, has often proved of more than verbal importance. Clear
thinking on the subject of Probability is not possible without a
symbol which takes an explicit account of the premiss of the argument
as well as of its conclusion; and endless confusion has arisen through
discussions about the probability of a conclusion without reference to
the argument as a whole. I claim, therefore, the introduction of the
symbol a/h as an essential step towards any progress in the subject.

4.

Inasmuch as relations of Probability cannot be assumed
to possess the properties of numbers, the terms addition and
multiplication of probabilities have to be given appropriate meanings
by definition. It is convenient to employ these familiar expressions,
rather than to invent new ones, because the properties which arise
out of our definitions of addition and multiplication in Probability are
analogous to those of addition and multiplication in Arithmetic. But
the process of establishing these properties is a little complicated and
occupies the greater part of Chapter XII.

The most important of the definitions of Chapter XII. are the
following (the numbers referring to the numbers of Chapter XII.):

II. The Definition of Certainty: a/h = 1.

III. The Definition of Impossibility: a/h = 0.

pt. ii

A TREATISE ON PROBABILITY

132

VI. The Definition of Inconsistency: ah is inconsistent if a/h = 0.

VII. The Definition of a Group: the class of propositions a such
that a/h = 1 is the group h.

VIII. The Definition of Equivalence: if b/ah = 1 and a/bh = 1

(a ≡ b)/h = 1.

IX. The Definition of Addition: ab/h + a¯b/h∗ = a/h.

X. The Definition of Multiplication: ab/h = a/bh b/h = b/ah a/h.

The symbolical development of the subject largely proceeds out of
these definitions of Addition and Multiplication. It is to be observed
that they give a meaning, not to the addition and multiplication of
any pairs of probabilities, but only to pairs which satisfy a certain
form. The definition of Multiplication may be read: ‘the probability
of both a and b given h is equal to the probability of a given bh,
multiplied by the probability of b given h.’

XI.

The Definition of Independence: if a1/a2h = a1/h and
a2/a1h = a2/h, a1/h and a2/h are independent.

XII.

The Definition of Irrelevance: if a1/a2h = a1/h, a2 is
irrelevant to a1/h.

5.

In Chapter XIII. these definitions, supplemented by a few
axioms, are employed to demonstrate the fundamental theorems of

Certain or Necessary Inference. The interest of this chiefly lies in the
fact that these theorems include those which the traditional Logic has
termed the Laws of Thought, as for example the Law of Contradiction
and the Law of Excluded Middle. These are here exhibited as a part
of the generalised theory of Inference or Rational Argument, which
includes probable Inference as well as certain Inference. The object of
this chapter is to show that the ordinarily accepted rules of Inference
can in fact be deduced from the definitions and axioms of Chapter

XII.

6.

In Chapter XIV. I proceed to the fundamental Theorems of

Probable Inference, of which the following are the most interesting:

∗¯
b stands for the contradictory of b.

ch. x

FUNDAMENTAL THEOREMS

133

Addition Theorem: (a + b)/h = a/h + b/h − ab/h, which reduces
to (a + b)/h = a/h + b/h, where a and b are mutually exclusive; and,
if p1p2 . . . pn form, relative to h a set of exclusive and exhaustive
n
alternatives, a/h =
pra/h.

1

Theorem of Irrelevance: If a/h

¯

1h2 = a/h1 then a/h1h2 = a/h1; i.e.

if a proposition is irrelevant, its contradictory also is irrelevant.

Theorem of Independence: If a2/a1h = a2/h, a1/a2h = a1/h; i.e. if
a1 is irrelevant to a2/h, it follows that a2 is irrelevant to a1/h and
that a1/h and a2/h are independent.

Multiplication Theorem:

If a1/h and a2/h are independent,
a1a2/h = a1/h a2/h.

b/a1h a1/h

Theorem of Inverse Probability: a1/bh =

. Further, if
a2/bh
b/a2h a2/h
a1/h = p1, a2/h = p2, b/a1h = q1, b/a2h = q2, and a1/bh + a2/bh = 1,
then
p
q
a

1q1

1

1/bh =

; and if a1/h = a2/h, a1/bh =

, which
p1q1 + p2q2
q1 + q2
is equivalent to the statement that the probability of a1 when we
know b is equal to
q1

, where q1 is the probability of b when
q1 + q2
we know a1 and q2 its probability when we know a2. This theorem
enunciated with varying degrees of inaccuracy appears in all Treatises
on Probability, but is not generally proved.

Chapter XIV. concludes with some elaborate theorems on the
combination of premisses based on a technical symbolic device, known
as the Cumulative Formula, which is the work of Mr. W. E. Johnson.

7.

In Chapter XV. I bring the non-numerical theory of probability
developed in the preceding chapters into connection with the usual
numerical conception of it, and demonstrate how and in what class of
cases a meaning can be given to a numerical measure of a relation
of probability. This leads on to what may be termed numerical
approximation, that is to say, the relating of probabilities, which are
not themselves numerical, to probabilities, which are numerical, by
means of greater and less, by which in some cases numerical limits
pt. ii

A TREATISE ON PROBABILITY

134
may be ascribed to probabilities which are not capable of numerical
measures.

CHAPTER XI
the theory of groups, with special reference to
logical consistence, inference, and logical
priority

1.

The Theory of Probability deals with the relation between
two sets of propositions, such that, if the first set is known to be true,
the second can be known with the appropriate degree of probability
by argument from the first.∗ The relation, however, also exists when
the first set is not known to be true and is hypothetical.

In a symbolical treatment of the subject it is important that we
should be free to consider hypothetical premisses, and to take account
of relations of probability as existing between any pair of sets of
propositions, whether or not the premiss is actually part of knowledge.

But in acting thus we must be careful to avoid two possible sources
of error.

2.

The first is that which is liable to arise wherever variables
are concerned. This was mentioned in passing in § 18 of Chapter IV.

We must remember that whenever we substitute for a variable some
particular value of it, this may so affect the relevant evidence as to
modify the probability. This danger is always present except where,
as in the first half of Chapter XIII., the conclusions respecting the
variable are certain.

3.

The second difficulty is of a different character. Our premisses
may be hypothetical and not actually the subject of knowledge. But
must they not be possible subjects of knowledge? How are we to deal
with hypothetical premisses which are self-contradictory or formally
inconsistent with themselves, and which cannot be the subject of
rational belief of any degree?

∗Or more strictly, “perception of which, together with knowledge of the first
set, justifies an appropriate degree of rational belief about the second.”

135
pt. ii

A TREATISE ON PROBABILITY

136

Whether or not a relation of probability can be held to exist
between a conclusion and a self-inconsistent premiss, it will be
convenient to exclude such relations from our scheme, so as to avoid
having to provide for anomalies which can have no interest in an
account of the actual processes of valid reasoning. Where a premiss
is inconsistent with itself it cannot be required.

4.

Let us term the collection of propositions, which are logically
involved in the premisses in the sense that they follow from them,
or, in other words, stand to them in the relation of certainty,∗ the
group specified by the premisses. That is to say, we define a group
as containing all the propositions logically involved in any of the
premisses or in any conjunction of them; and as excluding all the
propositions the contradictories of which are logically involved in any
of the premisses or in any conjunction of them.† To say, therefore,
that a proposition follows from a premiss, is the same thing as to say
that it belongs to the group which the premiss specifies.

The idea of a ‘group’ will then enable us to define ‘logical
consistency.’ If any part of the premisses specifies a group containing
a proposition, the contradictory of which is contained in a group
specified by some other part, the premisses are logically inconsistent;
otherwise they are logically consistent.

In short, premisses are
inconsistent if a proposition ‘follows from’ one part of them, and its
contradictory from another part.

5.

We have still, however, to make precise what we mean in this
definition by one proposition following from or being logically involved

∗‘a can be inferred from b,’ ‘a follows from b,’ ‘a is certain in relation
to b,’ ‘a is logically involved in b,’ I regard as equivalent expressions, the precise
meaning of which will be defined in succeeding paragraphs. ‘a is implied by b,’

I use in a different sense, namely, in Mr. Russell’s sense, as the equivalent of

‘b or not-a.’

†For the conception of a group, and for many other notions and definitions
in the course of this chapter—those, for example, of a real group and of logical
priority—I am largely indebted to Mr. W. E. Johnson. The origination of the
theory of groups is due to him.

ch. xi

FUNDAMENTAL THEOREMS

137
in the truth of another. We seem to intend by these expressions
some kind of transition by means of a logical principle. A logical
principle cannot be better defined, I think, than in terms of what
in Mr. Russell’s Logic of Implication is termed a formal implication.

‘p implies q’ is a formal implication if ‘not-p or q’ is formally true;
and a proposition is formally true, if it is a value of a propositional
function, in which all the constituents other than the arguments are
logical constants, and of which all the values are true.

We might define a group in such a way that all logical principles
belonged to every group. In this case all formally true propositions
would belong to every group. This definition is logically precise and
would lead to a coherent theory. But it possesses the defect of not
closely corresponding to the methods of reasoning we actually employ,
because all logical principles are not in fact known to us. And even in
the case of those which we do know, there seems to be a logical order

(to which on the above definition we cannot give a sense) amongst
propositions, which are about logical constants and are formally true,
just as there is amongst propositions which are not formally true.

Thus, if we were to assume the premisses in every argument to include
all formally true propositions, the sphere of probable argument would
be limited to what (in contradistinction to formally true propositions)

we may term empirical propositions.

6.

For this reason, therefore, I prefer a narrower definition—which
shall correspond more exactly to what we seem to mean when we say
that one proposition follows from another. Let us define a group of
propositions as a set of propositions such that:

(i.) if the proposition ‘p is formally true’ belongs to the group,
all propositions which are instances of the same formal propositional
function also belong to it;

(ii.) if the proposition p and the proposition ‘p implies q’ both
belong to it, then the proposition q also belongs to it;

(iii.) if any proposition p belongs to it, then the contradictory of p
is excluded from it.

pt. ii

A TREATISE ON PROBABILITY

138

According to this definition all processes of certain inference are
wholly composed of steps each of which is of one of two simple types

(and if we like we might perhaps regard the first as comprehending
the other). I do not feel certain that these conditions may not
be narrower than what we mean when we say that one proposition
follows from another. But it is not necessary for the purpose of
defining a group, to dogmatise as to whether any other additional
methods of inference are, or are not, open to us. If we define a group
as the propositions logically involved in the premisses in the above
sense, and prescribe that the premisses of an argument in probability
must specify a group not less extensive than this, we are placing
the minimum amount of restriction upon the form of our premisses.

If, sometimes or as a rule, our premisses in fact include some more
powerful principle of argument, so much the better.

In the formal rules of probability which follow, it will be postulated
that the set of propositions, which form the premiss of any argument,
must not be inconsistent. The premiss must, that is to say, specify
a ‘group’ in the sense that no part of the premiss must exclude a
proposition which follows from another part. But for this purpose we
do not need to dogmatise as to what the criterion is of inference or
certainty.

7.

It will be convenient at this point to define a term which
expresses the relation converse to that which exists between a set
of propositions and the group which they specify. The propositions
p1, p2 . . . pn are said to be fundamental to the group h if (i.) they
themselves belong to the group (which involves their being consistent
with one another); (ii.) if between them they completely specify the
group; and (iii.) if none of them belong to the group specified by the
rest (for if pr belongs to the group specified by the rest, this term is
redundant).

When the fundamental set is uniquely determined, a group h is a
subgroup to the group h, if the set fundamental to h is included in
the set fundamental to h.

ch. xi

FUNDAMENTAL THEOREMS

139

Logically there can be more than one distinct set of propositions
fundamental to a given group; and some extra-logical test must be
applied before the fundamental set is determined uniquely. On the
other hand, a group is completely determined when the constituent
propositions of the fundamental set are given. Further, any consistent
set of propositions evidently specifies some group, although such a set
may contain propositions additional to those which are fundamental
to the group it specifies. It is clear also that only one group can
be specified by a given set of consistent propositions. The members
of a group are, we may say, rationally bound up with the set of
propositions fundamental to it.

8.

If Mr. Bertrand Russell is right, the whole of pure mathematics
and of formal logic follows, in the sense defined above, from a small
number of primitive propositions. The group, therefore, which is
specified by these primitive propositions, includes the most remote
deductions not only amongst those known to mathematicians, but
amongst those which time and skill have not yet served to solve. If
we define certainty in a logical and not a psychological sense, it seems
necessary, if our premisses include the essential axioms, to regard as
certain all propositions which follow from these, whether or not they
are known to us. Yet it seems as if there must be some logical
sense in which unproved mathematical theorems—some of those, for
instance, which deal with the theory of numbers—can be likely or
unlikely, and in which a proposition of this kind, which has been
suggested to us by analogy or supported by induction, can possess an
intermediate degree of probability.

There can be no doubt, I think, that the logical relation of
certainty does exist in these cases in which lack of skill or insight
prevents our apprehending it, in spite of the fact that sufficient
premisses, including sufficient logical principles, are known to us. In
these cases we must say, what we are not permitted to say when the
indeterminacy arises from lack of premisses, that the probability is
unknown. There is still a sense, however, in which in such a case the
pt. ii

A TREATISE ON PROBABILITY

140
knowledge we actually possess can be, in a logical sense, only probable.

While the relation of certainty exists between the fundamental axioms
and every mathematical hypothesis (or its contradictory), there are
other data in relation to which these hypotheses possess intermediate
degrees of probability. If we are unable through lack of skill to discover
the relation of probability which an hypothesis does in fact bear
towards one set of data, this set is practically useless, and we must fix
our attention on some other set in relation to which the probability
is not unknown. When Newton held that the binomial theorem
possessed for empirical reasons sufficient probability to warrant a
further investigation of it, it was not in relation to the axioms of
mathematics, whether he knew them or not, that the probability
existed, but in relation to his empirical evidence combined, perhaps,
with some of the axioms. There is, in short, an exception to the rule
that we must always consider the probability of any conclusion in
relation to the whole of the data in our possession. When the relation
of the conclusion to the whole of our evidence cannot be known,
then we must be guided by its relation to some part of the evidence.

When, therefore, in later chapters I speak of a formal proposition as
possessing an intermediate degree of probability, this will always be
in relation to evidence from which the proposition does not logically
follow in the sense defined in § 6.

9.

It follows from the preceding definitions that a proposition is
certain in relation to a given premiss, or, in other words, follows from
this premiss if it is included in the group which that premiss specifies.

It is impossible if it is excluded from the group—if, that is to say,
its contradictory follows from the premiss. We often say, somewhat
loosely, that two propositions are contradictory to one another, when
they are inconsistent in the sense that, relative to our evidence, they
cannot belong to the same group. On the other hand, a proposition,
which is not itself included in the group specified by the premiss
and whose contradictory is not included either, has in relation to the
premiss an intermediate degree of probability.

ch. xi

FUNDAMENTAL THEOREMS

141

If a follows from h and is, therefore, included in the group specified
by h, this is denoted by a/h = 1. The relation of certainty, that is to
say, is denoted by the symbol of unity. The reason why this notation
is useful and has been adopted by common consent will appear when
the meaning of the product of a pair of relations of probability has
been explained. If we represent the relation of certainty by γ and
any other probability by α, the product α γ = α. Similarly, if
a is excluded from the group specified by h and is impossible in
relation to it, this is denoted by a/h = 0. The use of the symbol
zero to denote impossibility arises out of the fact that, if ω denotes
impossibility and α any other relation of probability, then, in the
senses of multiplication and addition to be defined later, the product
α ω = ω, and the sum α + ω = α. Lastly, if a is not included in the
group specified by h, this is written a/h =| 1 or a/h < 1; and if it is
not excluded, this is written a/h =| 0 or a/h > 0.

10.

The theory of groups now enables us to give an account,
with the aid of some further conceptions, of logical priority and of the
true nature of inference. The groups, to which we refer the arguments
by which we actually reason, are not arbitrarily chosen. They are
determined by those propositions of which we have direct knowledge.

Our group of reference is specified by those direct judgments in
which we personally rationally certify the truth of some propositions
and the falsity of others. So long as it is undetermined, or not
determined uniquely, which propositions are fundamental, it is not
possible to discover a necessary order amongst propositions or to show
in what way a true proposition ‘follows from’ one true premiss rather
than another. But when we have determined what propositions are
fundamental, by selecting those which we know directly to be true,
or in some other way, then a meaning can be attached to priority
and to the distinction between inference and implication. When the
propositions which we know directly are given, there is a logical order
amongst those other propositions which we know indirectly and by
pt. ii

A TREATISE ON PROBABILITY

142
argument.

11.

It will be useful to distinguish between those groups which
are hypothetical and those of which the fundamental set is known to
be true. We will term the former hypothetical groups and the latter
real groups. To the real group, which contains all the propositions
which are known to be true, we may assign the old logical term

Universe of Reference. While knowledge is here taken as the criterion
of a real group, what follows will be equally valid whatever criterion
is taken, so long as the fundamental set is in some manner or other
determined uniquely.

If it is impossible for us to know a proposition p except by
inference from a knowledge of q, so that we cannot know p to be
true unless we already know q, this may be expressed by saying that

‘p requires q’. More precisely requirement is defined as follows:

p does not require q if there is some real group to which p belongs
and q does not belong, i.e. if there is a real group h such that
p/h = 1, q/h =| 1; hence
p requires q if there is no real group to which p belongs and q does
not belong.

p does not require q within the group h, if the group h, to which
p belongs, contains a subgroup∗ h
to which p belongs and q does
not belong; i.e. if there is a group h such that h /h = 1, p/h = 1,
q/h =

| 1. This reduces to the proposition next but one above if h is
the Universe of Reference. In § 13 these definitions will be generalised
to cover intermediate degrees of probability.

12.

Inference and logical priority can be defined in terms of
requirement and real groups. It is convenient to distinguish two types
of inference corresponding to hypothetical and real groups—i.e. to
cases where the argument is only hypothetical, and cases where the
conclusion can be asserted:

Hypothetical Inference.—‘If p, q,’ which may also be read ‘q is

∗Subgroups have only been defined, it must be noticed (see § 7 above) when
the fundamental set of the group has been, in some way, uniquely determined.

ch. xi

FUNDAMENTAL THEOREMS

143
hypothetically inferrible from p,’ means that there is a real group h
such that q/ph = 1, and q/h =| 1. In order that this may be the
case, ph must specify a group; i.e. p/h =| 0, or in other words p must
not be excluded from h. Hypothetical inference is also equivalent to:

‘p implies q’ and ‘p implies q’ does not require ‘q’. In other words,
q is hypothetically inferrible from p, if we know that q is true or p is
false and if we can know this without first knowing either that q is
true or that p is false.

Assertoric Inference.—‘p ∴ q,’ which may be read ‘p therefore q’

or ‘q may be asserted by inference from p,’ means that ‘if p, q’ is
true, and in addition ‘p’ belongs to a real group; i.e. there are proper
groups h and h such that p/h = 1, q/ph = 1, q/h =| 1, and p/h =| 0.

p is prior to q when p does not require q, and q requires p, when,
that is to say, we can know p without knowing q, but not q unless we
first know p.

p is prior to q within the group h when p does not require q within
the group, and q does require p within the group.

It follows from this and from the preceding definitions that, if
a proposition is fundamental in the sense that we can only know
it directly, there is no proposition prior to it; and, more generally,
that, if a proposition is fundamental to a given group, there is no
proposition prior to it within the group.

13.

We can now apply the conception of requirement to
intermediate degrees of probability. The notation adopted is, it will
be remembered, as follows:

p/h = α means that the proposition p has the probable relation
of degree α to the proposition h; while it is postulated that h is
self-consistent and therefore specifies a group.

p/h = 1 means that p follows from h and is, therefore, included in
the group specified by h.

p/h = 0 means that p is excluded from the group specified by h.

If h specifies the Universe of Reference, i.e. if its group
comprehends the whole of our knowledge, p/h is called the absolute
pt. ii

A TREATISE ON PROBABILITY

144
probability of p, or (for short) the probability of p; and if p/h = 1 and
h specifies any real group, p is said to be absolutely certain or (for
short) certain. Thus p is ‘certain’ if it is a member of a real group,
and a ‘certain’ proposition is one which we know to be true. Similarly
if p/h = 0 under the same conditions, p is absolutely impossible, or

(for short) impossible. Thus an ‘impossible’ proposition is one which
we know to be false.

The definition of requirement, when it is generalised so as to take
account of intermediate degrees of probability, becomes, it will be
seen, equivalent to that of relevance:

The probability of p does not require q within the group h, if there
is a subgroup h such that, for every subgroup h which includes h
and is included in h (i.e. h /h = 1, h /h = 1), p/h = p/h , and
q/h =

| q/h.

When p is included in the group h, this definition reduces to the
definition of requirement given in § 11.

14.

The importance of the theory of groups arises as soon as
we admit that there are some propositions which we take for granted
without argument, and that all arguments, whether demonstrative
or probable, consist in the relating of other conclusions to these as
premisses.

The particular propositions, which are in fact fundamental to the

Universe of Reference, vary from time to time and from person to
person. Our theory must also be applicable to hypothetical Universes.

Although a particular Universe of Reference may be defined by
considerations which are partly psychological, when once the Universe
is given, our theory of the relation in which other propositions stand
towards it is entirely logical.

The formal development of the theory of argument from imposed
and limited premisses, which is attempted in the following chapters,
resembles in its general method other parts of formal logic. We
seek to establish implications between our primitive axioms and the
derivative propositions, without specific reference to what particular
ch. xi

FUNDAMENTAL THEOREMS

145
propositions are fundamental in our actual Universe of Reference.

It will be seen more clearly in the following chapters that the
laws of inference are the laws of probability, and that the former is
a particular case of the latter. The relation of a proposition to a
group depends upon the relevance to it of the group, and a group is
relevant in so far as it contains a necessary or sufficient condition of
the proposition, or a necessary or sufficient condition of a necessary
or sufficient condition, and so on; a condition being necessary if every
hypothetical group, which includes the proposition together with the

Universe of Reference, includes the condition, and sufficient if every
hypothetical group, which includes the condition together with the

Universe of Reference, includes the proposition.

CHAPTER XII
the definitions and axioms of inference and
probability

1.

It is not necessary for the validity of what follows to
decide in what manner the set of propositions is determined, which
is fundamental to our Universe of Reference, or to make definite
assumptions as to what propositions are included in the group which
is specified by the data. When we are investigating an empirical
problem, it will be natural to include the whole of our logical
apparatus, the whole body, that is to say, of formal truths which
are known to us, together with that part of our empirical knowledge
which is relevant. But in the following formal developments, which
are designed to display the logical rules of probability, we need only
assume that our data always include those logical rules, of which the
steps of our proofs are instances, together with the axioms relating to
probability which we shall enunciate.

The object of this and the chapters immediately following is to
show that all the usually assumed conclusions in the fundamental
logic of inference and probability follow rigorously from a few axioms,
in accordance with the fundamental conceptions expounded in Part

I. This body of axioms and theorems corresponds, I think, to what logicians have termed the Laws of Thought, when they have meant by
this something narrower than the whole system of formal truth. But
it goes beyond what has been usual, in dealing at the same time with
the laws of probable, as well as of necessary, inference.

2.

This and the following chapters of Part II. are largely
independent of many of the more controversial issues raised in the
preceding chapters. They do not prejudge the question as to whether
or not all probabilities are theoretically measurable; and they are
not dependent on our theories as to the part played by direct
judgment in establishing relations of probability or inference between

146
ch. xii

FUNDAMENTAL THEOREMS

147
particular propositions. Their premisses are all hypothetical. Given
the existence of certain relations of probability, others are inferred.

Of the conclusions of Chapter III., of the criteria of equiprobability
and of inequality discussed in Chapters IV. and V., and of the criteria
of inference discussed in §§ 5, 6 of Chapter XI., they are, I think,
wholly independent. They deal with a different part of the subject,
not so closely connected with epistemology.

3.

In this chapter I confine myself to Definitions and Axioms.

Propositions will be denoted by small letters, and relations by
capital letters. In accordance with common usage, a disjunctive
combination of propositions is represented by the sign of addition,
and a conjunctive combination by simple juxtaposition (or, where it
is necessary for clearness, by the sign of multiplication): e.g. ‘a or
b or c’ is written ‘a + b + c,’ and ‘a and b and c’ is written ‘abc.’

‘a + b’ is not so interpreted as to exclude ‘a and b.’ The contradictory
of a is written ¯a.

4.

Preliminary Definitions:

I.

If there exists a relation of probability P between the
proposition a and the premiss h
a/h = P.

Def.

II. If P is the relation of certainty∗

P = 1.

Def.

III. If P is the relation of impossibility∗

P = 0.

Def.

IV. If P is a relation of probability, but not the relation of
certainty

P < 1.

Def.

∗These symbols were first employed by Leibnitz. See p. 155 below.

pt. ii

A TREATISE ON PROBABILITY

148

V. If P is a relation of probability, but not the relation of
impossibility

P > 0.

Def.

VI. If a/h = 0, the conjunction ah is inconsistent.

Def.

VII. The class of propositions a such that a/h = 1 is the group
specified by h or (for short) the group h.

Def.

VIII. If b/ah = 1 and a/bh = 1, (a ≡ b)/h = 1.

Def.

This may be regarded as the definition of Equivalence. Thus we
see that equivalence is relative to a premiss h. a is equivalent to b,
given h, if b follows from ah, and a from bh.

5.

Preliminary Axioms:

We shall assume that there is included in every premiss with
which we are concerned the formal implications which allow us to
assert the following axioms:

(i.) Provided that a and h are propositions or conjunctions of
propositions or disjunctions of propositions, and that h is not an
inconsistent conjunction, there exists one and only one relation of
probability P between a as conclusion and h as premiss. Thus any
conclusion a bears to any consistent premiss h one and only one
relation of probability.

(ii.) If (a ≡ b)/h = 1, and x is a proposition, x/ah = x/bh. This
is the Axiom of Equivalence.

(iii.)

( a + b ≡ ¯
a¯
b )/h = 1

(aa ≡ a)/h = 1

( ¯
a ≡ a )/h = 1

(ab + ¯
ab ≡ b)/h = 1.

If a/h = 1, ah ≡ h.

That is to say, if a is included in the group specified by h, h and ah
are equivalent.

6.

Addition and Multiplication.—If we were to assume that
probabilities are numbers or ratios, these operations could be given
ch. xii

FUNDAMENTAL THEOREMS

149
their usual arithmetical signification.

In adding or multiplying
probabilities we should be simply adding or multiplying numbers.

But in the absence of such an assumption, it is necessary to give a
meaning by definition to these processes. I shall define the addition
and multiplication of relations of probabilities only for certain types
of such relations. But it will be shown later that the limitation thus
placed on our operations is not of practical importance.

We define the sum of the probable relations ab/h and a¯b/h as
being the probable relation a/h; and the product of the probable
relations a/bh and b/h as being the probable relation ab/h. That is
to say:

IX.

ab/h + a¯
b/h = ah.

Def.

X.

ab/h = a/bh b/h = b/ah a/h.

Def.

Before we proceed to the axioms which will make these symbols
operative, the definitions may be restated in more familiar language.

IX. may be read: “The sum of the probabilities of ‘both a and b’

and of ‘a but not b,’ relative to the same hypothesis, is equal to the
probability of ‘a’ relative to this hypothesis.” X. may be read: “The
probability of ‘both a and b,’ assuming h, is equal to the product of
the probability of b, assuming h, and the probability of a, assuming
both b and h.” Or in the current terminology∗ we should have:

“The probability that both of two events will occur is equal to the
probability of the first multiplied by the probability of the second,
assuming the occurrence of the first.” It is, in fact, the ordinary
rule for the multiplication of the probabilities of events which are not

‘independent.’ It has, however, a much more central position in the
development of the theory than has been usually recognised.

Subtraction and division are, of course, defined as the inverse
operations of addition and multiplication:

XI. If

R

PQ = R, P =

.

Def.

Q

∗E.g. Bertrand, Calcul des probabilités, p. 26.

pt. ii

A TREATISE ON PROBABILITY

150

XII. If P + Q = R, P = R − Q.

Def.

Thus we have to introduce as definitions what would be axioms
if the meaning of addition and multiplication were already defined.

In this latter case we should have been able to apply the ordinary
processes of addition and multiplication without any further axioms.

As it is, we need axioms in order to make these symbols, to which we
have given our own meaning, operative. When certain properties are
associated, it is often more or less arbitrary which we take as defining
properties and which we associate with these by means of axioms. In
this case I have found it more convenient, for the purposes of formal
development, to reverse the arrangement which would come most
natural to common sense, full of preconceptions as to the meaning of
addition and multiplication. I define these processes, for the theory
of probability, by reference to a comparatively unfamiliar property,
and associate the more familiar properties with this one by means of
axioms. These axioms are as follows:

(iv.) If P, Q, R are relations of probability such that the products

PQ, PR and the sums P + Q, P + R exist, then:

(iv.a) If PQ exists, QP exists, and PQ = QP. If P + Q exists,

Q + P exists and P + Q = Q + P.

(iv.b) PQ < P unless Q = 1 or P = 0; P + Q > P unless Q = 0.

PQ = P
if

Q = 1 or P = 0; P + Q = P
if

Q = 0.

(iv.c) If PQ

PR, then Q

R unless P = 0. If P + Q

P + R,
then Q

R and conversely.

A meaning has not been given, it is important to notice, to the
signs of addition and multiplication between probabilities in all cases.

According to the definitions we have given, P + Q and PQ have not
an interpretation whenever P and Q are relations of probability, but
in certain conditions only. Furthermore, if P + Q = R and Q = S + T,
it does not follow that P + S + T = R, since no meaning has been
assigned to such an expression as P + S + T. The equation must
be written P + (S + T) = R, and we cannot infer from the foregoing
axioms that (P + S) + T = R. The following axioms allow us to make
ch. xii

FUNDAMENTAL THEOREMS

151
this and other inferences in cases in which the sum P + S exists, i.e.

when P + S = A and A is a relation of probability.

(v.)

[±P ± Q] + [±R ± S] = [±P ± R] − [∓Q ∓ S]

= [±P ± R] + [±Q ± S]

= [±P ± Q] − [∓R ∓ S]

in every case in which the probabilities [±P ± Q], [±R ± S], [±P ± R],
etc., exist, i.e., in which these sums satisfy the conditions necessary
in order that a meaning may be given to them in the terms of our
definition.

(vi.) P(R ± S) = PR ± PS, if the sum R ± S and the products

PR and PS exist as probabilities.

7.

From these axioms it is possible to derive a number of
propositions respecting the addition and multiplication of probabilities.

They enable us to prove, for instance, that if P + Q = R + S then

P−R = S−Q, provided that the differences P−R and S−Q exist; and
that (P+Q)(R+S) = (P+Q)R+(P+Q)S = [PR+QR]+[PS+QS] =

[PR+QS]+[QR+PS], provided that the sums and products in question
exist. In general any rearrangement which would be legitimate in
an equation between arithmetic quantities is also legitimate in an
equation between probabilities, provided that our initial equation and
the equation which finally results from our symbolic operations can
both be expressed in a form which contains only products and sums
which have an interpretation as probabilities in accordance with the
definitions. If, therefore, this condition is observed, we need not
complicate our operations by the insertion of brackets at every stage,
and no result can be obtained as a result of leaving them out, if it is
of the form prescribed above, which could not be obtained if they had
been rigorously inserted throughout. We can only be interested in
our final results when they deal with actually existent and intelligible
probabilities—for our object is, always, to compare one probability
with another—and we are not incommoded, therefore, in one symbolic
pt. ii

A TREATISE ON PROBABILITY

152
operations by the circumstance that sums and products do not exist
between every pair of probabilities.

8.

Independence:

XIII. If a1/a2h = a1/h and a2/a1h = a2/h, the probabilities
a1/h and a2/h are independent.

Def.

Thus the probabilities of two arguments having the same premisses
are independent, if the addition to the premisses of the conclusion of
either leaves them unaffected.

Irrelevance:∗

XIV. If a1/a2h = a1/h, a2 is irrelevant on the whole, or, for short,
irrelevant to a1/h.

Def.

∗This is repeated for convenience of reference from Chapter IV. § 14. It is
only necessary here to take account of irrelevance on the whole, not of the more
precise sense.

CHAPTER XIII
the fundamental theorems of necessary
inference

1.

In this chapter we shall be mainly concerned with deducing
the existence of relations of certainty or impossibility, given other
relations of certainty or impossibility,—with the rules, that is to
say, of Certain or, as De Morgan termed it, of Necessary Inference.

But it will be convenient to include here a few theorems dealing
with intermediate degrees of probability.

Except in one or two
important cases I shall not trouble to translate these theorems from
the symbolism in which they are expressed, since their interpretation
presents no difficulty.

2.

(1) a/h + ¯a/h = 1.

For
ab/h + ¯
ab/h = b/h
by IX.,
a/bh b/h + ¯
a/bh b/h = b/h
by X.

Put
b/h = 1, then a/bh + ¯a/bh = 1
by (iv. b),
since
b/h = 1,
bh ≡ h
by (iii.).

Thus
a/h + ¯
a/h = 1
by (ii.).

(1.1) If a/h = 1, ¯a/h = 0,
a/h + ¯
a/h = 1
by (1),

∴ a/h + ¯a/h = a/h = a/h + 0
by (iv. b),

∴ ¯a/h = 0
by (iv. c).

(1.2) Similarly, if ¯a/h = 1, a/h = 0.

(1.3) If a/h = 0, ¯a/h = 1,
a/h + ¯
a/h = 1
by (1),

∴ 0 + ¯a/h = 0 + 1
by (iv. b),

∴ ¯a/h = 1
by (iv. c).

153
pt. ii

A TREATISE ON PROBABILITY

154

(1.4) Similarly, if ¯a/h = 0, a/h = 1.

(2)

a/h < 1 or a/h = 1
by IV.

(3)

a/h > 0 or a/h = 0
by V.,
i.e. there are no negative probabilities.

(4)

ab/h < b/h or ab/h = b/h
by X. and (iv. b).

If P and Q are relations of probability and P + Q = 0, then P = 0
and Q = 0.

P + Q > P unless Q = 0
by (iv. b),
and

P > 0 unless P = 0
by V.

∴ P + Q > 0 unless Q = 0.

Hence, if P + Q = 0, Q = 0 and similarly P = 0.

(6) If PQ = 0, P = 0 or Q = 0,

Q > 0 unless Q = 0
by V.

Hence

PQ > P 0 unless Q = 0 or P = 0
by (iv. c),
i.e.

PQ > 0 unless Q = 0 or P = 0
by (iv. b).

Whence, if PQ = 0, the result follows.

(7) If PQ = 1, P = 1 and Q = 1,

PQ < P unless P = 0 or Q = 1
by (iv. b),

PQ = P if P = 0 or Q = 1
by (iv. b),
and

P < 1 unless P = 1
by IV.,

∴ PQ < 1 unless P = 1.

ch. xiii

FUNDAMENTAL THEOREMS

155

Hence P = 1; similarly Q = 1.

(8) If a/h = 0, ab/h = 0 and a/bh = 0 if bh is not inconsis-
tent.

For
ab/h = b/ah a/h = a/bh b/h
by X.,
and since a/h = 0,
b/ah a/h = 0
by (iv. b),

∴ ab/h = 0 and a/bh b/h = 0,

∴ unless b/h = 0, a/bh = 0
by (5),
whence the result by VI.

Thus, if a conclusion is impossible, we may add to the conclusion
or add consistently to the premisses without affecting the argument.

(9) If a/h = 1, a/bh = 1 if bh is not inconsistent.

Since a/h = 1,

¯
a/h = 0
by (1.1),

∴ ¯a/bh = 0 by (8) if bh is not inconsistent,
whence
a/bh = 1
by (1.4).

Thus we may add to premisses, which make a conclusion certain,
any other premisses not inconsistent with them, without affecting the
result.

(10) If a/h = 1, ab/h = b/ah = b/h,
ab/h = b/ah a/h = a/bh b/h
by X.

Since a/h = 1, a/bh = 1 by (9) unless b/h = 0,

∴ b/ah a/h = b/ah and a/bh b/h = b/h
by (iv. b),
pt. ii

A TREATISE ON PROBABILITY

156
whence the result, unless b/h = 0. If b/h = 0, the result follows from

(8).(11) If ab/h = 1, a/h = 1.

For
ab/h = b/ah a/h
by X.,

∴ a/h = 1
by (7).

(12) If (a ≡ b)/h = 1, a/h = b/h,
b/ah a/h = a/bh b/h
by X.

and
b/ah = 1, a/bh = 1
by VIII.,

∴ a/h = b/h
by (iv. b).

(12.1) If (a ≡ b)/h = 1 and hx is not inconsistent,
a/hx = b/hx.

a/hx x/h = x/ah a/h,
and
b/hx x/h = x/bh b/h
by X.,
x/ah = x/bh
by (ii.),
and
a/h = b/h
by (12),

∴ a/hx = b/hx unless x/h = 0.

This is the principle of equivalence. In virtue of it and of axiom

(ii.), if (a ≡ b)/h = 1, we can substitute a for b and vice versa,
wherever they occur in a probability whose premisses include h.

(13) a/a = 1, unless a is inconsistent.

For
a/a = aa/a = a/aa a/a
by (iii.), (12), and X.,
whence a/aa = 1 by (ii.), unless a/a = 0,
i.e.

a/a = 1, unless a is inconsistent
by (iii.), (12), and VI.

(13.1) ¯a/a = 0, unless a is inconsistent.

This follows from

(13) and (1.1).

ch. xiii

FUNDAMENTAL THEOREMS

157

(13.2) a/¯a = 0, unless ¯a is inconsistent. This follows from (iii.) by
writing ¯a for a in (13.1).

(14) If a/b = 0 and a is not inconsistent, b/a = 0.

Let f be the group of assumptions, common to a and b, which we
have supposed to be included in every real group;
then
a/b = a/bf and b/a = b/af
by (iii.) and (12),
and
a/bf b/f = b/af a/f
by X.

Since
a/bf = 0 by hypothesis,
and
a/f =

| 0, since a is not inconsistent,

∴ b/af = 0,
whence
b/a = 0.

Thus, if a is impossible given b, then b is impossible given a.

(15) If h1/h2 = 0, h1h2/h = 0,
h1/h2/h = h1/h2h h2/h
by X.,
and since h1/h2 = 0, h/h2h = 0 by (8), unless h/h2 = 0, whence the
result by (iv. b), unless h/h2 = 0.

If
h/h2 = 0,
h2/h = 0
by (14),
since we assume that h is not inconsistent, and hence
h1h2/h = 0
by (8).

Thus, if h1 is impossible given h2, h1h2 is always impossible and is
excluded from every group.

(15.1) If h1h2/h = 0 and h2h is not inconsistent, h1/h2h = 0. This,
which is the converse of (15), follows from X. and (6).

pt. ii

A TREATISE ON PROBABILITY

158

(16) If h1/h2 = 1, (h1 + ¯h2)/h = 1,

¯
h1/h2 = 0
by (1),

∴ ¯h1h2/h = 0
by (15),

∴ h1h2/h = 1
by (1.3),

∴ (h1 + ¯h2)/h = 1
by (12) and (iii.).

(16.1) We may write (16):

If h1/h2 = 1, (h2 ⊃ h1)/h = 1, where ‘⊃’ symbolises ‘implies.’

Thus if h1 follows from h2, then it is always certain that h2 implies h1.

(16.2) If (h1 + ¯h2)/h = 1 and h2h is not inconsistent, h1/h2h =

1.

¯
h1h2/h = 0,
as in (16),

∴ ¯h1/h2h = 0 by (15.1), since h2h is not inconsistent,

∴ h1/h2h = 1
by (1.4).

This is the converse of (14).

(16.3) We may write (16.2):

If (h2 ⊃ h1)/h = 1 and h2h is not inconsistent, h1/h2h = 1. Thus,
if we define a ‘group’ as a set of propositions, which follow from and
are certain relatively to the proposition which specifies them, this
proposition proves that, if h2 ⊃ h1 and h2 belong to a group h2h,
then h1 also belongs to this group.

(17) If (h1 ⊃: a ≡ b)/h = 1 and h1h is not inconsistent,
a/h1h = b/h1h. This follows from (16.3) and (12).

(18) a/a = 1 or ¯a/¯a = 1.

¯
a/a = 1, unless a is inconsistent,
by (13).

If a is inconsistent, a/h = 0, where h is not inconsistent, and
therefore

¯
a/h = 1
by (1.3).

ch. xiii

FUNDAMENTAL THEOREMS

159

Thus unless a is inconsistent, ¯a is not inconsistent, and therefore

¯
a/¯
a = 1
by (13).

(19) a¯a/h = 0,

¯
a/¯
a = 1 or a/a = 1
by (18),

∴ a/¯a = 0 or ¯a/a = 0
by (1.1) and (1.2).

In either case a¯a/h = 0 by (15).

Thus it is impossible that both a and its contradictory should be
true. This is the Law of Contradiction.

(20) (a + ¯a)/h = 1.

Since

(a¯
a ≡ a + ¯
a)/h = 1
by (iii.),
a + ¯
a/h = 0
by (19) and (12),

∴ (a + ¯a)/h = 1
by (1.3).

Thus it is certain that either a or its contradictory is true. This is
the Law of Excluded Middle.

(21) If a/h1 = 1 and a/h2 = 0, h1h2/h = 0.

For
a/h1h2 h1/h2 = h1/ah2 ah2,
and

¯
a/h1h2 h2/h1 = h2/¯
ah1 ¯
a/h1
by X.,

∴ a/h1h2 h1/h2 = 0 and ¯a/h1h2 h2/h1 = 0,
since, by hypothesis and (1), ¯a/h1 = 0 and a/h2 = 0,

∴ a/h1h2 = 0 or h1/h2 = 0,
and
a/h1h2 = 1 or h2/h1 = 0,

∴ h1/h2 = 0 or h2/h1 = 0.

In either case h1h2/h = 0 by (15).

Thus, if a proposition is certain relatively to one set of premisses,
and impossible relatively to another set, the two sets are incompatible.

pt. ii

A TREATISE ON PROBABILITY

160

(22) If a/h1 = 0 and h1/h = 1, a/h = 0,
ah1/h = 0 by (15),

∴ h1/ah a/h = 0,
h1/ah = 1 by (9), unless a/h = 0.

∴ in any case a/h = 0.

(23) If b/a = 0 and b/¯a = 0, b/h = 0.

ab/h = 0 and ¯ab/h = 0
by (15),

∴ a/bh = 0 or b/h = 0,
and

¯
a/bh = 0 or b/h = 0
by II. and (iv.),
whence
b/h = 0
by (1.4).

CHAPTER XIV
the fundamental theorems of probable
inference

1.

I shall give proofs in this chapter of most of the fundamental
theorems of Probability, with very little comment. The bearing of
some of them will be discussed more fully in Chapter XVI.

2.

The Addition Theorems:

(24) (a + b)/h = a/h + b/h − ab/h.

In IX. write (a + b) for a, and ¯ab for b.

Then

(a + b)¯
ab/h + (a + b)¯
ab/h = (a + b)/h,
whence

¯
ab/h + (a + b)(a + ¯
b)/h = (a + b)/h
by (iii.),

¯
a/bh b/h + a/h = (a + b)/h
by (iii.) and IX.

That is to say, (a + b)/h = a/h + (1 − a/bh) b/h,

= a/h + b/h − ab/h.

In accordance with the principles of Chapter XII. § 6, this should
be written, strictly, in the form a/h + (b/h − ab/h), or in the form
b/h + (a/h − ab/h).

The argument is valid, since the probability

(b/h − ab/h) is equal to ¯ab/h, as appears from the preceding proof,
and, therefore, exists. This important theorem gives the probability
of ‘a or b’ relative to a given hypothesis in terms of the probabilities
of ‘a,’ ‘b,’ and ‘a and b’ relative to the same hypothesis.

(24.1) If ab/h = 0, i.e. if a and b are exclusive alternatives relative
to the hypothesis, then

(a + b)/h = a/h + b/h.

This is the ordinary rule for the addition of the probabilities of
exclusive alternatives.

161
pt. ii

A TREATISE ON PROBABILITY

162

(24.2) ab/h + ¯ab/h = b/h,
since
ab + ¯
ab ≡ b
by (iii.),
and
a¯
ab/h = 0
by (19) and (8).

(24.3) (a + b)/h = a/h + b¯a/h. This follows from (24) and (24.2).

(24.4) (a + b + c)/h = (a + b)/h + c/h − (ac + bc)/h

= a/h + b/h + c/h

− ab/h − bc/h − ca/h + abc/h.

(24.5) And in general

(p1 + p2 + . . . + pn)/h =
pr/h −
pspt/h +
prpspt/h . . .

+ (−1)n − 1p1p2 . . . pn/h.

(24.6) If pspt/h = 0 for all pairs of values of s and t, it follows by
repeated application of X. that
n

(p1 + p2 + . . . + pn)/h =
pr/h.

1

(24.7) If pspt/h = 0, etc., and (p1 + p2 + . . . + pn)/h = 1, i.e. if
p1p2 . . . pn form, relatively to h, a set of exclusive and exhaustive
alternatives, then
n
pr/h = 1.

1

(25) If p1p2 . . . pn form, relative to h, a set of exclusive and
exhaustive alternatives,
n
a/h =
pra/h.

1
ch. xiv

FUNDAMENTAL THEOREMS

163

Since (p1 + p2 + . . . + pn)/h = 1 by hypothesis,

∴ (p1 + p2 + . . . + pn)/ah = 1
by (9) if ah is not inconsistent;
and since pspt/h = 0 by hypothesis,

∴ pspt/ah = 0 by (9), if ah is not inconsistent.

n

Hence
pr/ah = (p1 + p2 + . . . + pn)/ah
by (24.6)

1

= 1

Also
pra/h = pr/ah a/h.

n
n

Summing
pra/h = a/h
pr/ah,

1

1
n

∴ a/h =
pra/h,
if ah is not inconsistent.

1

If ah is inconsistent, i.e. if a/h = 0 (for h is by hypothesis
consistent), the result follows at once by (8).

(25.1) If pra/h = Xr, the above may be written

X
p
r
r /ah =

.

n

Xr

1

(26) a/h = (a + ¯h)/h.

For

(a + ¯
h)/h = a/h + ¯
h/h − a¯
h/h
by (24),

= a/h
by (13.1) and (8).

(26.1) This may be written
a/h = (h ⊃ a)/h.

pt. ii

A TREATISE ON PROBABILITY

164

(27) If (a + b)/h = 0, a/h = 0.

a/h + [b/h − ab/h] = 0,
by (24) and hypothesis

∴ a/h = 0.

by (5)

(27.1) If a/h = 0 and b/h = 0, (a + b)/h = 0. This follows
from (24).

(28) If a/h = 1, (a + ¯b)/h = 1,

(a + ¯
b)/h = a/h + ¯
b¯
a/h,
by (24.3)

whence (a + ¯b)/h = a/h = 1 by (1.1) and (8), together with the
hypothesis. That is to say, a certain proposition is implied by every
proposition.

(28.1) If a/h = 0, (¯a + b)/h = 1 by substituting ¯a for a and b for ¯b
in (28). That is to say, a certainly false proposition implies every
proposition.

(29) If a/(h1 + h2) = 1, a/h1 = 1, a/h2 = 1.

¯
a/(h1 + h2) = 0,
and

∴ ¯a(h1 + h2)/h1 = 0
by (15).

Hence

¯
ah1/h1 = 0
by (27),
whence the result.

(29.1) Ifa/h1 = 1 and a/h2 = 1, a/(h1 + h2) = 1.

As in (20)

¯
ah1/(h1 + h2) = 0 and ¯ah2/(h1 + h2) = 0.

Hence

¯
a(h1 + h2)/(h1 + h2) = 0
by (27.1),
whence the result.

(29.2) If a/(h1 + h2) = 0, a/h1 = 0. This follows from (29).

(29.3) If a/h1 = 0 and a/h2 = 0, a/(h1 + h2) = 0. This follows
from (29.1).

3.

Irrelevance and Independence:

ch. xiv

FUNDAMENTAL THEOREMS

165

(30) If a/h

¯

¯

1h2 = a/h1,
then a/h1h2 = a/h1, if h1h2 is not
inconsistent.

a/h1 = ah2/h1 + a¯
h2/h1
by (24.2),

= a/h

¯

¯

1h2
h2/h1 + a/h1h2 h2/h1,

= a/h

¯

¯

1
h2/h1 + a/h1h2 h2/h1,

∴ a/h

¯

¯

¯

1
h2/h1 = a/h1h2 h2/h1,
whence a/h

¯

¯

1 = a/h1h2,
unless ¯h2/h1 = 0, i.e. if h1h2 is not
inconsistent.

Thus, if a proposition is irrelevant to an argument, then the
contradictory of the proposition is also irrelevant.

(31) If a2/a1h = a2/h and a2h is not inconsistent, a1/a2h = a1/h.

This follows by (iv. c), since a2/a1h a1/h = a1/a2h a2/h by X.

If, that is to say, a1 is irrelevant to the argument a2/h (see XIV.),
and a2 is not inconsistent with h: then a2 is irrelevant to the
argument a1/h; and a1/h and a2/h are independent (see XIII.).

4.

Theorems of Relevance:

(32) If a/hh1 > a/h, h1/ah > h1/h.

ah is consistent since, otherwise, a/hh1 = a/h = 0.

Therefore
a/h h1/ah = a/hh1 h1/h
by X.,

> a/h h1/h
by hypothesis;
so that
h1/ah > h1/h.

Thus if h1 is favourably relevant to the argument a/h, a is
favourably relevant to the argument h1/h.

This constitutes a formal demonstration of the generally accepted
principle that if a hypothesis helps to explain a phenomenon, the fact
of the phenomenon supports the reality of the hypothesis.

In the following theorems p will be said to be more favourable
to
b/qh
a/h, than q is to b/h, if a/ph >

, i.e. if, in the language of
a/h
b/h
pt. ii

A TREATISE ON PROBABILITY

166

§ 8 below, the coefficient of influence of p on a/h is greater than the
coefficient of influence of q on b/h.

(33) If x is favourable to a/h, and h1 is not more favourable
to a/hx than x is to a/hh1, then h1 is favourable to a/h.

For
a/hx
a/hh
a/hh
a/hh

1x

1

1 = a/h

; and by hypothesis the
a/h
a/hx
a/hh1x
second term on the right is greater than unity and the product of the
third and fourth terms is greater than or equal to unity.

(33.1) A fortiori, if x is favourable to a/h and not favourable
to a/hh1, and if h1 is not unfavourable to a/hx, then h1 is favourable
to a/h.

(34) If x is favourable to a/h, and h1 is not less favourable to x/ha
than x is to h1/ha, then h1 is favourable to a/h.

This follows by the same reasoning as (33), since by an application
of the Multiplication Theorem
a/hh1x
a/hh1
x/hh
h

=

1a

1/ha .

a/hx
a/hh1x
x/ha
h1/hax

(35) If x is favourable to a/h, but not more favourable to it
than h1x is, and not less favourable to it than to a/hh1, then h1 is
favourable to a/h.

For
a/h
a/hh
a/hx
a/hh
a/hh

1x

1

1 = a/h

.

a/hx
a/h
a/h
a/hh1x

This result is a little more substantial than the two preceding. By
judging the influence of x and h1x on the arguments a/h and a/hh1,
we can infer the influence of h1 by itself on the argument a/h.

5.

The Multiplication Theorems:

(36) If a1/h and a2/h are independent, a1a2/h = a1/h
a2/h.

For
a1a2/h = a1/a2h a2/h = a2/a1h a1/h
by X.,
ch. xiv

FUNDAMENTAL THEOREMS

167
and since a1/h and a2/h are independent,
a1/a2h = a1/h and a2/a1h = a2/h
by XIII.

Therefore
a1a2/h = a1/h a2/h.

Hence, when a1/h and a2/h are independent, we can arrive at the
probability of a1 and a2 jointly on the same hypothesis by simple
multiplication of the probabilities a1/h and a2/h taken separately.

(37) If p1/h = p2/p1h = p3/p1p2h = . . .,
p1p2p3 . . . pn/h = {p1/h}n .

For p1p2p3 . . . /h = p1/h p2/p1h p3/p1p2h . . . by repeated applica-
tions of X.

6.

The Inverse Principle:

(38) a1/bh
b/a
a

=

1h

1/h , provided bh, a1h, and a2h are each
a2/bh
b/a2h
a2/h
consistent.

For
a1/bh b/h = b/a1h a1/h,
and
a2/bh b/h = b/a2h a2/h
by X.,
whence the result follows, since b/h =| 0, unless bh is inconsistent.

(38.1) If
a1/h = p1, a2/h = p2, b/a1h = q1, b/a2h = q2,
and a1/bh + a2/bh = 1 then it easily follows that
p
a

1q1

1/bh =

,
p1q1 + p2q2
and
p
a

2q2

2/bh =

.

p1q1 + p2q2

(38.2) If a1/h = a2/h the above reduces to
q
a

1

1/bh =

,
q1 + q2
and
q
a

2

2/bh =

,
q1 + q2
pt. ii

A TREATISE ON PROBABILITY

168
since a1/h =| 0, unless a1h is inconsistent.

The proposition is easily extended to the cases in which the
number of a’s is greater than two.

It will be worth while to translate this theorem into familiar
language. Let b represent the occurrence of an event B, a1 and a2
the hypotheses of the existence of two possible causes A1 and A2
of B, and h the general data of the problem. Then p1 and p2 are the
à priori probabilities of the existence of A1 and A2 respectively, when
it is not known whether or not the event B has occurred; q1 and q2
the probabilities that each of the causes A1 and A2, if it exists, will
be followed by the event B. Then
p1q1
and
p2q2
are
p1q1 + p2q2
p1q1 + p2q2
the probabilities of the existence of A1 and A2 respectively after the
event, i.e. when, in addition to our other data, we know that the
event B has occurred. The initial condition, that bh must not be
inconsistent, simply ensures that the problem is a possible one, i.e.

that the occurrence of the event B is on the initial data at least
possible.

The reason why this theorem has generally been known as the

Inverse Principle of Probability is obvious. The causal problems to
which the Calculus of Probability has been applied are naturally
divided into two classes—the direct in which, given the cause, we
deduce the effect; the indirect or inverse in which, given the effect,
we investigate the cause. The Inverse Principle has been usually
employed to deal with the latter class of problem.

7.

Theorems on the Combination of Premisses:

The Multiplication Theorems given above deal with the combina-
tion of conclusions; given a/h1 and a/h2 we considered the relation
of a1a2/h to these probabilities. In this paragraph the corresponding
problem of the combination of premisses will be treated; given
a/h1 and a/h2 we shall consider the relation of a/h1h2 to these
ch. xiv

FUNDAMENTAL THEOREMS

169
probabilities.

(39)

ah
ah
a/h

1h2/h

1h2/h

1h2h =

=
by X. and (24.2)

h1h2/h
ah1h2/h + ¯
ah1h2/h
u

=

,
u + v
where u is the à priori probability of the conclusion a and both
hypotheses h1 and h2 jointly, and v is the à priori probability of
the contradictory of the conclusion and both hypotheses h1 and h2
jointly.

(40)

ah
h
a/h

1/h2

1/ah2
q

1h2 =

=

,
ah1/h2 + ¯
ah1/h2
h1/ah2 q + h1/¯
ah2 (1 − q)

h

=

2/ah1
p

,
h2/ah1 p + h2/¯
ah1 (1 − p)

where p = a/h1 and q = a/h2.

(40.1) If

1
h
p =

, a/h

2/ah1

1h2 =

,

2
h2/ah1 + h2/¯
ah1
and increases with
h2/ah1 .

h2/¯
ah1

These results are not very valuable and show the need of an
original method of reduction. This is supplied by Mr. W. E. Johnson’s

Cumulative Formula, which is at present unpublished but which I
have his permission to print below.∗

8.

It is first of all necessary to introduce a new symbol. Let us
write

XV.

a/bh = {ahb}a/h

Def.

∗The substance of propositions (41) to (49) below is derived in its entirety
from his notes,—the exposition only is mine.

pt. ii

A TREATISE ON PROBABILITY

170

We may call {ahb} the coefficient of influence of b upon a on
hypothesis h.

XVI.

{ahb} {abhc} = {ahbhc}

Def.

and similarly

{ahb} {abhcdhe} = {ahbhcdhe}.

These coefficients thus belong by definition to a general class of
operators, which we may call separative factors.

(41)

ab/h = {ahb} a/h b/h,
since
ab/h = a/bh b/h.

Thus we may also call {ahb} the coefficient of dependence between
a and b on hypothesis h.

(41.1)

abc/h = {ahbhc} a/h b/h c/h.

For
abc/h = {abhc}ab/h c/h
by (41),

= {abhc} {ahb} a/h b/h c/h
by (41).

(41.2) And in general
abcd . . . /h = {ahbhchdh . . .} a/h b/h c/h d/h . . .

(42)

{ahb} = {bha},
since
a/bh b/h = b/ah a/h.

(42.1)

{ahbhc} = ahchb,
since
a/h b/h c/h = a/h c/h b/h.

(42.2) And in general we have a commutative rule, by which the
order of the terms may be always commuted—
e.g.

{ahbchdef hg} = {bchahghdef }

{ahbchdef hg} = {ahcbhf edhg}.

ch. xiv

FUNDAMENTAL THEOREMS

171

(43) As a multiplier the separative factor operates so as to separate
the terms that may be associated (or joined) in the multiplicand.

Thus

{abhcdhe} {ahb} = {ahbhcdhe},
for
abcde/h = {abhcdhe} ab/h cd/h e/h

= {abhcdhe} {ahb} a/h b/h cd/h e/h,
and also abcde/h = {ahbhcdhe} a/h b/h cd/h e/h.

Similarly (for example)

{abchdhef } {abhc} {ahb} = {ahbhchdhef }.

(44)

{ahb} {ab} = {ahb}.

For
ab/h = {ab}ab/h.

By a symbolic convention, therefore, we may put {ab} = 1.

(44.1) If {ahb} = 1, it follows that a/h and b/h are independent
arguments; and conversely.

(45) Rule of Repetition {aahb} = {ahb}.

For
aab/h = ab/h
by (vi.) and (12).

(46) The Cumulative Formula:

x/ah : x /ah : x /ah : . . .

= x/h a/xh : x /h a/x h : x /h a/x h : . . .

by (38).

Take n + 1 propositions a, b, c . . .

Then by repetition
x/ah x/bh x/ch . . . : x /a x /b x /c . . . : x /a x /b x /c . . . : . . .

= (x/h)n+1a/xh b/xh . . . : (x /h)n+1a/x h b/x h . . .

: (x /h)n+1a/x h b/x h . . .

pt. ii

A TREATISE ON PROBABILITY

172
which may be written
n+1
n+1
n+1
x/ah :

x /ah :

x /ah : . . .

n+1
n+1

= (x/h)n+1
a/xh : (x /h)n+1
a/x h : . . .

Now
x/habc . . . : x /habc . . . : x /habc . . .

= x/h (abc . . .)/xh : x /h (abc . . .)/x h : . . . by (38),
and
n+1
abc . . . /xh = {axhbxhc . . .}

a/xh
by (41.2),

∴ (x/h)n x/habc . . . : (x /h)n x /habc . . . : (x /h)n x /habc . . . : . . .

= {axhbxhc . . .}x/ah x/bh x/ch . . . : {ax hbx hc . . .}x /ah x /bh
x /ch . . . : . . .

which may be written

(x/h)n x/habc . . . ∝ {axhbxhc . . .} x/ah x/bh x/ch . . .

where variations of x are involved.

The cumulative formula is to be applied when, having accumulated
the evidence a, b, c . . ., we desire to know the comparative probabilities
of the various possible inferences x, x . . . which may be drawn, and
already know determinately the force of each of the items a, b, c . . .

separately as evidence for x, x . . ..

Besides the factors x/ah, x/bh, etc., we require to know two other
sets of values, viz.: (1) x/h, etc., i.e. the à priori probabilities of x,
etc., and (2) {axhbxhc . . .}, etc., i.e. the coefficients of dependence
ch. xiv

FUNDAMENTAL THEOREMS

173
between a, b, and c . . . on hypotheses xh, etc. It may be remarked
that the values {axhbxhc . . .}, {ax hbx hc . . .} . . . are not in any way
related, even when x ≡ ¯x.

What corresponds to the cumulative formula has been employed,
sometimes, by mathematicians in a simplified form which is, except
under special conditions, incorrect. First, it has been tacitly assumed
that {axhbxhc . . .}, {ax hbx hc . . .} . . . are all unity: so that

(x/h)nx/habc . . . ∝ x/ah x/bh x/ch . . .

Secondly, the factor (x/h)n has been omitted, so that
x/habc . . . ∝ x/ah x/bh x/ch . . .

It is this second incorrect statement of the formula which leads
to the fallacious rule for the combination of the testimonies of
independent witnesses ordinarily given in the text-books.∗

(46.1). If abc . . . /xh = {axhbxhc . . .}a/xh b/xh c/xh . . .

then
x/habc . . . ∝ {axhbxhc . . .}x/ah x/bh x/ch . . . .

This result is exceedingly interesting. Mr. Johnson is the first to
arrive at the simple relation, expressed above, between the direct
and the inverse formulæ: viz. that the same coefficient is required
for correcting the simple formulæ of multiplication in both cases.

As he remarks, however, while the direct formula gives the required
probability directly by multiplication, the inverse formula gives only
the comparative probability.

∗See p. 207 below.

pt. ii

A TREATISE ON PROBABILITY

174

(46.2) If x, x , x . . . are exclusive and exhaustive alternatives,
n−1

(x/h)−n {axhbxhc . . .}

x/ah
x/habc . . . =

,
n−1

(x /h)−n {ax hbx hc . . .}

x /ah
n−1
since
x/habc . . . ∝ (x/h)−n{axhbxhc . . .}

x/ah,
and
x /habc . . . = 1
by (24.7).

(47). x/habc . . .

a/h b/h c/h . . .

=
x/h
abc . . . /h
abc . . . /xh
x/ah
x/bh . . . .

a/xh b/xh c/xh . . .

x/h
x/h

For
abc . . . x/h = x/h abc . . . /xh,
abc . . . x/h
abc . . . /xh
a/h b/h c/h . . .

∴

=

=
abc . . . /h x/h
abc . . . /h
abc . . . /h
abc . . . /xh
a/xh
b/xh . . . ,
a/xh b/xh c/xh . . .

a/h
b/h
whence the result, since a/xh
x/ah

=

, etc.

a/h
x/h

(47.1) The above formula may be written in the condensed form

1

{abc . . .h x} =

{axhbxhcxh . . .} [{ahx} {bhx} {chx} . . .].

{ahbhch . . .}

{

{

(48.)

x/h}nx/habc . . .

axhbxhcxh . . .}

x/ah x/bh x/ch . . .

=

.

{¯
x/h}n ¯
x/habc . . .

{a¯xhb¯xhc¯xh . . .}

¯
x/ah ¯
x/bh ¯
x/ch . . .

ch. xiv

FUNDAMENTAL THEOREMS

175

This follows at once from (46.2), since x and ¯x are exclusive and
exhaustive alternatives. (It is assumed that xh, ¯xh, and ah, etc., are
not inconsistent.)

This formula gives x/habc . . . in terms of x/ah, x/bh, etc., together
with the three values x/h, {axhbxhcxh . . .}, and {a¯xhb¯xhc¯xh . . .}.

{

(48.1)

x/habcd . . .

x/hbcd . . .

axhbcd . . .} x/ah
x/h

:

=

:

.

¯
x/habcd . . .

¯
x/hbcd . . .

{a¯xhbcd . . .} ¯
x/ah

¯
x/h

This gives the effect on the odds (prob. x : prob. ¯x) of the extra
knowledge a.

(49) When several data co-operate as evidence in favour of a
proposition, they continually strengthen their own mutual probabil-
ities, on the assumption that when the proposition is known to be
true or to be false the data jointly are not counterdependent.

I.e. if {axhbxhc . . .} and {a¯xhb¯xhc . . .} are not less than unity, and
x/kh > x/h where k is any of the data a, b, c . . ., then {ahbhchd . . .}

beginning with unity, continually increases, as the number of its terms
is increased.

abc . . . /h = xabc . . . /h + ¯
xabc . . . /h
by (24.2).

= x/h abc . . . /xh + ¯
x/h abc . . . /¯
xh.

≥ x/h
a/xh b/xh . . . + ¯
xh
a/¯
xh b/¯
xh . . .

(since {axhbxhc . . .} and {a¯xhb¯xhc . . .} are not less than unity),
ax/h
bx/h
a¯
x/h
b¯
x/h

≥ x/h

. . . + ¯
x/h

. . . ,
x/h
x/h

¯
x/h

¯
x/h
abc . . . /h
x/ah
x/bx

∴

≥ x/h

. . .

[a/h b/h . . .]

x/h
x/h

¯
x/ah

¯
x/bh

+ ¯
x/h

. . . .

¯
x/h

¯
x/h
pt. ii

A TREATISE ON PROBABILITY

176

We can show that each additional piece of evidence a, b, c . . .

increases the value of this expression. For let x/h G + ¯x/h G be its
value when all the evidence up to k exclusive is taken, so that
x/kh G + ¯
x/kh G
is its value when k is taken. Now G > G since x/ah > x/h, etc.,
and ¯x/ah < ¯x/h, etc., by the hypothesis that the evidence favours x;
and for the same reason x/kh − x/h, which is equal to ¯x/h − ¯x/kh, is
positive.

∴ G(x/kh − x/h) > G (¯
x/h − ¯
x/kh),
i.e.

x/kh G + ¯
x/kh G > x/h G + ¯
x/h G ,
whence the result.

(49.1) The above proposition can be generalised for the case of
exclusive alternatives x, x , x . . . (in place of x, ¯x).

For {ahbhch . . .}

= x/h.{axhbxhc . . .}{ahx}{bhx}{chx} . . .

+ x /h.{ax hbx hc . . .}{ahx }{bhx }{chx } . . .

+ x /h.{ax hbx hc . . .}{ahx }{bhx }{chx } . . . + . . . ;
from which, it follows that, if {axhbxhc . . .}etc. <| 1, and if {ahx − 1},

{bhx − 1}, {chx − 1}, etc., have the same sign, then {ahbhc . . .} is
increasing (with the number of letters) from unity.

Mr. Johnson describes this result as a generalisation of the
corrected “middle term fallacy” (see Chap. V. § 4).

APPENDIX

ON SYMBOLIC TREATMENTS OF PROBABILITY

The use of the symbol 0 for impossibility and 1 for certainty
was first introduced by Leibnitz in a very early pamphlet, entitled
ch. xiv

FUNDAMENTAL THEOREMS

177

Specimen certitudinis seu demonstrationum in jure, exhibitum in
doctrina conditionum, published in 1665 (vide Couturat, Logique
de Leibnitz, p. 553).

Leibnitz represented intermediate degrees of
probability by the sign 1, meaning, however, by this symbol a variable

2
between 0 and 1.

Several modern writers have made some attempt at a symbolic
treatment of Probability. But with the exception of Boole, whose
methods I have discussed in detail in Chapters XV., XVI., and XVII.,
no one has worked out anything very elaborate.

Mr. McColl published a number of brief notes on Probability of
considerable interest—see especially his Symbolic Logic, Sixth Paper
on the Calculus of Equivalent Statements, and On the Growth and Use
of a Symbolical Language. The conception of probability as a relation
between propositions underlies his symbolism, as it does mine.∗ The
probability of a, relative to the à priori premiss h, he writes a;
and the probability, given b in addition to the à priori premiss, he
writes a. Thus a
a
a

=

, and a = a/bh. The difference a − , i.e.

b
h
b
b
the change in the probability of a brought about by the addition of b
to the evidence, he calls ‘the dependence of the statement a upon
the statement
a
a
b,’ and denotes it by δ . Thus δ

= 0, where, in my
b
b
terminology, b is irrelevant to a on evidence h. The multiplication
and addition formulæ he gives as follows: ab
a
b
b
a

=

=

.

a
b
a + b
a
b
ab

=

+

−

.

Also
a

A b
a
δ

=
δ , where A = .

b

B a

It is surprising how little use he succeeds in making of these good

∗I did not come across these notes until my own method was considerably
developed.

Mr. McColl has been the first to use the fundamental symbol of

Probability.

pt. ii

A TREATISE ON PROBABILITY

178
results. He arrives, however, at the inverse formula in the shape—
cr v
cr
c

=
r

,
v
r=n cr
v
r=1
cr
where c1 . . . cn are a series of mutually exclusive causes of the event v
and include all possible causes of it; reaching it as a generalisation of
the proposition
a
b
a =
a
b
a
b

¯
a
b

+
a

¯
a

In a paper entitled “Operations in Relative Number with Applica-
tions to the Theory of Probabilities,”∗ Mr. B. I. Gilman attempted
a symbolic treatment based on a frequency theory similar to Venn’s,
but made more precise and more consistent with itself: “Probability
has to do, not with individual events, but with classes of events; and
not with one class, but with a pair of classes,—the one containing,
the other contained. The latter being the one with which we are
principally concerned, we speak, by an ellipsis, of its probability
without mentioning the containing class; but in reality probability
is a ratio, and to define it we must have both correlates given.”

But Mr. Gilman’s symbolic treatment leads to very little. More
recently R. Laemmel, in his Untersuchungen über die Ermittlung von

Wahrscheinlichkeiten, made a beginning on somewhat similar lines;
but in his case also the symbolic treatment leads to no substantial
results.

Apart from the writers mentioned above, there are a few who have
incidentally made use of a probability symbol. It will be sufficient
to cite Czuber.† He denotes the probability of an event E by W(E),

∗Published in the volume of Johns Hopkins Studies in Logic.

†Wahrscheinlichkeitsrechnung, vol. i. pp. 43–48.

ch. xiv

FUNDAMENTAL THEOREMS

179
and the probability of the event E given the occurrence of an event F
by WF(E). He uses this symbol to give WF(E) = W¯F(E) as the
criterion of the independence of the events E and F (¯F denoting the
non-occurrence of F); WF(E) = 1, as the expression of the fact that E
is a necessary consequence of F; and one or two other similar results.

Finally there is in the Bulletin of the Physico-mathematical Society
of Kazan for 1887 a memoir in Russian by Platon S. Porotzki entitled

“A Solution of the General Problem of the Theory of Probability by

Means of Mathematical Logic.” I have seen it stated that Schröder
intended to publish ultimately a symbolic treatment of Probability.

Whether he had prepared any manuscript on the subject before his
death I do not know.

CHAPTER XV
numerical measurement and approximation of
probabilities

1.

The possibility of numerical measurement, mentioned at the
close of Chapter III., arises out of the Addition Theorem (24.1). In
introducing the definitions and the axiom, which are required in order
to make the convention of numerical measurement operative, we may
appear, as in the case of the original definitions of Addition and

Multiplication, to be arguing in an artificial way. This appearance
is due, here as in Chapter XII., to our having given the names
of addition and multiplication to certain processes of compounding
probabilities in advance of postulating that the processes in question
have the properties commonly associated with these names. As
common sense is hasty to impute the properties as soon as it hears
the names, it may overlook the necessity of formally introducing
them.

2.

The definitions and the axiom which are needed in order to
give a meaning to numerical measurement are the following:—

XVII. a/h + {a/h + [a/h + (a/h + . . . r terms)]} = r a/h. Def.

XVIII. If

1
r a/h = b/f , then a/h =
b/f .

Def.

r

XIX. If
q
b/f = q c/g, then 1 b/f =
c/g.

Def.

r
r

Thus if b/h = a/h + a/h + . . . to r terms, then the probability b/h
is said to be r times the probability a/h; hence if ab/h = 0 and
a/h = b/h, the probability (a + b)/h is twice the probability a/h. If
a and b are exhaustive as well as exclusive alternatives relatively to h,
so that (a + b)/h = 1, since we take the relation of certainty as our
unit, then a/h = b/h = 1.

2

180
ch. xv

FUNDAMENTAL THEOREMS

181

We also need the following axiom postulating the existence of
relations of probability corresponding to all proper fractions:

(vii.) If q and r are any finite integers and q < r, there exists
a relation of probability which can be expressed, by means of the
convention of the foregoing definitions, as q .

r

3.

From these axioms and definitions combined with those
of Chapter XII., it is easy to show (certainty being represented by
unity and impossibility by zero) that we can manipulate according to
the ordinary laws of arithmetic the “numbers” which by means of a
special convention we have thus introduced to represent probabilities.

Of the kind of proofs necessary for the complete demonstration of
this the following is given as an example:

(50)

If

1

1
m + n
a/f =
and b/h = , a/f + b/h =

.

m
n
mn

Let the probability 1 = P, which exists by (vii.),
mn
then

1
n P =

= a/f
by (XIX.),
m
and

1
m P =

= b/h,
n

∴ a/f + b/h = n P + m P, if this probability exists,

= P + P . . . to n terms + P + P . . . to m terms,

= P + P . . . to m + n terms,
m + n

= (m + n)P =
by (XIX.).

mn

This probability exists in virtue of (vii.).

4.

Many probabilities—in fact all those which are equal to
the probability of some other argument which has the same premiss
pt. ii

A TREATISE ON PROBABILITY

182
and of which the conclusion is incompatible with that of the original
argument—are numerically measurable in the sense that there is some
other probability with which they are comparable in the manner
described above. But they are not numerically measurable in the
most usual sense, unless the probability with which they are thus
comparable is the relation of certainty. The conditions under which a
probability a/h is numerically measurable and equal to q are easily
r
seen. It is necessary that there should exist probabilities a1/h1,
a2/h2 . . ., aq/hq . . . ar/hr, such that
a1/h1 = a2/h2 = . . . = aq/hq = . . . = ar/hr,
q
r
a/h =
as/hs, and
as/hs = 1.

1

1

If
q
q
q
a/h = 1 and b/h = 2 , it follows from (32) that ab/h = 1q2
r1
r2
r1r2
only if a/h and b/h are independent arguments. Unless, therefore,
we are dealing with independent arguments, we cannot apply detailed
mathematical reasoning even when the individual probabilities are
numerically measurable. The greater part of mathematical probability,
therefore, is concerned with arguments which are both independent
and numerically measurable.

5.

It is evident that the cases in which exact numerical
measurement is possible are a very limited class, generally dependent
on evidence which warrants a judgment of equiprobability by an
application of the Principle of Indifference. The fuller the evidence
upon which we rely, the less likely is it to be perfectly symmetrical
in its bearing on the various alternatives, and the more likely is it
to contain some piece of relevant information favouring one of them.

In actual reasoning, therefore, perfectly equal probabilities, and hence
exact numerical measures, will occur comparatively seldom.

The sphere of inexact numerical comparison is not, however, quite
so limited. Many probabilities, which are incapable of numerical
measurement, can be placed nevertheless between numerical limits.

ch. xv

FUNDAMENTAL THEOREMS

183

And by taking particular non-numerical probabilities as standards a
great number of comparisons or approximate measurements become
possible. If we can place a probability in an order of magnitude with
some standard probability, we can obtain its approximate measure by
comparison.

This method is frequently adopted in common discourse. When
we ask how probable something is, we often put our question in the
form—Is it more or less probable than so and so?—where ‘so and
so’ is some comparable and better known probability. We may thus
obtain information in cases where it would be impossible to ascribe
any number to the probability in question. Darwin was giving a
numerical limit to a non-numerical probability when he said of a
conversation with Lyell that he thought it no more likely that he
should be right in nearly all points than that he should toss up
a penny and get heads twenty times running.∗ Similar cases and
others also, where the probability which is taken as the standard of
comparison is itself non-numerical and not, as in Darwin’s instance, a
numerical one, will readily occur to the reader.

A specially important case of approximate comparison is that
of ‘practical certainty.’ This differs from logical certainty since its
contradictory is not impossible, but we are in practice completely
satisfied with any probability which approaches such a limit. The
phrase has naturally not been used with complete precision; but in its
most useful sense it is essentially non-numerical—we cannot measure
practical certainty in terms of logical certainty. We can only explain
how great practical certainty is by giving instances. We may say, for
instance, that it is measured by the probability of the sun’s rising
to-morrow. The type which we shall be most likely to take will be
that of a well-verified induction.

6.

Most of such comparisons must be based on the principles of

Chapter V. It is possible, however, to develop a systematic method of

∗Life and Letters, vol. ii. p. 240.

pt. ii

A TREATISE ON PROBABILITY

184
approximation which may be occasionally useful. The theorems given
below are chiefly suggested by some work of Boole’s. His theorems
were introduced for a different purpose, and he does not seem to
have realised this interesting application of them; but analytically his
problem is identical with that of approximation.∗ This method of
approximation is also substantially the same analytically as that dealt
with by Mr. Yule under the heading of Consistence.†

(51) xy/h always lies between‡ x/h and x/h + y/h − 1 and between
y/h and x/h + y/h − 1.

For
xy/h = x/h − x¯
y/h
by (24.2),

= x/h − ¯
y/h x/¯
y/h
by X.

Now
x/¯
yh lies between 0 and 1
by (2) and (3),

∴ xy/h lies between x/h and x/h − ¯
y/h,
i.e. between x/h and x/h + y/h − 1.

As xy/h <| 0, the above limits may be replaced by x/h and 0, if
x/h + y/h − 1 < 0.

∗In Boole’s Calculus we are apt to be left with an equation of the second or
of an even higher degree from which to derive the probability of the conclusion;
and Boole introduced these methods in order to determine which of the several
roots of his equation should be taken as giving the true solution of the problem
in probability.

In each case he shows that that root must be chosen which
lies between certain limits, and that only one root satisfies this condition. The
general theory to be applied in such cases is expounded by him in Chapter XIX.

of The Laws of Thought, which is entitled “On Statistical Conditions.”

But
the solution given in that chapter is awkward and unsatisfactory, and he
subsequently published a much better method in the Philosophical Magazine for

1854 (4th series, vol. viii.) under the title “On the Conditions by which the

Solutions of Questions in the Theory of Probabilities are limited.”

†Theory of Statistics, chap. ii.

‡In this and the following theorems the term ‘between’ includes the limits.

ch. xv

FUNDAMENTAL THEOREMS

185

We thus have limits for xy/h, close enough sometimes to be useful,
which are available whether or not x/h and y/h are independent
arguments. For instance, if y/h is nearly certain, xy/h = x/h nearly,
quite independently of whether or not x and y are independent. This
is obvious; but it is useful to have a simple and general formula for
all such cases.

n+1

(52) x1x2 . . . xn+1/h is always greater than
xr/h − n.

1

For by (51) x1x2 . . . xn+1/h > x1x2 . . . xn/h + xn+1/h − 1

> x1x2 . . . xn−1/h + xn/h + xn+1/h − 2,
and so on.

(53) xy/h + ¯x¯y/h is always less than x/h − y/h + 1, and less than
y/h − x/h + 1.

For as in (51)

xy/h = x/h − x¯
y/h
and

¯
x¯
y/h = ¯
y/h − x¯
y/h,

∴ xy/h + ¯
x¯
y/h = x/h − y/h + 1 − 2x¯
y/h,
whence the required result.

(54) xy/h − ¯x¯y/h = x/h + y/h − 1.

This proposition, which follows immediately from the above, is
really out of place here. But its close connection with conclusions

(51) and (53) is obvious. It is slightly unexpected, perhaps, that the
difference of the probabilities that both of two events will occur and
that neither of them will, is independent of whether or not the events
themselves are independent.

7.

It is not worth while to work out more of these results here.

Some less systematic approximations of the same kind are given in
the course of the solutions in Chapter XVII.

In seeking to compare the degree of one probability with that of
another we may desire to get rid of one of the terms, on account of its
not being comparable with any of our standard probabilities. Thus
pt. ii

A TREATISE ON PROBABILITY

186
our object in general is to eliminate a given symbol of quantity from
a set of equations or inequations. If, for instance, we are to obtain
numerical limits within which our probability must lie, we must
eliminate from the result those probabilities which are non-numerical.

This is the general problem for solution.

(55) A general method of solving these problems when we can throw
our equations into a linear shape so far as all symbols of probability
are concerned, is best shown in the following example:—

Suppose we have
λ + ν = a

(i.)

λ + σ = b

(ii.)

λ + ν + σ = c

(iii.)

λ + µ + ν + ρ = d

(iv.)

λ + µ + σ + τ = e

(v.)

λ + µ + ν + ρ + σ + τ + υ = 1

(vi.)

where λ, µ, ν, ρ, σ, τ, υ represent probabilities which are to be
eliminated, and limits are to be found for c in terms of the standard
probabilities a, b, d, e, and 1.

λ, µ, etc., must all lie between 0 and 1.

From (i.) and (iii.) σ = c − a; from (ii.) and (iii.) ν = c − b.

From (i.), (ii.), and (iii.) λ = a + b − c,
whence
c − a

0,
c − b

0,
a + b − c

0,
substituting for σ, ν, λ in (iv.), (v.), and (vi.)

µ + ρ = d − a,
µ + τ = e − b,
µ + ρ + τ + υ = 1 − c,
whence
ρ = d − a − µ,
τ = e − b − µ,
υ = 1 − c − d + a − e + b + µ,

∴ d − a − µ

0,
e − b − µ

0,

1 − c − d + a − e + b + µ

0.

ch. xv

FUNDAMENTAL THEOREMS

187

We have still to eliminate µ.

µ
d − a,
µ
e − b,
µ
c + d + e − a − b − 1,

∴ d − a
c + d + e − a − b − 1 and e − b
c + d + e − a − b − 1.

Hence we have:

Upper limits of c:—b + 1 − e, a + 1 − d, a + b (whichever is least),

Lower limits of c:—a, b (whichever is greatest).

This example, which is only slightly modified from one given by

Boole, represents the actual conditions of a well-known problem in
probability.

CHAPTER XVI
observations on the theorems of chapter xiv.

and their developments, including testimony

1.

In Definition XIII. of Chapter XII. a meaning was given to
the statement that a1/h and a2/h are independent arguments. In

Theorem (33) of Chapter XIV. it was shown that, if a1/h and a2/h
are independent, a1a2/h = a1/h a2/h. Thus where on given evidence
there is independence between a1 and a2, the probability on this
evidence of a1a2 jointly is the product of the probabilities of a1 and a2
separately. It is difficult to apply mathematical reasoning to the

Calculus of Probabilities unless this condition is fulfilled; and the
fulfilment of the condition has often been assumed too lightly. A good
many of the most misleading fallacies in the theory of Probability
have been due to a use of the Multiplication Theorem in its simplified
form in cases where this is illegitimate.

2.

These fallacies have been partly due to the absence of a
clear understanding as to what is meant by Independence. Students
of Probability have thought of the independence of events, rather
than of the independence of arguments or propositions. The one
phraseology is, perhaps, as legitimate as the other; but when we speak
of the dependence of events, we are led to believe that the question
is one of direct causal dependence, two events being dependent if
the occurrence of one is a part cause or a possible part cause of
the occurrence of the other. In this sense the result of tossing a
coin is dependent on the existence of bias in the coin or in the
method of tossing it, but it is independent of the actual results of
other tosses; immunity from smallpox is dependent on vaccination,
but is independent of statistical returns relating to immunity; while
the testimonies of two witnesses about the same occurrence are
independent, so long as there is no collusion between them.

This sense, which it is not easy to define quite precisely, is at

188
ch. xvi

FUNDAMENTAL THEOREMS

189
any rate not the sense with which we are concerned when we deal
with independent probabilities. We are concerned, not with direct
causation of the kind described above, but with ‘dependence for
knowledge,’ with the question whether the knowledge of one fact or
event affords any rational ground for expecting the existence of the
other. The dependence for knowledge of two events usually arises, no
doubt, out of causal connection, or what we term such, of some kind.

But two events are not independent for knowledge merely because
there is an absence of direct causal connection between them; nor,
on the other hand, are they necessarily dependent because there is
in fact a causal train which brings them into an indirect connection.

The question is whether there is any known probable connection,
direct or indirect. A knowledge of the results of other tossings of
a coin may be hardly less relevant than a knowledge of the bias of
the coin; for a knowledge of these results may be a ground for a
probable knowledge of the bias. There is a similar connection between
the statistics of immunity from smallpox and the causal relations
between vaccination and smallpox. The truthful testimonies of two
witnesses about the same occurrence have a common cause, namely
the occurrence, however independent (in the legal sense of the absence
of collusion) the witnesses may be. For the purposes of probability
two facts are only independent if the existence of one is no indication
of anything which might be a part cause of the other.

3.

While dependence and independence may be thus connected
with the conception of causality, it is not convenient to found our
definition of independence upon this connection. A partial or possible
cause involves ideas which are still obscure, and I have preferred
to define independence by reference to the conception of relevance,
which has been already discussed. Whether there really are material
external causal laws, how far causal connection is distinct from logical
connection, and other such questions, are profoundly associated with
the ultimate problems of logic and probability and with many of
the topics, especially those of Part III., of this treatise. But I have pt. ii

A TREATISE ON PROBABILITY

190
nothing useful to say about them. Nearly everything with which I
deal can be expressed in terms of logical relevance. And the relations
between logical relevance and material cause must be left doubtful.

4.

It will be useful to give a few examples out of writers who, as

I conceive, have been led into mistakes through misapprehending the
significance of Independence.

Cournot,∗ in his work on Probability, which after a long period of
neglect has come into high favour with a modern school of thought
in France, distinguishes between ‘subjective probability’ based on
ignorance and ‘objective probability’ based on the calculation of

‘objective possibilities,’ an ‘objective possibility’ being a chance event
brought about by the combination or convergence of phenomena
belonging to independent series.

The existence of objectively
chance events depends on his doctrine that, as there are series
of phenomena causally dependent, so there are others between the
causal developments of which there is independence. These objective
possibilities of Cournot’s, whether they be real or fantastic, can have,
however, small importance for the theory of probability. For it is
not known to us what series of phenomena are thus independent. If
we had to wait until we knew phenomena to be independent in this
sense before we could use the simplified multiplication theorem, most
mathematical applications of probability would remain hypothetical.

5.

Cournot’s ‘objective probability,’ depending wholly on
objective fact, bears some resemblances to the conception in the
minds of those who adopt the frequency theory of probability. The
proper definition of independence on this theory has been given most
clearly by Mr. Yule† as follows:

“Two attributes A and B are usually defined to be independent,
within any given field of observation or ‘universe,’ when the chance of
finding them together is the product of the chances of finding either of

∗For some account of Cournot, see Chapter XXIV. § 3.

†“Notes on the Theory of Association of Attributes in Statistics,” Biometrika,
vol. ii. p. 125.

ch. xvi

FUNDAMENTAL THEOREMS

191
them separately. The physical meaning of the definition seems rather
clearer in a different form of statement, viz. if we define A and B
to be independent when the proportion of A’s amongst the B’s of
the given universe is the same as in that universe at large. If, for
instance, the question were put, ‘What is the test for independence of
smallpox attack and vaccination?’ the natural reply would be, ‘The
percentage of vaccinated amongst the attacked should be the same as
in the general population.’. . . ”

This definition is consistent with the rest of the theory to which
it belongs, but is, at the same time, open to the general objections
to it.∗ Mr. Yule admits that A and B may be independent in the
world at large but not in the world of C’s. The question therefore
arises as to what world given evidence specifies, and whether any step
forward is possible when, as is generally the case, we do not know
for certain what the proportions in a given world actually are. As in
the case of Cournot’s independent series, it is in general impossible
that we should know whether A and B are or are not independent in
this sense. The logical independence for knowledge which justifies our
reasoning in a certain way must be something different from either of
these objective forms of independence.

6.

I come now to Boole’s treatment of this subject. The
central error in his system of probability arises out of his giving two
inconsistent definitions of ‘independence.’† He first wins the reader’s

∗See Chapter VIII.

†Boole’s mistake was pointed out, accurately though somewhat obscurely, by

H. Wilbraham in his review “On the Theory of Chances developed in Professor

Boole’s Laws of Thought ” (Phil. Mag. 4th series, vol. vii., 1854). Boole failed
to understand the point of Wilbraham’s criticism, and replied hotly, challenging
him to impugn any individual results (“Reply to some Observations published
by Mr. Wilbraham,” Phil. Mag. 4th series, vol. viii., 1854).

He returned to
the same question in a paper entitled “On a General Method in the Theory
of Probabilities,” Phil. Mag. 4th series, vol. viii., 1854, where he endeavours to
support his theory by an appeal to the Principle of Indifference. McColl, in his

“Sixth Paper on Calculus of Equivalent Statements,” saw that Boole’s fallacy
pt. ii

A TREATISE ON PROBABILITY

192
acquiescence by giving a perfectly correct definition: “Two events are
said to be independent when the probability of the happening of
either of them is unaffected by our expectation of the occurrence or
failure of the other.”∗ But a moment later he interprets the term
in quite a different sense; for, according to Boole’s second definition,
we must regard the events as independent unless we are told either
that they must concur or that they cannot concur. That is to say,
they are independent unless we know for certain that there is, in fact,
an invariable connection between them. “The simple events, x, y, z,
will be said to be conditioned when they are not free to occur in
every possible combination; in other words, when some compound
event depending upon them is precluded from occurring. . . . Simple
unconditioned events are by definition independent.”† In fact as
long as xz is possible, x and z are independent. This is plainly
inconsistent with Boole’s first definition, with which he makes no
attempt to reconcile it. The consequences of his employing the term
independence in a double sense are far-reaching. For he uses a
method of reduction which is only valid when the arguments to which
it is applied are independent in the first sense, and assumes that it is
valid if they are independent in the second sense. While his theorems
are true if all the propositions or events involved are independent in
the first sense, they are not true, as he supposes them to be, if the
events are independent only in the second sense. In some cases this
mistake involves him in results so paradoxical that they might have
led him to detect his fundamental error.‡ Boole was almost certainly
turned on his definition of Independence; but I do not think he understood, at
least he does not explain, where precisely Boole’s mistake lay.

∗Laws of Thought, p. 255. The italics in this quotation are mine.

†Op. cit. p. 258.

‡There is an excellent instance of this, Laws of Thought, p. 286.

Boole
discusses the problem:

Given the probability p of the disjunction ‘either Y
is true, or X and Y are false,’ required the probability of the conditional
proposition, ‘If X is true, Y is true.’

The two propositions are formally
equivalent; but Boole, through the error pointed out above, arrives at the result
ch. xvi

FUNDAMENTAL THEOREMS

193
led into this error through supposing that the data of a problem can
be of the form, “Prob. x = p,” i.e. that it is sufficient to state that
the probability of a proposition is such and such, without stating to
what premisses this probability is referred.∗

It is interesting that De Morgan should have given, incidentally,
a definition of independence almost identical with Boole’s second
definition: “Two events are independent if the latter might have
existed without the former, or the former without the latter, for
anything that we know to the contrary”†

7.

In many other cases errors have arisen, not through a
misapprehension of the meaning of independence, but merely through
careless assumptions of it, or through enunciating the Theorem
of Multiplication without its qualifying condition. Mathematicians
have been too eager to assume the legitimacy of those complicated
processes of multiplying probabilities, for which the greater part of
the mathematics of probability is engaged in supplying simplifications
cp

, where c is the probability of ‘If either Y is true, or X and Y false,

1 − p + cp

X is true.’ His explanation of the paradox amounts to an assertion that, so long
as two propositions, which are formally equivalent when true, are only probable,
they are not necessarily equivalent.

∗In studying and criticising Boole’s work on Probability, it is very important
to take into account the various articles which he contributed to the Philosophical

Magazine during 1854, in which the methods of The Laws of Thought are
considerably improved and modified. His last and most considered contribution
to Probability is his paper “On the application of the Theory of Probabilities
to the question of the combination of testimonies or judgments,” to be found in
the Edin. Phil. Trans. vol. xxi., 1857. This memoir contains a simplification and
general summary of the method originally proposed in The Laws of Thought,
and should be regarded as superseding the exposition of that book. In spite of
the error already alluded to, which vitiates many of his conclusions, the memoir
is as full as are his other writings of genius and originality.

†“Essay on Probabilities” in the Cabinet Encyclopaedia, p. 26. De Morgan is
not very consistent with himself in his various distinct treatises on this subject,
and other definitions may be found elsewhere.

Boole’s second definition of

Independence is also adopted by Macfarlane, Algebra of Logic, p. 21.

pt. ii

A TREATISE ON PROBABILITY

194
and approximate solutions. Even De Morgan was careless enough in
one of his writings∗ to enunciate the Multiplication Theorem in the
following form: “The probability of the happening of two, three, or
more events is the product of the probabilities of their happening
separately (p. 398). . . . Knowing the probability of a compound event,
and that of one of its components, we find the probability of the
other by dividing the first by the second. This is a mathematical
result of the last too obvious to require further proof (p. 401).”

An excellent and classic instance of the danger of wrongful
assumptions of independence is given by the problem of determining
the probability of throwing heads twice in two consecutive tosses of
a coin. The plain man generally assumes without hesitation that
the chance is (1)2. For the à priori chance of heads at the first

2
toss is 1, and we might naturally suppose that the two events are

2
independent,—since the mere fact of heads having appeared once can
have no influence on the next toss. But this is not the case unless
we know for certain that the coin is free from bias. If we do not
know whether there is bias, or which way the bias lies, then it is
reasonable to put the probability somewhat higher than (1)2. The

2
fact of heads having appeared at the first toss is not the cause of
heads appearing at the second also, but the knowledge, that the coin
has fallen heads already, affects our forecast of its falling thus in the
future, since heads in the past may have been due to a cause which
will favour heads in the future. The possibility of bias in a coin, it
may be noticed, always favours ‘runs’; this possibility increases the
probability both of ‘runs’ of heads and of ‘runs’ of tails.

This point is discussed at some length in Chapter XXIX. and
further examples will be given there. In this chapter, therefore, I will
do no more than refer to an investigation by Laplace and to one real
and one supposed fallacy of Independence of a type with which we

∗“Theory of Probabilities” in the Encyclopaedia Metropolitana.

ch. xvi

FUNDAMENTAL THEOREMS

195
shall not be concerned in Chapter XXIX.

8.

Laplace, in so far as he took account at all of the
considerations explained in § 7, discussed them under the heading of

Des inégalités inconnues qui peuvent exister entre les chances que l’on
suppose égales.∗ In the case, that is to say, of the coin with unknown
bias, he held that the true probability of heads even at the first toss
differed from 1 by an amount unknown. But this is not the correct

2
way of looking at the matter. In the supposed circumstances the
initial chances for heads and tails respectively at the first toss really
are equal. What is not true is that the initial probability of ‘heads
twice’ is equal to the probability of ‘heads once’ squared.

Let us write ‘heads at first toss’ = h1; ‘heads at second toss’

= h2. Then h1/h = h2/h = 1, and h

2

1h2/h = h2/h1h
h1/h. Hence
h1h2/h = {h1/h}2 only if h2/h1h = h2/h, i.e. if the knowledge that
heads has fallen at the first toss does not affect in the least the
probability of its falling at the second.

In general, it is true
that h2/h1h will not differ greatly from h2/h (for relative to most
hypotheses heads at the first toss will not much influence our
expectation of heads at the second), and 1 will, therefore, give a

4
good approximation to the required probability. Laplace suggests an
ingenious method by which the divergence may be diminished. If we
throw two coins and define ‘heads’ at any toss as the face thrown by
the second coin, he discusses the probability of ‘heads twice running’

with the first coin. The solution of this problem involves, of course,
particular assumptions, but they are of a kind more likely to be
realised in practice than the complete absence of bias. As Laplace
does not state them, and as his proof is incomplete, it may be worth
while to give a proof in detail.

Let h1, t1, h2, t2 denote heads and tails respectively with the
first and second coins respectively at the first toss, and h , t , h , t

1

1

2

2

∗Essai philosophique, p. 49. See also “Mémoire sur les Probabilités,” Mém. de
l’Acad. p. 228, and cp. D’Alembert, “Sur le calcul des probabilités,” Opuscules
mathématiques (1780), vol. vii.

pt. ii

A TREATISE ON PROBABILITY

196
the corresponding events at the second toss, then the probability

(with the above convention) of ‘heads twice running,’ i.e. agreement
between the two coins twice running, is

(h2h +t

)(h

+t

)/h = (h

+t

)/(h

+t

, h) (h

+t

)/h.

2

2t2

1h1

1t1

2h2

2t2

1h1

1t1

1h1

1t1

Since h2h /(h

+ t

, h) = t

/(h

+ t

, h) by the Principle of

2

1h1

1t1

2t2

1h1

1t1

Indifference, and h2h t /h = 0.

2 2t2

∴ (h2h + t

)/(h

+ t

, h) = 2 h

/(h

+ t

, h) by (24.1).

2

2t2

1h1

1t1

2h2

1h1

1t1

Similarly

(h1h + t

)/h = 2h

/h.

1

1t1

1h1

We may assume that h1/h h = h

1

1/h, i.e. that heads with one
coin is irrelevant to the probability of heads with the other; and
h1/h = h /h = 1 by the Principle of Indifference, so that

1

2

2

(h1h + t

)/h = 2 1

= 1 .

1

1t1

2

2

∴ (h2h + t

)(h

+ t

)/h = 2h

/(h

+ t

, h) 1

2

2t2

1h1

1t1

2h2

1h1

1t1

2

= h2h /(h

+ t

, h)

2

1h1

1t1

= 1 h

, h

+ t

, h),

2

2/(h2

1h1

1t1
since, (h1h + t ) being irrelevant to h /h, h /(h

+ t

, h) =

1

1t1

2

2

1h1

1t1
h /h = 1 .

2

2

Now h2/(h , h

+ t

, h) is greater than 1, since the fact of the

2

1h1

1t1

2
coins having agreed once may be some reason for supposing they
will agree again. But it is less than h2/h1h: for we may assume
that h2/(h , h

+ t

, h) is less than h

, h

, h), and also that

2

1h1

1t1

2/(h2

1h1
h2/(h , h

, h) = h

2

1h1

2/h1h,
i.e. that heads twice running with one
coin does not increase the probability of heads twice running with a
different coin. Laplace’s method of tossing, therefore, yields with these
assumptions, more or less legitimate according to the content of h, a
ch. xvi

FUNDAMENTAL THEOREMS

197
probability nearer to 1 than is h

, h

+ t

, h) = 1 ,

4

1h2/h. If h2/(h2

1h1

1t1

2
then the probability is exactly 1.

4

9.

Two other examples will complete this rather discursive
commentary. It has been supposed that by the Principle of Indifference
the probability of the existence of iron upon Sirius is 1, and that

2
similarly the probability of the existence there of any other element
is also 1. The probability, therefore, that not one of the 68 terrestrial

2
elements will be found on Sirius is (1)68, and that at least one will be

2
found there is

68

1 − 1
or approximately certain. This argument, or

2
a similar one, has been seriously advanced. It would seem to prove
also, amongst many other things, that at least one college exactly
resembling some college at either Oxford or Cambridge will almost
certainly be found on Sirius. The fallacy is partly due, as has been
pointed out by Von Kries and others, to an illegitimate use of the

Principle of Indifference. The probability of iron on Sirius is not 1.

2

But the result is also due to the fallacy of false independence. It
is assumed that the known existence of 67 terrestrial elements on

Sirius would not increase the probability of the sixty-eighth’s being
found there also, and that their known absence would not decrease
the sixty-eighth’s probability.∗

10.

The other example is that of Maxwell’s classic mistake in the
theory of gases.† According to this theory molecules of gas move with

∗See Von Kries, Die Principien der Wahrscheinlichkeitsrechnung, p. 10.

Stumpf (Über den Begriff der mathem. Wahrscheinlichkeit, pp. 71–74) argues
that the fallacy results from not taking into account the fact that there might
be as many metals as atomic weights, and that therefore the chance of iron

1
is

, where z is the number of possible atomic weights. A. Nitsche (Vierteljsch.

z
f. wissensch. Philos., 1892) thinks that the real alternatives are 0, or only 1,
or only 2 . . . or 68 terrestrial elements on Sirius, and that these are equally

1
probable, the chance of each being

.

69

†I take the statement of this from Bertrand’s Calcul des probabilités, p. 30.

Let me here quote a precocious passage on Probability regarded as a branch of

Logic, from a letter written by Maxwell in his nineteenth year (1850), before
pt. ii

A TREATISE ON PROBABILITY

198
great velocity in every direction. Both the directions and velocities
are unknown, but the probability that a molecule has a given velocity
is a function of that velocity and is independent of the direction. The
maximum velocity and the mean velocity vary with the temperature.

Maxwell seeks to determine, on these conditions alone, the probability
that a molecule has a given velocity. His argument is as follows:

If φ(x) represents the probability that the component of velocity
parallel to the axis of X is x, the probability that the velocity has
components x, y, z parallel to the three axes is φ(x)φ(y)φ(z). Thus
if F(v) represents the probability of a total velocity v, we have
φ(x)φ(y)φ(z) = F(v), where v2 = x2 + y2 + z2. It is not difficult to
deduce from this (assuming that the functions are analytical) that
φ(x) must be of the form Ge−k2x2.

It is generally agreed at the present time that this result is
erroneous. But the nature of the error is, I think, quite different from
what it is commonly supposed to be.

Bertrand,∗ Poincaré,† and Von Kries,‡ all cite this argument of

Maxwell’s as an illustration of the fallacy of Independence; and
argue that φ(x), φ(y), and φ(z) cannot, as he assumes, represent
independent probabilities, if, as he also assumes, the probability of a
velocity is a function of that velocity. But it is not in this way that the
error in the result really arises. If we do not know what function of the
velocity the probability of that velocity is, a knowledge of the velocity
parallel to the axes of x and y tells us nothing about the velocity
he came up to Cambridge: “They say that Understanding ought to work by
the rules of right reason. These rules are, or ought to be, contained in Logic;
but the actual science of logic is conversant at present only with things either
certain, impossible, or entirely doubtful, none of which (fortunately) we have to
reason on. Therefore the true logic for this world is the calculus of Probabilities,
which takes account of the magnitude of the probability which is, or ought to
be, in a reasonable man’s mind” (Life, page 143).

∗Calcul des probabilités, p. 30.

†Calcul des probabilités (2nd ed.), pp. 41–44.

‡Wahrscheinlichkeitsrechnung, p. 199.

ch. xvi

FUNDAMENTAL THEOREMS

199
parallel to the axis of z. Maxwell was, I think, quite right to hold that
a mere assumption that the probability of a velocity is some function
of that velocity, does not interfere with the mutual independence of
statements as to the velocity parallel to each of the three axes. Let
us denote the proposition, ‘the velocity parallel to the axis of X is x’

by X(x), the corresponding propositions relative to the axes of Y
and Z by Y(y) and Z(z), and the proposition ‘the total velocity is v’

by V(v); and let h represent our à priori data. Then if X(x)/h = φ(x)

it is a justifiable inference from the Principle of Indifference that

Y(y)/h = φ(y) and Z(z)/h = φ(z). Maxwell infers from this that

X(x)Y(y)Z(z)/h = φ(x)φ(y)φ(z). That is to say, he assumes that

Y(y)/X(x) h = Y(y)/h and that Z(z)/Y(y) X(x) h = Z(z)/h. I do
not agree with the authorities cited above that this is illegitimate.

So long as we do not know what function of the total velocity the
probability of that velocity is, a knowledge of the velocities parallel
to the axes of x and y has no bearing on the probability of a given
velocity parallel to the axis of z. But Maxwell goes on to infer that

X(x)Y(y)Z(z)/h = V(v)/h where v2 = x2 + y2 + z2. It is here, and
in a very elementary way, that the error creeps in. The propositions

X(x)Y(y)Z(z) and V(v) are not equivalent. The latter follows from
the former, but the former does not follow from the latter. There
is more than one set of values x, y, z, which will yield the same
value v. Thus the probability V(v)/h is much greater than the
probability X(x)Y(y)Z(z)/h. As we do not know the direction of
the total velocity v, there are many ways, not inconsistent with our
data, of resolving it into components parallel to the axes. Indeed I
think it is a legitimate extension of the preceding argument to put

V(v)/h = φ(v); for there is no reason for thinking differently about
the direction V from what we think about the direction X.

A difficulty analogous to this occurs in discussing the problem of
the dispersion of bullets over a target—a subject round which, on
account of a curiosity which it seems to have raised in the minds of
many students of probability, a literature has grown up of a bulk
pt. ii

A TREATISE ON PROBABILITY

200
disproportionate to its importance.

11.

I now pass to the Principle of Inverse Probability, a theorem
of great importance in the history of the subject. With various
arguments which have been based upon it I shall deal in Chapter

XXX. But it will be convenient to discuss here the history of the

Principle itself and of attempts at proving it.

It first makes its appearance somewhat late in the history of the
subject. Not until 1763, when Bayes’s theorem was communicated
to the Royal Society,∗ was a rule for the determination of inverse
probabilities explicitly enunciated.

It is true that solutions to
inductive problems requiring an implicit and more or less fallacious
use of the inverse principle had already been propounded, notably by

Daniel Bernoulli in his investigations into the statistical evidence in
favour of inoculation.† But the appearance of Bayes’s Memoir marks
the beginning of a new stage of development. It was followed in 1767
by a contribution from Michell‡ to the Philosophical Transactions on
the distribution of the stars, to which further reference will be made
in Chapter XXV. And in 1774 the rule was clearly, though not quite
accurately, enunciated by Laplace in his “Mémoire sur la probabilité
des causes par les évènemens” (Mémoires présentés à l’Académie des

Sciences, vol. vi., 1774). He states the principle as follows (p. 623):

∗Published in the Phil. Trans. vol. liii., 1763, pp. 376–398. This Memoir
was communicated by Price after Bayes’s death; there was a second Memoir in
the following year (vol. liv. pp. 298–310), to which Price himself made some
contributions. See Todhunter’s History, pp. 299 et seq. Thomas Bayes was a
dissenting minister of Tunbridge Wells, who was a Fellow of the Royal Society
from 1741 until his death in 1761. A German edition of his contributions to

Probability has been edited by Timerding.

†“Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et
des avantages de l’inoculation pour la prévenir,” Hist. de l’Acad., Paris, 1760

(published 1766).

Bernoulli argued that the recorded results of inoculation
rendered it a probable cause of immunity. This is an inverse argument, though

Bayes’s theorem is not used in the course of it. See also D. Bernoulli’s Memoir
on the Inclinations of the Planetary Orbits.

‡Michell’s argument owes more, perhaps, to Daniel Bernoulli than to Bayes.

ch. xvi

FUNDAMENTAL THEOREMS

201

“Si un évènement peut être produit par un nombre n de causes
différentes, les probabilités de l’existence de ces causes prises de
l’évènement sont entre elles comme les probabilités de l’évènement
prises de ces causes; et la probabilité de l’existence de chacune d’elles
est égale à la probabilité de l’évènement prise de cette cause, divisée
par la somme de toutes les probabilités de l’évènement prises de
chacune de ces causes.”

He speaks as if he intended to prove this principle, but he only
gives explanations and instances without proof. The principle is not
strictly true in the form in which he enunciates it, as will be seen on
reference to theorems (38) of Chapter XIV.; and the omission of the
necessary qualification has led to a number of fallacious arguments,
some of which will be considered in Chapter XXX.

12.

The value and originality of Bayes’s Memoir are considerable,
and Laplace’s method probably owes much more to it than is generally
recognised or than was acknowledged by Laplace. The principle,
often called by Bayes’s name, does not appear in his Memoir in the
shape given it by Laplace and usually adopted since; but Bayes’s
enunciation is strictly correct and his method of arriving at it shows
its true logical connection with more fundamental principles, whereas

Laplace’s enunciation gives it the appearance of a new principle
specially introduced for the solution of causal problems. The following
passage∗ gives, in my opinion, a right method of approaching the
problem: “If there be two subsequent events, the probability of the
second b and the probability of both together P , and, it being first

N

N
discovered that the second event has happened, from hence I guess
that the first event has also happened, the probability I am in the
right is P.” If the occurrence of the first event is denoted by a and of
b
the second by b, this corresponds to ab/h = a/bh b/h and therefore

∗Quoted by Todhunter, op. cit. p. 296. Todhunter underrates the importance
of this passage, which he finds unoriginal, yet obscure.

pt. ii

A TREATISE ON PROBABILITY

202
ab/h

P
b

P
a/bh =

; for ab/h = , b/h = , a/bh = . The direct and
b/h

N

N
b
indeed fundamental dependence of the inverse principle on the rule
for compound probabilities was not appreciated by Laplace.

13.

A number of proofs of the theorem have been attempted
since Laplace’s time, but most of them are not very satisfactory,
and are generally couched in such a form that they do no more
than recommend the plausibility of their thesis. Mr. McColl∗ gave a
symbolic proof, closely resembling theorem (38) when differences of
symbolism are allowed for; and a very similar proof has also been
given by A. A. Markoff.† I am not acquainted with any other rigorous
discussion of it.

Von Kries‡ presents the most interesting and careful example of a
type of proof which has been put forward in one shape or another
by a number of writers. We have initially, according to this view, a
certain number of hypothetical possibilities, all equally probable, some
favourable and some unfavourable to our conclusion. Experience, or
rather knowledge that the event has happened, rules out a number of
these alternatives, and we are left with a field of possibilities narrower
than that with which we started. Only part of the original field or

Spielraum of possibility is now admissible (zulässig). Causes have
à posteriori probabilities which are proportional to the extent of their
occurrence in the now restricted field of possibility.

There is much in this which seems to be true, but it hardly
amounts to a proof. The whole discussion is in reality an appeal
to intuition. For how do we know that the possibilities admissible
à posteriori are still, as they were assumed to be à priori, equal
possibilities? Von Kries himself notices that there is a difficulty; and

∗“Sixth Paper on the Calculus of Equivalent Statements,” Proc. Lond. Math.

Soc., 1897, vol. xxviii. p. 567. See also p. 155 above.

†Wahrscheinlichkeitsrechnung, p. 178.

‡Die Principien der Wahrscheinlichkeitsrechnung, pp. 117–121. The above
account of Von Kries’s argument is much condensed.

ch. xvi

FUNDAMENTAL THEOREMS

203

I do not see how he is to avoid it, except by the introduction of an
axiom.

This was in fact the course taken by Professor Donkin in 1851, in
an article which aroused some interest in the Philosophical Magazine
at the time, but which has since been forgotten. Donkin’s theory
is, however, of considerable interest. He laid down as one of the
fundamental principles of probability the following:∗

“If there be any number of mutually exclusive hypotheses h1h2h3 . . .

of which the probabilities relative to a particular state of information
are p1p2p3 . . ., and if new information be gained which changes
the probabilities of some of them, suppose of hm+1 and all that
follow, without having otherwise any reference to the rest, then the
probabilities of these latter have the same ratios to one another, after
the new information, that they had before.”†

Donkin goes on to say that the most important case is where
the new information consists in the knowledge that some of the
hypotheses must be rejected, without any further information as to
those of the original set which are retained. This is the proposition
which Von Kries requires.

As it stands, the phrase “without having otherwise any reference
to the rest” obviously lacks precision. An interpretation, however,
can be put upon it, with which the principle is true. If, given the
old information and the truth of one of the hypotheses h1 . . . hm to
the exclusion of the rest, the probability of what is conveyed by the
new information is the same whichever of the hypotheses h1 . . . hm
has been taken, then Donkin’s principle is valid. For let a be the old

∗“On certain Questions relating to the Theory of Probabilities,” Phil. Mag.

4th series, vol. i., 1851.

†It is interesting to notice that an axiom, practically equivalent to this,
has been laid down more lately by A. A. Markoff (Wahrscheinlichkeitsrechnung,
p. 8) under the title ‘Unabhängigkeitsaxiom.’

pt. ii

A TREATISE ON PROBABILITY

204
information, a the new, and let hr/a = pr, hr/aa = p ; then
r
h
a /h
p = h
r a /a =
r a
pr ,
r
r /aa =
a/ a
a /a
p

∴ r = p /ps, etc., if a /hra = a /hsa, which is the condition already
p
s
r
explained.

14.

Difficulties connected with the Inverse Principle have arisen,
however, not so much in attempts to prove the principle as in those
to enunciate it—though it may have been the lack of a rigorous
proof that has been responsible for the frequent enunciation of an
inaccurate principle.

It will be noticed that in the formula (38.2) the à priori
probabilities of the hypotheses a1 and a2 drop out if p1 = p2, and
the results can then be expressed in a much simpler shape. This
is the shape in which the principle is enunciated by Laplace for the
general case,∗ and represents the uninstructed view expressed with
great clearness by De Morgan:† “Causes are likely or unlikely, just
in the same proportion that it is likely or unlikely that observed
events should follow from them. The most probable cause is that
from which the observed event could most easily have arisen.” If this
were true the principle of Inverse Probability would certainly be a
most powerful weapon of proof, even equal, perhaps, to the heavy
burdens which have been laid on it. But the proof given in Chapter

XIV. makes plain the necessity in general of taking into account the
à priori probabilities of the possible causes. Apart from formal proof
this necessity commends itself to careful reflection. If a cause is very
improbable in itself, the occurrence of an event, which might very
easily follow from it, is not necessarily, so long as there are other
possible causes, strong evidence in its favour. Amongst the many
writers who, forgetting the theoretic qualification, have been led into

∗See the passage quoted above, p. 201.

†“Essay on Probabilities,” in the Cabinet Encyclopædia, p. 27.

ch. xvi

FUNDAMENTAL THEOREMS

205
actual error, are philosophers as diverse as Laplace, De Morgan,

Jevons, and Sigwart, Jevons∗ going so far as to maintain that the
fallacious principle he enunciates is “that which common sense leads
us to adopt almost instinctively.”

15.

The theory of the combination of premisses dealt with in

§§ 7, 8 of Chapter XIV. has not often been discussed, and the history
of it is meagre. Archbishop Whately† was led astray by a superficial
error, and De Morgan, adopting the same mistaken rule, pushed it

∗Principles of Science, vol. i. p. 280.

†Logic, 8th ed. p. 211: “As in the case of two probable premisses, the
conclusion is not established except upon the supposition of their being both
true, so in the case of two distinct and independent indications of the truth of
some proposition, unless both of them fail, the proposition must be true: we
therefore multiply together the fractions indicating the probability of the failure
of each—the chances against it—and, the result being the total chances against
the establishment of the conclusion by these arguments, this fraction being
deducted from unity, the remainder gives the probability for it. E.g. a certain
book is conjectured to be by such and such an author, partly, 1st, from its
resemblance in style to his known works; partly, 2nd, from its being attributed
to him by some one likely to be pretty well informed. Let the probability of the
conclusion, as deduced from one of these arguments by itself, be supposed 2 ,

5
and in the other case 3 ; then the opposite probabilities will be 3 and 4 , which

7

5

7
multiplied together give 12 as the probability against the conclusion. . . .”

35

The Archbishop’s error, in that a negative can always be turned into an
affirmative by a change of verbal expression, was first pointed out by a mere
diocesan, Bishop Terrot, in the Edin. Phil. Trans. vol. xxi. The mistake is well
explained by Boole in the same volume of the Edin. Phil. Trans.: “A confusion
may here be noted between the probability that a conclusion is proved, and
the probability in favour of a conclusion furnished by evidence which does not
prove it. In the proof and statement of his rule, Archbishop Whately adopts
the former view of the nature of the probabilities concerned in the data. In the
exemplification of it, he adopts the latter.”

pt. ii

A TREATISE ON PROBABILITY

206
to the point of absurdity.∗ Bishop Terrot† approached the question
more critically. Boole’s‡ last and most considered contribution to
the subject of probability dealt with the same topic. I know of no
discussion of it during the past sixty years.

Boole’s treatment is full and detailed. He states the problem
as follows: “Required the probability of an event z, when two
circumstances x and y are known to be present,—the probability
of the event z, when we know only of the existence of the
circumstances x, being p, and the probability, when we only know of
the existence of y, being q.”§ His solution, however, is vitiated by the
fundamental error examined in § 6 above. Two of his conclusions may
be mentioned for their plausibility, but neither is valid.

“If the causes in operation, or the testimonies borne,” he argues,

“are, separately, such as to leave the mind in a state of equipoise as
respects the event whose probability is sought, united they will but
produce the same effect.” If, that is to say, a/h1 = 1 and a/h

,

2

2 = 1

2
he concludes that a/h1h2 = 1. The plausibility of this is superficial.

2

∗“Theory of Probabilities,” Encyclopædia Metropolitana, p. 400. He shows by
means of it that “if any assertion appear neither likely nor unlikely in itself,
then any logical argument in favour of it, however weak the premisses, makes it
in some degree more likely than not—a theorem which will be readily admitted
on its own evidence.” He then gives an example: “à priori vegetation on the
planets is neither likely nor unlikely; suppose argument from analogy makes
it 3 ; then the total probability is 1 + 1

3
or 13 .” De Morgan seems to accept

10

2

2

10

20
without hesitation the conclusion to be derived from this, that everything which
is not impossible is as probable as not.

†“On the Possibility of combining two or more Probabilities of the same

Event, so as to form one definite Probability,” Edin. Phil. Trans., 1856, vol. xxi.

‡“On the Application of the Theory of Probabilities to the Question of the

Combination of Testimonies or Judgments,” Edin. Phil. Trans., 1857, vol. xxi.

§Loc. cit. p. 631. Boole’s principle (loc. cit. p. 620) that “the mean strength
of any probabilities of an event which are founded upon different judgments
or observations is to be measured by that supposed probability of the event
à priori which those judgments or observations following thereupon would not
tend to alter,” is not correct if it means more than that the mean strength of
z/x and z/y is to be measured by z/xy.

ch. xvi

FUNDAMENTAL THEOREMS

207

Consider, for example, the following instance: h1 = A is black and

B is black or white, h2 = B is black and A is black or white, a =
both A and B are black. Boole also concluded without valid reason
that a/h1h2 increases, the greater the à priori improbability of the
combination h1h2.

16.

The theory of “Testimony” itself, the theory, that is to
say, of the combination of the evidence of witnesses, has occupied so
considerable a space in the traditional treatment of Probability that
it will be worth while to examine it briefly. It may, however, be
safely said that the principal conclusions on the subject set out by

Condorcet, Laplace, Poisson, Cournot, and Boole, are demonstrably
false. The interest of the discussion is chiefly due to the memory of
these distinguished failures.

It seems to have been generally believed by these and other
logicians and mathematicians∗ that the probability of two witnesses
speaking the truth, who are independent in the sense that there is
no collusion between them, is always the product of the probabilities
that each of them separately will speak the truth.† On this basis
conclusions such as the following, for example, are arrived at:

X and Y are independent witnesses (i.e. there is no collusion
between them). The probability that X will speak the truth is x, that

Y will speak the truth is y. X and Y agree in a particular statement.

∗Perhaps M. Bertrand should be registered as an honourable exception. At
least he points out a precisely analogous fallacy in an example where two
meteorologists prophesy the weather, Calcul des Probabilités, p. 31.

†E.g., Boole, Laws of Thought, p. 279.

De Morgan, Formal Logic, p. 195.

Condorcet, Essai, p. 4.

Lacroix, Traité, p. 248.

Cournot, Exposition, p. 354.

Poisson, Recherches, p. 323.

This list could be greatly extended.

pt. ii

A TREATISE ON PROBABILITY

208

The chance that this statement is true is
xy

.

xy + (1 − x)(1 − y)

For the chance that they both speak the truth is xy, and the
chance that they both speak falsely is (1 − x)(1 − y). As, in this
case, our hypothesis is that they agree, these two alternatives are
exhaustive; whence the above result, which may be found in almost
every discussion of the subject.

The fallacy of such reasoning is easily exposed by a more exact
statement of the problem. For let a1 stand for “X1 asserts a,” and
let a/a1h = x1, where h, our general data, is by itself irrelevant to a,
i.e., x1 is the probability that a statement is true of which we only
know that X1 has asserted it. Similarly let us write b/b2h = x2
where b2 stands for “X2 asserts b.” The above argument then assumes
that, if X1 and X2 are witnesses who are causally independent in
the sense there is no collusion between them direct or indirect,
ab/a1b2h = a/a1h b/b2h = x1x2.

But ab/a1b2h = a/a1bb2h b/a1b2h, and this is not equal to x1x2
unless a/a1bb2h = a/a1h and b/a1b2h = b/b2h. It is not a sufficient
condition for this, as seems usually to be supposed, that X1 and X2
should be witnesses causally independent of one another. It is also
necessary that a and b, i.e. the propositions asserted by the witnesses,
should be irrelevant to one another and also each of them irrelevant
to the fact of the assertion of the other by a witness. If a knowledge
of a affects the probability either of b or of b1, it is evident that the
formula breaks down. In the one extreme case, where the assertions
of the two contradict one another, ab/a1b2h = 0.

In the other
extreme, where the two agree in the same assertion, i.e. where a ≡ b,
a/a1bb2h = 1 and not = a/a1h.

17.

The special problem of the agreement of witnesses, who
make the same statement, can be best attacked as follows, a certain
amount of simplification being introduced. Let the general data h of
ch. xvi

FUNDAMENTAL THEOREMS

209
the problem include the hypothesis that X1 and X2 are each asked
and reply to a question to which there is only one correct answer.

Let ai = “Xi asserts a in reply to the question,” and mi = “Xi gives
the correct answer to the question.” Then
m1/a1h = x1 and m2/a2h = x2,
x1 and x2 being, in the conventional language of this problem, the

“credibilities” of the witnesses. We have, since the witnesses agree
and since a follows from miai and mi follows from aai,
m1/a1a2h = m1m2/a1a2h = m2/a1a2h;
a/aih = mi/aih;
a/aimih = 1;
mi/aaih = 1.

Also, since the witnesses are, in the ordinary sense, “independent”

witnesses, a2/a1ah = a2/ah and a2/a1¯ah = a2/¯ah; that is to say, the
probability of X2’s asserting a is independent of the fact of X1’s
having asserted a, given we know that a is, in fact, true or false, as
the case may be.

The probability that, if the witnesses agree, their assertion is true
is
m
a/a

1a2/a1h

1a2h = m1/a1a2h =
a2/a1h
a
a

=

2/a1m1h.m1/a1h =

2/a1ah.x1

.

a2a/a1h + a2¯
a/a1h
a2/a1ah.x1 + a2/a1¯
ah.(1 − x2)

If this is to be equal to
x1x2

, we must have
x1x2 + (1 − x1)(1 − x2)

a2/a1ah
x

=

2

.

a2/a1¯
ah

1 − x2
pt. ii

A TREATISE ON PROBABILITY

210

Now
a2/a1ah
a

=

2/ah by the hypothesis of “independence”

a2/a1¯
ah
a2/¯
ah
aa

¯
a/h
a/a

¯
a/h

=

2/h .

=

2h .

¯
aa2/h a/h

¯
a/a2h a/h
x

¯
a/h

=

2

.

.

1 − x2 a/h

This then is the assumption which has tacitly slipped into the
conventional formula,—that a/h = ¯a/h = 1. It is assumed, that is

2
to say, that any proposition taken at random is as likely as not to
be true, so that any answer to a given question is, à priori, as
likely as not to be correct. Thus the conventional formula ought
to be employed only in those cases where the answer which the

“independent” witnesses agree in giving is, à priori and apart from
their agreement, as likely as not.

18.

A somewhat similar confusion has led to the controversy
as to whether and in what manner the à priori improbability of a
statement modifies its credibility in the mouth of a witness whose
degree of reliability is known. The fallacy of attaching the same
weight to a testimony regardless of the character of what is asserted,
is pointed out, of course, by Hume in the Essay on Miracles, and his
argument, that the great à priori improbability of some assertions
outweighs the force of testimony otherwise reliable, depends on the
avoidance of it. The correct is also taken by Laplace in his Essai
philosophique (pp. 98–102), where he argues that a witness is less
to be believed when he asserts an extraordinary fact, declaring the
opposite view (taken by Diderot in the article on “Certitude” in the

Encyclopédie) to be inconceivable before “le simple bon sens.”

The manner in which the resultant probability is affected depends
upon the precise meaning we attach to “degree of reliability” or

“coefficient of credibility.” If a witness’s credibility is represented
by x, do we mean that, if a is the true answer, the probability of his
ch. xvi

FUNDAMENTAL THEOREMS

211
giving it is x, or do we mean that if he answers a the probability of
a’s being true is x? These two things are not equivalent.

Let a1 stand for “a is asserted by the witness”; h1 for our evidence
bearing on the witness’s veracity; and h2 for other evidence bearing
on the truth of a. Let a/h1h2, i.e. the à priori probability of a apart
from our knowledge of the fact that the witness has asserted it, be
represented by p.

Let
a/h
a/a

1

1h1 = x1 and a1/ah1 = x2; so that x1 =
x2. In
a1/h1
general a/h1 = a1/h1. Do we mean by the witness’s credibility x1
or x2?

We require a/a1h1h2.

Let a1/¯ah1 = r, i.e. the probability, apart from our special
knowledge concerning a, that, if a is false, the witness will hit on that
particular falsehood.

a
x
a/a

1/ah1h2
a/h1h2

2p

1h1h2 =

=
a1/h1h2
a1a/h1h2 + a1¯
a/h1h2
x
x

=

2p

=

2p

;
x2p + a1/¯
ah1h2 (1 − p)

x2p + r(1 − p)

for a1/ah1h2 = a1/ah1 and a1/¯ah1h2 = a1/¯ah1, since, given certain
knowledge concerning a, h2 is irrelevant to the probability of a1.

19.

Generally speaking, all problems, in regard to the
combination of testimonies or to the combination of evidence derived
from testimony with evidence derived from other sources, may be
treated as special instances of the general problem of the combination
of arguments. Beyond pointing out the above plausible fallacies, there
is little to add. Mr. W. E. Johnson, however, has proposed a method
of defining credibility, which is sometimes valuable, because it regards
the witness’s credibility not absolutely, but with reference to a given
type of question, so that it enables us to measure the force of the
witness’s testimony under special circumstances. If a represents the
fact of A’s testimony regarding x, then we may define A’s credibility
pt. ii

A TREATISE ON PROBABILITY

212
for x as α, where α is given by the equation
x/ah = x/h + α
x/h ¯
x/h;
so that α x/h ¯x/h measures the amount by which A’s assertion
of x increases its probability.

20.

One of the most ancient problems in probability is concerned
with the gradual diminution of the probability of a past event, as the
length of the tradition increases by which it is established. Perhaps
the most famous solution of it is that propounded by Craig in his

Theologiae Christianae Principia Mathematica, published in 1699.∗

He proves that suspicions of any history vary in the duplicate ratio
of the times taken from the beginning of the history in a manner
which has been described as a kind of parody of Newton’s Principia.

“Craig,” says Todhunter, “concluded that faith in the Gospel so far
as it depended on oral tradition expired about the year 880, and
that so far as it depended on written tradition it would expire in
the year 3150. Peterson by adopting a different law of diminution
concluded that faith would expire in 1789.”† About the same time

∗See Todhunter’s History, p. 54. It has been suggested that the anonymous
essay in the Phil. Trans. for 1699 entitled “A Calculation of the Credibility of

Human Testimony” is due to Craig. In this it is argued that, if the credibilities
of a set of witnesses are p1 . . . pn, then if they are successive the resulting
probability is the product p1p2 . . . pn; if they are concurrent, it is:

1 − (1 − p1)(1 − p2) . . . (1 − pn).

This last result follows from the supposition that the first witness leaves
an amount of doubt represented by 1 − p1; of this the second removes the
fraction p2, and so on. See also Lacroix, Traité élémentaire, p. 262. The above
theory was actually adopted by Bicquilley.

†In the Budget of Paradoxes De Morgan quotes Lee, the Cambridge

Orientalist, to the effect that Mahometan writers, in reply to the argument that
the Koran has not the evidence derived from Christian miracles, contend that,
as evidence of Christian miracles is daily weaker, a time must at last arrive
when it will fail of affording assurance that they were miracles at all: whence
the necessity of another prophet and other miracles.

ch. xvi

FUNDAMENTAL THEOREMS

213

Locke raised the matter in chap. xvi. bk. iv. of the Essay Concerning

Human Understanding: “Traditional testimonies the farther removed,
the less their proof. . . . No Probability can rise higher than its first
original.” This is evidently intended to combat the view that the
long acceptance by the human race of a reputed fact is an additional
argument in its favour and that a long tradition increases rather than
diminishes the strength of an assertion. “This is certain,” says Locke,

“that what in one age was affirmed upon slight grounds, can never
after come to be more valid in future ages, by being often repeated.”

In this connection he calls attention to “a rule observed in the law of

England, which is, that though the attested copy of a record be good
proof, yet the copy of a copy never so well attested, and by never
so credible witnesses, will not be admitted as a proof in Judicature.”

If this is still a good rule of law, it seems to indicate an excessive
subservience to the principle of the decay of evidence.

But, although Locke affirms sound maxims, he gives no theory
that can afford a basis for calculation. Craig, however, was the
more typical professor of probability, and in attempting an algebraic
formula he was the first of a considerable family. The last grand
discussion of the problem took place in the columns of the Educational

Times.∗ Macfarlane† mentions that four different solutions have been
put forward by mathematicians of the problem: “A says that B says
that a certain event took place; required the probability that the event
did take place, p1 and p2 being A’s and B’s respective probabilities of
speaking the truth.” Of these solutions only Cayley’s is correct.

∗Reprinted in Mathematics from the Educational Times, vol. xxvii.

†Algebra of Logic, p. 151. Macfarlane attempts a solution of the general
problem without success. Its solution is not difficult, if enough unknowns are
introduced, but of very little interest.

CHAPTER XVII
some problems in inverse probability, including
averages

1.

The present chapter deals with ‘problems’—that is to
say, with applications to particular abstract questions of some of
the fundamental theorems demonstrated in Chapter XIV.

It is
without philosophical interest and should probably be omitted by
most readers. I introduce it here in order to show the analytical
power of the method developed above and its advantage in ease and
especially in accuracy over other methods which have been employed.∗

§ 2 is mainly based upon some problems discussed by Boole. §§ 3–7
deal with the fundamental theory connecting averages and laws of
error. §§ 8–11 treat discursively the Arithmetic Average, the Method
of Least Squares, and Weighting.

2.

In the following paragraph solutions are given of some
problems posed by Boole in chapter xx. of his Laws of Thought.

Boole’s own method of solving them is constantly erroneous,† and the
difficulty of his method is so great that I do not know of any one
but himself who has ever attempted to use it. The term ‘cause’ is
frequently used in these examples where it might have been better
to use the term ‘hypothesis.’ For by a possible cause of an event no
more is here meant than an antecedent occurrence, the knowledge of
which is relevant to our anticipation of the event; it does not mean
an antecedent from which the event in question must follow.

(56) The à priori probabilities of two causes A1 and A2 are
c1 and c2 respectively. The probability that if the cause A1 occur, an

∗Such examples as these might sometimes be set to test the wits of students.

The problems on Probability usually given are simply problems on mathematical
combinations. These, on the other hand, are really problems in logic.

†For the reason given in § 6 of Chapter XVI. The solutions of problems

I.–VI., for example, in the Laws of Thought, chap. xx., are all erroneous.

214
ch. xvii

FUNDAMENTAL THEOREMS

215
event E will accompany it (whether as a consequence of A1 or not),
is p1, and the probability that E will accompany A2, if A2 present
itself, is p2. Moreover, the event E cannot appear in the absence of
both the causes A1 and A2. Required the probability of the event E.

This problem is of great historical interest and has been called

Boole’s ‘Challenge Problem.’ Boole originally proposed it for solution
to mathematicians in 1851 in the Cambridge and Dublin Mathematical

Journal. A result was given by Cayley∗ in the Philosophical Magazine,
which Boole declared to be erroneous.† He then entered the field
with his own solution.‡ “Several attempts at its solution,” he says,

“have been forwarded to me, all of them by mathematicians of
great eminence, all of them admitting of particular verification, yet
differing from each other and from the truth.”§ After calculations of
considerable length and great difficulty he arrives at the conclusion
that u is the probability of the event E where u is that root of the

∗Phil. Mag. 4th series, vol. vi.

†Cayley’s solution was defended against Boole by Dedekind (Crelle’s Journal,
vol. 1. p. 268). The difference arises out of the extreme ambiguity as to the
meaning of the terms as employed by Cayley.

‡“Solution of a Question in the Theory of Probabilities,” Phil. Mag. 4th series,
vol. vii., 1854.

This solution is the same as that printed by Boole shortly
afterwards in the Laws of Thought, pp. 321–326. In the Phil. Mag. Wilbraham
gave as the solution u = c1p1 + c2p2 − z, where z is necessarily less than either
c1p1 or c2p2. This solution is correct so far as it goes, but is not complete. The
problem is also discussed by Macfarlane, Algebra of Logic, p. 154.

§In proposing the problem Boole had said: “The motives which have led
me, after much consideration, to adopt, with reference to this question, a course
unusual in the present day, and not upon slight grounds to be revived, are the
following: First, I propose the question as a test of the sufficiency of received
methods. Secondly, I anticipate that its discussion will in some measure add to
our knowledge of an important branch of pure analysis.”

When printing his
own solution in the Laws of Thought, he adds, that the above “led to some
interesting private correspondence, but did not elicit a solution.”

pt. ii

A TREATISE ON PROBABILITY

216
equation

[1 − c1(1 − p1) − u] [1 − c2(1 − p2) − u]

(u − c

=

1p1)(u − c2p2)

1 − u
c1p1 + c2p2 − u
which is not less than c1p1 and c2p2 and not greater than 1−c1(1−p1),

1 − c2(1 − p2) or c1p1 + c2p2.

This solution can easily be seen to be wrong. For in the case
where A1 and A2 cannot both occur, the solution is u = c1p1 + c2p2;
whereas Boole’s equations do not reduce to this simplified form. The
mistake which Boole has made is the one general to his system,
referred to in Chapter XVI., § 6.∗

The correct solution, which is very simple, can be reached as
follows:

Let a1, a2, e assert the occurrences of the two causes and the
event respectively, and let h be the data of the problem.

Then we have a1/h = c1, a2/h = c2, e/a1h = p1, e/a2h = p2: we
require e/h. Let e/h = u, and let a1a2/eh = z. Since the event
cannot occur in the absence of both the causes,
e/¯
a1¯
a2h = 0.

It follows from this that ¯a1¯a2/eh = 0, unless e/h = 0,
i.e.

(a1 + a2)/eh = 1,
whence
a1/eh + a2/eh = 1 + a1a2/eh
by (24).

Now
c
c
a

1p1

2p2

1/eh =
and a2/eh =

,
u
u
c1p1 + c2p2

∴ u =

,

1 + z
where z is the probability after the event that both the causes were
present.

∗Boole’s error is pointed out and a correct solution given in Mr. McColl’s

“Sixth Article on the Calculus of Equivalent Statements” (Proc. Lond. Math.

Soc. vol. xxviii. p. 562).

ch. xvii

FUNDAMENTAL THEOREMS

217

If we write ea1a2/h = y,
y = a1a2/eh e/h = uz
so that
u = (c1p1 + c2p2) − y.

Boole’s solution fails by attempting to be independent of y or z.

(56.1) Suppose that we wish to find limits for the solution which
are independent of y and z: then, since y

0, u
c1p1 + c2p2.

Again
e/h = e¯
a1/h+ea1/h

¯
a1/h+ea1/h

1−c2 +c2p2
by (24.2) and (4).

Similarly e/h

−c2 + c2p2. From the same equations it appears that
e/h
c1p1 and c2p2.

∴ u lies between

 c1p1 + c2p2
c




the greatest of

1p1 and the least of

1 − c1(1 − p1)

c2p2



 1 − c2(1 − p2).

It will be seen that these numerical limits are the same as the
limits obtained by Boole for the roots of his equations.

(56.2) Suppose that the à priori probabilities of the causes
c1 and c2 are to be eliminated.

The only limit we then have is
u < p1 + p2.

(56.3) Suppose that one of the à priori probabilities c2 is to be
eliminated. We then have limits c1p1
u

1 − c1 + c1p1. If, therefore,
c1 is large, u does not differ widely from c1p1.

(56.4) Suppose p2 is to be eliminated. We then have
c1p1
u
c1p1 + c2
c1p1 + 1 − c1.

If therefore c1 is large or c2 small, u does not differ widely
from c1p1.

pt. ii

A TREATISE ON PROBABILITY

218

(56.5) If a1/a2h = a1/h, i.e. if our knowledge of each of the causes
is independent, we have a closer approximation. For
y = ea1a2/h = e/a1a2h a1/a2h a2/h = e/a1a2h c1c2,

∴ u = c1p1 + c2p2 − c1c2 e/a1a2h,

∴ u > c1p1 + c2p2 − c1c2.

(57) We may now generalise (56) and discuss the case of n causes.

If an event can only happen as a consequence of one or more of
certain causes A1, A2, . . . An, and if c1 is the à priori probability of
the cause A1 and p1 the probability that, if the cause A1 be known
to exist, the event E will occur: required the probability of E.

This is Boole’s problem VI. (Laws of Thought, p. 336). As the
result of ten pages of mathematics, he finds the solution to be the
root lying between certain limits of an equation of the nth degree
which he cannot solve. I know no other discussion of the problem.

The solution is as follows:

e/h = e¯
a1/h + ea1/h = e¯
a1/h + e/a1h a1/h = e¯
a1/h + c1p1

(i.)

e¯
a1/h = e¯
a1¯
a2/h + e¯
a1/a2h a2/h = e¯
a1¯
a2/h + c2 e¯
a1/a2h,

1
e¯
a1/a2h = e/a2h − ea1/a2h = p2 −
ea1a2/h,
c2

∴ e/h = e¯a1¯a2/h + c1p1 + c2p2 − ea1a2/h,
e¯
a1¯
a2/h = e¯
a1¯
a2¯
a3/h + e¯
a1¯
a2a3/h,
and
e¯
a1¯
a2a3/h = e¯
a1¯
a2/a3h c3 = c3 {e/a3h − e¯
a1¯
a2/a3/h}

= c3p3 − e¯
a1¯
a2a3/h,

∴ e/h = e¯a1¯a2¯a3/h + c1p1 + c2p2 + c3p3 − e¯a1a2/h − e¯a1¯a2a3/h.

ch. xvii

FUNDAMENTAL THEOREMS

219

In general
e¯
a1¯
a2 . . . ¯
ar−1/h = e¯
a1¯
a2 . . . ¯
ar−1¯
ar/h + e¯
a1¯
a2 . . . ¯
ar−1ar/h

= e¯
a1 . . . ¯
ar/h + e¯
a1 . . . ¯
ar−1/arh cr

= e¯
a1 . . . ¯
ar/h + cr {e/arh − e¯
a1 . . . ¯
ar−1/arh}

= e¯
a1 . . . ¯
ar/h + crpr − e¯
a1 . . . ¯
ar−1ar/h.

n
n

∴ finally we have e/h = e¯a1 . . . ¯an/h +
crpr −
e¯
a . . . ¯
ar−1ar/h.

1

2

But since the n causes are supposed to be exhaustive
e¯
a1 . . . ¯
an/h = 0,
n
n

∴ e/h =
crpr −
e¯
a1 . . . ¯
ar−1ar/h

(ii.).

1

2

Let
e¯
a1 . . . ¯
ar−1ar/h = nr;
n
n
then
e/h =
crpr −
nr

(iii.).

1

2

(57.1) If our knowledge of the several causes is independent, if,
that is to say, our knowledge of the existence of any one of them is
not relevant to the probability of the existence of any other, so that
ar/ash = ar/h = cr, then
e¯
a1 . . . ¯
ar−1ar/h = e¯
a1 . . . ¯
ar−1/arh cr

= cr e¯
a1 . . . ¯
ar−1arh{1 − ¯
a1 . . . ¯
ar−1/arh}

r−1

= cr 1 −

(1 − c1) . . . (1 − cr−1) e/¯
a1 . . . ¯
ar−1arh.

1

Let
e/¯
a1 . . . ¯
ar−1arh = mr,
r=n
r=n
s=r−1
then
e/h =
crpr −
cr 1 −

(1 − cs) mr.

r=1
r=2
s=1
pt. ii

A TREATISE ON PROBABILITY

220

These results do not look very promising as they stand, but they
lead to some useful approximations on the elimination of mr and nr
and to some interesting special cases.

(57.2) From equation (i.) it follows that e/h
c1p1 and e/h
n

1 − c1(1 − p1); and from equation (ii.) that e/h
crpr;

1

∴ e/h lies between


n


c




r pr
c





1



1p1










the greatest of

... and the least of 1 − c1(1 − p1)

.









.

 c

.

npn









 1 − cn(1 − pn).

(57.3) Further, if the causes are independent it follows from (57.1)

that
n
n
r−1
e/h
crpr −
cr 1 −

(1 − cs) ,

1

2

1
so that e/h lies between


n
n
r−1


n


c
c

(1 − c


c


r pr −
r 1 −
s)


r pr




the 



1



1

2

1










greatest c
and the

1 − c

1p1

1(1 − p1)

of
least of

.



.





.



.



.



.













 c

 1 − c
npn
n(1 − pn).

(57.4) Now consider the case in which p1 = p2 = . . . = pn = 1,
i.e. in which any of the causes would be sufficient, and in which the
causes are independent. Then mr = 1; so that
r=n
r=n
s=r−1
e/h =
cr −
cr 1 −

(1 − cs)

r=1
r=2
s=1

= 1 − (1 − c1)(1 − c2) . . . (1 − cn).

ch. xvii

FUNDAMENTAL THEOREMS

221

(57.5) Let c1, c2 . . . cn be small quantities so that their squares
and products may be neglected.

Then
e/h =
crpr,
i.e. the smaller the probabilities of the causes the more do they
approach the condition of being mutually exclusive.∗

(57.6) The à posteriori probability of a particular cause ar after
the event has been observed is
e/a
a
r h
ar/h
r /eh =
e/h
p

= rcr .

e/h

(This is Boole’s problem IX., p. 357).

(58) The probability of the occurrence of a certain natural
phenomenon under given circumstances is p.

There is also a
probability a of a permanent cause of the phenomenon, i.e. of a
cause which would always produce the event under the circumstances
supposed. What is the probability that the phenomenon, being
observed n times, will occur the n + 1th?

This is Boole’s problem X. (Laws of Thought, p. 358). Boole
arrives by his own method at the same result as that given below.

It is necessary first of all to state the assumption somewhat more
precisely. If xr asserts the occurrence of the event at the rth trial and
t the existence of the ‘permanent cause’ we have
xr/h = p,
t/h = a,
xr/th = 1,
and we require
xn+1/x1 . . . xnh = yn+1.

It is also assumed that if there is no permanent cause the probability

∗Boole arrives at this result, Laws of Thought, p. 345, but I doubt his proof.

pt. ii

A TREATISE ON PROBABILITY

222
of xs is not affected by the observations xr, etc., i.e.

x

¯
s/xr . . . xtth = xs/¯
th, ∗
x ¯
t/h
x
p − a
x
s
s/h − xst/h
s/¯
th =

=

=

,

¯
t/h

¯
t/h

1 − a
x

¯
r /x1 . . . xr−1h = xr t/x1 . . . xr−1h + xr t/x1 . . . xr−1h

= t/x1 . . . xr−1h + xr/¯
tx1 . . . xr−1h ¯
t/x1 . . . xr−1h
x
p − a
x

=

1 . . . xr−1t/h +

1 . . . xr−1/¯
th ¯
t/h
x1 . . . xr−1/h

1 − a
x1 . . . xr−1/h
p − a r−1 (1 − a)

a
p − a

1 − a

=

+

,
y1y2 . . . yr−1

1 − a
y1y2 . . . yr−1
p − a r−1
a + (p − a)

1 − a
i.e.

yr =

.

y1y2 . . . yr−1
p − a
a + (p − a)

Also

1 − a
y1 = p and y2 =

,
y1
p − a n
a + (p − a)

so that

1 − a
yn+1 =

.

p − a n−1
a + (p − a)

1 − a

(58.1) If p = a, yn = 1; for if an event can only occur as the result
of a permanent cause, a single occurrence makes future occurrences
certain under similar conditions.

∗This assumption, which is tacitly introduced by Boole, is not generally
justifiable. I use it here, as my main purpose is to illustrate a method. The
same problem, without this assumption, will be discussed in dealing with Pure

Induction.

ch. xvii

FUNDAMENTAL THEOREMS

223

(58.2)

p − a n−2
p − a
a(p − a)

1 −

1 − a

1 − a
yn+1 − yn =
p − a n−1
p − 1 n−2
a + (p − a)

a + (p − a)

1 − a

1 − a

(by easy algebra);
and p is always > a and < 1.

r

So that
p − a

(p − a)

is positive and decreases as r increases,

1 − a

∴ yn+1 > yn.

As n increases yn = 1 − , where
p − a n−2
p − a

1 − a

= (p − a) 1 −

,

1 − a
p − a n−2
a + (p − a)

1 − a
so that for any positive value of η however small a value of n can be
found such that < η so long as a is not zero.

(58.3) tn the à posteriori probability of a permanent cause after
n successful observations is
x
a
t/x

1 . . . xn/th
t/h

1 . . . xnh =

=

,
x1 . . . xn/h
y1y2 . . . yn
a
i.e.

tn =
p − a n ,
a + (p − a)

1 − a
p − a n

(p − a)

1 − a
tn = 1 − , where

=
p − a n .

a + (p − a)

1 − a
pt. ii

A TREATISE ON PROBABILITY

224

So that tn approaches the limit unity as n increases, so long as a is
not zero.

3.

The following is a common type of statistical problem.∗

We are given a series of measurements, or observations, or estimates
of the true value of a given quantity; and we wish to determine
what function of these measurements will yield us the most probable
value of the quantity, on the basis of this evidence. The problem
is not determinate unless we have some good ground for making
an assumption as to how likely we are in each case to make errors
of given magnitudes. But such an assumption, with or without
justification, is frequently made.

The functions of the original measurements which we commonly
employ, in order to yield us approximations to the most probable
value of the quantity measured, are the various kinds of means or
averages—the arithmetic mean, for example, or the median. The
relation, which we assume, between errors of different magnitudes
and the probabilities that we have made errors of those magnitudes,
is called a law of error. Corresponding to each law of error which
we might assume, there is some function of the measurements which
represents the most probable value of the quantity. The object of the
following paragraphs is to discover what laws of error, if we assume
them, correspond to each of the simple types of average, and to
discover this by means of a systematic method.

4.

Let us assume that the real value of the quantity is either
b1, . . . br . . . bn, and let ar represent the conclusion that the value is,
in fact, br. Further let xr represent the evidence that a measurement
has been made of magnitude yr.

If a measurement yp has been made, what is the probability that
the real value is bs? The application of the theorem of inverse

∗The substance of §§ 3–7 has been printed in the Journal of the Royal

Statistical Society, vol. lxxiv. p. 323 (February 1911).

ch. xvii

FUNDAMENTAL THEOREMS

225
probability yields the following result:

x
a
p/ash
as/h
s/xph = r=n
xp/arh ar/h
r=1

(the number of possible values of the quantity being n), where
h stands for any other relevant evidence which we may have, in
addition to the fact that a measurement xp has been made.

Next, let us suppose that a number of measurements y1 . . . ym have
been made; what is now the probability that the real value is bs? We
require the value of as/x1x2 . . . xmh. As before,
x
a

1 . . . xm/ash
as/h
s/x1x2 . . . xmh =

.

r=x
x1 . . . xm/arh ar/h
r=1

At this point we must introduce the simplifying assumption that,
if we knew the real value of the quantity, the different measurements
of it would be independent, in the sense that a knowledge of what
errors have actually been made in some of the measurements would
not affect in any way our estimate of what errors are likely to be made
in the others. We assume, in fact, that xr/xp . . . xsarh = xr/arh.

This assumption is exceedingly important. It is tantamount to the
assumption that our law of error is unchanged throughout the series
of observations in question. The general evidence h, that is to say,
which justifies our assumption of the particular law of error which we
do assume, is of such a character that a knowledge of the actual errors
made in a number of measurements, not more numerous than those
in question, are absolutely or approximately irrelevant to the question
of what form of law we ought to assume. The law of error which we
assume will be based, presumably, on an experience of the relative
frequency with which errors of different magnitudes have been made
under analogous circumstances in the past. The above assumption
pt. ii

A TREATISE ON PROBABILITY

226
will not be justified if the additional experience, which a knowledge
of the errors in the new measurements would supply, is sufficiently
comprehensive, relatively to our former experience, to be capable of
modifying our assumption as to the shape of the law of error, or if it
suggests that the circumstances, in which the measurements are being
carried out, are not so closely analogous as was originally supposed.

With this assumption, i.e. that x1, etc., are independent of one
another relatively to evidence arh, etc., it follows from the ordinary
rule for the multiplication of independent probabilities that
q=m
x1 . . . xm/ash =
xq/ash.

q=1
q=m
as/h
xq/ash

Hence
q=1
as/x1x2 . . . xmh =

.

r=n
q=m
xq/arh ar/h
r=1
q=1

The most probable value of the quantity under measurement, given
the m measurements y1, etc.—which is our quaesitum—is therefore
that value which makes the above expression a maximum. Since
the denominator is the same for all values of bs, we must find the
value which makes the numerator a maximum. Let us assume that
a1/h = a2/h = . . . = an/h. We assume, that is to say, that we have
no reason à priori (i.e. before any measurements have been made)

for thinking any one of the possible values of the quantity more likely
than any other. We require, therefore, the value of bs which makes
q=m
the expression
xq/ash a maximum. Let us denote this value by y.

q=1

We can make no further progress without a further assumption. Let
us assume that xq/ash—namely, the probability of a measurement yq
assuming the real value to be bs—is an algebraic function f of yq
and bs, the same function for all values of yq and bs within the limits
ch. xvii

FUNDAMENTAL THEOREMS

227
of the problem.∗ We assume, that is to say, xq/ash = f(yq, bs), and
q=m
we have to find the value of bs, namely y, which makes
f (yq, y)

q=1
a maximum. Equating to zero the differential coefficient of this
q=m
expression with respect to
f (y
y, we have
q , y) = 0,† where
q=1 f (yq , y)

df
f =

. This equation may be written for brevity in the form
dy
fq = 0.

fq

If we solve this equation for y, the result gives us the value of the
quantity under observation, which is most probable relatively to the
measurements we have made.

The act of differentiation assumes that the possible values of y
are so numerous and so uniformly distributed within the range
in question, that we may, without sensible error, regard them as
continuous.

5.

This completes the prolegomena of the inquiry.

We
are now in a position to discover what laws of error correspond
to given assumptions respecting the algebraic relation between the
measurements and the most probable value of the quantity, and vice
versa. For the law of error determines the form of f (yq, y). And the
f
form f(y
q
q , y) determines the algebraic relation

= 0 between the
fq
measurements and the most probable value. It may be well to repeat

∗Gauss, in obtaining the normal law of error, made, in effect, the more
special assumption that xq/ash is a function of eq only, where eq is the error
and eq = bs − yq.

We shall find in the sequel that all symmetrical laws of
error, such that positive and negative errors of the same absolute magnitude
are equally likely, satisfy this condition—the normal law, for example, and the
simplest median law. But other laws, such as those which lead to the geometric
mean, do not satisfy it.

†Since none of the measurements actually made can be impossible, none of
the expressions f (yq, y) can vanish.

pt. ii

A TREATISE ON PROBABILITY

228
that f(yq, y) denotes the probability to us that an observer will make
a measurement yq in observing a quantity whose true value we know
to be y. A law of error tells us what this probability is for all possible
values of yq and y within the limits of the problem.

(i.) If the most probable value of the quantity is equal to the
arithmetic mean of the measurements, what law of error does this
imply?

fq = 0 must be equivalent to (y − yq) = 0, since the
fq
q=m
most probable value y must equal 1
yq.

m q=1
f

∴ q = φ (y)(y − yq) where φ (y) is some function which
fq
is not zero and is independent of yq.

Integrating,
log fq =
φ (y)(y − yq) dy + ψ(yq) where ψ(yq) is some
function independent of y.

= φ (y)(y − yq) − φ(y) + ψ(yq).

So that fq = eφ (y)(y−yq)−φ(y)+ψ(yq).

Any law of error of this type, therefore, leads to the arithmetic
mean of the measurements as the most probable value of the quantity
measured.

If we put φ(y) = −k2y2 and ψ(yq) = −k2y2 + log A, we obtain
q
fq = Ae−k2(y−yq)2, the form normally assumed,

= Ae−k2z2q,
where zq is the absolute magnitude of the error in the measurement yq.

This is, clearly, only one amongst a number of possible solutions.

But with one additional assumption we can prove that this is the
ch. xvii

FUNDAMENTAL THEOREMS

229
only law of error which leads to the arithmetic mean. Let us assume
that negative and positive errors of the same absolute amount are
equally likely.

In this case fq must be of the form Beθ(y−yq)2,

∴ φ (y)(y − yq) − φ(y) + ψ(yq) = θ(y − yq)2.

Differentiating with respect to y,
d
φ (y) = 2
θ(y − yq)2.

d(y − yq)2

But φ (y) is, by hypothesis, independent of yq.

d

∴
θ(y − yq)2 = −k2 where k is constant; integrating,
d(y − yq)2
θ(y − yq)2 = −k2(y − yq)2 + log C and we have fq = Ae−k2(y−yq)2 (where

A = BC).

(ii.) What is the law of error, if the geometric mean of the
measurements leads to the most probable value of the quantity?

f
q=m

In this case
q = 0 must be equivalent to
yq = ym, i.e. to
fq
q=1
y
log q = 0.

y

Proceeding as before, we find that the law of error is
yq
f

+
φ (y) dy+ψ(y
y
y
q )

q = Aeφ (y) log

.

There is no solution of this which satisfies the condition that negative
and positive errors of the same absolute magnitude are equally likely.

For we must have
y
φ (y)

φ (y) log q +
dy + ψ(yq) = φ(y − yq)2
y
y
or
y
d
φ (y) log q =
φ(y − yq)2,
y
dy
pt. ii

A TREATISE ON PROBABILITY

230
which is impossible.

The simplest law of error, which leads to the geometric mean,
seems to be obtained by putting φ (y) = −ky, ψ(yq) = 0. This gives
y
ky
fq = A
e−ky.

yq

A law of error, which leads to the geometric mean of the
observations as the most probable value of the quantity, has been
previously discussed by Sir Donald McAlister (Proceedings of the

Royal Society, vol. xxix. (1879) p. 365). His investigation depends
upon the obvious fact that, if the geometric mean of the observations
yields the most probable value of the quantity, the arithmetic mean
of the logarithms of the observations must yield the most probable
value of the logarithm of the quantity. Hence, if we suppose that the
logarithms of the observations obey the normal law of error (which
leads to their arithmetic mean as the most probable value of the
logarithms of the quantity), we can by substitution find a law of error
for the observations themselves which must lead to the geometric
mean of them as the most probable value of the quantity itself.

If, as before, the observations are denoted by yq, etc., and the
quantity by y, let their logarithms be denoted by lq, etc., and by l.

Then, if lq, etc., obey the normal law of error, f(lq, l) = Ae−k2(lq−l)2.

Hence the law of error for yq, etc., is determined by
f (yq, y) = Ae−k2(log yq−log y)2
yq

= Ae−k2(log

)2
y

,
and the most probable value of y must, clearly, be the geometric
mean of yq, etc.

This is the law of error which was arrived at by Sir Donald

McAlister. It can easily be shown that it is a special case of the
generalised form which I have given above of all laws of error leading
to the geometric mean. For if we put ψ(yq) = −k2(log yq)2, and
ch. xvii

FUNDAMENTAL THEOREMS

231
φ (y) = 2k2 log y, we have
yq
f

+ 2k2 log y dy−k2(log y
y
y
q )2
q = Ae2k2 log y log

= Ae2k2 log y log yq−2k2(log y)2+k2(log y)2−k2(log yq)2
yq

= Ae−k2(log

)2
y

.

A similar result has been obtained by Professor J. C. Kapteyn.∗

But he is investigating frequency curves, not laws of error, and
this result is merely incidental to his main discussion. His method,
however, is not unlike a more generalised form of Sir Donald

McAlister’s. In order to discover the frequency curve of certain
quantities y, he supposes that there are certain other quantities z,
functions of the quantities y, which are given by z = F(y), and that
the frequency curve of these quantities z is normal. By this device
he is enabled in the investigation of a type of skew frequency curve,
which is likely to be met with often, to utilise certain statistical
constants corresponding to those which have been already calculated
for the normal curve.

In fact the main advantage both of Sir Donald McAlister’s
law of error and of Professor Kapteyn’s frequency curves lies in
the possibility of adapting without much trouble to unsymmetrical
phenomena numerous expressions which have been already calculated
for the normal law of error and the normal curve of frequency.†

This method of proceeding from arithmetic to geometric laws
of error is clearly capable of generalisation. We have dealt with
the geometric law which can be derived from the normal arithmetic
law.

Similarly if we start from the simplest geometric law of

∗Skew Frequency Curves, p. 22, published by the Astronomical Laboratory
at Groningen (1903).

†It may be added that Professor Kapteyn’s monograph brings forward
considerations which would be extremely valuable in determining the types of
phenomena to which geometric laws of error are likely to be applicable.

pt. ii

A TREATISE ON PROBABILITY

232
k2y
error, namely,
y
fq = A
e−k2y, we can easily find, by writing
yq
log y = l and log yq = lq, the corresponding arithmetic law, namely,
fq = Aek2el(l−lq)−k2el, which is obtained from the generalised arithmetic
law by putting φ(l) = k2el and ψ(lq) = 0.

And, in general,
corresponding to the arithmetic law
fq = Aeφ (y)(y−yq)−φ(y)+ψ(yq),
we have the geometric law
zq
φl(z)

f

(z) log

+
dz+ψl(zq)

q = Aeφl
z
z

,
where
φ (z)

y = log z, y

1
q = log zq ,
dz = φ(log z) and ψl(zq) = ψ(log zq).

z

(iii.) What law of error does the harmonic mean imply?

f

In this case,
q

1

= 0 must be equivalent to

1 −

= 0.

fq
yq
y

Proceeding as before, we find that
φ (y)

1 − 1 − φ (y) dy+ψ(y
f
y
q )

q
y
y2
q = Ae

.

A simple form of this is obtained by putting φ (y) = −k2y2 and
k2
z2

(y−y

−k2 q
ψ(y
y
q )2
q
yq
q ) = −k2yq .

Then fq = Ae

= Ae

. With this law,
positive and negative errors of the same absolute magnitude are not
equally likely.

(iv.) If the most probable value of the quantity is equal to the
median of the measurements, what is the law of error?

The median is usually defined as the measurement which occupies
the middle position when the measurements are ranged in order of
magnitude. If the number of measurements m is odd, the most
probable value of the quantity is the m + 1th, and, if the number

2
is even, all values between the mth and the m + 1 th are equally

2

2
ch. xvii

FUNDAMENTAL THEOREMS

233
probable amongst themselves and more probable than any other. For
the present purpose, however, it is necessary to make use of another
property of the median, which was known to Fechner (who first
introduced the median into use) but which seldom receives as much
attention as it deserves. If y is the median of a number of magnitudes,
the sum of the absolute differences (i.e. the difference always reckoned
positive) between y and each of the magnitudes is a minimum. The
m
median y of y1y2 . . . ym is found, that is to say, by making

|yq − y|

1
a minimum where |yq − y| is the difference always reckoned positive
between yq and y.

We can now return to the investigation of the law of error
corresponding to the median.

m

Write |y − yq| = zq. Then since
zq is to be a minimum we must

1
m
have
y − yq = 0. Whence, proceeding as before, we have

1
zq
y−yq φ (y) dy+ψ(y
f
z
q ).

q
q = Ae

The simplest case of this is obtained by putting
φ (y) = −k2,
y − y
ψ(y
q
q ) =
k2yq,
zq
whence
fq = Ae−k2|y−yq| = Ae−k2zq .

This satisfies the additional condition that positive and negative
errors of equal magnitude are equally likely. Thus in this important
respect the median is as satisfactory as the arithmetic mean, and the
law of error which leads to it is as simple. It also resembles the
normal law in that it is a function of the error only, and not of the
magnitude of the measurement as well.

pt. ii

A TREATISE ON PROBABILITY

234

The median law of error, fq = Ae−k2zq, where zq is the absolute
amount of the error always reckoned positive, is of some historical
interest, because it was the earliest law of error to be formulated.

The first attempt to bring the doctrine of averages into definite
relation with the theory of probability and with laws of error was
published by Laplace in 1774 in a memoir “sur la probabilité des
causes par les événemens.”∗

This memoir was not subsequently
incorporated in his Théorie analytique, and does not represent his
more mature view. In the Théorie he drops altogether the law
tentatively adopted in the memoir, and lays down the main lines of
investigation for the next hundred years by the introduction of the
normal law of error. The popularity of the normal law, with the
arithmetic mean and the method of least squares as its corollaries, has
been very largely due to its overwhelming advantages, in comparison
with all other laws of error, for the purposes of mathematical
development and manipulation. And in addition to these technical
advantages, it is probably applicable as a first approximation to a
larger and more manageable group of phenomena than any other
single law. So powerful a hold indeed did the normal law obtain
on the minds of statisticians, that until quite recent times only a
few pioneers have seriously considered the possibility of preferring in
certain circumstances other means to the arithmetic and other laws
of error to the normal. Laplace’s earlier memoir fell, therefore, out of
remembrance. But it remains interesting, if only for the fact that a
law of error there makes its appearance for the first time.

Laplace sets himself the problem in a somewhat simplified form:

“Déterminer le milieu que l’on doit prendre entre trois observations
données d’un même phénomène.” He begins by assuming a law of
error z = φ(y), where z is the probability of an error y; and finally, by
means of a number of somewhat arbitrary assumptions, arrives at the
result
m
φ(y) =
e−my. If this formula is to follow from his arguments,

2

∗Mémoires présentés à l’Académie des Sciences, vol. vi.

ch. xvii

FUNDAMENTAL THEOREMS

235
y must denote the absolute error, always taken positive. It is not
unlikely that Laplace was led to this result by considerations other
than those by which he attempts to justify it.

Laplace, however, did not notice that his law of error led to the
median. For, instead of finding the most probable value, which would
have led him straight to it, he seeks the “mean of error”—the value,
that is to say, which the true value is as likely to fall short of as
to exceed. This value is, for the median law, laborious to find and
awkward in the result. Laplace works it out correctly for the case
where the observations are no more than three.

6.

I do not think that it is possible to find by this method a law
of error which leads to the mode. But the following general formulae
are easily obtained:

(v.) If
θ(yq, y) is the law of relation between the measurements
and the most probable value of the quantity, then the law of error
fq(yq, y) is given by fq = Ae θ(yqy)φ (y) dy+ψ(yq). Since fq lies between

0 and 1,
θ(yqy)φ (y) dy + ψ(yq) + log A must be negative for all
values of yq and y that are physically possible; and, since the values
of yq are between them exhaustive,

Ae θ(yqy)φ (y) dy+ψ(yq) = 1,
where the summation is for all terms that can be formed by giving yq
every value à priori possible.

(vi.) The most general form of the law of error, when it is
assumed that positive and negative errors of the same magnitude are
equally probable, is Ae−k2f(y−yq)2, where the most probable value of
the quantity is given by the equation
d

(y − yq)f (y − yq)2 = 0, where f (y − yq)2 =
f (y − yq)2.

d(y − yq)2

The arithmetic mean is a special case of this obtained by putting
f (y − yq)2 = (y − yq)2; and the median is a special case obtained by
putting f(y − yq)2 = + (y − yq)2.

pt. ii

A TREATISE ON PROBABILITY

236

We can obtain other special cases by putting
f (y − yq)2 = (y − yq)4,
when the law of error is Ae−k2(y−yq)2 and the most probable values
are the roots of my3 − 3y2
yq + 3y
y2 −
y3 = 0; and by putting
q
q
f (y − yq)2 = log(y − yq)2, when the law of error is

A

= 0 and

(y − yq)

the most probable values the roots of

1

= 0. In all these

(y − yq)

cases the law is a function of the error only.

7.

These results may be summarised thus. We have assumed:

(a) That we have no reason, before making measurements, for
supposing that the quantity we measure is more likely to have any
one of its possible values than any other.

(b) That the errors are independent, in the sense that a knowledge
of how great an error has been made in one case does not affect our
expectation of the probable magnitude of the error in the next.

(c) That the probability of a measurement of given magnitude,
when in addition to the à priori evidence the real value of the
quantity is supposed known, is an algebraic function of this given
magnitude of the measurement and of the real value of the quantity.

(d) That we may regard the series of possible values as continuous,
without sensible error.

(e) That the à priori evidence permits us to assume a law of error
of the type specified in (c); i.e. that the algebraic function referred to
in (c) is known to us à priori.

Subject to these assumptions, we have reached the following
conclusions:

(1) The most general form of the law of error is
fq = Ae φ (y)θ(yqy) dy+ψ(yq),
leading to the equation
θ(yqy) = 0, connecting the most probable
value and the actual measurements, where y is the most probable
value and yq, etc., the measurements.

ch. xvii

FUNDAMENTAL THEOREMS

237

(2) Assuming that positive and negative errors of the same
absolute magnitude are equally likely, the most general form is
fq = Ae−k2f(y−yq)2, leading to the equation

(y − yq)f (y − yq)2 = 0,
where
d
f z =
f z. Of the special cases to which this form gives rise,
dz
the most interesting were

(3) fq = Ae−k2(y−yq)2 = Ae−k2z2q, where zq = |y − yq|, leading to the
arithmetic mean of the measurements as the most probable value of
the quantity; and

(4) fq = Ae−k2zq, leading to the median.

(5) The most general form leading to the arithmetic mean is
fq = Aeφ (y−yq)−φ(y)+ψ(yq), with the special cases (3), and

(6) fq = Aek2ey(y−yq)−k2ey.

(7) The most general form leading to the geometric mean is
yq
f

+
φ (y) dy+ψ(y
y
y
q )

q = Aeφ (y) log

, with the special cases:

k2y

(8) fq = A y
e−k2y, and
yq
y

(9)

q
f

)2
y
q = Ae−k2(log

.

(10) The most general form leading to the harmonic mean is
φ (y)

1 − 1 − φ (y dy+ψ(y
f
y
q )

q
y
y2
q = Ae

, with the special case
z2
q

(11)

−k2 (y−yq)2

−k2
f
yq
yq
q = Ae

= Ae

.

(12) The most general form leading to the median is
y−y
φ (y)

q +ψ(y
f
z
q )

q
q = Ae

,
with the special case (4).

In each of these expressions, fq is the probability of a measure-
ment yq, given that the true value is y.

8.

The doctrine of Means and the allied theory of Least Squares
comprise so extensive a subject-matter that they cannot be adequately
treated except in a volume primarily devoted to them. As, however,
they are one of the important practical applications of the theory of
pt. ii

A TREATISE ON PROBABILITY

238
probability, I am unwilling to pass them by entirely; and the following
discursive observations, chiefly relating to the Normal Law of Error,
will serve, taken in conjunction with the paragraphs immediately
preceding, to illustrate the connection between the theories of this
treatise and the general treatment of averages.

9.

The Claims of the Arithmetic Average.—By definition the
arithmetic average of a number of quantities is nothing more than
their arithmetic sum divided by their number. But the utility of
an average generally consists in our supposed right to substitute, in
certain cases, this single measure for the varying measures of which
it is a function. Sometimes this requires no justification; the word

“average” is in these cases used for the sake of shortness, and merely
to summarise a set of facts: as, for instance, when we say that the
birth-rate in England is greater than the birth-rate in France.

But there are other cases in which the average makes a more
substantial claim to add to our knowledge. After a number of
examiners of equal capacity have given varying marks to a candidate
for the same paper, it may be thought fair to allow the candidate
the average of the different marks allotted: and in general if several
estimates of a magnitude have been made, between the accuracy of
which we have no reason to discriminate, we often think it reasonable
to act as if the true magnitude were the average of the several
measurements. Perhaps De Witt, in his report on Annuities to the

States General in 1671,∗ was the first to use it scientifically. But as

Leibniz points out: “Our peasants have made use of it for a long time
according to their natural mathematics. For example, when some
inheritance or land is to be sold, they form three bodies of appraisers;
these bodies are called Schurzen in Low Saxon, and each body makes
an estimate of the property in question. Suppose, then, that the
first estimates its value to be 1000 crowns, the second, 1400, the
third, 1500; the sum of these three estimates is taken, viz. 3900, and

∗De vardye van de lif-renten na proportie van de los-renten. The Hague,

1671.

ch. xvii

FUNDAMENTAL THEOREMS

239
because they were three bodies, the third, i.e. 1300, is taken as the
mean value asked for. This is the axiom: aequalibus aequalia, equal
suppositions must have equal consideration.”∗

But this is a very inadequate axiom. Equal suppositions would
have equal consideration, if the three estimates had been multiplied
together instead of being added. The truth is that at all times the
arithmetic mean has had simplicity to recommend it. It is always
easier to add than to multiply. But simplicity is a dangerous criterion:

“La nature,” says Fresnel, “ne s’est pas embarassée des difficultés
d’analyse, elle n’a évité que la complication des moyens.”

With Laplace and Gauss there began a series of attempts to
prove the worth of the arithmetic mean.

It was discovered that
its use involved the assumption of a particular type of law of error
for the à priori probabilities of given errors. It was also found
that the assumption of this law led on to a more complicated rule,
known as the Method of Least Squares for combining the results of
observations which contain more than one doubtful quantity. In spite
of a popular belief that, whilst the Arithmetic Mean is intuitively
obvious, the Method of Least Squares depends upon doubtful and
arbitrary assumptions, it can be demonstrated that the two stand
and fall together.†

The analytical theorems of Laplace and Gauss are complicated, but
the special assumptions upon which they are based are easily stated.‡

∗Nouveaux Essais. Engl. transl. p. 540.

†Venn (Logic of Chance, p. 40) thinks that the Normal Law of Error and
the Method of Least Squares “are not only totally distinct things, but they have
scarcely even any necessary connection with each other. The Law of Error is
the statement of a physical fact. . . . The Method of Least Squares, on the other
hand, is not a law at all in the scientific sense of the term. It is simply a rule
or direction. . . .”

‡For an account of the three principal methods of arriving at the Method of

Least Squares and the Arithmetic Mean, see Ellis, Least Squares. Gauss’s first
method is in the Theoria Motus, and his second in the Theoria Combinationis

Observationum.

Laplace’s investigations are in chap. iv. of the second Book
pt. ii

A TREATISE ON PROBABILITY

240

Gauss supposes (a) that the probability of a given error is a function
of the error only and not also of the magnitude of the observation,

(b) that the errors are so small that their cubes and higher powers
may be neglected. Assumption (a) is arbitrary,∗ and Gauss did not
state it explicitly. These two assumptions, together with certain
others, lead us to the result. For let φ(z) be the law of error where
z is the error, and let us assume, as it always is assumed in these
proofs, that φ(z) can be expanded by Maclaurin’s Theorem. Then
z2
z3
φ(x) = φ(0) + zφ (0) +
φ (0) +
φ (0) + . . .. It is also supposed that

2!

3!

positive and negative errors are equally probable, i.e. φ(z) = φ(−z),
so that φ (0) and φ (0) vanish. Since we may neglect z4 in comparison
with z2, φz = φ(0) + 1z2φ (0). But (neglecting z4 and higher powers)

2
bz2
bz2
a + bz2 = ae a , so that φ(z) = ae a .

Gauss’s proof looks much more complicated than this, but he
obtains the form
bz2
ae a by neglecting higher powers of z, so that this
expression is really equivalent to a + bz2. By this approximation he
has reduced all the possible laws to an equivalent form.† It is true,
therefore, that the normal law of error is, to the second power of the
error, equivalent to any law of error, which is a function of the error
only, and for which positive and negative errors are equally probable.

Laplace also introduces assumptions equivalent to these.

While mathematicians have endeavoured to establish the normal
law of error and the arithmetic mean as a law of logic, others have
claimed for it the testimony of experience and have deemed it a law
of nature.‡
of the Théorie analytique.

Laplace’s method was improved by Poisson in the

Connaissance des temps for 1827 and 1832.

∗It does not follow, as G. Hagen argues (Grundzüge der Wahrscheinlichkeit-
srechnung, p. 29), that, because a larger error is less probable than a smaller,
therefore the probability of a given error is a function of its magnitude only.

†This is pointed out by Bertrand, Calcul des probabilités, p. 267.

‡This is, of course, a very common point of view indeed. Cf. Bertrand,
ch. xvii

FUNDAMENTAL THEOREMS

241

That this cannot be so, is evident. For suppose that x1x2 . . . xn
are a set of observations of an unknown quantity x. Then, by
this principle,

1
x =
xr gives the most probable value of x. But
n
suppose we had wished to determine x2, our observations, assuming
that we can multiply correctly, would be x2, x2 . . . x2, and the most

1

2
n
probable value of

1

1

1
x2 =
x2. But (
xr)2 =
x2. And in
n
r
n
n
r
general, 1

1
f (xr) =

| f (
xr). Nor is this a consideration which
n
n
can safely be ignored in practice. For our “observations” are often the
result of some manipulation, and the particular shape in which we
get them is not necessarily fixed for us. It is not easy to say what
the direct observation is. In particular if any such law of sensation,
as that enunciated by Fechner, is true (i.e. that sensation varies as
the logarithm of the stimulus), the arithmetic mean must break down
as a practical rule in all cases where human sensation is part of the
instrument by means of which the observations are recorded.∗

Apart, however, from theoretical refutations, statisticians now
recognise that the arithmetic mean and the normal law of error can
only be applied to certain special classes of phenomena. Quetelet†
was, I think, the first to point this out. In England, Galton drew
attention to the fact many years ago, and Professor Pearson‡ has
shown “that the Gaussian-Laplace normal distribution is very far
from being a general law of frequency distribution either for errors
of observation or for the distribution of deviations from type such as
occur in organic populations. . . . It is not even approximately correct,
for example, in the distribution of barometric variations, of grades of
op. cit. p. 183: “Malgré les objections précédentes, la formule de Gauss doit être
adoptée. L’observation la confirme: cela doit suffire dans les applications.”

∗This was noticed by Galton.

†E.g. Letters on the Theory of Probabilities, p. 114.

‡On “Errors of Judgment, etc.,” Phil. Trans. A, vol. cxcviii. pp. 235–299.

The following quotation is from his memoir On the General Theory of Skew

Correlation and Non-linear Regression, where further references are given.

pt. ii

A TREATISE ON PROBABILITY

242
fertility and incidence of disease.”

The Arithmetic Mean occupies, therefore, no unique position; and
it is worth while, from the point of view of probability, to discuss the
properties of other possible means and laws of error, as, for example,
on the lines indicated in the earlier part of this chapter.

10.

The Method of Least Squares.—The problem, to which
this method is applied, is no more than the application of the
same considerations, as those which we have just been discussing, to
cases where the relation between the observed measurements and the
quantity whose most probable value we require, involves more than
one unknown.

Owing to the surprising character of its conclusions, if they
could be accepted as universally valid, and to the obscurity of the
mathematical fabric that has been reared on and about it, this
method has been surrounded by an unnecessary air of mystery. It
is true that in recent times scepticism has grown at the expense of
mystery. It is also true that just views have been held by individuals
for sixty years past, notably by Leslie Ellis. But the old mistakes are
not always corrected in the current text-books, and even so useful
and generally used a treatise on Least Squares, as Professor Mansfield

Merriman’s, opens with a series of very fallacious statements.

The controversial side of the Method of Least Squares is purely
logical; in the later developments there is much elaborate mathematics
of whose correctness no one is in doubt. What it is important to
state with the utmost possible clearness is the precise assumptions on
which the mathematics is based; when these assumptions have been
set forth, it remains to determine their applicability in particular
cases.

In dealing with averages we supposed ourselves to be presented
with a number of direct observations of some quantity which it is
desired to determine. But it is obvious that direct observations will be
in many cases either impracticable or inconvenient; and our natural
course will be to measure certain other quantities which we know
ch. xvii

FUNDAMENTAL THEOREMS

243
to bear fixed and invariable relations to the unknowns we wish to
determine. In surveying, for instance, or in astronomy, we constantly
prefer to take measurements of angles or distances in which we are
not interested for their own sakes, but which bear known geometrical
relationships to the set of ultimate unknowns.

If we wish to determine the most probable values of a set of
unknowns x1, x2, . . . xr, instead of obtaining a number of sets of
direct observations of each, we may obtain a number of equations of
observation of the following type:

a1x1 + a2x2 + . . . + arxr = V1,
b1x1 + b2x2 + . . . + brxr = V2,

.

.

.

.

.

.

.

k1x1 + k2x2 + . . . + krxr = Vn,
where V1, etc., are the quantities directly observed, and the a s,
b s, etc., are supposed known (n > r).

We have in such a case n equations to determine r unknowns, and
since the observations are likely to be inexact, there may be no precise
solution whatever. In these circumstances we wish to know the most
probable set of values of the x’s warranted by these observations.

The problem is precisely similar in kind to that dealt with by
averages and differs only in the degree of its complexity. It is
the problem of finding the most probable solution of such a set of
discrepant equations of observation that the Method of Least Squares
claims to solve.

By 1750 the astronomers were obtaining such equations of
observation in the course of their investigations, and the question
arose as to the proper manner of their solution. Boscovich in Italy,

Mayer and Lambert in Germany, Laplace in France, Euler in Russia,
and Simpson in England proposed different methods of solution.

Simpson, in 1757, was the first to introduce, by way of simplification,
the assumption or axiom that positive and negative errors are equally
pt. ii

A TREATISE ON PROBABILITY

244
probable.∗ The Method of Least Squares was first definitely stated
by Legendre in 1805, who proposed it as an advantageous method
of adjusting observations. This was soon followed by the ‘proofs’ of

Laplace and Gauss. But it is easily shown that these proofs involve
the normal law of error y = ke−k2x2, and the theory of Least Squares
simply develops the mathematical results of applying to equations of
observation, which involve more than one unknown, that law of error
which leads to the Arithmetic Mean in the case of a single unknown.

11.

The Weighting of Averages.—It is necessary to recur to the
distinction made at the beginning of § 9 between the two types to
which our average, or, as it is generally termed in social inquiries,
our index number, may belong. The average or index number may
simply summarise a set of facts and give us the actual value of a
composite quantity, as, for example, the index number of the cost
of living. In such cases the composite quantity, in which we are
interested, need not contain precisely the same number of units of
each of the elementary quantities of which it is composed, so that
the ‘weights,’ which denote the numbers of each elementary quantity
appropriate to the composite quantity, are part of the definition of
the composite quantity, and can no more be dispensed with than
the magnitudes of the elementary quantities themselves. Nor in such
cases is the rejection of discordant observations permissible; if, that is
to say, some of the elementary quantities are subject to much wider
variation, or to variations of a different type than the majority, that
is no reason for rejecting them.

On the other hand, the individual items, out of which the average
is composed, may each be indications or approximate estimates of
some one single quantity; and the average, instead of representing

∗See Merriman’s Method of Least Squares, p. 181, for an historical sketch,
from which the above is taken. In 1877 Merriman published in the Transactions
of the Connecticut Academy a list of writings relating to the Method of Least

Squares and the theory of accidental errors of observation, which comprised

408 titles—classified as 313 memoirs, 72 books, 23 parts of books.

ch. xvii

FUNDAMENTAL THEOREMS

245
the measure of a composite quantity, may be selected as furnishing
the most probable value of the single quantity, given, as evidence of
its magnitude, the values of the various terms which make up the
average.

If this is the character of our average, the problem of weighting
depends upon what we know about the individual observations or
samples or indications, out of which our average is to be built up.

The units in question may be known to differ in respects relevant to
the probable value of the quaesitum. Thus there may be reasons,
quite apart from the actual results of the individual observations or
samples, for trusting some of them more than others. Our knowledge
may indicate to us, in fact, that the constants of the laws of error
appropriate to the several instances, even if the type of the law can
be assumed to be constant, should be varied according to the data
we possess about each. It may also indicate to us that the condition
of independence between the instances, which the method of averages
presumes, is imperfectly satisfied, and consequently that our mode of
combining the instances in an average must be modified accordingly.

Some modern statisticians, who, really influenced perhaps by prac-
tical considerations, have been inclined to deprecate the importance
of weighting on theoretical grounds, have not always been quite clear
what kind of average they supposed themselves to be dealing with.

In particular, discussions of the question of weighting in connection
with index numbers of the value of money have suffered from this
confusion.

It has not been clear whether such index numbers
really represent measures of a composite quantity or whether they
are probable estimates of the value of a single quantity formed by
combining a number of independent approximations towards the value
of this quantity. The original Jevonian conception of an index number
of the value of money was decidedly of the latter type. Modern
work on the subject has been increasingly dominated by the other
conception. A discussion of where the truth lies would lead me too
far into the field of a subject-matter alien to that of this treatise.

pt. ii

A TREATISE ON PROBABILITY

246

Theoretical arguments against weighting have sometimes been
based on the fact that to weight the items of the average in an
irrelevant manner, or, as it is generally expressed, in a random
manner, is not likely, provided the variations between the weights
are small compared with the variations between the items, to affect
the result very much. But why should any one wish to weight an
average “at random”? Such observations overlook the real meaning
and significance of weights. They are probably inspired by the fact
that a superficial treatment of statistics would sometimes lead to the
introduction of weights which are irrelevant. In drawing a conclusion,
for example, from the vital statistics of various towns, the figures of
population for the different towns may or may not be relevant to our
conclusion. It depends on the character of the argument. If they are
relevant, it may be right to employ them as weights. If they are
irrelevant, it must be wrong and unnecessary to do so. The fact that
what is a more important article of consumption than pins may, on
certain assumptions, be irrelevant to the usefulness of variations in the
price of each article as indications of variation in the value of money.

With other assumptions, it may be extremely relevant. Or again, we
may know that observations with a particular instrument tend to be
too large and must, therefore, be weighted down. It is contrary both
to theory and to common sense to suppose that the possession of
information as to the relative reliability of different statistics is not
useful. There is no place, therefore, in my judgment, for a generalised
argument as to the propriety or impropriety of weighting an average.

It should be added that, where we seek to build up an index
number of a conception, which is quantitative but is not itself
numerically measurable in any defined or unambiguous sense, by
combining a number of numerical quantities, which, while they do not
measure our quaesitum are nevertheless indications of its quantitative
variations and tend to fluctuate in the same sense, as, for example,
by means of what are sometimes called economic barometers of the
state of business, or the prosperity of the country or the like, some
ch. xvii

FUNDAMENTAL THEOREMS

247
very confusing questions can arise both as to what sort of a thing
our resulting index really is, and as to the mode of compilation
appropriate to it.

These confusing questions always arise when, instead of measuring
a quantity directly, we seek an index to fluctuations in its magnitude
by combining in an average the fluctuations of a series of magnitudes,
which are, each of them in a different way, to some extent (but only
to some extent), correlated with fluctuations in our quaesitum. I
must not burden this book with a discussion of the problems of Index

Numbers. But I venture to think that they would be sooner cleared
up if the natures and purposes of differing index numbers were more
sharply distinguished—those, namely, which are simply descriptive
of a composite commodity, those which seek to combine results
differing from one another in a way analogous to the variations of an
instrument of precision, and those which combine results, not of the
quaesitum itself, but of various other quantities, variations in which
are partly due to variations in the quaesitum, but which we well know
to be also due to other distinguishable influences. Index numbers
of the third type are often treated by methods and arguments only
appropriate to those of the second type.

12.

The Rejection of Discordant Observations.—This differs
from the problem just discussed, because we have supposed so far
that our system of weighting is determined by data which we possess
prior to and apart from our knowledge of the actual magnitude of
the items of our average. The principle of the rejection of discordant
observations comes in when it is argued that, if one or more of our
observations show great discrepancies from the results of the greater
number, these ought to be partly or entirely neglected in striking the
average, even if there is no reason, except their discrepancy from the
rest, for attributing less weight to them than to the others. By some
this practice has been thought to be in accordance with the dictates
of common sense; by others it is denounced as savouring even of
pt. ii

A TREATISE ON PROBABILITY

248
forgery.∗

This controversy, like so many others in Probability, is due
to a failure to understand the meaning of ‘independence.’ The
mathematics of the orthodox theory of Averages and Least Squares
depend, as we have seen, upon the assumption that the observations
are ‘independent’; but this has sometimes been interpreted to mean
a physical independence. In point of fact, the theory requires that
the observations shall be independent, in the sense that a knowledge
of the result of some does not affect the probability that the others,
when known, involve given errors.

Clearly there may be initial data in relation to which this
supposition is entirely or approximately accurate. But in many cases
the assumption will be inadmissible. A knowledge of the results of a
number of observations may lead us to modify our opinion as to the
relative reliabilities of others.

The question, whether or not discordant observations should be
specially weighted down, turns, therefore, upon the nature of the
preliminary data by which we have been guided in initially adopting
a particular law of error as appropriate to the observations. If the
observations are, relevant to these data, strictly ‘independent,’ in the
sense required for probability, then rejection is not permissible. But
if this condition is not fulfilled, a bias against discordant observations
may be well justified.

∗E.g. G. Hagen’s Grundzüge der Wahrscheinlichkeitsrechnung, p. 63: “Die

Täuschung, die man durch Verschweigen von Messungen begeht, lässt sich eben
so wenig entschuldigen, als wenn man Messungen fälschen oder fingiren wollte.”

PART III
induction and analogy

CHAPTER XVIII
introduction

Nothing so like as eggs; yet no one, on account of this apparent similarity,
expects the same taste and relish in all of them. ’Tis only after a long course
of uniform experiments in any kind, that we attain a firm reliance and security
with regard to a particular event. Now where is that process of reasoning, which
from one instance draws a conclusion, so different from that which it infers from
a hundred instances, that are no way different from that single instance? This
question I propose as much for the sake of information, as with any intention of
raising difficulties. I cannot find, I cannot imagine any such reasoning. But I
keep my mind still open to instruction, if any one will vouchsafe to bestow it
on me.—Hume.∗

1.

I have described Probability as comprising that part of logic
which deals with arguments which are rational but not conclusive. By
far the most important types of such arguments are those which are
based on the methods of Induction and Analogy. Almost all empirical
science rests on these. And the decisions dictated by experience in the
ordinary conduct of life generally depend on them. To the analysis
and logical justification of these methods the following chapters are
directed.

Inductive processes have formed, of course, at all times a vital,
habitual part of the mind’s machinery.

Whenever we learn by
experience, we are using them. But in the logic of the schools they
have taken their proper place slowly. No clear or satisfactory account
of them is to be found anywhere. Within and yet beyond the scope
of formal logic, on the line, apparently, between mental and natural
philosophy, Induction has been admitted into the organon of scientific
proof, without much help from the logicians, no one quite knows

∗Philosophical Essays concerning Human Understanding.

250
ch. xviii

INDUCTION AND ANALOGY

251
when.

2.

What are its distinguishing characteristics? What are the
qualities which in ordinary discourse seem to afford strength to an
inductive argument?

I shall try to answer these questions before I proceed to the
more fundamental problem—What ground have we for regarding such
arguments as rational?

Let the reader remember, therefore, that in the first of the
succeeding chapters my main purpose is no more than to state in
precise language what elements are commonly regarded as adding
weight to an empirical or inductive argument. This requires some
patience and a good deal of definition and special terminology. But

I do not think that the work is controversial. At any rate, I am
satisfied myself that the analysis of Chapter XIX. is fairly adequate.

In the next section, Chapters XX. and XXI., I continue in part
the same task, but also try to elucidate what sort of assumptions, if
we could adopt them, lie behind and are required by the methods
just analysed. In Chapter XXII. the nature of these assumptions is
discussed further, and their possible justification is debated.

3.

The passage quoted from Hume at the head of this chapter
is a good introduction to our subject. Nothing so like as eggs, and
after a long course of uniform experiments we can expect with a firm
reliance and security the same taste and relish in all of them. The
eggs must be like eggs, and we must have tasted many of them. This
argument is based partly upon Analogy and partly upon what may
be termed Pure Induction. We argue from Analogy in so far as we
depend upon the likeness of the eggs, and from Pure Induction when
we trust the number of the experiments.

It will be useful to call arguments inductive which depend in any
way on the methods of Analogy and Pure Induction. But I do not
mean to suggest by the use of the term inductive that these methods
are necessarily confined to the objects of phenomenal experience and
to what are sometimes called empirical questions; or to preclude from
pt. iii

A TREATISE ON PROBABILITY

252
the outset the possibility of their use in abstract and metaphysical
inquiries. While the term inductive will be employed in this general
sense, the expression Pure Induction must be kept for that part of
the argument which arises out of the repetition of instances.

4.

Hume’s account, however, is incomplete. His argument
could have been improved. His experiments should not have been
too uniform, and ought to have differed from one another as much
as possible in all respects save that of the likeness of the eggs. He
should have tried eggs in the town and in the country, in January
and in June. He might then have discovered that eggs could be good
or bad, however like they looked.

This principle of varying those of the characteristics of the
instances, which we regard in the conditions of our generalisation as
non-essential, may be termed Negative Analogy.

It will be argued later on that an increase in the number of
experiments is only valuable in so far as, by increasing, or possibly
increasing, the variety found amongst the non-essential characteristics
of the instances, it strengthens the Negative Analogy. If Hume’s
experiments had been absolutely uniform, he would have been right to
raise doubts about the conclusion. There is no process of reasoning,
which from one instance draws a conclusion different from that
which it infers from a hundred instances, if the latter are known to
be in no way different from the former. Hume has unconsciously
misrepresented the typical inductive argument.

When our control of the experiments is fairly complete, and the
conditions in which they take place are well known, there is not much
room for assistance from Pure Induction. If the Negative Analogies
are known, there is no need to count the instances. But where our
control is incomplete, and we do not know accurately in what ways
the instances differ from one another, then an increase in the mere
number of the instances helps the argument. For unless we know for
certain that the instances are perfectly uniform, each new instance
may possibly add to the Negative Analogy.

ch. xviii

INDUCTION AND ANALOGY

253

Hume might also have weakened his argument. He expects no
more than the same taste and relish from his eggs. He attempts no
conclusion as to whether his stomach will always draw from them the
same nourishment. He has conserved the force of his generalisation
by keeping it narrow.

5.

In an inductive argument, therefore, we start with a number
of instances similar in some respects AB, dissimilar in others C. We
pick out one or more respects A in which the instances are similar,
and argue that some of the other respects B in which they are
also similar are likely to be associated with the characteristics A
in other unexamined cases. The more comprehensive the essential
characteristics A, the greater the variety amongst the non-essential
characteristics C, and the less comprehensive the characteristics B
which we seek to associate with A, the stronger is the likelihood or
probability of the generalisation we seek to establish.

These are the three ultimate logical elements on which the
probability of an empirical argument depends,—the Positive and the

Negative Analogies and the scope of the generalisation.

6.

Amongst the generalisations arising out of empirical
argument we can distinguish two separate types.

The first of
these may be termed universal induction. Although such inductions
are themselves susceptible of any degree of probability, they affirm
invariable relations. The generalisations which they assert, that is to
say, claim universality, and are upset if a single exception to them
can be discovered. Only in the more exact sciences, however, do we
aim at establishing universal inductions. In the majority of cases
we are content with that other kind of induction which leads up to
laws upon which we can generally depend, but which does not claim,
however adequately established, to assert a law of more than probable
connection.∗ This second type may be termed Inductive Correlation.

If, for instance, we base upon the data, that this and that and those

∗What Mill calls ‘approximate generalisations.’

pt. iii

A TREATISE ON PROBABILITY

254
swans are white, the conclusion that all swans are white, we are
endeavouring to establish a universal induction. But if we base upon
the data that this and those swans are white and that swan is black,
the conclusion that most swans are white, or that the probability of
a swan’s being white is such and such, then we are establishing an
inductive correlation.

Of these two types, the former—universal induction—presents
both the simpler and the more fundamental problem. In this part of
my treatise I shall confine myself to it almost entirely. In Part V.,
on the Foundations of Statistical Inference, I shall discuss, so far as I
can, the logical basis of inductive correlation.

7.

The fundamental connection between Inductive Method and

Probability deserves all the emphasis I can give it. Many writers,
it is true, have recognised that the conclusions which we reach
by inductive argument are probable and inconclusive. Jevons, for
instance, endeavoured to justify inductive processes by means of the
principles of inverse probability. And it is true also that much of
the work of Laplace and his followers was directed to the solution of
essentially inductive problems. But it has been seldom apprehended
clearly, either by these writers or by others, that the validity of every
induction, strictly interpreted, depends, not on a matter of fact, but
on the existence of a relation of probability. An inductive argument
affirms, not that a certain matter of fact is so, but that relative to
certain evidence there is a probability in its favour. The validity of
the induction, relative to the original evidence, is not upset, therefore,
if, as a fact, the truth turns out to be otherwise.

The clear apprehension of this truth profoundly modifies our
attitude towards the solution of the inductive problem. The validity
of the inductive method does not depend on the success of its
predictions. Its repeated failure in the past may, of course, supply
us with new evidence, the inclusion of which will modify the force
of subsequent inductions. But the force of the old induction relative
to the old evidence is untouched.

The evidence with which our
ch. xviii

INDUCTION AND ANALOGY

255
experience has supplied us in the past may have proved misleading,
but this is entirely irrelevant to the question of what conclusion
we ought reasonably to have drawn from the evidence then before
us. The validity and reasonable nature of inductive generalisation is,
therefore, a question of logic and not of experience, of formal and not
of material laws. The actual constitution of the phenomenal universe
determines the character of our evidence; but it cannot determine
what conclusions given evidence rationally supports.

CHAPTER XIX
the nature of argument by analogy

All kinds of reasoning from causes or effects are founded on two particulars,
viz. the constant conjunction of any two objects in all past experience,
and the resemblance of a present object to any of them.

Without some
degree of resemblance, as well as union, ’tis impossible there can be any
reasoning.—Hume.∗

1.

Hume rightly maintains that some degree of resemblance must
always exist between the various instances upon which a generalisation
is based. For they must have this, at least, in common, that they are
instances of the proposition which generalises them. Some element of
analogy must, therefore, lie at the base of every inductive argument.

In this chapter I shall try to explain with precision the meaning of

Analogy, and to analyse the reasons, for which, rightly or wrongly,
we usually regard analogies as strong or weak, without considering at
present whether it is possible to find a good reason for our instinctive
principle that likeness breeds the expectation of likeness.

2.

There are a few technical terms to be defined. We mean
by a generalisation a statement that all of a certain definable class
of propositions are true. It is convenient to specify this class in the
following way. If f(x) is true for all those values of x for which φ(x) is
true, then we have a generalisation about φ and f which we may
write g(φ, f). If, for example, we are dealing with the generalisation,

“all swans are white,” this is equivalent to the statement “ ‘x is white’

is true for all those values of x for which ‘x is a swan’ is true.” The
proposition φ(a) f(a) is an instance of the generalisation g(φ, f).

By thus defining a generalisation in terms of propositional
functions, it becomes possible to deal with all kinds of generalisations
in a uniform way; and also to bring generalisation into convenient

∗A Treatise of Human Nature.

256
ch. xix

INDUCTION AND ANALOGY

257
connection with our definition of Analogy.

If some one thing is true about both of two objects, if, that is
to say, they both satisfy the same propositional function, then to
this extent there is an analogy between them. Every generalisation
g(φ, f ), therefore, asserts that one analogy is always accompanied by
another, namely, that between all objects having the analogy φ there
is also the analogy f. The set of propositional functions, which are
satisfied by both of the two objects, constitute the positive analogy.

The analogies, which would be disclosed by complete knowledge, may
be termed the total positive analogy; those which are relative to
partial knowledge, the known positive analogy.

As the positive analogy measures the resemblances, so the negative
analogy measures the differences between the two objects. The set of
functions, such that each is satisfied by one and not by the other of
the objects, constitutes the negative analogy. We have, as before, the
distinction between the total negative analogy and the known negative
analogy.

This set of definitions is soon extended to the cases in which
the number of instances exceeds two. The functions which are true
of all of the instances constitute the positive analogy of the set of
instances, and those which are true of some only, and are false of
others, constitute the negative analogy. It is clear that a function,
which represents positive analogy for a group of instances taken out of
the set, may be a negative analogy for the set as a whole. Analogies
of this kind, which are positive for a sub-class of the instances, but
negative for the whole class, we may term sub-analogies. By this it is
meant that there are resemblances which are common to some of the
instances, but not to all.

A simple notation, in accordance with these definitions, will be
useful. If there is a positive analogy φ between a set of instances
a1, . . . an, whether or not this is the total analogy between them, let
us write this—

A (φ).∗
a1...an
pt. iii

A TREATISE ON PROBABILITY

258

And if there is a negative analogy φ , let us write this—

¯

A (φ ).∗
a1...an

Thus A (φ) expresses the fact that there is a set of character-
a1...an
istics φ which are common to all the instances, and

¯

A (φ ) that
a1...an
there is a set of characteristics φ which is true of at least one of the
instances and false of at least one.

3.

In the typical argument from analogy we wish to generalise
from one part to another of the total analogy which experience
has shown to exist between certain selected instances. In all the
cases where one characteristic φ has been found to exist, another
characteristic f has been found to be associated with it. We argue
from this that any instance, which is known to share the first
analogy φ, is likely to share also the second analogy f. We have
found in certain cases, that is to say, that both φ and f are true of
them; and we wish to assert f as true of other cases in which we
have only observed φ. We seek to establish the generalisation g(φ, f),
on the ground that φ and f constitute between them an observed
positive analogy in a given set of experiences.

But while the argument is of this character, the grounds, upon
which we attribute more or less weight to it, are often rather complex;
and we must discuss them, therefore, in a systematic manner.

4.

According to the view suggested in the last chapter, the
value of such an argument depends partly upon the nature of the
conclusion which we seek to draw, partly upon the evidence which
supports it. If Hume had expected the same degree of nourishment
x=an

∗Hence

A (φ) ≡ φ(a1) φ(a2) . . . φ(an) ≡
φ(x).

a1...an
x=a1
x=a
x=a
s
s

∗Hence

¯

A (φ ) ≡
φ (x)

φ (x).

a1...an
x=ar
x=ar
ch. xix

INDUCTION AND ANALOGY

259
as well as the same taste and relish from all of the eggs, he would
have drawn a conclusion of weaker probability. Let us consider,
then, this dependence of the probability upon the scope of the
generalisation g(φ, f),—upon the comprehensiveness, that is to say, of
the condition φ and the conclusion f respectively.

The more comprehensive the condition φ and the less comprehen-
sive the conclusion f, the greater à priori probability do we attribute
to the generalisation g. With every increase in φ this probability
increases, and with every increase in f it will diminish.

The condition φ(≡ φ1φ2) is more comprehensive than the con-
dition φ1, relative to the general evidence h, if φ2 is a condition
independent of φ1 relative to h, φ2 being independent of φ1, if
g(φ1, φ2)/h =

| 1, i.e. if, relative to h, the satisfaction of φ2 is not
inferrible from that of φ1.

Similarly the conclusion f(≡ f1f2) is more comprehensive than the
conclusion f1, relative to the general evidence h, if f2 is a conclusion
independent of f1, relative to h, i.e. if g(f1, f2)/h =| 1.

If φ ≡ φ1φ2 and f ≡ f1f2, where φ1 and φ2 are independent and
f1 and f2 are independent relative to h, we have—
g(φ

¯

1, f )/h = g(φ1φ2, f )

g(φ1φ2, f )/h
g(φ, f )/h,
and
g(φ, f )/h = g(φ, f1f2)/h

= g(φf1, f2)/h g(φ, f1)/h

≤ g(φ, f1)/h,
so that
g(φ, f1)/h ≥ g(φ, f )/h ≥ g(φ1, f )/h.

This proves the statement made above. It will be noticed that
we cannot necessarily compare the à priori probabilities of two
generalisations in respect of more and less, unless the condition of the
first is included in the condition of the second, and the conclusion of
the second is included in that of the first.

pt. iii

A TREATISE ON PROBABILITY

260

We see, therefore, that some generalisations stand initially in a
stronger position than others. In order to attain a given degree of
probability, generalisations require, according to their scope, different
amounts of favourable evidence to support them.

5.

Let us now pass from the character of the generalisation
à priori to the evidence by which we support it. Since, whenever
the conclusion f is complex, i.e. resolvable into the form f1f2 where
g(f1, f2)/h =

| 1, we can express the probability of the generalisation
g(φ, f ) as the product of the probabilities of the two generalisations
g(φf1, f2) and g(φ, f ), we may assume in what follows, that the
conclusion f is simple and not capable of further analysis, without
diminishing the generality of our argument.

We will begin with the simplest case, namely, that which arises
in the following conditions. First, let us assume that our knowledge
of the examined instances is complete, so that we know of every
statement, which is about the examined instances, whether it is true
or false of each.∗ Second, let us assume that all the instances which
are known to satisfy the condition φ, are also known to satisfy the
conclusion f of the generalisation. And third let us assume that there
is nothing which is true of all the examined instances and yet not
included either in φ or in f, i.e. that the positive analogy between the
instances is exactly coextensive with the analogy φf which is covered
by the generalisation.

Such evidence as this constitutes what we may term a perfect
analogy. The argument in favour of the generalisation cannot be
further improved by a knowledge of additional instances. Since the
positive analogy between the instances is exactly coextensive with the
analogy covered by the generalisation, and since our knowledge of the
examined instances is complete, there is no need to take account of
the negative analogy.

An analogy of this kind, however, is not likely to have much

∗If ψ(a) is a proposition and ψ(a) = h θ(a), where h is a proposition not
involving a, then we must regard θ(a), not ψ(a), as the statement about a.

ch. xix

INDUCTION AND ANALOGY

261
practical utility; for if the analogy covered by the generalisation,
covers the whole of the positive analogy between the instances it
is difficult to see to what other instances the generalisation can be
applicable. Any instance, about which everything is true which is
true of all of a set of instances, must be identical with one of them.

Indeed, an argument from perfect analogy can only have practical
utility, if, as will be argued later on, there are some distinctions
between instances which are irrelevant for the purposes of analogy,
and if, in a perfect analogy, the positive analogy, of which we must
take account, need cover only those distinctions which are relevant.

In this case a generalisation based on perfect analogy might cover
instances numerically distinct from those of the original set.

The law of the Uniformity of Nature appears to me to amount
to an assertion that an analogy which is perfect, except that mere
differences of position in time and space are treated as irrelevant,
is a valid basis for a generalisation, two total causes being regarded
as the same if they only differ in their positions in time or space.

This, I think, is the whole of the importance which this law has
for the theory of inductive argument. It involves the assertion of a
generalised judgment of irrelevance, namely, of the irrelevance of mere
position in time and space to generalisations which have no reference
to particular positions in time and space. It is in respect of such
position in time or space that ‘nature’ is supposed ‘uniform.’ The
significance of the law and the nature of its justification, if any, are
further discussed in Chapter XXII.

6.

Let us now pass to the type which is next in order of
simplicity. We will relax the first condition and no longer assume that
the whole of the positive analogy between the instances is covered
by the generalisation, though retaining the assumption that our
knowledge of the examined instances is complete. We know, that is
to say, that there are some respects in which the examined instances
are all alike, and yet which are not covered by the generalisation. If
φ1 is the part of the positive analogy between the instances which is
pt. iii

A TREATISE ON PROBABILITY

262
not covered by the generalisation, then the probability of this type of
argument from analogy can be written—
g(φ, f )

A (φφ1f ).

a1...an

The value of this probability turns on the comprehensiveness of φ1.

There are some characteristics φ1 common to all the instances, which
the generalisation treats as unessential, but the less comprehensive
these are the better. φ1 stands for the characteristics in which
all the instances resemble one another outside those covered by the
generalisation. To reduce these resemblances between the instances
is the same thing as to increase the differences between them. And
hence any increase in the Negative Analogy involves a reduction in
the comprehensiveness of φ1. When, however, our knowledge of the
instances is complete, it is not necessary to make separate mention of
the negative analogy ¯A (φ ) in the above formula. For φ simply
a1...an
includes all those functions about the instances, which are not
included in φφ1f, and of which the contradictories are not included in
them; so that in stating A (φφ1f), we state by implication ¯A (φ )

a1...an
a1...an
also.The whole process of strengthening the argument in favour of
the generalisation g(φ, f) by the accumulation of further experience
appears to me to consist in making the argument approximate as
nearly as possible to the conditions of a perfect analogy, by steadily
reducing the comprehensiveness of those resemblances φ1 between the
instances which our generalisation disregards. Thus the advantage of
additional instances, derived from experience, arises not out of their
number as such, but out of their tendency to limit and reduce the
comprehensiveness of φ1, or, in other words, out of their tendency to
increase the negative analogy φ , since φ1φ comprise between them
whatever is not covered by φf. The more numerous the instances,
the less comprehensive are their superfluous resemblances likely to
be. But a single additional instance which greatly reduced φ1 would
ch. xix

INDUCTION AND ANALOGY

263
increase the probability of the argument more than a large number of
instances which affected φ1 less.

7.

The nature of the argument examined so far is, then, that
the instances all have some characteristics in common which we have
ignored in framing our generalisation; but it is still assumed that
our knowledge about the examined instances is complete. We will
next dispense with this latter assumption, and deal with the case in
which our knowledge of the characteristics of the examined instances
themselves is or may be incomplete.

It is now necessary to take explicit account of the known negative
analogy. For when the known positive analogy falls short of the total
positive analogy, it is not possible to infer the negative analogy from
it. Differences may be known between the instances which cannot
be inferred from the known positive analogy. The probability of the
argument must, therefore, be written—
g(φ, f )

A (φφ1f ) ¯

A (φ ),
a1...an
a1...an
where φφ1f stands for the characteristics in which all n instances
a1 . . . an are known to be alike, and φ stands for the characteristics
in which they are known to differ.

This argument is strengthened by any additional instance or by any
additional knowledge about the former instances which diminishes
the known superfluous resemblances φ1 or increases the negative
analogy φ . The object of the accumulation of further experience is
still the same as before, namely, to make the form of the argument
approximate more and more closely to that of perfect analogy. Now,
however, that our knowledge of the instances is no longer assumed
to be complete, we must take account of the mere number n of the
instances, as well as of our specific knowledge in regard to them; for
the more numerous the instances are, the greater the opportunity
for the total negative analogy to exceed the known negative analogy.

But the more complete our knowledge of the instances, the less
pt. iii

A TREATISE ON PROBABILITY

264
attention need we pay to their mere number, and the more imperfect
our knowledge the greater the stress which must be laid upon the
argument from number. This part of the argument will be discussed
in detail in the following chapter on Pure Induction.

8.

When our knowledge of the instances is incomplete, there may
exist analogies which are known to be true of some of the instances
and are not known to be false of any. These sub-analogies (see § 2) are
not so dangerous as the positive analogies φ1, which are known to be
true of all the instances, but their existence is, evidently, an element
of weakness, which we must endeavour to eliminate by the growth of
knowledge and the multiplication of instances. A sub-analogy of this
kind between the instances ar . . . as may be written A (ψk); and the
ar...as
formula, if it is to take account of all the relevant information, ought,
therefore, to be written—
g(φ, f )

A (φφ1f ) ¯

A (φ )

A (ψk) ,
a1...an
a1...an
ar...as
where the terms of

A (ψk)

stand for the various sub-analogies
ar...as
between sub-classes of the instances, which are not included in φφ1f
or in φ .

9.

There is now another complexity to be introduced. We
must dispense with the assumption that the whole of the analogy
covered by the generalisation is known to exist in all the instances.

For there may be some instances within our experience, about which
our knowledge is incomplete, but which show part of the analogy
required by the generalisation and nothing which contradicts it; and
such instances afford some support to the generalisation. Suppose
that bφ and bf are part of φ and f respectively, then we may have a
set of instances b1 . . . bm which show the following analogies:

A (bφ bφ1 bf ) ¯

A (bφ )

A (bψk) ,
b1...bm
b1...bm
br...bs
ch. xix

INDUCTION AND ANALOGY

265
where bφ1 is the analogy not covered by the generalisation, and so on,
as before.

The formula, therefore, is now as follows:

g(φ, f )

A (aφ aφ1 af ) ¯

A (aφ )

A (ψk) .

a,b...

a1...an
a1...an
ar bs...

In this expression aφ, af are the whole or part of φ, f; the product
is composed of the positive and negative analogies for each of the
a,b...

sets of instances a1 . . . an, b1 . . . bm, etc.; and the product
contains
the various sub-analogies of different sub-classes of all the instances
a1 . . . an, b1 . . . bm, etc., regarded as one set.∗

10.

This completes our classification of the positive evidence
which supports a generalisation; but the probability may also be
affected by a consideration of the negative evidence. We have taken
account so far of that part of the evidence only which shows the
whole or part of the analogy we require, and we have neglected those
instances of which φ, the condition of the generalisation, or f, its
conclusion, or part of φ or of f is known to be false. Suppose that
there are instances of which φ is true and f false, it is clear that the
generalisation is ruined. But cases in which we know part of φ to
be true and f to be false, and are ignorant as to the truth or falsity
of the rest of φ, weaken it to some extent. We must take account,
therefore, of analogies

A (

¯
a φ a f ),
a ...a

1
n
where a φ, part of φ, is true of all the set, and a f, part of f, is false
of all the set, while the truth or falsity of some part of φ and f is
unknown. The negative evidence, however, can strengthen as well as

∗Even if we want to distinguish between the sub-analogies of the a set
and the sub-analogies of the b set, this information can be gathered from the
product

.

pt. iii

A TREATISE ON PROBABILITY

266
weaken the evidence. We deem instances favourably relevant in which
φ and f are both false together.∗

Our final formula, therefore, must include terms, similar to those
in the formula which concludes § 9, not only for sets of instances
which show analogies aφ af where aφ and af are parts of φ and f,
but also for sets which show analogies ¯

¯ ¯
aφ af , or analogies aφ af , where
aφ and af
are the whole or part of φ and f, and ¯φ, ¯
f are the
contradictories of φ and f.†

It should be added, perhaps, that the theoretical classification of
most empirical arguments in daily use is complicated by the account
which we reasonably take of generalisations previously established. We
often take account indirectly, therefore, of evidence which supports in
some degree other generalisations than that which we are concerned
to establish or refute at the moment, but the probability of which is
relevant to the problem under investigation.

11.

The argument will be rendered unnecessarily complex,
without much benefit to its theoretical interest, if we deal with the
most general case of all. What follows, therefore, will deal with the
formula of the third degree of generality, namely—
g(φ, f )

A (φφ1f ) ¯

A (φ )

A (ψk) ,
a1...an
a1...an
ar...as
in which no partial instances occur, i.e. no instances in which part
only of the analogy, required by the generalisation, is known to exist.

In this third degree of generality, it will be remembered, our knowledge
of the characteristics of the instances is incomplete, there is more
analogy between the instances than is covered by the generalisation,
and there are some sub-analogies to be reckoned with. In the above

∗I am disposed to think that we need not pay attention to instances for
which part of φ is known to be false, and part of f to be true.

But the
question is a little perplexing.

†Where the conclusion f is simple and not complex (see § 5), some of these
complications cannot, of course, arise.

ch. xix

INDUCTION AND ANALOGY

267
formula the incompleteness of our knowledge is implicitly recognised
in that φφ1fφ are not between them entirely comprehensive. It is also
supposed that all the evidence we have is positive, no knowledge is
assumed, that is to say, of instances characterised by the conjunctions

¯

¯

¯ ¯
aφ af , aφ af , or aφ af , where aφ and af are part of φ and f .

An argument, therefore, from experience, in which, on the basis of
examined instances, we establish a generalisation applicable beyond
these instances, can be strengthened, if we restrict our attention to
the simpler type of case, by the following means:

(1) By reducing the resemblances φ1 known to be common to all
the instances, but ignored as unessential by the generalisation.

(2) By increasing the differences φ known to exist between the
instances.

(3) By diminishing the sub-analogies or unessential resemblances ψk
known to be common to some of the instances and not known to be
false of any.

These results can generally be obtained in two ways, either by
increasing the number of our instances or by increasing our knowledge
of those we have.

The reasons why these methods seem to common sense to
strengthen the argument are fairly obvious. The object of (1) is to
avoid the possibility that φ1 as well as φ is a necessary condition of f.

The object of (2) is to avoid the possibility that there may be some
resemblances additional to φ, common to all the instances, which have
escaped our notice. The object of (3) is to get rid of indications that
the total value of φ1 may be greater than the known value. When
φφ1f is the total positive analogy between the instances, so that the
known value of φ1 is its total value, it is (1) which is fundamental;
and we need take account of (2) and (3) only when our knowledge of
the instances is incomplete. But when our knowledge of the instances
is incomplete, so that φ1 falls short of its total value and we cannot
infer φ from it, it is better to regard (2) as fundamental; in any case
pt. iii

A TREATISE ON PROBABILITY

268
every reduction of φ1 must increase φ .

12.

I have now attempted to analyse the various ways in which
common practice seems to assume that considerations of Analogy can
yield us presumptive evidence in favour of a generalisation.

It has been my object, in making a classification of empirical
arguments, not so much to put my results in forms closely similar
to those in which problems of generalisation commonly present
themselves to scientific investigators, as to inquire whether ultimate
uniformities of method can be found beneath the innumerable modes,
superficially differing from another, in which we do in fact argue.

I have not yet attempted to justify this way of arguing. After
turning aside to discuss in more detail the method of Pure Induction,

I shall make this attempt; or rather I shall try to see what sort of
assumptions are capable of justifying empirical reasoning of this kind.

CHAPTER XX
the value of multiplication of instances, or pure
induction

1.

It has often been thought that the essence of inductive
argument lies in the multiplication of instances. “Where is that
process of reasoning,” Hume inquired, “which from one instance draws
a conclusion, so different from that which it infers from a hundred
instances, that are no way different from that single instance?”

I repeat that by emphasising the number of the instances Hume
obscured the real object of the method. If it were strictly true that
the hundred instances are no way different from the single instance,

Hume would be right to wonder in what manner they can strengthen
the argument. The object of increasing the number of instances arises
out of the fact that we are nearly always aware of some difference
between the instances, and that even where the known difference
is insignificant we may suspect, especially when our knowledge of
the instances is very incomplete, that there may be more. Every
new instance may diminish the unessential resemblances between the
instances and by introducing a new difference increase the Negative

Analogy. For this reason, and for this reason only, new instances are
valuable.

If our premisses comprise the body of memory and tradition which
has been originally derived from direct experience, and the conclusion
which we seek to establish is the Newtonian theory of the Solar

System, our argument is one of Pure Induction, in so far as we
support the Newtonian theory by pointing to the great number of
consequences which it has in common with the facts of experience.

The predictions of the Nautical Almanack are a consequence of the

Newtonian theory, and these predictions are verified many thousand
times a day. But even here the force of the argument largely depends,
not on the mere number of these predictions, but on the knowledge

269
pt. iii

A TREATISE ON PROBABILITY

270
that the circumstances in which they are fulfilled differ widely from
one another in a vast number of important respects. The variety of
the circumstances, in which the Newtonian generalisation is fulfilled,
rather than the number of them, is what seems to impress our
reasonable faculties.

2.

I hold, then, that our object is always to increase the

Negative Analogy, or, which is the same thing, to diminish the
characteristics common to all the examined instances and yet not
taken account of by our generalisation.

Our method, however,
maybe one which certainly achieves this object, or it may be
one which possibly achieves it.

The former of these, which is
obviously the more satisfactory, may consist either in increasing our
definite knowledge respecting instances examined already, or in finding
additional instances respecting which definite knowledge is obtainable.

The second of them consists in finding additional instances of the
generalisation, about which, however, our definite knowledge may be
meagre; such further instances, if our knowledge about them were
more complete, would either increase or leave unchanged the Negative

Analogy; in the former case they would strengthen the argument and
in the latter case they would not weaken it; and they must, therefore,
be allowed some weight. The two methods are not entirely distinct,
because new instances, about which we have some knowledge but not
much, may be known to increase the Negative Analogy a little by the
first method, and suspected of increasing it further by the second.

It is characteristic of advanced scientific method to depend on the
former, and of the crude unregulated induction of ordinary experience
to depend on the latter. It is when our definite knowledge about
the instances is limited, that we must pay attention to their number
rather than to the specific differences between them, and must fall
back on what I term Pure Induction.

In this chapter I investigate the conditions and the manner in
which the mere repetition of instances can add to the force of the
argument.

The chief value of the chapter, in my judgment, is
ch. xx

INDUCTION AND ANALOGY

271
negative, and consists in showing that a line of advance, which might
have seemed promising, turns out to be a blind alley, and that we are
thrown back on known Analogy. Pure Induction will not give us any
very substantial assistance in getting to the bottom of the general
inductive problem.

3.

The problem of generalisation∗ by Pure Induction can be
stated in the following symbolic form:

Let h represent the general à priori data of the investigation;
let g represent the generalisation which we seek to establish; let
x1x2 . . . xn represent instances of g.

Then x1/gh = 1, x2/gh = 1 . . . xn/gh = 1; given g, that is to
say, the truth of each of its instances follows. The problem is to
determine the probability g/hx1x2 . . . xn, i.e. the probability of the
generalisation when n instances of it are given. Our analysis will be
simplified, and nothing of fundamental importance will be lost, if we
introduce the assumption that there is nothing in our à priori data
which leads us to distinguish between the à priori likelihood of the
different instances; we assume, that is to say, that there is no reason
à priori for expecting the occurrence of any one instance with greater
reliance than any other, i.e.

x1/h = x2/h = . . . = xn/h.

Write
g/hx1x2 . . . xn = pn
and
xn+1/hx1x2 . . . xn = yn+1;

∗In the most general sense we can regard any proposition as the generalisation
of all the propositions which follow from it. For if h is any proposition, and we
put φ(x) ≡ ‘x can be inferred from h’ and, f (x) ≡ x, then g(φ, f ) ≡ h.

Since

Pure Induction consists in finding as many instances of a generalisation as
possible, it is, in the widest sense, the process of strengthening the probability
of any proposition by adducing numerous instances of known truths which
follow from it.

The argument is one of Pure Induction, therefore, in so far
as the probability of a conclusion is based upon the number of independent
consequences which the conclusion and the premisses have in common.

pt. iii

A TREATISE ON PROBABILITY

272
then
pn
g/hx
gx

=

1 . . . xn

=
n/hx1 . . . xn−1
pn−1
g/hx1 . . . xn−1
g/hx1 . . . xn−1 xn/hx1 . . . xn−1
x

=
n/ghx1 . . . xn−1
xn/hx1 . . . xn−1

1

=

.

yn
pn

1

1

∴

=

, and hence pn =
p0, where p0 = g/h, i.e. p0
pn−1
yn
y1y2 . . . yn
is the à priori probability of the generalisation.

It follows, therefore, that pn > pn−1 so long as yn > 1.

Further,
x1x2 . . . xn/h = xn/hx1x2 . . . xn−1 x1x2 . . . xn−1/h

= yn x1x2 . . . xn−1/h

= ynyn−1 . . . y1.

p0
p0

∴ pn =

=
y1y2 . . . yn
x1x2 . . . xn/h
p

=

0
x1x2 . . . xng/h + x1x2 . . . xn¯
g/h
p

=

0
g/h + x1x2 . . . xn/¯
gh ¯
g/h
p

=

0
p0 + x1x2 . . . xn/¯
gh(1 − p0)

This approaches unity as a limit, if

1
x1x2 . . . xn/¯
gh
approaches
p0
zero as a limit, when n increases.

4.

We may now stop to consider how much this argument
has proved. We have shown that if each of the instances necessarily
follows from the generalisation, then each additional instance increases
the probability of the generalisation, so long as the new instance
ch. xx

INDUCTION AND ANALOGY

273
could not have been predicted with certainty from a knowledge of the
former instances.∗ This condition is the same as that which came
to light when we were discussing Analogy. If the new instance were
identical with one of the former instances, a knowledge of the latter
would enable us to predict it. If it differs or may differ in analogy,
then the condition required above is satisfied.

The common notion, that each successive verification of a doubtful
principle strengthens it, is formally proved, therefore, without any
appeal to conceptions of law or of causality.

But we have not
proved that this probability approaches certainty as a limit, or even
that our conclusion becomes more likely than not, as the number of
verifications or instances is indefinitely increased.

5.

What are the conditions which must be satisfied in order that
the rate, at which the probability of the generalisation increases, may
be such that it will approach certainty as a limit when the number of
independent instances of it are indefinitely increased? We have already
shown, as a basis for this investigation, that pn approaches the limit
of certainty for a generalisation g, if, as n increases, x1x2 . . . xn/¯gh
becomes small compared with p0, i.e. if the à priori probability of so
many instances, assuming the falsehood of the generalisation, is small
compared with the generalisation’s à priori probability. It follows,
therefore, that the probability of an induction tends towards certainty
as a limit, when the number of instances is increased, provided that
xr/x1x2 . . . xr−1¯
gh < 1 −
for all values of r, and p0 > η, where and η are finite probabilities,
separated, that is to say, from impossibility by a value of some finite
amount, however small. These conditions appear simple, but the
meaning of a ‘finite probability’ requires a word of explanation.†

∗Since pn > pn−1 so long as yn =| 1.

†The proof of these conditions, which is obvious, is as follows:

x1x2 . . . xn/¯
gh = xn−1¯
gh x1x2 . . . xn−1/¯
gh < (1 − )n,
pt. iii

A TREATISE ON PROBABILITY

274

I argued in Chapter III. that not all probabilities have an exact
numerical value, and that, in the case of some, one can say no
more about their relation to certainty and impossibility than that
they fall short of the former and exceed the latter. There is one
class of probabilities, however, which I called the numerical class,
the ratio of each of whose members to certainty can be expressed
by some number less than unity; and we can sometimes compare a
non-numerical probability in respect of more and less with one of
these numerical probabilities. This enables us to give a definition of

‘finite probability’ which is capable of application to non-numerical
as well as to numerical probabilities. I define a ‘finite probability’

as one which exceeds some numerical probability, the ratio of which
to certainty can be expressed by a finite number.∗ The principal
method, in which a probability can be proved finite by a process
of argument, arises either when its conclusion can be shown to
be one of a finite number of alternatives, which are between them
exhaustive or, at any rate, have a finite probability, and to which
the Principle of Indifference is applicable; or (more usually), when its
conclusion is more probable than some hypothesis which satisfies this
first condition.

6.

The conditions, which we have now established in order that
the probability of a pure induction may tend towards certainty as
the number of instances is increased, are (1) that xr/x1x2 . . . xr−1¯gh
falls short of certainty by a finite amount for all values of r, and

(2) that p0, the à priori probability of our generalisation, exceeds
impossibility by a finite amount. It is easy to see that we can
where
is finite and p0 > η where η is finite. There is always, under these

(1 − )n
conditions, some finite value of n such that both (1 − )n and
are less
η
than any given finite quantity, however small.

∗Hence a series of probabilities p1p2 . . . pr approaches a limit L, if, given any
positive finite number
however small, a positive integer n can always be found
such that for all values of r greater than n the difference between L and pr is
less than
γ, where γ is the measure of certainty.

ch. xx

INDUCTION AND ANALOGY

275
show by an exactly similar argument that the following more general
conditions are equally satisfactory:

(1) That xr/x1x2 . . . xr−1¯gh falls short of certainty by a finite
amount for all values of r beyond a specified value s.

(2) That ps, the probability of the generalisation relative to a
knowledge of these first s instances, exceeds impossibility by a finite
amount.

In other words Pure Induction can be usefully employed to
strengthen an argument if, after a certain number of instances have
been examined, we have, from some other source, a finite probability
in favour of the generalisation, and, assuming the generalisation is
false, a finite uncertainty as to its conclusion being satisfied by the
next hitherto unexamined instance which satisfies its premiss. To take
an example, Pure Induction can be used to support the generalisation
that the sun will rise every morning for the next million years,
provided that with the experience we have actually had there are
finite probabilities, however small, derived from some other source,
first, in favour of the generalisation, and, second, in favour of the
sun’s not rising to-morrow assuming the generalisation to be false.

Given these finite probabilities, obtained otherwise, however small,
then the probability can be strengthened and can tend to increase
towards certainty by the mere multiplication of instances provided
that these instances are so far distinct that they are not inferrible one
from another.

7.

Those supposed proofs of the Inductive Principle, which are
based openly or implicitly on an argument in inverse probability, are
all vitiated by unjustifiable assumptions relating to the magnitude of
the à priori probability p0. Jevons, for instance, avowedly assumes
that we may, in the absence of special information, suppose any
unexamined hypothesis to be as likely as not. It is difficult to see
how such a belief, if even its most immediate implications had been
properly apprehended, could have remained plausible to a mind of
so sound a practical judgment as his. The arguments against it and
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276
the contradictions to which it leads have been dealt with in Chapter

IV. The demonstration of Laplace, which depends upon the Rule of

Succession, will be discussed in Chapter XXX.

8.

The prior probability, which must always be found, before
the method of pure induction can be usefully employed to support a
substantial argument, is derived, I think, in most ordinary cases—
with what justification it remains to discuss—from considerations of

Analogy. But the conditions of valid induction as they have been
enunciated above, are quite independent of analogy, and might be
applicable to other types of argument. In certain cases we might
feel justified in assuming directly that the necessary conditions are
satisfied.

Our belief, for instance, in the validity of a logical scheme is
based partly upon inductive grounds—on the number of conclusions,
each seemingly true on its own account, which can be derived from
the axioms—and partly on a degree of self-evidence in the axioms
themselves sufficient to give them the initial probability upon which
induction can build. We depend upon the initial presumption that, if
a proposition appears to us to be true, this is by itself, in the absence
of opposing evidence, some reason for its being as well as appearing
true. We cannot deny that what appears true is sometimes false,
but, unless we can assume some substantial relation of probability
between the appearance and the reality of truth, the possibility of
even probable knowledge is at an end.

The conception of our having some reason, though not a conclusive
one, for certain beliefs, arising out of direct inspection, may prove
important to the theory of epistemology. The old metaphysics has
been greatly hindered by reason of its having always demanded
demonstrative certainty. Much of the cogency of Hume’s criticism
arises out of the assumption of methods of certainty on the part of
those systems against which it was directed. The earlier realists were
hampered by their not perceiving that lesser claims in the beginning
might yield them what they wanted in the end. And transcendental
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philosophy has partly arisen, I believe, through the belief that there
is no knowledge on these matters short of certain knowledge, being
combined with the belief that such certain knowledge of metaphysical
questions is beyond the power of ordinary methods.

When we allow that probable knowledge is, nevertheless, real,
a new method of argument can be introduced into metaphysical
discussions. The demonstrative method can be laid on one side,
and we may attempt to advance the argument by taking account
of circumstances which seem to give some reason for preferring one
alternative to another. Great progress may follow if the nature
and reality of objects of perception,∗ for instance, can be usefully
investigated by methods not altogether dissimilar from those employed
in science and with the prospect of obtaining as high a degree of
certainty as that which belongs to some scientific conclusions; and it
may conceivably be shown that a belief in the conclusions of science,
enunciated in any reasonable manner however restricted, involves a
preference for some metaphysical conclusions over others.

9.

Apart from analysis, careful reflection would hardly lead us
to expect that a conclusion which is based on no other than grounds
of pure induction, defined as I have defined them as consisting of
repetition of instances merely, could attain in this way to a high
degree of probability. To this extent we ought all of us to agree
with Hume. We have found that the suggestions of common sense
are supported by more precise methods. Moreover, we constantly
distinguish between arguments, which we call inductive, upon other
grounds than the number of instances upon which they are based; and
under certain conditions we regard as crucial an insignificant number
of experiments. The method of pure induction may be a useful means
of strengthening a probability based on some other ground. In the

∗A paper by Mr. G. E. Moore entitled, “The Nature and Reality of Objects
of Perception,” which was published in the Proceedings of the Aristotelian Society
for 1906, seems to me to apply for the first time a method somewhat resembling
that which is described above.

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278
case, however, of most scientific arguments, which would commonly
be called inductive, the probability that we are right, when we
make predictions on the basis of past experience, depends not so
much on the number of past experiences upon which we rely, as on
the degree in which the circumstances of these experiences resemble
the known circumstances in which the prediction is to take effect.

Scientific method, indeed, is mainly devoted to discovering means of
so heightening the known analogy that we may dispense as far as
possible with the methods of pure induction.

When, therefore, our previous knowledge is considerable and the
analogy is good, the purely inductive part of the argument may take
a very subsidiary place. But when our knowledge of the instances is
slight, we may have to depend upon pure induction a good deal. In
an advanced science it is a last resort,—the least satisfactory of the
methods. But sometimes it must be our first resort, the method upon
which we must depend in the dawn of knowledge and in fundamental
inquiries where we must presuppose nothing.

CHAPTER XXI
the nature of inductive argument continued

1.

In the enunciation, given in the two preceding chapters,
of the Principles of Analogy and Pure Induction there has been no
reference to experience or causality or law. So far, the argument has
been perfectly formal and might relate to a set of propositions of any
type. But these methods are most commonly employed in physical
arguments where material objects or experiences are the terms of the
generalisation. We must consider, therefore, whether there is any
good ground, as some logicians seem to have supposed, for restricting
them to this kind of inquiry.

I am inclined to think that, whether reasonably or not, we
naturally apply them to all kinds of argument alike, including formal
arguments as, for example, about numbers. When we are told that

Fermat’s formula for a prime, namely, 22a + 1 for all values of a,
has been verified in every case in which verification is not excessively
laborious—namely, for a = 1, 2, 3, and 4, we feel that this is
some reason for accepting it, or, at least, that it raises a sufficient
presumption to justify a further examination of the formula.∗ Yet
there can be no reference here to the uniformity of nature or physical
causation. If inductive methods are limited to natural objects, there
can no more be an appreciable ground for thinking that 22a + 1 is
a true formula for primes, because empirical methods show that it
yields primes up to a = 4, or even if they showed that it yielded primes
for every number up to a million million, than there is to think that
any formula which I may choose to write down at random is a true
source of primes. To maintain that there is no appreciable ground in
such a case is paradoxical. If, on the other hand, a partial verification

∗This formula has, in fact, been disproved in recent times, e.g. 225 + 1 =

4, 294, 967, 297 = 641 × 6, 700, 417. Thus it is no longer so good an illustration
as it would have been a hundred years ago.

279
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280
does raise some just appreciable presumption in the formula’s favour,
then we must include numbers, at any rate, as well as material objects
amongst the proper subjects of the inductive method. The conclusion
of the previous chapter indicates, however, that, if arguments of this
kind have force, it can only be in virtue of there being some finite
à priori probability for the formula based on other than inductive
grounds.

There are some illustrations in Jevons’s Principles of Science,∗
which are relevant to this discussion. We find it to be true of the
following six numbers:

5, 15, 35, 45, 65, 95
that they all end in five, and are all divisible by five without
remainder. Would this fact, by itself, raise any kind of presumption
that all numbers ending in five are divisible by five without remainder?

Let us also consider the six numbers,

7, 17, 37, 47, 67, 97.

They all end in seven and also agree in being primes. Would this
raise a presumption in favour of the generalisation that all numbers
are prime, which end in seven? We might be prejudiced in favour of
the first argument, because it would lead us to a true conclusion; but
we ought not to be prejudiced against the second because it would
lead us to a false one; for the validity of empirical arguments as the
foundation of a probability cannot be affected by the actual truth or
falsity of their conclusions. If, on the evidence, the analogy is similar
and equal, and if the scope of the generalisation and its conclusion is
similar, then the value of the two arguments must be equal also.

Whether or not the use of empirical argument appears plausible
to us in these particular examples, it is certainly true that

∗Pp. 229–231 (one volume edition). Jevons uses these illustrations, not for
the purpose to which I am here putting them, but to demonstrate the fallibility
of empirical laws.

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many mathematical theorems have actually been discovered by such
methods.

Generalisations have been suggested nearly as often,
perhaps, in the logical and mathematical sciences, as in the physical,
by the recognition of particular instances, even where formal proof
has been forthcoming subsequently. Yet if the suggestions of analogy
have no appreciable probability in the formal sciences, and should be
permitted only in the material, it must be unreasonable for us to
pursue them. If no finite probability exists that a formula, for which
we have empirical verification, is in fact universally true, Newton was
acting fortunately, but not reasonably, when he hit on the Binomial

Theorem by methods of empiricism.∗

2.

I am inclined to believe, therefore, that, if we trust the
promptings of common sense, we have the same kind of ground for
trusting analogy in mathematics that we have in physics, and that
we ought to be able to apply any justification of the method, which
suits the latter case, to the former also. This does not mean that
the à priori probabilities, from some other source than induction,
which the inductive method requires as its foundation, may not be
sought and found differently in the two types of inquiry. A reason
why it has been thought that analogy ought to be confined to natural
laws may be, perhaps, that in most of those cases, in which we
could support a mathematical theorem by a very strong analogy, the
existence of a formal proof has done away with the necessity for the
limping methods of empiricism; and because in most mathematical
investigations, while in our earliest thoughts we are not ashamed
to consult analogy, our later work will be more profitably spent in
searching for a formal proof than in establishing analogies which must,
at the best, be relatively weak. As the modern scientist discards, as a
rule, the method of pure induction, in favour of experimental analogy,
where, if he takes account of his previous knowledge, one or two
cases may prove immensely significant; so the modern mathematician

∗See Jevons, loc. cit. p. 231.

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282
prefers the resources of his analysis, which may yield him certainty,
to the doubtful promises of empiricism.

3.

The main reason, however, why it has often been held that
we ought to limit inductive methods to the content of the particular
material universe in which we live, is, most probably, the fact that
we can easily imagine a universe so constructed that such methods
would be useless. This suggests that analogy and induction, while
they happen to be useful to us in this world, cannot be universal
principles of logic, on the same footing, for instance, as the syllogism.

In one sense this opinion may be well founded. I do not deny
or affirm at present that it may be necessary to confine inductive
methods to arguments about certain kinds of objects or certain
kinds of experiences. It may be true that in every useful argument
from analogy our premisses must contain fundamental assumptions,
obtained directly and not inductively, which some possible experiences
might preclude. Moreover, the success of induction in the past can
certainly affect its probable usefulness for the future. We may discover
something about the nature of the universe—we may even discover it
by means of induction itself—the knowledge of which has the effect of
destroying the further utility of induction. I shall argue later on that
the confidence with which we ourselves use the method does in fact
depend upon the nature of our past experience.

But this empirical attitude towards induction may, on the other
hand, arise out of either one of two possible confusions. It may
confuse, first, the reasonable character of arguments with their
practical usefulness. The usefulness of induction depends, no doubt,
upon the actual content of experience. If there were no repetition of
detail in the universe, induction would have no utility. If there were
only a single object in the universe, the laws of addition would have
no utility. But the processes of induction and addition would remain
reasonable. It may confuse, secondly, the validity of attributing
probability to the conclusion of an argument with the question of the
actual truth of the conclusion. Induction tells us that, on the basis
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of certain evidence, a certain conclusion is reasonable, not that it is
true. If the sun does not rise to-morrow, if Queen Anne still lives,
this will not prove that it was foolish or unreasonable of us to have
believed the contrary.

4.

It will be worth while to say a little more in this connection
about the not infrequent failure to distinguish the rational from the
true. The excessive ridicule, which this mistake has visited on the
supposed irrationality of barbarous and primitive peoples, affords
some good examples. “Reflection and enquiry should satisfy us,”

says Dr. Frazer in the Golden Bough, “that to our predecessors we
are indebted for much of what we thought most our own, and that
their errors were not wilful extravagances or the ravings of insanity,
but simply hypotheses, justifiable as such at the time when they
were propounded, but which a fuller experience has proved to be
inadequate. . . . Therefore, in reviewing the opinions and practices of
ruder ages and races we shall do well to look with leniency upon
their errors as inevitable slips made in the search for truth. . . .” The
first introduction of iron ploughshares into Poland, he tells in another
passage, having been followed by a succession of bad harvests, the
farmers attributed the badness of the crops to the iron ploughshares,
and discarded them for the old wooden ones. The method of reasoning
of the farmers is not different from that of science, and may, surely,
have had for them some appreciable probability in its favour. “It
is a curious superstition,” says a recent pioneer in Borneo, “this of
the Dusuns, to attribute anything—whether good or bad, lucky or
unlucky—that happens to them to something novel which has arrived
in their country. For instance, my living in Kindram has caused the
intensely hot weather we have experienced of late.”∗ What is this
curious superstition but the Method of Difference?

The following passage from Jevons’s Principles of Science well
illustrates the tendency, to which he himself yielded, to depreciate

∗Golden Bough, p. 174.

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284
the favourite analogies of one age, because the experience of their
successors has confuted them. Between things which are the same
in number, he points out, there is a certain resemblance, namely in
number; and in the infancy of science men could not be persuaded
that there was not a deeper resemblance implied in that of number.

“Seven days are mentioned in Genesis; infants acquire their teeth at
the end of seven months; they change them at the end of seven years;
seven feet was the limit of man’s height; every seventh year was a
climacteric or critical year, at which a change of disposition took
place. In natural science there were not only the seven planets, and
the seven metals, but also the seven primitive colours, and the seven
tones of music. So deep a hold did this doctrine take that we still
have its results in many customs, not only in the seven days of the
week, but the seven years’ apprenticeship, puberty at fourteen years,
the second climacteric, and legal majority at twenty-one years, the
third climacteric.” Religious systems from Pythagoras to Comte have
sought to derive strength from the virtue of seven. “And even in
scientific matters the loftiest intellects have occasionally yielded, as
when Newton was misled by the analogy between the seven tones of
music and the seven colours of his spectrum. . . . Even the genius of

Huyghens did not prevent him from inferring that but one satellite
could belong to Saturn, because, with those of Jupiter and the earth,
it completed the perfect number of six.” But is it certain that Newton
and Huyghens were only reasonable when their theories were true,
and that their mistakes were the fruit of a disordered fancy? Or
that the savages, from whom we have inherited the most fundamental
inductions of our knowledge, were always superstitious when they
believed what we now know to be preposterous?

It is important to understand that the common sense of the race
has been impressed by very weak analogies and has attributed to
them an appreciable probability, and that a logical theory, which is to
justify common sense, need not be afraid of including these marginal
cases. Even our belief in the real existence of other people, which we
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285
all hold to be well established, may require for its justification the
combination of experience with a just appreciable à priori possibility
for Animism generally.∗ If we actually possess evidence which renders
some conclusion absurd, it is very difficult for us to appreciate
the relation of this conclusion to data which are different and less
complete; but it is essential that we should realise arguments from
analogy as relative to premisses, if we are to approach the logical
theory of Induction without prejudice.

5.

While we depreciate the former probability of beliefs which
we no longer hold, we tend, I think, to exaggerate the present degree
of certainty of what we still believe. The preceding paragraph is not
intended to deny that savages often greatly overestimate the value of
their crude inductions, and are to this extent irrational. It is not easy
to distinguish between a belief’s being the most reasonable of those
which it is open to us to believe, and its being more probable than
not. In the same way we, perhaps, put an excessive confidence in
those conclusions—the existence of other people, for instance, the law
of gravity, or to-morrow’s sunrise—of which, in comparison with many
other beliefs, we are very well assured. We may sometimes confuse
the practical certainty, attaching to the class of beliefs upon which it
is rational to act with the utmost confidence, with the more wholly
objective certainty of logic. We might rashly assert, for instance, that

∗“This is animism, or that sense of something in Nature which to the
enlightened or civilised man is not there, and in the civilised man’s child, if it
be admitted that he has it at all, is but a faint survival of a phase of the
primitive mind. And by animism I do not mean the theory of a soul in nature,
but the tendency or impulse or instinct, in which all myth originates, to animate
all things; the projection of ourselves into nature; the sense and apprehension of
an intelligence like our own, but more powerful in all visible things” (Hudson,

Far Away and Long Ago, pp. 224–5). This ‘tendency or impulse or instinct,’

refined by reason and enlarged by experience, may be required, in the shape
of an intuitive à priori probability, if some of those universal conclusions of
common sense, which the most sceptical do not kick away, are to be supported
with rational foundations.

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286
to-morrow’s sunrise is as likely to us as failure, and the special virtue
of the number seven as unlikely, even to Pythagoras, as success, in
an attempt to throw heads a hundred times in succession with an
unbiassed coin.∗

6.

As it has often been held upon various grounds, with reason or
without, that the validity of Induction and Analogy depends in some
way upon the character of the actual world, logicians have sought
for material laws upon which these methods can be founded. The

Laws of Universal Causation and the Uniformity of Nature, namely,
that all events have some cause and that the same total cause always
produces the same effect, are those which commonly do service. But
these principles merely assert that there are some data from which
events posterior to them in time could be inferred. They do not seem
to yield us much assistance in solving the inductive problem proper,
or in determining how we can infer with probability from partial
data. It has been suggested in Chapter XIX. that the Principle of
the Uniformity of Nature amounts to an assertion that an argument
from perfect analogy (defined as I have defined it) is valid when
applied to events only differing in their positions in time or space.† It
has also been pointed out that ordinary inductive arguments appear
to be strengthened by any evidence which makes them approximate
more closely in character to a perfect analogy. But this, I think, is
the whole extent to which this principle, even if its truth could be
assumed, would help us. States of the universe, identical in every
particular, may never recur, and, even if identical states were to recur,
we should not know it.

The kind of fundamental assumption about the character of
material laws, on which scientists appear commonly to act, seems to

∗Yet if every inhabitant of the world, Grimsehl has calculated, were to toss
a coin every second, day and night, this latter event would only occur once on
the average in every twenty billion years.

†Is this interpretation of the Principle of the Uniformity of Nature affected
by the Doctrine of Relativity?

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me to be much less simple than the bare principle of Uniformity. They
appear to assume something much more like what mathematicians
call the principle of the superposition of small effects, or, as I prefer
to call it, in this connection, the atomic character of natural law.

The system of the material universe must consist, if this kind of
assumption is warranted, of bodies which we may term (without
any implication as to their size being conveyed thereby) legal atoms,
such that each of them exercises its own separate, independent, and
invariable effect, a change of the total state being compounded of a
number of separate changes each of which is solely due to a separate
portion of the preceding state. We do not have an invariable relation
between particular bodies, but nevertheless each has on the others
its own separate and invariable effect, which does not change with
changing circumstances, although, of course, the total effect may be
changed to almost any extent if all the other accompanying causes
are different. Each atom can, according to this theory, be treated as a
separate cause and does not enter into different organic combinations
in each of which it is regulated by different laws.

Perhaps it has not always been realised that this atomic uniformity
is in no way implied by the principle of the Uniformity of Nature.

Yet there might well be quite different laws for wholes of different
degrees of complexity, and laws of connection between complexes
which could not be stated in terms of laws connecting individual
parts. In this case natural law would be organic and not, as it is
generally supposed, atomic. If every configuration of the Universe
were subject to a separate and independent law, or if very small
differences between bodies—in their shape or size, for instance,—led
to their obeying quite different laws, prediction would be impossible
and the inductive method useless. Yet nature might still be uniform,
causation sovereign, and laws timeless and absolute.

The scientist wishes, in fact, to assume that the occurrence
of a phenomenon which has appeared as part of a more complex
phenomenon, may be some reason for expecting it to be associated
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288
on another occasion with part of the same complex. Yet if different
wholes were subject to different laws quâ wholes and not simply
on account of and in proportion to the differences of their parts,
knowledge of a part could not lead, it would seem, even to presumptive
or probable knowledge as to its association with other parts. Given,
on the other hand, a number of legally atomic units and the laws
connecting them, it would be possible to deduce their effects pro tanto
without an exhaustive knowledge of all the coexisting circumstances.

We do habitually assume, I think, that the size of the atomic unit is
for mental events an individual consciousness, and for material events
an object small in relation to our perceptions. These considerations
do not show us a way by which we can justify Induction. But they
help to elucidate the kind of assumptions which we do actually make,
and may serve as an introduction to what follows.

CHAPTER XXII
the justification of these methods

1.

The general line of thought to be followed in this chapter
may be indicated, briefly, at the outset.

A system of facts or propositions, as we ordinarily conceive it,
may comprise an indefinite number of members. But the ultimate
constituents or indefinables of the system, which all the members
of it are about, are less in number than these members themselves.

Further, there are certain laws of necessary connection between the
members, by which it is meant (I do not stop to consider whether
more than this is meant) that the truth or falsity of every member
can be inferred from a knowledge of the laws of necessary connection
together with a knowledge of the truth or falsity of some (but not
all) of the members.

The ultimate constituents together with the laws of necessary
connection make up what I shall term the independent variety of
the system. The more numerous the ultimate constituents and the
necessary laws, the greater is the system’s independent variety. It is
not necessary for my present purpose, which is merely to bring before
the reader’s mind the sort of conception which is in mine, that I
should attempt a complete definition of what I mean by a system.

Now it is characteristic of a system, as distinguished from a
collection of heterogeneous and independent facts or propositions,
that the number of its premisses, or, in other words, the amount
of independent variety in it, should be less than the number of its
members. But it is not an obviously essential characteristic of a
system that its premisses or its independent variety should be actually
finite. We must distinguish, therefore, between systems which may
be termed finite and infinite respectively, the terms finite and infinite
referring not to the number of members in the system but to the
amount of independent variety in it.

289
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290

The purpose of the discussion, which occupies the greater part of
this chapter, is to maintain that, if the premisses of our argument
permit us to assume that the facts or propositions, with which
the argument is concerned, belong to a finite system, then probable
knowledge can be validly obtained by means of an inductive argument.

I now proceed to approach the question from a slightly different
standpoint, the controlling idea, however, being that which is outlined
above.

2.

What is our actual course of procedure in an inductive
argument? We have before us, let us suppose, a set of n instances
which have r known qualities, a1a2 . . . ar in common, these r qualities
constituting the known positive analogy. From these qualities three

(say) are picked out, namely, a1, a2, a3 and we inquire with what
probability all objects having these three qualities have also certain
other qualities which we have picked out, namely, ar−1, ar. We wish
to determine, that is to say, whether the qualities ar−1, ar are bound
up with the qualities a1, a2, a3. In thus approaching this question we
seem to suppose that the qualities of an object are bound together
in a limited number of groups, a sub-class of each group being an
infallible symptom of the coexistence of certain other members of it
also.Three possibilities are open, any of which would prove destructive
to our generalisation. It may be the case (1) that ar−1 or ar is
independent of all the other qualities of the instances—they may not
overlap, that is to say, with any other groups; or (2) that a1a2a3 do
not belong to the same groups as ar−1ar; or (3) that a1a2a3, while
they belong to the same group as ar−1ar, are not sufficient to specify
this group uniquely—they belong, that is to say, to other groups
also which do not include ar−1 and ar. The precautions we take are
directed towards reducing the likelihood, so far as we can, of each of
these possibilities. We distrust the generalisation if the terms typified
by ar−1ar are numerous and comprehensive, because this increases
the likelihood that some at least of them fall under heading (1), and
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291
also because it increases the likelihood of (3). We trust it if the
terms typified by a1a2a3 are numerous and comprehensive, because
this decreases the likelihood both of (2) and of (3). If we find a new
instance which agrees with the former instances in a1a2a3ar−1ar but
not in a4, we welcome it, because this disposes of the possibility that
it is a4, alone or in combination, that is bound up with ar−1ar. We
desire to increase our knowledge of the properties, lest there be some
positive analogy which is escaping us, and when our knowledge is
incomplete we multiply instances, which we do not know to increase
the negative analogy for certain, in the hope that they may do so.

If we sum up the various methods of Analogy, we find, I think, that
they are all capable of arising out of an underlying assumption, that if
we find two sets of qualities in coexistence there is a finite probability
that they belong to the same group, and a finite probability also
that the first set specifies this group uniquely. Starting from this
assumption, the object of the methods is to increase the finite
probability and make it large. Whether or not anything of this sort is
explicitly present to our minds when we reason scientifically, it seems
clear to me that we do act exactly as we should act, if this were the
assumption from which we set out.

In most cases, of course, the field is greatly simplified from the
first by the use of our pre-existing knowledge. Of the properties
before us we generally have good reason, derived from prior analogies,
for supposing some to belong to the same group and others to belong
to different groups. But this does not affect the theoretical problem
confronting us.

3.

What kind of ground could justify us in assuming the
existence of these finite probabilities which we seem to require? If we
are to obtain them, not directly, but by means of argument, we must
somehow base them upon a finite number of exhaustive alternatives.

The following line of argument seems to me to represent, on the
whole, the kind of assumption which is obscurely present to our
minds. We suppose, I think, that the almost innumerable apparent
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292
properties of any given object all arise out of a finite number of
generator properties, which we may call φ1φ2φ3 . . .. Some arise out
of φ1 alone, some out of φ1 in conjunction with φ2, and so on.

The properties which arise out of φ1 alone form one group; those
which arise out of φ1φ2 in conjunction form another group, and so
on. Since the number of generator properties is finite, the number
of groups also is finite. If a set of apparent properties arise (say)

out of three generator properties φ1φ2φ3, then this set of properties
may he said to specify the group φ1φ2φ3. Since the total number
of apparent properties is assumed to be greater than that of the
generator properties, and since the number of groups is finite, it
follows that, if two sets of apparent properties are taken, there is, in
the absence of evidence to the contrary, a finite probability that the
second set will belong to the group specified by the first set.

There is, however, the possibility of a plurality of generators.

The first set of apparent properties may specify more than one
group,—there is more than one group of generators, that is to say,
which are competent to produce it; and some only of these groups
may contain the second set of properties. Let us, for the moment,
rule out this possibility.

When we argue from an analogy, and the instances have two
groups of characters in common, namely φ and f, either f belongs
to the group φ or it arises out of generators partly distinct from
those out of which φ arises. For the reason already explained there
is a finite probability that f and φ belong to the same group. If
this is the case, i.e. if the generalisation g(φf) is valid, then f will
certainly be true of all other cases in which φ is true; if this is not
the case, then f will not always be true when φ is true. We have,
therefore, the preliminary conditions necessary for the application of
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293
pure induction. If xr, etc., are the instances,
g/h = p0, where p0 is finite,
xr/gh = 1, etc.,
and
xr/x1x2 . . . xr−1¯
gh = 1 − , where
is finite.

And hence, by the argument of Chapter XX., the probability of
a generalisation, based on such evidence as this, is capable, under
suitable conditions, of tending towards certainty as a limit, when the
number of instances is increased.

If φ is complex and includes a number of characters which are not
always found together, it must include a number of separate generator
properties and specify a large group; hence the initial probability that
f belongs to this group is relatively large. If, on the other hand,
f is complex, there will be, for the same reasons mutatis mutandis, a
relatively smaller initial probability than otherwise that f belongs to
any other given group.

When the argument is mainly by analogy, we endeavour to obtain
evidence which makes the initial probability p0 relatively high; when
the analogy is weak and the argument depends for its strength
upon pure induction, p0 is small and pm, which is based upon
numerous instances, depends for its magnitude upon their number.

But an argument from induction must always involve some element
of analogy, and, on the other hand, few arguments from analogy
can afford to ignore altogether the strengthening influence of pure
induction.

4.

Let us consider the manner in which the methods of analogy
increase the initial likelihood that two characters belong to the same
group. The numerous characters of an object which are known to
us may be represented by a1a2 . . . an. We select two sets of these,
ar and as, and seek to determine whether as always belongs to the
group specified by ar. Our previous knowledge will enable us, in
general, to rule out many of the object’s characters as being irrelevant
pt. iii

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294
to the groups specified by ar and as, although this will not be possible
in the most fundamental inquiries. We may also know that certain
characters are always associated with ar or with as. But there will be
left a residuum of whose connection with ar or as we are ignorant.

These characters, whose relevance is in doubt, may be represented by
ar+1 . . . as−1. If the analogy is perfect, these characters are eliminated
altogether. Otherwise, the argument is weakened in proportion to the
comprehensiveness of these doubtful characters. For it may be the
case that some of ar+1 . . . as−1 are necessary as well as ar, in order to
specify all the generators which are required to produce as.

5.

We may possibly be justified in neglecting certain of the
characters ar+1 . . . as−1 by direct judgments of irrelevance. There are
certain properties of objects which we rule out from the beginning
as wholly or largely independent and irrelevant to all, or to some,
other properties. The principal judgments of this kind, and those
alone about which we seem to feel much confidence, are concerned
with absolute position in time and space, this class of judgments of
irrelevance being summed up, I have suggested, in the Principle of the

Uniformity of Nature. We judge that mere position in time and space
cannot possibly affect, as a determining cause, any other characters;
and this belief appears so strong and certain, although it is hard to
see how it can be based on experience, that the judgment by which
we arrive at it seems perhaps to be direct. A further type of instance
in which some philosophers seem to have trusted direct judgments of
relevance in these matters arises out of the relation between mind
and matter. They have believed that no mental event can possibly be
a necessary condition for the occurrence of a material event.

The Principle of the Uniformity of Nature, as I interpret it,
supplies the answer, if it is correct, to the criticism that the instances,
on which generalisations are based, are all alike in being past, and
that any generalisation, which is applicable to the future, must be
based, for this reason, upon imperfect analogy. We judge directly
that the resemblance between instances, which consists in their being
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past, is in itself irrelevant, and does not supply a valid ground for
impugning a generalisation.

But these judgments of irrelevance are not free from difficulty,
and we must be suspicious of using them. When I say that position
is irrelevant, I do not mean to deny that a generalisation, the
premiss of which specifies position, may be true, and that the same
generalisation without this limitation might be false. But this is
because the generalisation is incompletely stated; it happens that
objects so specified have the required characters, and hence their
position supplies a sufficient criterion. Position may be relevant as
a sufficient condition but never as a necessary condition, and the
inclusion of it can only affect the truth of a generalisation when we
have left out some other essential condition. A generalisation which is
true of one instance must be true of another which only differs from
the former by reason of its position in time or space.

6.

Excluding, therefore, the possibility of a plurality of
generators, we can justify the method of perfect analogy, and other
inductive methods in so far as they can be made to approximate to
this, by means of the assumption that the objects in the field, over
which our generalisations extend, do not have an infinite number
of independent qualities; that, in other words, their characteristics,
however numerous, cohere together in groups of invariable connection,
which are finite in number. This does not limit the number of
entities which are only numerically distinct. In the language used
at the beginning of this chapter, the use of inductive methods can
be justified if they are applied to what we have reason to suppose a
finite system.∗

7.

Let us now take account of a possible plurality of generators.

I mean by this the possibility that a given character can arise in
more than one way, can belong to more than one distinct group, and
can arise out of more than one generator. φ might, for instance, be

∗Mr. C. D. Broad, in two articles “On the Relation between Induction and

Probability” (Mind, 1918 and 1920), has been following a similar line of thought.

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296
sometimes due to a generator α1, and α1 might invariably produce f.

But we could not generalise from φ to f, if φ might be due in other
cases to a different generator α2 which would not be competent to
produce f.

If we were dealing with inductive correlation, where we do not
claim universality for our conclusions, it would be sufficient for us
to assume that the number of distinct generators, to which a given
property φ can be due, is always finite. To obtain validity for universal
generalisations it seems necessary to make the more comprehensive
and less plausible assumption that a finite probability always exists
that there is not, in any given case, a plurality of causes. With this
assumption we have a valid argument from pure induction on the
same lines, nearly, as before.

8.

We have thus two distinct difficulties to deal with, and we
require for the solution of each a separate assumption. The point
may be illustrated by an example in which only one of the difficulties
is present. There are few arguments from analogy of which we are
better assured than the existence of other people. We feel indeed so
well assured of their existence that it has been thought sometimes
that our knowledge of them must be in some way direct. But analogy
does not seem to me unequal to the proof. We have numerous
experiences in our own person of acts which are associated with
states of consciousness, and we infer that similar acts in others are
likely to be associated with similar states of consciousness. But this
argument from analogy is superior in one respect to nearly all other
empirical arguments, and this superiority may possibly explain the
great confidence which we feel in it. We do seem in this case to have
direct knowledge, such as we have in no other case, that our states of
consciousness are, sometimes at least, causally connected with some
of our acts. We do not, as in other cases, merely observe invariable
sequence or coexistence between consciousness and act; and we do
believe it to be vastly improbable in the case of some at least of our
own physical acts that they could have occurred without a mental
ch. xxii

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act to support them. Thus, we seem to have a special assurance of
a kind not usually available for believing that there is sometimes a
necessary connection between the conclusion and the condition of the
generalisation; we doubt it only from the possibility of a plurality of
causes.

The objection to this argument on the ground that the analogy is
always imperfect, in that all the observed connections of consciousness
and act are alike in being mine, seems to me to be invalid on the
same ground as that on which I have put on one side objections to
future generalisations, which are based on the fact that the instances
which support them are all alike in being past. If direct judgments
of irrelevance are ever permissible, there seems some ground for
admitting one here.

9.

As a logical foundation for Analogy, therefore, we seem
to need some such assumption as that the amount of variety in the
universe is limited in such a way that there is no one object so
complex that its qualities fall into an infinite number of independent
groups (i.e. groups which might exist independently as well as in
conjunction); or rather that none of the objects about which we
generalise are as complex as this; or at least that, though some objects
may be infinitely complex, we sometimes have a finite probability that
an object about which we seek to generalise is not infinitely complex.

To meet a possible plurality of causes some further assumption is
necessary. If we were content with Inductive Correlations and sought
to prove merely that there was a probability in favour of any instance
of the generalisation in question, without inquiring whether there
was a probability in favour of every instance, it would be sufficient
to suppose that, while there may be more than one sufficient cause
of a character, there is not an infinite number of distinct causes
competent to produce it. And this involves no new assumption; for
if the aggregate variety of the system is finite, the possible plurality
of causes must also be finite. If, however, our generalisation is to
be universal, so that it breaks down if there is a single exception
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298
to it, we must obtain, by some means or other, a finite probability
that the set of characters, which condition the generalisation, are
not the possible effect of more than one distinct set of fundamental
properties. I do not know upon what ground we could establish
a finite probability to this effect. The necessity for this seemingly
arbitrary hypothesis strongly suggests that our conclusions should
be in the form of inductive correlations, rather than of universal
generalisations. Perhaps our generalisations should always run: ‘It is
probable that any given φ is f,’ rather than, ‘It is probable that all φ
are f.’ Certainly, what we commonly seem to hold with conviction is
the belief that the sun will rise to-morrow, rather than the belief that
the sun will always rise so long as the conditions explicitly known to
us are fulfilled. This will be matter for further discussion in Part V.,
when Inductive Correlation is specifically dealt with.

10.

There is a vagueness, it may be noticed, in the number
of instances, which would be required on the above assumptions to
establish a given numerical degree of probability, which corresponds to
the vagueness in the degree of probability which we do actually attach
to inductive conclusions. We assume that the necessary number of
instances is finite, but we do not know what the number is. We
know that the probability of a well-established induction is great, but,
when we are asked to name its degree, we cannot. Common sense
tells us that some inductive arguments are stronger than others, and
that some are very strong. But how much stronger or how strong we
cannot express. The probability of an induction is only numerically
definite when we are able to make definite assumptions about the
number of independent equiprobable influences at work. Otherwise,
it is non-numerical, though bearing relations of greater and less to
numerical probabilities according to the approximate limits within
which our assumption as to the possible number of these causes lies.

11.

Up to this point I have supposed, for the sake of simplicity,
that it is necessary to make our assumptions as to the limitation of
independent variety in an absolute form, to assume, that is to say,
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299
the finiteness of the system, to which the argument is applied, for
certain. But we need not in fact go so far as this.

If our conclusion is C and our empirical evidence is E, then, in
order to justify inductive methods, our premisses must include, in
addition to E, a general hypothesis H such that C/H, the à priori
probability of our conclusion, has a finite value. The effect of E
is to increase the probability of C above its initial à priori value,

C/HE being greater than C/H. But the method of strengthening C/H
by the addition of evidence E is valid quite apart from the particular
content of H. If, therefore, we have another general hypothesis H
and other evidence E , such that H/H has a finite value, we can,
without being guilty of a circular argument, use evidence E by
the same method as before to strengthen the probability H/H . If
we call H, namely, the absolute assertion of the finiteness of the
system under consideration, the inductive hypothesis, and the process
of strengthening C/H by the addition E the inductive method, it is
not circular to use the inductive method to strengthen the inductive
hypothesis itself, relative to some more primitive and less far-reaching
assumption. If, therefore, we have any reason (H ) for attributing
à priori a finite probability to the Inductive Hypothesis (H), then
the actual conformity of experience à posteriori with expectations
based on the assumption of H can be utilised by the inductive
method to attribute an enhanced value to the probability of H. To
this extent, therefore, we can support the Inductive Hypothesis by
experience. In dealing with any particular question we can take the

Inductive Hypothesis, not at its à priori value, but at the value to
which experience in general has raised it. What we require à priori,
therefore, is not the certainty of the Inductive Hypothesis, but a finite
probability in its favour.∗

∗I have implicitly assumed in the above argument that if H supports H, it
strengthens an argument which H would strengthen. This is not necessarily the
case for the reasons given on pp. 74 and 165. In these passages the necessary conditions for the above are elucidated. I am, therefore, assuming that in the
pt. iii

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300

Our assumption, in its most limited form, then, amounts to this,
that we have a finite à priori probability in favour of the Inductive

Hypothesis as to there being some limitation of independent variety

(to express shortly what I have already explained in detail) in the
objects of our generalisation. Our experience might have been such
as to diminish this probability à posteriori. It has, in fact, been such
as to increase it. It is because there has been so much repetition
and uniformity in our experience that we place great confidence in
it. To this extent the popular opinion that Induction depends upon
experience for its validity is justified and does not involve a circular
argument.

12.

I think that this assumption is adequate to its purpose
and would justify our ordinary methods of procedure in inductive
argument.

It was suggested in the previous chapter that our
theory of Analogy ought to be as applicable to mathematical as
to material generalisations, if it is to justify common sense. The
above assumptions of the limitation of independent variety sufficiently
satisfy this condition. There is nothing in these assumptions which
gives them a peculiar reference to material objects. We believe, in
fact, that all the properties of numbers can be derived from a limited
number of laws, and that the same set of laws governs all numbers.

To apply empirical methods to such things as numbers renders it
necessary, it is true, to make an assumption about the nature of
numbers. But it is the same kind of assumption as we have to make
about material objects, and has just about as much, or as little,
plausibility. There is no new difficulty.

The assumption, also, that the system of Nature is finite is
in accordance with the analysis of the underlying assumption of
scientists, given at the close of the previous chapter. The hypothesis
of atomic uniformity, as I have called it, while not formally equivalent
to the hypothesis of the limitation of independent variety, amounts
case now in question these conditions actually are fulfilled.

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to very much the same thing. If the fundamental laws of connection
changed altogether with variations, for instance, in the shape or size
of bodies, or if the laws governing the behaviour of a complex had
no relation whatever to the laws governing the behaviour of its parts
when belonging to other complexes, there could hardly be a limitation
of independent variety in the sense in which this has been defined.

And, on the other hand, a limitation of independent variety seems
necessarily to carry with it some degree of atomic uniformity. The
underlying conception as to the character of the System of Nature is
in each case the same.

13.

We have now reached the last and most difficult stage of the
discussion. The logical part of our inquiry is complete, and it has left
us, as it is its business to leave us, with a question of epistemology.

Such is the premiss or assumption which our logical processes need to
work upon. What right have we to make it? It is no sufficient answer
in philosophy to plead that the assumption is after all a very little
one.I do not believe that any conclusive or perfectly satisfactory
answer to this question can be given, so long as our knowledge of
the subject of epistemology is in so disordered and undeveloped a
condition as it is in at present. No proper answer has yet been
given to the inquiry—of what sorts of things are we capable of direct
knowledge? The logician, therefore, is in a weak position, when he
leaves his own subject and attempts to solve a particular instance of
this general problem. He needs guidance as to what kind of reason we
could have for such an assumption as the use of inductive argument
appears to require.

On the one hand, the assumption may be absolutely à priori in
the sense that it would be equally applicable to all possible objects.

On the other hand, it may be seen to be applicable to some classes
of objects only. In this Case it can only arise out of some degree
of particular knowledge as to the nature of the objects in question,
and is to this extent dependent on experience. But if it is experience
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302
which in this sense enables us to know the assumption as true of
certain amongst the objects of experience, it must enable us to know
it in some manner which we may term direct and not as the result of
an inference.

Now an assumption, that all systems of fact are finite (in the
sense in which I have defined this term), cannot, it seems perfectly
plain, be regarded as having absolute, universal validity in the sense
that such an assumption is self-evidently applicable to every kind of
object and to all possible experiences. It is not, therefore, in quite the
same position as a self-evident logical axiom, and does not appeal to
the mind in the same way. The most which can be maintained is that
this assumption is true of some systems of fact, and, further, that
there are some objects about which, as soon as we understand their
nature, the mind is able to apprehend directly that the assumption in
question is true.

In Chapter II. § 7, I wrote: “By some mental process of which
it is difficult to give an account, we are able to pass from direct
acquaintance with things to a knowledge of propositions about the
things of which we have sensations or understand the meaning.”

Knowledge, so obtained, I termed direct knowledge. From a sensation
of yellow and from an understanding of the meaning of ‘yellow’ and
of ‘colour,’ we could, I suggested, have direct knowledge of the fact or
proposition ‘yellow is a colour;’ we might also know that colour cannot
exist without extension, or that two colours cannot be perceived at
the same time in the same place. Other philosophers might use terms
differently and express themselves otherwise; but the substance of
what I was there trying to say is not very disputable. But when we
come to the question as to what kinds of propositions we can come
to know in this manner, we enter upon an unexplored field where no
certain opinion is discoverable.

In the case of logical terms, it seems to be generally agreed that
if we understand their meaning we can know directly propositions
about them which go far beyond a mere expression of this meaning;—
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propositions of the kind which some philosophers have termed
synthetic. In the case of non-logical or empirical entities, it seems
sometimes to be assumed that our direct knowledge must be confined
to what may be regarded as an expression or description of the
meaning or sensation apprehended by us. If this view is correct the

Inductive Hypothesis is not the kind of thing about which we can
have direct knowledge as a result of our acquaintance with objects.

I suggest, however, that this view is incorrect, and that we
are capable of direct knowledge about empirical entities which goes
beyond a mere expression of our understanding or sensation of them.

It may be useful to give the reader two examples, more familiar than
the Inductive Hypothesis, where, as it appears to me, such knowledge
is commonly assumed. The first is that of the causal irrelevance of
mere position in time and space, commonly called the Uniformity of

Nature. We do believe, and yet have no adequate inductive reason
whatever for believing, that mere position in time and space cannot
make any difference. This belief arises directly, I think, out of our
acquaintance with the objects of experience and our understanding of
the concepts of ‘time’ and ‘space.’ The second is that of the Law of

Causation. We believe that every object in time has a ‘necessary’

connection∗ with some set of objects at a previous time. This belief
also, I think, arises in the same way. It is to be noticed that neither
of these beliefs clearly arises, in spite of the directness which may
be claimed for them, out of any one single experience. In a way
analogous to these, the validity of assuming the Inductive Hypothesis,
as applied to a particular class of objects, appears to me to be
justified.

Our justification for using inductive methods in an argument about
numbers arises out of our perceiving directly, when we understand
the meaning of a number, that they are of the required character.†

∗I do not propose to define the meaning of this.

†Since numbers are logical entities, it may be thought less unorthodox to
make such an assumption in their case.

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304

And when we perceive the nature of our phenomenal experiences,
we have a direct assurance that in their case also the assumption
is legitimate. We are capable, that is to say, of direct synthetic
knowledge about the nature of the objects of our experience. On the
other hand, there may be some kinds of objects, about which we have
no such assurance and to which inductive methods are not reasonably
applicable. It may be the case that some metaphysical questions are
of this character and that those philosophers have been right who
have refused to apply empirical methods to them.

14.

I do not pretend that I have given any perfectly adequate
reason for accepting the theory I have expounded, or any such theory.

The Inductive Hypothesis stands in a peculiar position in that it
seems to be neither a self-evident logical axiom nor an object of
direct acquaintance; and yet it is just as difficult, as though the
inductive hypothesis were either of these, to remove from the organon
of thought the inductive method which can only be based on it or on
something like it.

As long as the theory of knowledge is so imperfectly understood
as now, and leaves us so uncertain about the grounds of many of
our firmest convictions, it would be absurd to confess to a special
scepticism about this one. I do not think that the foregoing argument
has disclosed a reason for such scepticism. We need not lay aside the
belief that this conviction gets its invincible certainty from some valid
principle darkly present to our minds, even though it still eludes the
peering eyes of philosophy.

CHAPTER XXIII
some historical notes on induction

1.

The number of books, which deal with inductive∗ theory, is
extraordinarily small. It is usual to associate the subject with the
names of Bacon, Hume, and Mill. In spite of the modern tendency to
depreciate the first and the last of these, they are the principal names,

I think, with which the history of induction ought to be associated.

The next place is held by Laplace and Jevons. Amongst contemporary
logicians there is an almost complete absence of constructive theory,
and they content themselves for the most part with the easy task of
criticising Mill, or with the more difficult one of following him.

That the inductive theories of Bacon and of Mill are full of errors
and even of absurdities, is, of course, a commonplace of criticism.

But when we ignore details, it becomes clear that they were really
attempting to disentangle the essential issues. We depreciate them
partly, perhaps, as a reaction from the view once held that they
helped the progress of scientific discovery. For it is not plausible to
suppose that Newton owed anything to Bacon, or Darwin to Mill.

But with the logical problem their minds were truly occupied, and in
the history of logical theory they should always be important.

It is true, nevertheless, that the advancement of science was the
main object which Bacon himself, though not Mill, believed that
his philosophy would promote. The Great Instauration was intended
to promulgate an actual method of discovery entirely different from
any which had been previously known.† It did not do this, and
against such pretensions Macaulay’s well-known essay was not unjustly
directed. Mill, however, expressly disclaimed in his preface any other

∗See note at the end of this chapter on “The Use of the Term Induction.”

†He speaks of himself as being “in hac re plane protopirus, et vestigia
nullius sequutus”; and in the Praefatio Generalis he compares his method to the
mariner’s compass, until the discovery of which no wide sea could be crossed

(see Spedding and Ellis, vol. i. p. 24).

305
pt. iii

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306
object than to classify and generalise the practices “conformed to
by accurate thinkers in their scientific inquiries.” Whereas Bacon
offered rules and demonstrations, hitherto unknown, with which any
man could solve all the problems of science by taking pains, Mill
admitted that “in the existing state of the cultivation of the sciences,
there would be a very strong presumption against any one who
should imagine that he had effected a revolution in the theory of the
investigation of truth, or added any fundamentally new process to the
practice of it.”

2.

The theories of both seem to me to have been injured, though
in different degrees, by a failure to keep quite distinct the three
objects: (1) of helping the scientist, (2) of explaining and analysing
his practice, and (3) of justifying it. Bacon was really interested in
the second as well as in the first, and was led to some of his methods
by reflecting upon what distinguished good arguments from bad in
actual investigations. To logicians his methods were as new as he
claimed, but they had their origin, nevertheless, in the commonest
inferences of science and daily life. But his main preoccupation was
with the first, which did injury to his treatment of the third. He
himself became aware as the work progressed that, in his anxiety to
provide an infallible mode of discovery, he had put forth more than
he would ever be able to justify.∗ His own mind grew doubtful,
and the most critical parts of the description of the new method
were never written. No one who has reflected much upon Induction
need find it difficult to understand the progress and development of

Bacon’s thoughts. To the philosopher who first distinguished some of
the complexities of empirical proof in a generalised, and not merely a
particular, form, the prospects of systematising these methods must
have seemed extraordinarily hopeful. The first investigator could not

∗This view is taken in the edition of James Spedding and Leslie Ellis. Their
introductions to Bacon’s philosophical works seem to me to be very greatly
superior to the accounts to be found elsewhere. They make intelligible, what
seems, according to other commentaries, fanciful and without sense or reason.

ch. xxiii

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have anticipated that Induction, in spite of its apparent certainty,
would prove so elusive to analysis.

Mill also was led, in a not dissimilar way, to attempt a too simple
treatment, and, in seeking for ease and certainty, to treat far too
lightly the problem of justifying what he had claimed. Mill shirks,
almost openly, the difficulties; and scarcely attempts to disguise from
himself or his readers that he grounds induction upon a circular
argument.

3.

Some of the most characteristic errors both of Bacon
and of Mill arise, I think, out of a misapprehension, which it has
been a principal object of this book to correct. Both believed,
without hesitation it seems, that induction is capable of establishing
a conclusion which is absolutely certain, and that an argument is
invalid if the generalisation, which it supports, admits of exceptions
in fact.

“Absolute certainty,” says Leslie Ellis,∗ “is one of the
distinguishing characters of the Baconian induction.”

It was, in
this respect, mainly that it improved upon the older induction
per enumerationem simplicem. “The induction which the logicians
speak of,” Bacon argues in the Advancement of Learning, “is utterly
vicious and incompetent. . . . For to conclude upon an enumeration
of particulars, without instance contradictory, is no conclusion but a
conjecture.” The conclusions of the new method, unlike those of the
old, are not liable to be upset by further experience. In the attempt
to justify these claims and to obtain demonstrative methods, it was
necessary to introduce assumptions for which there was no warrant.

Precisely similar claims were made by Mill, although there are
passages in which he abates them,† for his own rules of procedure.

An induction has no validity, according to him as according to Bacon,
unless it is absolutely certain. The following passage‡ is significant
of the spirit in which the subject was approached by him: “Let us

∗Op. cit. vol. i. p. 23.

†When he deals with Plurality of Causes, for instance.

‡Bk. iii. chap. iii. 3 (the italics are mine).

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308
compare a few cases of incorrect inductions with others which are
acknowledged to be legitimate. Some, we know, which were believed
for centuries to be correct, were nevertheless incorrect. That all swans
are white, cannot have been a good induction, since the conclusion
has turned out erroneous.

The experience, however, on which the
conclusion rested was genuine.” Mill has not justly apprehended the
relativity of all inductive arguments to the evidence, nor the element
of uncertainty which is present, more or less, in all the generalisations
which they support.∗ Mill’s methods would yield certainty, if they
were correct, just as Bacon’s would. It is the necessity, to which

Mill had subjected himself, of obtaining certainty that occasions their
want of reality. Bacon and Mill both assume that experiment can
shape and analyse the evidence in a manner and to an extent which
is not in fact possible. In the aims and expectations with which
they attempt to solve the inductive problem, there is on fundamental
points an unexpectedly close resemblance between them.

4.

Turning from these general criticisms to points of greater
detail, we find that the line of thought pursued by Mill was essentially
the same as that which had been pursued by Bacon, and, also,
that the argument of the preceding chapters is, in spite of some
real differences, a development of the same fundamental ideas which
underlie, as it seems to me, the theories of Mill and Bacon alike.

We have seen that all empirical arguments require an initial
probability derived from analogy, and that this initial probability
may be raised towards certainty by means of pure induction or the
multiplication of instances. In some arguments we depend mainly
upon analogy, and the initial probability obtained by means of it

(with the assistance, as a rule, of previous knowledge) is so large

∗This misapprehension may be connected with Mill’s complete failure to
grasp with any kind of thoroughness the nature and importance of the theory
of probability. The treatment of this topic in the System of Logic is exceedingly
bad. His understanding of the subject was, indeed, markedly inferior to the best
thought of his own time.

ch. xxiii

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that numerous instances are not required. In other arguments pure
induction predominates. As science advances and the body of pre-
existing knowledge is increased, we depend increasingly upon analogy;
and only at the earlier stages of our investigations is it necessary
to rely, for the greater part of our support, upon the multiplication
of instances. Bacon’s great achievement, in the history of logical
theory, lay in his being the first logician to recognise the importance
of methodical analogy to scientific argument and the dependence
upon it of most well-established conclusions. The Novum Organum is
mainly concerned with explaining methodical ways of increasing what

I have termed the Positive and Negative Analogies, and of avoiding
false Analogies. The use of exclusions and rejections, to which Bacon
attached supreme importance, and which he held to constitute the
essential superiority of his method over those which preceded it,
entirely consists in the determination of what characters (or natures
as he would call them) belong to the positive and negative analogies
respectively. The first two tables with which the investigation begins
are, first, the table essentiae et praesentiae, which contains all known
instances in which the given nature is present, and, second, the
table declinationis sive absentiae in proximo, which contains instances
corresponding in each case to those of the first table, but in which,
notwithstanding this correspondence, the given nature is absent.∗ The
doctrine of prerogative instances is concerned no less plainly with
the methodical determination of Analogy. And the doctrine of idols
is expounded for the avoidance of false analogies, standing, he says,
in the same relation to the interpretation of Nature, as the doctrine
of fallacies to ordinary logic.† Bacon’s error lay in supposing that,
because these methods were new to logic, they were therefore new
to practice. He exaggerated also their precision and their certainty;
and he underestimated the importance of pure induction. But there
was, at bottom, nothing about his rules impracticable or fantastic, or

∗Ellis, vol. i. p. 33.

†Ellis, vol. i. p. 89.

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310
indeed unusual.

5.

Almost the whole of the preceding paragraph is equally
applicable to Mill. He agreed with Bacon in depreciating the part
played in scientific inquiry by pure induction, and in emphasising
the importance of analogy to all systematic investigators. But he
saw further than Bacon in allowing for the Plurality of Causes, and
in admitting that an element of pure induction was therefore made
necessary. “The Plurality of Causes,” he says,∗ “is the only reason
why mere number of instances is of any importance in inductive
inquiry. The tendency of unscientific inquirers is to rely too much
on number, without analysing the instances. . . . Most people hold
their conclusions with a degree of assurance proportioned to the mere
mass of the experience on which they appear to rest; not considering
that by the addition of instances to instances, all of the same kind,
that is, differing from one another only in points already recognised
as immaterial, nothing whatever is added to the evidence of the
conclusion. A single instance eliminating some antecedent which
existed in all the other cases, is of more value than the greatest
multitude of instances which are reckoned by their number alone.”

Mill did not see, however, that our knowledge of the instances is
seldom complete, and that new instances, which are not known to
differ from the former in material respects, may add, nevertheless,
to the negative analogy, and that the multiplication of them may,
for this reason, strengthen the evidence. It is easy to see that his
methods of Agreement and Difference closely resemble Bacon’s, and
aim, like Bacon’s, at the determination of the Positive and Negative

Analogies. By allowing for Plurality of Causes Mill advanced beyond

Bacon. But he was pursuing the same line of thought which alike
led to Bacon’s rules and has been developed in the chapters of this
book. Like Bacon, however, he exaggerated the precision with which

∗Book iv. chap. x. 2.

ch. xxiii

INDUCTION AND ANALOGY

311
his canons of inquiry could be used in practice.

6.

No more need be said respecting method and analysis.

But in both writers the exposition of method is closely intermingled
with attempts to justify it. There is nothing in Bacon which at all
corresponds to Mill’s appeals to Causation or to the Uniformity of

Nature, and, when they seek for the ground of induction, there is
much that is peculiar to each writer. It is my purpose, however, to
consider in this place the details common to both, which seem to me
to be important and which exemplify the only line of investigation
which seems likely to be fruitful; and I shall pursue no further,
therefore, their numerous points of difference.

The attempt, which I have made to justify the initial probability
which Analogy seems to supply, primarily depends upon a certain
limitation of independent variety and upon the derivation of all the
properties of any given object from a limited number of primary
characters. In the same way I have supposed that the number of
primary characters which are capable of producing a given property
is also limited. And I have argued that it is not easy to see how a
finite probability is to be obtained unless we have in each case some
such limitation in the number of the ultimate alternatives.

It was in a manner which bears fundamental resemblances to this
that Bacon endeavoured to demonstrate the cogency of his method.

He considers, he says, “the simple forms or difference of things which
are few in number, and the degrees and co-ordinations whereof make
all this variety,” And in Valerius Terminus he argues “that every
particular that worketh any effect is a thing compounded more or less
of diverse single natures, more manifest and more obscure, and that
it appeareth not to which of the natures the effect is to be ascribed.”∗

It is indeed essential to the method of exclusions that the matter to
which it is applied should be somehow resolvable into a finite number
of elements. But this assumption is not peculiar, I think, to Bacon’s

∗Quoted by Ellis, vol. i. p. 41.

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312
method, and is involved, in some form or other, in every argument
from Analogy. In making it Bacon was initiating, perhaps obscurely,
the modern conception of a finite number of laws of nature out of
the combinations of which the almost boundless variety of experience
ultimately arises. Bacon’s error was double and lay in supposing,
first, that these distinct elements lie upon the surface and consist in
visible characters, and second, that their natures are, or easily can
be, known to us, although the part of the Instauration, in which
the manner of conceiving simple natures was to be explained, he
never wrote. These beliefs falsely simplified the problem as he saw
it, and led him to exaggerate the ease, certainty, and fruitfulness of
the new method. But the view that it is possible to reduce all the
phenomena of the universe to combinations of a limited number of
simple elements—which is, according to Ellis,∗ the central point of

Bacon’s whole system—was a real contribution to philosophy.

7.

The assumption that every event can be analysed into
a limited number of ultimate elements, is never, so far as I am
aware, explicitly avowed by Mill. But he makes it in almost every
chapter, and it underlies, throughout, his mode of procedure. His
methods and arguments would fail immediately, if we were to suppose
that phenomena of infinite complexity, due to an infinite number of
independent elements, were in question, or if an infinite plurality of
causes had to be allowed for.

In distinguishing, therefore, analogy from pure induction, and
in justifying it by the assumption of a limited complexity in the
problems which we investigate, I am, I think, pursuing, with numerous
differences, the line of thought which Bacon first pursued and which

Mill popularised. The method of treatment is dissimilar, but the
subject-matter and the underlying beliefs are, in each case, the same.

8.

Between Bacon and Mill came Hume. Hume’s sceptical
criticisms are usually associated with causality; but argument by

∗Vol. i. p. 28.

ch. xxiii

INDUCTION AND ANALOGY

313
induction—inference from past particulars to future generalisations—
was the real object of his attack. Hume showed, not that inductive
methods were false, but that their validity had never been established
and that all possible lines of proof seemed equally unpromising. The
full force of Hume’s attack and the nature of the difficulties which
it brought to light were never appreciated by Mill, and he makes no
adequate attempt to deal with them. Hume’s statement of the case
against induction has never been improved upon; and the successive
attempts of philosophers, led by Kant, to discover a transcendental
solution have prevented them from meeting the hostile arguments on
their own ground and from finding a solution along lines which might,
conceivably, have satisfied Hume himself.

9.

It would not be just here to pass by entirely the name of the
great Leibniz, who, wiser in correspondence and fragmentary projects
than in completed discourses, has left to us sufficient indications
that his private reflections on this subject were much in advance
of his contemporaries’. He distinguished three degrees of conviction
amongst opinions, logical certainty (or, as we should say, propositions
known to be formally true), physical certainty which is only logical
probability, of which a well-established induction, as that man is a
biped, is the type, and physical probability (or, as we should say,
an inductive correlation), as for example that the south is a rainy
quarter.∗ He condemned generalisations based on mere repetition
of instances, which he declared to be without logical value, and
he insisted on the importance of Analogy as the basis of a valid
induction.† He regarded a hypothesis as more probable in proportion
to its simplicity and its power, that is to say, to the number of
the phenomena it would explain and the fewness of the assumptions
it involved. In particular a power of accurate prediction and of
explaining phenomena or experiments previously untried is a just
ground of secure confidence, of which he cites as a nearly perfect

∗Couturat, Opuscules et fragments inédits de Leibniz, p. 232.

†Couturat, La Logique de Leibniz d’après des documents inédits, pp. 262, 267.

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A TREATISE ON PROBABILITY

314
example the key to a cryptogram.∗

10.

Whewell and Jevons furnished logicians with a storehouse
of examples derived from the practice of scientists. Jevons, partly
anticipated by Laplace, made an important advance when he
emphasised the close relation between Induction and Probability.

Combining insight and error, he spoilt brilliant suggestions by erratic
and atrocious arguments. His application of Inverse Probability to
the inductive problem is crude and fallacious, but the idea which
underlies it is substantially good. He, too, made explicit the element
of Analogy, which Mill, though he constantly employed it, had seldom
called by its right name. There are few books, so superficial in
argument yet suggesting so much truth, as Jevons’s Principles of

Science.

11.

Modern text-books on Logic all contain their chapters
on Induction, but contribute little to the subject. Their recognition
of Mill’s inadequacy renders their exposition, which, in spite of
criticisms, is generally along his lines, nerveless and confused. Where

Mill is clear and offers a solution, they, confusedly criticising, must
withhold one. The best of them, Sigwart and Venn, contain criticism
and discussion which is interesting, but constructive theory is lacking.

Hitherto Hume has been master, only to be refuted in the manner of

Diogenes or Dr. Johnson.

∗Letter to Conring, 19th March 1678.

NOTES ON PART III

(i.) On the Use of the Term Induction

1.

Induction is in origin a translation of the Aristotelian
ἐπαγωγή.

This term was used by Aristotle in two quite distinct
senses—first, and principally, for the process by which the observation
of particular instances, in which an abstract notion is exemplified,
enables us to realise and comprehend the abstraction itself; secondly,
for the type of argument in which we generalise after the complete
enumeration and assertion of all the particulars which the generali-
sation embraces. From this second sense it was sometimes extended
to cases in which we generalise after an incomplete enumeration. In
post-Aristotelian writers the induction per enumerationem simplicem
approximates to induction in Aristotle’s second sense, as the number
of instances is increased. To Bacon, therefore, “the induction of which
the logicians speak” meant a method of argument by multiplication
of instances. He himself deliberately extended the use of the term so
as to cover all the systematic processes of empirical generalisation.

But he also used it, in a manner closely corresponding to Aristotle’s
first use, for the process of forming scientific conceptions and correct
notions of “simple natures.”∗

2.

The modern use of the term is derived from Bacon’s.

Mill defines it as “the operation of discovering and proving general
propositions.” His philosophical system required that he should define
it as widely as this; but the term has really been used, both by
him and by other logicians, in a narrower sense, so as to cover those
methods of proving general propositions, which we call empirical, and
so as to exclude generalisations, such as those of mathematics, which
have been proved formally. Jevons was led, partly by the linguistic

∗See Ellis’s edition of Bacon’s Works, vol. i. p. 37. On the first occasion
on which Induction is mentioned in the Novum Organum, it is used in this
secondary sense.

315
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A TREATISE ON PROBABILITY

316
resemblance, partly because in the one case we proceed from the
particular to the general and in the other from the general to the
particular, to define Induction as the inverse process of Deduction.

In contemporary logic Mill’s use prevails; but there is, at the same
time, a suggestion—arising from earlier usage, and because Bacon
and Mill never quite freed themselves from it—of argument by mere
multiplication of instances. I have thought it best, therefore, to use
the term pure induction to describe arguments which are based upon
the number of instances, and to use induction itself for all those types
of arguments which combine, in one form or another, pure induction
with analogy.

(ii.) On the Use of the Term Cause

1.

Throughout the preceding argument, as well as in Part II., I have been able to avoid the metaphysical difficulties which surround
the true meaning of cause. It was not necessary that I should inquire
whether I meant by causal connection an invariable connection in
fact merely, or whether some more intimate relation was involved. It
has also been convenient to speak of causal relations between objects
which do not strictly stand in the position of cause and effect, and
even to speak of a probable cause, where there is no implication of
necessity and where the antecedents will sometimes lead to particular
consequents and sometimes will not. In making this use of the
term, I have followed a practice not uncommon amongst writers on
probability, who constantly use the term cause, where hypothesis
might seem more appropriate.∗

One is led, almost inevitably, to use ‘cause’ more widely than

‘sufficient cause’ or than ‘necessary cause,’ because, the necessary

∗Cf. Czuber, Wahrscheinlichkeitsrechnung, p. 139. In dealing with Inverse

Probability Czuber explains that he means by possible cause the various

Bedingungskomplexe from which the cause can result.

notes

INDUCTION AND ANALOGY

317
causation of particulars by particulars being rarely apparent to us,
the strict sense of the term has little utility. Those antecedent
circumstances, which we are usually content to accept as causes, are
only so in strictness under a favourable conjunction of innumerable
other influences.

2.

As our knowledge is partial, there is constantly, in our use of
the term cause, some reference implied or expressed to a limited body
of knowledge. It is clear that, whether or not, as Cournot∗ maintains,
there are such things as independent series in the order of causation,
there is often a sense in which we may hold that there is a closer
intimacy between some series than between others. This intimacy is
relative, I think, to particular information, which is actually known to
us, or which is within our reach. It will be useful, therefore, to give
precise definitions of these wider senses in which it is often convenient
to use the expression cause.

We must first distinguish between assertions of law and assertions
of fact, or, in the terminology of Von Kries,† between nomologic
and ontologic knowledge. It may be convenient in dealing with
some questions to frame this distinction with reference to the special
circumstances. But the distinction generally applicable is between
propositions which contain no reference to particular moments of
time, and existential propositions which cannot be stated without
reference to specific points in the time series. The Principle of the

Uniformity of Nature amounts to the assertion that natural laws are
all, in this sense, timeless. We may, therefore, divide our data into
two portions k and l, such that k denotes our formal and nomologic
evidence, consisting of propositions whose predication does not involve
a particular time reference, and l denotes the existential or ontologic
propositions.

3.

Let us now suppose that we are investigating two existential
propositions a and b, which refer two events A and B to particular

∗See Chapter XXIV. § 3.

†Die Principien der Wahrscheinlichkeitsrechnung, p. 86.

pt. iii

A TREATISE ON PROBABILITY

318
moments of time, and that A is referred to moments which are all
prior to those at which B occurred. What various meanings can we
give to the assertion that A and B are causally connected?

(i.) If b/ak = 1, A is a sufficient cause of B. In this case A is a
cause of B in the strictest sense. b can be inferred from a, and no
additional knowledge consistent with k can invalidate this.

(ii.) If b/¯ak = 0, A is a necessary cause of B.

(iii.) If k includes all the laws of the existent universe, then A is
not a sufficient cause of B unless b/ak = 1. The Law of Causation,
therefore, which states that every existent has to some other previous
existent the relation of effect to sufficient cause, is equivalent to the
proposition that, if k is the body of natural law, then, if b is true,
there is always another true proposition a, which asserts existences
prior to B, such that b/ak = 1. No use has been made so far
of our existential knowledge l, which is irrelevant to the definitions
preceding.

(iv.) If b/akl = 1 and b/kl =| 1, A is a sufficient cause of B under
conditions l.

(v.) If b/¯akl = 0 and b/kl =| 0, A is a necessary cause of B under
conditions l.

(vi.) If there is any existential proposition h such that b/ahk = 1
and b/hk =| 1, A is, relative to k, a possible sufficient cause of B.

(vii.) If there is an existential proposition h such that b/¯ahk = 0
and b/hk =| 0, A is, relative to k, a possible necessary cause of B.

(viii.) If b/ahkl = 1, b/hk =| 1, and h/akl =| 0, A is, relative to k,
a possible sufficient cause of B under conditions l.

(ix.) If b/¯ahkl = 0, b/hkl =| 0, h/¯akl = 0, and h/akl =| 0, A is,
relative to k, a possible necessary cause of B under conditions l. Thus
an event is a possible necessary cause of another, relative to given
nomologic data, if circumstances can arise, not inconsistent with our
existential data, in which the first event will be indispensable if the
second is to occur.

(x.) Two events are causally independent if no part of either
notes

INDUCTION AND ANALOGY

319
is, relative to our nomologic data, a possible cause of any part of
the other under the conditions of our existential knowledge. The
greater the scope of our existential knowledge, the greater is the
likelihood of our being able to pronounce events causally dependent
or independent.

4.

These definitions preserve the distinction between ‘causally
independent’ and ‘independent for probability,’—the distinction be-
tween causa essendi and causa cognoscendi. If b/ahkl =| b/¯ahkl, where
a and b may be any propositions whatever and are not limited as they
were in the causal definitions, we have ‘dependence for probability,’

and a is a causa cognoscendi for b, relative to data kl. If a and b are
causally dependent, according to definition (x.), b is a possible causa
essendi, relative to data kl.

But, after all, the essential relation is that of ‘independence for
probability.’ We wish to know whether knowledge of one fact throws
light of any kind upon the likelihood of another. The theory of
causality is only important because it is thought that by means
of its assumptions light can be thrown by the experience of one
phenomenon upon the expectation of another.

PART IV
some philosophical applications
of probability

CHAPTER XXIV
the meanings of objective chance, and of
randomness

1.

Many important differences of opinion in the treatment
of Probability have been due to confusion or vagueness as to what
is meant by Randomness and by Objective Chance, as distinguished
from what, for the purposes of this chapter, may be termed Subjective

Probability. It is agreed that there is a sort of Probability which
depends upon knowledge and ignorance, and is relative, in some
manner, to the mind of the subject; but it is supposed that there
is also a more objective Probability which is not thus dependent,
or less completely so, though precisely what this conception stands
for is not plain. The relation of Randomness to the other concepts
is also obscure. The problem of clearing up these distinctions is of
importance if we are to criticise certain schools of opinion intelligently,
as well as to the treatment of the foundations of Statistical Inference
which is to be attempted in Part V.

There are at least three distinct issues to be kept apart. There is
the antithesis between knowledge and ignorance, between events, that
is to say, which we have some reason to expect, and events which we
have no reason to expect, which gives rise to the theory of subjective
probability and subjective chance; and, connected with this, the
distinction between ‘random’ selection and ‘biassed’ selection. There
are next objective probability and objective chance, which are as yet
obscure, but which are commonly held to arise out of the antithesis
between ‘cause’ and ‘chance,’ between events, that is to say, which are
causally connected and events which are not causally connected. And
there is, lastly, the antithesis between chance and design, between

‘blind causes’ and ‘final causes,’ where we oppose a ‘chance’ event to
one, part of whose cause is a volition following on a conscious desire

321
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A TREATISE ON PROBABILITY

322
for the event.∗

2.

The method of this treatise has been to regard subjective
probability as fundamental and to treat all other relevant conceptions
as derivative from this. That there is such a thing as probability in
this sense has been admitted by all sensible philosophers since the
middle of the eighteenth century at least.† But there is also, many
writers have supposed, something else which may be fitly described as
objective probability; and there is, besides, a long tradition in favour
of the view that it is this (whatever it may be) which is logically and
philosophically important, subjective probability being a vague and
mainly psychological conception about which there is very little to be
said.The distinction exists already in Hume: “Probability is of two
kinds, either when the object is really in itself uncertain, and to be
determined by chance; or when, though the object be already certain,
yet ’tis uncertain to our judgment, which finds a number of proofs
on each side of the question.”‡ But the distinction is not elucidated,
and one can only infer from other passages that Hume did not intend
to imply in this passage the existence of objective chance in a sense
contradictory to a determinist theory of the Universe. In Condorcet
all is confused; and in Laplace nearly all. In the nineteenth century
the distinction begins to grow explicit in the writings of Cournot.

“Les explications que j’ai données. . . ,” he writes in the preface to
his Exposition, “sur le double sens du mot de probabilité, qui tantôt
se rapporte à une certaine mesure de nos connaissances, et tantôt
à une mesure de la possibilité des choses, indépendamment de la

∗This is discussed in Chapter XXV. § 4.

†D’Alembert, collecting (largely from Hume, many passages being translated
almost verbatim) in the Encyclopédie méthodique the most up-to-date common-
places of the subject, found it natural to write: “Il n’y a point de hasard à
proprement parler; mais il y a son équivalent: l’ignorance, où nous sommes des
vraies causes des événemens, a sur notre esprit l’influence qu’on suppose au
hasard.” Compare also the sentences from Spinoza quoted on p. 117 above.

‡A Treatise of Human Nature, Book ii. part iii. section ix.

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PHILOSOPHICAL APPLICATIONS

323
connaissance que nous en avons: ces explications, dis-je, me semblent
propres à resoudre les difficultés qui ont rendu jusqu’ici suspecte à
de bons esprits toute la théorie de la probabilité mathématique.” It
will be worth while to pause for a moment to consider the ideas of

Cournot.

3.

Cournot, while admitting that there is such a thing as
subjective chance, was concerned to dispute the opinion that chance is
merely the offspring of ignorance, saying that in this case “le calcul des
chances” is merely “un calcul des illusions.” The chance, upon which

“le calcul des chances” is based, is something different, and depends,
according to him, on the combination or convergence of phenomena
belonging to independent series. By “independent series” he means
series of phenomena which develop as parallel or successive series
without any causal interdependence or link of solidarity whatever.∗

No one, he says by way of example, seriously believes that in striking
the ground with his foot he puts out the navigator in the Antipodes,
or disturbs the system of Jupiter’s satellites. Separate trains of
events, that is to say, have been set going by distinct initial acts
of creation, so to speak.† Every event is causally connected with
previous events belonging to its own series, but it cannot be modified
by contact with events belonging to a different series. A ‘chance’

∗“Le mot hasard,” Cournot writes in his Essai sur les fondemenis de nos
connaissances, “n’indique pas une cause substantielle, mais une idée: cette idée
est celle de la combinaison entre plusieurs séries de causes ou de faits qui se
développent chacun dans sa série propre, indépendamment les uns des autres.”

This is very like the definition given by Jean de la Placette in his Traité des
jeux de hasard, to which Cournot refers: “Pour moi, je suis persuadé que le
hasard renferme quelque chose de réal et de positif, savoir un concours de deux
ou plusieurs événements contingents, chacun desquels a ses causes, mais en sorte
que leur concours n’en a aucune que l’on connaisse.”

†Essai sur les fondements de nos connaissances, i. 134: “La nature ne se
gouverne pas par une loi unique. . . ses lois ne sont pas toutes dérivées les unes
des autres, on dérivées toutes d’une loi supérieure par une nécessité purement
logique. . . nous devons les concevoir au contraire comme ayant pu étre décrétées
séparément d’une infinité de manières.”

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A TREATISE ON PROBABILITY

324
event is a complex due to the concurrence in time or place of events
belonging to causally independent series.

This theory, as it stands, is evidently unsatisfactory. Even if there
are series of phenomena which are independent in Cournot’s sense,
it is not clear how we can know which they are, or how we can
set up a calculus which presumes an acquaintance with them. Just
as it is likely that we are all cousins if we go back far enough, so
there may be, after all, remote relationships between ourselves and

Jupiter. A remote connection or a reaction quantitatively small is a
matter of degree and not by any means the same thing as absolute
independence. Nevertheless Cournot has contributed something, I
think, to the stock of our ideas. He has hinted at, even if he has not
disentangled, one of the elements in a common conception of chance;
and of the notion, which he seems to have in his mind, we must in
due course take account.∗

4.

In the writings of Condorcet, I have said above, all is confused.

But in Bertrand’s criticism of him a relevant distinction, though not
elucidated, is brought before the mind. “The motives for believing,”

wrote Condorcet, “that, from ten million white balls mixed with one

∗Cournot’s work on Probability has been highly praised by authorities as
diverse and distinguished as Boole and Von Kries, and has been made the
foundation of a school by some recent French philosophers (see the special
number of the Revue de métaphysique et de morale, devoted to Cournot and
published in 1905, and the bibliography at the end of the present volume
passim). The best account with which I am acquainted, of Cournot’s theory of
probability, is to be found in A. Darbon’s Le Concept du hasard.

Cournot’s
philosophy of the subject is developed, not so much in his Exposition de la
théorie des chances, as in later works, especially in his Essai sur les fondements
de nos connaissances. Cournot never touched any subject without contributing
something to it, but, on the whole, his work on Probability is, in my opinion,
disappointing. No doubt his Exposition is superior to other French text-books
of the period, of which there is so large a variety, and his work, both here and
elsewhere, is not without illuminating ideas: but the philosophical treatment is
so confused and indefinite that it is difficult to make much of it beyond the one
specific point treated above.

ch. xxiv

PHILOSOPHICAL APPLICATIONS

325
black, it will not be the black ball which I shall draw at the first
attempt is of the same kind as the motive for believing that the sun
will not fail to rise to-morrow.” “The assimilation of the two cases,”

Bertrand writes in criticism of the above,∗ “is not legitimate: one of
the probabilities is objective, the other subjective. The probability
of drawing the black ball at the first attempt is

1

, neither

10,000,000
more nor less. Whoever evaluates it otherwise makes a mistake. The
probability that the sun will rise varies from one mind to another.

A scientist might hold on the basis of a false theory, without being
utterly irrational, that the sun will soon be extinguished; he would
be within his rights, just as Condorcet is within his; both would
exceed their rights in accusing of error those who think differently.”

Before commenting on this distinction, let us have before us also some
interesting passages by Poincaré.

5.

We certainly do not use the term ‘chance,’ Poincaré points
out, as the ancients used it, in opposition to determinism. For us
therefore the natural interpretation of ‘chance’ is subjective,—‘Chance
is only the measure of our ignorance. Fortuitous phenomena are, by
definition, those, of the laws of which we are ignorant.” But Poincaré
immediately adds: “Is this definition very satisfactory? When the first

Chaldaean shepherds followed with their eyes the movements of the
stars, they did not yet know the laws of astronomy, but would they
have dreamed of saying that the stars move by chance? If a modern
physicist is studying a new phenomenon, and if he discovers its law
on Tuesday, would he have said on Monday that the phenomenon was
fortuitous?”†

There is also another type of case in which “chance must be
something more than the name we give to our ignorance.” Among the

∗Calcul des probabilités, p. xix.

†Calcul des probabilités (2nd edition), p. 2. This passage also appears in an
article in the Revue du mois for 1907 and in the author’s Science et méthode, of
the English translation of which I have made use above,—at the cost of doing
incomplete justice to Poincaré’s most admirable style.

pt. iv

A TREATISE ON PROBABILITY

326
phenomena, of the causes of which we are ignorant, there are some,
such as those dealt with by the manager of a life insurance company,
about which the calculus of probabilities can give real information.

Surely it cannot be thanks to our ignorance, Poincaré urges, that
we are able to arrive at valuable conclusions. If it were, it would
be necessary to answer an inquirer thus: “You ask me to predict
the phenomena that will be produced. If I had the misfortune to
know the laws of these phenomena, I could not succeed except by
inextricable calculations, and I should have to give up the attempt to
answer you; but since I am fortunate enough to be ignorant of them,

I will give you an answer at once. And, what is more extraordinary
still, my answer will be right.” The ignorance of the manager of the
life insurance company as to the prospects of life of his individual
policy-holders does not prevent his being able to pay dividends to his
shareholders.

Both these distinctions seem to be real ones, and Poincaré
proceeds to examine further instances in which we seem to distinguish
objectively between events according as they are or are not due to

‘chance.’ He takes the case of a cone balanced upon its tip; we
know for certain that it will fall, but not on which side—chance will
determine. “A very small cause which escapes our notice determines
a considerable effect that we cannot fail to see, and then we say that
that effect is due to chance.” The weather, and the distribution of
the minor planets on the Zodiac, are analogous instances. And what
we term ‘games of chance’ afford, it has always been recognised, an
almost perfect example. “It may happen that small differences in the
initial conditions produce very great ones in the final phenomena.

A small error in the former will produce an enormous error in the
latter. Prediction becomes impossible, and we have the fortuitous
phenomenon.” “The greatest chance is the birth of a great man. It is
only by chance that the meeting occurs of two genital cells of different
sex that contain precisely, each on its side, the mysterious elements,
the mutual reaction of which is destined to produce genius. . . . How
ch. xxiv

PHILOSOPHICAL APPLICATIONS

327
little it would have taken to make the spermatozoid which carried
them deviate from its course. It would have been enough to deflect
it a hundredth part of an inch, and Napoleon would not have been
born and the destinies of a continent changed. No example can give
a better comprehension of the true character of chance.”

Poincaré calls attention next to another class of events, which we
commonly assign to ‘chance,’ the distinguishing characteristic of which
seems to be that their causes are very numerous and complex,—the
motions of molecules of gas, the distribution of drops of rain, the
shuffling of a pack of cards, or the errors of observation. Thirdly
there is the type, usually connected with one of the first two, and
specially emphasised, as we have seen above, by Cournot, in which
something comes about through the concurrence of events which we
regard as belonging to distinct causal trains,—a man is walking along
the street and is killed by the fall of a tile.

6.

When we attribute such events, as those illustrated by

Poincaré, to chance, we certainly do not mean merely to assert that
we do not know how they arose or that we had no special reason
for anticipating them à priori. So far from this being the case, we
mean to make a definite assertion as to the kind of way in which
they arose;—though exactly what we mean to assert about them it is
extremely difficult to say.

Now a careful examination of all the cases in which various
writers claim to detect the presence of ‘objective chance’ confirms
the view that ‘subjective chance,’ which is concerned with knowledge
and ignorance, is fundamental, and that so-called ‘objective chance,’

however important it may turn out to be from the practical or
scientific point of view, is really a special kind of ‘subjective chance’

and a derivative type of the latter. For none of the adherents
of ‘objective chance’ wish to question the determinist character of
natural order; and the possibility of this objective chance of theirs
seems always to depend on the possibility that a particular kind of
knowledge either is ours or is within our powers and capacity. Let me
pt. iv

A TREATISE ON PROBABILITY

328
try to distinguish as exactly as I can the criterion of objective chance.

7.

When we say that an event has happened by chance, we
do not mean that previous to its occurrence the event was, on the
available evidence, very improbable; this may or may not have been
the case. We say, for example, that if a coin falls heads it is ‘by
chance,’ whereas its falling heads is not at all improbable. The term

‘by chance’ has reference rather to the state of our information about
the concurrence of the event considered and the event premised.

The fall of the coin is a chance event if our knowledge of the
circumstances of the throw is irrelevant to our expectation of the
possible alternative results. If the number of alternatives is very
large, then the occurrence of the event is not only subject to chance
but is also very improbable. In general two events may be said to
have a chance connection, in the subjective sense, when knowledge of
the first is irrelevant to our expectation of the second, and produces
no additional presumption for or against it; when, that is to say, the
probabilities of the propositions asserting them are independent in
the sense defined in Chapter XII. § 8.

The above definition deals with chance in the widest sense. What
is the differentia of the narrower group of cases to which it is desired
to apply the term ‘objective chance’? The occurrence of an event may
be said to be subject to objective chance, I think, when it is not only
a chance event in the above sense, but when we also have good reason
to suppose that the addition of further knowledge of a given kind, if
it were procurable, would not affect its chance character. We must
consider, that is to say, the probability which is relative not to actual
knowledge but to the whole of a certain kind of knowledge. We may
be able to infer from our evidence that, even with certain kinds of
additions to our knowledge, the connections between the events would
still be subject to chance in the sense just defined, and we may be
able to infer this without actually having the additional information
in question. If, however complete our knowledge of certain kinds of
things might be, there would still exist independence between the
ch. xxiv

PHILOSOPHICAL APPLICATIONS

329
propositions, the conjunction of which we are investigating, then we
may say there is an objective sense in which the actual conjunction of
these propositions is due to chance.

8.

This is, I think, the right line of inquiry. It remains to decide,
what kinds of information must be irrelevant to the connection, in
order that the presence of objective chance may be established.

When we attribute a coincidence to objective chance, we mean
not only that we do not actually know a law of connection, but,
speaking roughly, that there is no law of connection to be known.

And when we say that the occurrence of one alternative rather than
another is due to chance, we mean not only that we know no principle
by which to choose between the alternatives, but also that no such
principle is knowable. This use of the term closely corresponds to
what Venn means by the term ‘casual’: “We call a coincidence casual,

I apprehend, when we mean to imply that no knowledge of one of
the two elements, which we can suppose to be practically attainable,
would enable us to expect the other.”∗

To make this more precise, we must revive our distinction,† between
nomologic knowledge and ontologic knowledge, between knowledge of
laws and knowledge of facts or existence. Given certain facts f(a)

about a and certain laws of connection, L, we can infer certainly or
probably other facts φ(a) about a. If a complete knowledge of laws
of connection together with f(a) yields no appreciable probability for
preferring φ(a) to other alternatives, then I suggest that an actual
connection between φ and f in a particular instance may be said
to be due to chance in a sense which usage justifies us in calling
objective. We do not, in fact, when we speak of objective chance,
always use it in so strict a sense as this, but this is, I think, the
underlying conception to which current usage approximates. Current
usage diverges from this sense mainly for two reasons. We speak of
objective chance if in the above conditions our grounds for preference,

∗Logic of Chance, p. 245.

†See Part III. Note (ii.) § 2, p. 316.

pt. iv

A TREATISE ON PROBABILITY

330
though appreciable, are very small; and we are not insistent to assert
the rule of chance if a comparatively slight addition to our ontologic
knowledge would render the probability or the grounds for preference
appreciable.

To sum up the above, an event is due to objective chance if in order
to predict it, or to prefer it to alternatives, at present equiprobable,
with any high degree of probability, it would be necessary to know
a great many more facts of existence about it than we actually do
know, and if the addition of a wide knowledge of general principles
would be little use.

It must be added that we make a distinction between facts of
existence which are highly variable from case to case and those which
are constant or nearly constant over a certain field of observation or
experience. Within the limits of this field we regard the permanent
facts of existence as being, from the standpoint of chance, in nearly
the same position as laws. A connection is not due to chance,
therefore, if a knowledge of the permanent facts of existence could
lead to their prediction.

To sum up again therefore,—if within a given field of observation
or experience a knowledge of those facts of existence which are
permanent or invariable within that field, together with a knowledge
of all the relevant fundamental causal laws or general principles, and
of a few other facts of existence, would not permit us, given f(a),
to attribute an appreciable probability to φ(a) (or an appreciable
probability to the alternative φ1(a) rather than φ2(a)); then the
conjunction of φ(a) (or of φ1(a) rather than φ2(a) with f(a)) is due
to objective chance.

9.

If we return to the examples of Poincaré, the above definition
appears to conform satisfactorily with the usages of common sense. It
is when an exact knowledge of fact, as distinguished from principle, is
required for even approximate prediction that the expression ‘objective
chance’ seems applicable. But neither our definition nor usage is
precise as to the amount of knowledge of fact which must be required
ch. xxiv

PHILOSOPHICAL APPLICATIONS

331
for prediction, in order that, in the absence of it, the event may be
regarded as subject to objective chance.

It may be added that the expression ‘chance’ can be used with
reference to general statements as well as to particular facts. We
say, for example, that it is a matter of chance if a man dies on his
birthday, meaning that, as a general principle and in the absence
of special information bearing on a particular case, there is no
presumption whatever in favour of his dying on his birthday rather
than on any other day. If as a general rule there were celebrations
on such a day such as would be not unlikely to accelerate death, we
should say that a man’s dying on his birthday was not altogether a
matter of chance. If we knew no such general rule but did not know
enough about birthdays to be assured that there was no such rule, we
could not call the chance ‘objective’; we could only speak of it thus,
if on the evidence before us there was a strong presumption against
the existence of any such general rule.

10.

The philosophical and scientific importance of objective
chance as defined above cannot be made plain, until Part V., on
the Foundations of Statistical Inference, has been reached. There it
will appear in more than one connection, but chiefly in connection
with the application of Bernoulli’s formula. In cases where the use
of this formula is valid, important inferences can be drawn; and it
will he shown that, when the conditions for objective chance are
approximately satisfied, it is probable that the conditions for the
application of Bernoulli’s formula will be approximately satisfied also.

11.

The term random has been used, it is well recognised, in
several distinct senses. Venn∗ and other adherents of the ‘frequency’

theory have given to it a precise meaning, but one which has avowedly
very little relation to popular usage. A random sample, says Peirce,†

∗Logic of Chance, chap. v., “The Conception Randomness and its Scientific

Treatment.”

†“A Theory of Probable Inference” (published in Johns Hopkins Studies in

Logic), p. 152.

pt. iv

A TREATISE ON PROBABILITY

332
is one “taken according to a precept or method, which, being applied
over and over again indefinitely, would in the long run result in the
drawing of any one set of instances as often as any other set of
the same number.” The same fundamental idea has been expressed
with greater precision by Professor Edgeworth in connection with his
investigations into the law of error.∗ It is a fatal objection, in my
opinion, to this mode of defining randomness, that in general we
can only know whether or not we have a random sample when our
knowledge is nearly complete. Its divergence from ordinary usage is
well illustrated by the fact that there would be perfect randomness in
the distribution of stars in the heavens, as Venn explicitly points out,
if they were disposed in an exact and symmetrical pattern.†

I do not believe, therefore, that this kind of definition is a useful
one. The term must be defined with reference to probability, not to
what will happen “in the long run”; though there may be two senses of
it, corresponding to subjective and objective probability respectively.

The most important phrase in which the term is used is that of ‘a
random selection’ or ‘taken at random.’ When we apply this term to
a particular member of a series or collection of objects, we may mean
one of two things. We may mean that our knowledge of the method
of choosing the particular member is such that à priori the member
chosen is as likely to be any one member of the series as any other.

We may also mean, not that we have no knowledge as to which
particular member is in question, but that such knowledge as we
have respecting the particular member, as distinguished from other
members of the series, is irrelevant to the question as to whether or
not this member has the characteristic under examination. In the
first case the particular member is a random member of the series for

∗“Law of Error,” Camb. Phil. Trans., 1904, p. 128.

†But it may be added that this seems inconsistent with Venn’s conception
of randomness as that of aggregate order and individual irregularity ; nor is it
concordant with Venn’s typically random diagram (p. 118). His usage, therefore,
is sometimes nearer than his definition to the popular usage.

ch. xxiv

PHILOSOPHICAL APPLICATIONS

333
all characteristics; in the second case it is a random member for some
only. As the second case is the more general, we had better take that
for the purpose of defining ‘random selection.’

The point will be brought out further if we discuss the more
difficult use of the term. What exactly do we mean by the statement:

“Any number, taken at random, is equally likely to be odd or
even”? According to the frequency theory, this simply means that
there are as many odd numbers as there are even. Taking it in
a sense corresponding to subjective chance (and to the explanations
given above), I propose as a definition the following: a is taken
at random from the class S for the purposes of the propositional
function S(x) φ(x), relative to evidence h, if ‘x is a’ is irrelevant to
the probability φ(x)/S(x) h. Thus ‘the number of the inhabitants
of France is odd’ is, relative to my knowledge, a random instance
of the propositional function ‘x is an odd number,’ since ‘a is the
number of the inhabitants of France’ is irrelevant to the probability
of ‘a is odd’.∗ Thus to say that a number taken at random is as
likely to be odd as even, means that there is a probability 1 that

2
any instance taken at random of the generalisation ‘all numbers are
odd’ (or of the corresponding generalisation ‘all numbers are even’)

is true; an instance being taken at random in respect of evenness
or oddness, if our knowledge about it satisfies the conditions defined
above. Whether or not a given instance is taken at random, depends,
therefore, upon what generalisation is in question.

12.

We may or may not have reason to believe that, if we take a
series of random selections, the proportionate number of occurrences
of one particular type of result will very probably lie within certain
limits. For reasons to be explained in Chapter XXIX., random
selection relative to such information may conveniently be termed

‘random selection under Bernoullian conditions.’ It is this kind of
random selection which is scientifically and statistically important.

∗In the above S(x) stands for ‘x is a number’, φ(x) stands for ‘x is odd,’

a stands for ‘the number of inhabitants of France.’

pt. iv

A TREATISE ON PROBABILITY

334

But, as this corresponds to ‘objective chance,’ it is convenient to have
a wider definition of ‘random selection’ unqualified, corresponding
to ‘subjective chance’; and it is this wider definition which is given
above.

The term opposite to ‘random selection’ in ordinary usage is

‘biassed selection.’ When I use this phrase without qualification

I shall use it as the opposite of ‘random selection’ in the wider
unqualified sense.

CHAPTER XXV
some problems arising out of the discussion of
chance

1.

There are two classical problems in which attempts have
been made to attribute certain astronomical phenomena to a specific
cause, rather than to objective chance in some such sense as has been
defined in the preceding chapter.

The first of these is concerned with the inclinations to the ecliptic
of the orbits of the planets of the solar system. This problem has a
long history, but it will be sufficient to take De Morgan’s statement of
it.∗ If we suppose that each of the orbits might have any inclination,
we obtain a vast number of combinations of which only a small
number are such that their sum is as small or smaller than the sum
of those of the actual system. But the very existence of ourselves
and our world can be shown to imply that one of this small number
has been selected, and De Morgan derives from this an enormous
presumption that “there was a necessary cause in the formation of the
solar system for the inclinations being what they are.”

The answer to this was pointed out by D’Alembert† in criticising

∗Article on Probabilities in Encyclopaedia Metropolitana, p. 412, § 46.

De Morgan takes this without acknowledgment from Laplace, Théorie analytique
des probabilités (1st edition), pp. 257, 258.

Laplace also allows for the fact
that all the planets move in the same sense as the earth. He concludes: “On
verra que l’existence d’une cause commune qui a dirigé tous ces mouvemens
dans le sens de la rotation du soleil, et sur des plans peu inclinés à celui de
son équateur, est indiquée avec une probabilité bien supérieure à celle du plus
grand nombre des faits historiques sur lesquels on ne se permet aucun doute.”

Laplace had in his turn borrowed the example, also without acknowledgment,
from Daniel Bernoulli. See also D’Alembert, Opuscules mathématiques, vol. iv.,

1768, pp. 89 and 292.

†Op. cit. p. 292. “Il y a certainement d’infini contre un à parier que les

Planètes ne devraient pas se trouver dans le même plan; ce n’est pas une raison
pour en conclure que cette disposition, si elle avoit lieu, auroit nécessairement

335
pt. iv

A TREATISE ON PROBABILITY

336

Daniel Bernoulli. De Morgan could have reached a similar result
whatever the configuration might have happened to be. Any arbitrary
disposition over the celestial sphere is vastly improbable à priori, that
is to say in the absence of known laws tending to favour particular
arrangements. It does not follow from this, as De Morgan argues, that
any actual disposition possesses à posteriori a peculiar significance.

2.

The second of these problems is known as Michell’s problem
of binary stars. Michell’s Memoir was published in the Philosophical

Transactions for 1767.∗

It deals with the question as to whether
stars which are optically double, i.e. which are so situated as to
appear close together to an observer on the earth—are also physically
so “either by an original act of the Creator, or in consequence of
some general law, such perhaps as gravity.” He argues that if the
stars “were scattered by mere chance as it might happen . . . it is
manifest . . . that every star being as likely to be in any one situation
as another, the probability that any one particular star should happen
to be within a certain distance (as, for example, one degree) of
any other given star would be represented . . . by a fraction whose
numerator would be to its denominator as a circle of one degree radius
to a circle whose radius is the diameter of a great circle . . . that
is, about 1 in 13131.” From this beginning he derives an immense
presumption against the scattering of the several contiguous stars
that may be observed “by mere chance as it might happen.” And
d’autre cause que le hasard; car il y auroit de même l’infini contre un à parier
que les Planètes pourroient n’avoir pas une certaine disposition déterminée à
volonté. . . .”

D’Alembert is employing the instance for his own purposes, in order to build
up an ad hominem argument in favour of his theory concerning ‘runs’ against

D. Bernoulli (see also p. 363).

∗See also Todhunter’s History, pp. 332–4; Venn, Logic of Chance, p. 260;

Forbes, “On the Alleged Evidence for a Physical Connexion between Stars
forming Binary or Multiple Groups, deduced from the Doctrine of Chances,”

Phil. Mag., 1850, and Boole, “On the Theory of Probabilities and in particular
on Michell’s Problem of the Distribution of the Fixed Stars,” Phil. Mag., 1851.

ch. xxv

PHILOSOPHICAL APPLICATIONS

337
he goes on to argue that, if there are causal laws directly tending to
produce the observed proximities, we may reasonably suppose that
the proximities are actual, and not merely optical and apparent. The
fact that Michell’s induction was confirmed by the later investigations
of Herschell adds interest to the speculation. But apart from this
the argument is evidently subtler than in the first example. Michell
argues that there are more stars optically contiguous, than would be
likely if there were no special cause acting towards this end, and
further that, if such a cause is in operation, it must be real, and not
merely optical, contiguity that results from it.

Let us analyse the argument more closely. By “mere chance as it
might happen” Michell cannot be supposed to mean “uncaused.” He
is thinking of objective chance in the sense in which I have defined
this in the preceding chapter. We speak of a chance occurrence when
it is brought about by the coincidence of forces and circumstances so
numerous and complex that knowledge sufficient for its prediction is
of a kind altogether out of our reach. Michell uses the term vaguely
but means, I think, something of this kind: An event is due to mere
chance when it can only occur if a large number of independent∗
conditions are fulfilled simultaneously. The alternatives which Michell
is discussing are therefore these: Are binary stars merely due to
the interaction of a vast variety of stellar laws and positions or are
they the result of a few fundamental tendencies, which might be the
subject of knowledge and which would lead us to expect such stars in
relative profusion?

The existence of numerous binary stars may give a real inductive
argument in favour of their arising out of the interaction of a relatively
small number of independent causes. But it is not possible to arrive
at such precise results as Michell’s. If there is some finite probability
à priori that binary stars, when they arise, do arise in this way,
then, since the frequent coincidence of a given set of independent

∗See § 3 of Note (ii.) to Part III.

pt. iv

A TREATISE ON PROBABILITY

338
causes relatively few in number is more likely than that of a set
relatively numerous, the observation of binary stars will raise this
probability à posteriori to an extent which depends upon the relative
profusion in which such stars appear. If, in short, the first of
the two alternatives proposed above is assumed, there is no greater
presumption for a distribution, covering a part of the heavens, in
which binary stars appear, than for any other distribution; if the
second is assumed, there is a greater presumption. The observation
of numerous distributions in which binary stars appear increases,
therefore, by the inverse principle, any à priori probability which
may exist in favour of the second hypothesis. But more than this the
argument cannot justify. That Michell’s argument is, as it stands, no
more valid than De Morgan’s, becomes plain when we notice that he
would still have a high probability for his conclusion even if only one
binary star had been observed. The valuable part of the argument
must clearly turn upon the observation of numerous binary stars.

Let us now turn to Michell’s second step. He argues that, if binary
stars arise out of the interaction of a small number of independent
forces, they must be physically and not merely optically double. The
force of this argument seems to depend upon our possessing previous
knowledge as to the nature of the principal natural laws, and upon
an assumption, arising out of this, that there are not likely to be
forces tending to arrange stars, in reality at great distances from
one another, so as to appear double from this particular planet.

But Michell, in arguing thus, was neglecting the possibility that the
optical connection between the stars might be due to the observer
and his means of observation. It was not impossible that there should
be a law, connected with the transmission of light for example, which
would cause stars to appear to an observer to be much nearer together
than they really are.

While, therefore, a relative profusion of binary stars constitutes
evidence favourably relevant to Michell’s conclusion, the argument
is more complex and much less conclusive than he seems to have
ch. xxv

PHILOSOPHICAL APPLICATIONS

339
supposed. This is a criticism which is applicable to many such
arguments. The simplicity of the evidence, which arises out of the
lack of much relevant information, is liable, unless we are careful,
to lead us into deceptive calculations and into assertions of high
numerical probabilities, upon which we should never venture in cases
where the evidence is full and complicated, but where, in fact, the
conclusion is established far more strongly. The enormously high
probability in favour of his conclusion, to which Michell’s calculations
led him, should itself have caused him to suspect the accuracy of the
reasoning by which he reached it.

3.

Some more recent problems of this type seem, however, so
far as I am acquainted with them, to follow safer lines of argument.

The most important are concerned with the existence of star drifts.

It seems to me not at all impossible to possess data on which a
valid argument can be constructed from the observation of optically
apparent star drifts to the probability of a real uniformity of motion
amongst certain sets of stars relatively to others.

Another problem, somewhat analogous to the preceding, has been
recently discussed by Professor Karl Pearson.∗ The title might prove
a little misleading, perhaps, until the explanation has been reached
of the sense in which the term ‘random’ is used in it. But Professor

Pearson uses the term in a perfectly precise sense. He defines a
random distribution as one in which spherical shells of equal volume
about the sun as centre contain the same number of stars.† He argues
that the observed facts render probable the following disjunction:

Either the distribution of stars is not random in the sense defined
above, or there is a correlation between their distance and their
brilliancy, such as might be produced, for example, by the absorption

∗“On the Improbability of a Random Distribution of the Stars in Space,”

Proceedings of Royal Society, series A, vol. 84, pp. 47–70, 1910.

†It is, therefore, independent of direction, and the distribution is random
even if the stars are massed in particular quarters of the heavens. The definition
is, therefore, exceedingly arbitrary.

pt. iv

A TREATISE ON PROBABILITY

340
of light in its transmission through space, or the space within which
they all lie is limited in volume and not spherical in form.∗ But it
is useless to employ the term random in this sense in such inquiries
as Michell’s. For there is no reason to suppose that a non-*random
distribution is more likely than a random distribution to depend
upon the interaction of a small number of independent forces, and
there might even exist a presumption the other way. This arbitrary
interpretation of randomness does not help us to the solution of any
interesting problem.

4.

The discussion of final causes and of the argument from design
has suffered confusion from its supposed connection with theology.

But the logical problem is plain and can be determined upon formal
and abstract considerations. The argument is in all cases simply
this—an event has occurred and has been observed which would be
very improbable à priori if we did not know that it had actually
happened; on the other hand, the event is of such a character that it
might have been not unreasonably predicted if we had assumed the
existence of a conscious agent whose motives are of a certain kind
and whose powers are sufficient.

Symbolically: Let h be our original data, a the occurrence of the
event, b the existence of the supposed conscious agent. Then a/h is
assumed very small in comparison with a/bh; and we require b/ah, the
probability, that is to say, of b after a is known. The inverse principle
of probability already demonstrated shows that
b/h
b/ah = a/bh

,
a/h
and b/ah is therefore not determinate in terms of a/bh and a/h alone.

Thus we cannot measure the probability of the conscious agent’s
existence after the event, unless we can measure its probability before
the event. And it is our ignorance of this, as a rule, that we are
endeavouring to remedy. The argument tells us that the existence of
the hypothetical agent is more likely after the event than before it;

∗This should run more correctly, I think, “not a sphere with the sun as
centre.”

ch. xxv

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341
but, as in the case of the general inductive problem dealt with in

Part III., unless there is an appreciable probability first, there cannot be an appreciable probability afterwards. No conclusion, therefore,
which is worth having, can be based on the argument from design
alone; like induction, this type of argument can only strengthen the
probability of conclusions, for which there is something to be said on
other grounds. We cannot say, for example, that the human eye is
due to design more probably than not, unless we have some reason,
apart from the nature of its construction, for suspecting conscious
workmanship. But the necessary à priori probability, derived from
some other source, may sometimes be forthcoming. The man who
upon a desert island picks up a watch, or who sees the symbol John

Smith traced upon the sand, can use with reason the argument from
design. For he has other grounds for supposing that beings, capable
of designing such objects, do exist, and that their presence on the
island, now or formerly, is appreciably possible.

5.

The most important problems at the present day,
in which arguments of this kind are employed, are those which
arise in connection with psychical research.∗ The analysis of the

‘cross-correspondences,’ which have played so large a part in recent
discussions, presents many points of difficulty which are not dissimilar
to those which arise in other scientific inquiries of great complexity
in which our initial knowledge is small. An important part of the
logical problem, therefore, is to distinguish the peculiarity of psychical
problems and to discover what special evidence they demand beyond
what is required when we deal with other questions. There is a certain

∗The probability that a remarkable success in naming playing cards is due
to psychic agency, was discussed by Professor Edgeworth in Metretike.

This
was, I think, the first application of probabilities to these questions. See also

Proceedings of the Society for Psychical Research, Parts VIII. and X.; Professor

Edgeworth’s article on Psychical Research and Statistical Method, Stat. Journ.

vol. lxxxii. (1919) p. 222; and Experiments in Psychical Research at Leland

Stanford Junior University, by J. Coover.

pt. iv

A TREATISE ON PROBABILITY

342
tendency, I think, arising out of the belief that psychical problems are
in some way peculiar, to raise sceptical doubts against them, which
are equally valid against all scientific proofs. Without entering into
any questions of detail, let us endeavour to separate those difficulties
which seem peculiar to psychical research from those which, however
great, are not different from the difficulties which confront students
of heredity, for instance, and which are not less likely than these to
yield ultimately to the patience and the insight of investigators.

For this purpose it is necessary to recur, briefly, to the analysis of

Part III. It was argued there that the methods of empirical proof, by which we strengthen the probability of our conclusions, are not
at all dissimilar, when we apply them to the discovery of formal
truth, and when we apply them to the discovery of the laws which
relate material objects, and that they may possibly prove useful even
in the case of metaphysics; but that the initial probability which
we strengthen by these means is differently obtained in each class
of problem. In logic it arises out of the postulate that apparent
self-evidence invests what seems self-evident with some degree of
probability; and in physical science, out of the postulate that there
is a limitation to the amount of independent variety amongst the
qualities of material objects. But both in logic and in physical
science we may wish to consider hypotheses which it is not possible
to invest with any à priori probability and which we entertain solely
on account of the known truth of many of their consequences. An
axiom which has no self-evidence, but which it seems necessary to
combine with other axioms which are self-evident in order to deduce
the generally accepted body of formal truth, stands in this category.

A scientific entity, such as the ether or the electron, whose qualities
have never been observed but whose existence we postulate for
purposes of explanation, stands in it also. If the analysis of Part

III. is correct, we can never attribute a finite probability∗ to the

∗I am assuming that there is no argument, arising either from self-evidence
or analogy, in addition to the argument arising from the truth of their
ch. xxv

PHILOSOPHICAL APPLICATIONS

343
truth of such axioms or to the existence of such scientific entities,
however many of their consequences we find to be true. They may
be convenient hypotheses, because, if we confine ourselves to certain
classes of their consequences, we are not likely to be led into error;
but they stand, nevertheless, in a position altogether different from
that of such generalisations as we have reason to invest with an initial
probability.

Let us now apply these distinctions to the problems of psychical
research. In the case of some of them we can obtain the initial
probability, I think, by the same kind of postulates as in physical
science, and our conclusions need not be open to a greater degree
of doubt than these. In the case of others we cannot; and these
must remain, unless some method is open to us peculiar to psychical
research, as tentative unproved hypotheses in the same category as
the ether.

The best example of the first class is afforded by telepathy. We
know that the consciousnesses which, if our hypothesis is correct, act
upon one another, do exist; and I see no logical difference between
the problem of establishing a law of telepathy and that of establishing
the law of gravitation. There is at present a practical difference on
account of the much narrower scope of our knowledge, in the case
of telepathy, of cognate matters. We can, therefore, be much less
certain; but there seems no reason why we should necessarily remain
less certain after more evidence has been accumulated. It is important
to remember that, in the case of telepathy, we are merely discovering
a relation between objects which we already know to exist.

The best example of the other class is afforded by attempts to
attribute psychic phenomena to the agency of ‘spirits’ other than
consequences, in favour of the truth of such axioms or the existence of such
objects; but I daresay that this may not certainly be the case. The reader may
be reminded also that, when I deny a finite probability this is not the same
thing as to affirm that the probability is infinitely small. I mean simply that it
is not greater than some numerically measurable probability.

pt. iv

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344
human beings. Such arguments are weakened at present by the fact
that no phenomena are known, so far as I am aware, which cannot be
explained, though improbably in some cases, in other ways. But even
if phenomena were to be observed of which no known agency could
afford even an improbable explanation, the hypothesis of ‘spirits’

would still lie in the same logical limbo as the hypothesis of the

‘ether,’ in which they might be supposed not inappropriately to move.

Such an hypothesis as the existence of ‘spirits’ could only become
substantial if some peculiar method of knowledge were within our
power which would yield us the initial probability which is demanded.

That such a method exists, it is not infrequently claimed. If we can
directly perceive these ‘spirits,’ as many of those who are described
in James’s Varieties of Religious Experience think they can, the
problem is, logically, altogether changed. We have, in fact, very much
the same kind of reason, though it may be with less probability,
that we have for believing in the existence of other people. The
preceding paragraph applies only to attempts at proving the existence
of ‘spirits’ from such evidence as is discussed by the Society for

Psychical Research.

In between these two extremes comes a class of cases, with regard
to which it is extremely difficult to come to a decision—that of
attempts to attribute psychic phenomena to the conscious agency of
the dead. I wish to discuss here, not the nature of the existing
evidence, but the question whether it is possible for any evidence
to be convincing. In this case the object whose existence we are
endeavouring to demonstrate resembles in many respects objects
which we know to exist. The question of epistemology, which is
before us, is this: Is it necessary, in order that we may have an
initial probability, that the object of our hypothesis should resemble
in every relevant particular some one object which we know to exist,
or is it sufficient that we should know instances of all its supposed
qualities, though never in combination? It is clear that some qualities
may be irrelevant—position in time and space, for example—and
ch. xxv

PHILOSOPHICAL APPLICATIONS

345
that ‘every relevant particular’ need not include these. But can the
initial probability exist if our hypothesis assumes qualities, which
have plainly some degree of relevance, in new combinations? If we
have no knowledge of consciousness existing apart from a living body,
can indirect evidence of whatever character afford us any probability
of such a thing? Could any evidence, for example, persuade us that
a tree felt the emotion of amusement, even if it laughed repeatedly
when we made jokes? Yet the analogy which we demand seems to
be a matter of degree; for it does not seem unreasonable to attribute
consciousness to dogs, although this constitutes a combination of
qualities unlike in many respects to any which we know to exist.

This discussion, however, is wandering from the subject of
probability to that of epistemology, and it will not be solved until
we possess a more comprehensive account of this latter subject than
we have at present. I wish only to distinguish between those cases
in which we obtain the initial probability in the same manner as in
physical science from those in which we must get it, if at all, in some
other way. The distinctions I have made are sufficiently summarised
by a recapitulation of the following comparisons: We compared the
proof of telepathy to the proof of gravitation, the proof of non-human

‘spirits’ to the proof of the ether, and, much less closely, the proof
of the consciousness of the dead to the proof of the consciousness of
trees, or, perhaps, of dogs.

Before passing to the next of the rather miscellaneous topics of
this chapter, it may be worth while to add that we should be very
chary of applying to problems of psychical research the calculus of
probabilities. The alternatives seldom satisfy the conditions for the
application of the Principle of Indifference, and the initial probabilities
are not capable of being measured numerically. If, therefore, we
endeavour to calculate the probability that some phenomenon is due
to ‘abnormal’ causes, our mathematics will be apt to lead us into
unjustifiable conclusions.

6.

Uninstructed common sense seems to be specially unreliable
pt. iv

A TREATISE ON PROBABILITY

346
in dealing with what are termed ‘remarkable occurrences.’ Unless
a ‘remarkable occurrence’ is simply one which produces on us a
particular psychological effect, that of surprise, we can only define
it as an event which before its occurrence is very improbable on
the available evidence. But it will often occur—whenever, in fact,
our data leave open the possibility of a large number of alternatives
and show no preference for any of them—that every possibility is
exceedingly improbable à priori. It follows, therefore, that what
actually occurs does not derive any peculiar significance merely from
the fact of its being ‘remarkable’ in the above sense. Something
further is required before we can build with success. Yet Michell’s
argument and the argument from design derive a good deal of their
plausibility, I think, from the ‘remarkable’ character of the actual
constitution whether of the heavens or of the universe, in forgetfulness
of the fact that it is impossible to propound any constitution which
would if it existed be other than ‘remarkable.’ It is supposed that
a remarkable occurrence is specially in need of an explanation, and
that any sufficient explanation has a high probability in its favour.

That an explanation is particularly required, possesses a measure of
truth; for it is likely that our original data were much lacking in
completeness, and the occurrence of the extraordinary event brings to
light this deficiency. But that we are not justified in adopting with
confidence any sufficient explanation, has been shown already.

Such arguments, however, get a part of their plausibility from
a quite different source. There is a general supposition that some
kinds of occurrences are more likely than others to be susceptible of
an explanation by us; and, therefore, any explanation which deals
with such cases falls in prepared soil. Results which, judging from
ourselves, conscious agents would be likely to produce fall into this
category.

Results which would be probable, supposing a direct
and predominant causal dependence between the elements whose
concomitance is remarked, belong to it also. There is, in fact, a sort
of argument from analogy as to whether certain sorts of phenomena
ch. xxv

PHILOSOPHICAL APPLICATIONS

347
are or are not likely to be due to ‘chance.’ This may explain, for
example, why the particular concurrence of atoms that go to compose
the human eye, why a series of correct guesses in naming playing
cards, why special symmetry or special asymmetry amongst the stars,
seem to require explanation in no ordinary degree. Prior to an
explanation these particular concurrences or series or distributions
are no more improbable than any other. But the causes of such
conjunctions as these are more likely to be discoverable by the human
mind than are the causes of others, and the attempt to explain them
deserves, therefore, to be more carefully considered. This supposition,
derived by analogy or induction from those cases in which we believe
the causes to be known to us, has, perhaps, some weight. But the
direct application of the Calculus of Probabilities can do no more in
these cases than suggest matter for investigation. The fact that a
man has made a long series of correct guesses in cases where he is cut
off from the ordinary channels of communication, is a fact worthy of
investigation, because it is more likely to be susceptible of a simple
causal explanation, which may have many applications, than a case
in which false and true guesses follow one another with no apparent
regularity.

7.

In the case of empirical laws, such as Bode’s law, which
have no more than a very slight connection with the general body
of scientific knowledge, it is sometimes thought that the law is more
probable if it is proposed before the examination of some or all of
the available instances than if it is proposed after their examination.

Supposing, for example, that Bode’s law is accurately true for seven
planets, it is held that the law would be more probable if it was
suggested after the examination of six and was confirmed by the
subsequent discovery of the seventh, than it would be if it had
not been propounded until after all seven had been observed. The
arguments in favour of such a conclusion are well put by Peirce:∗ “All

∗C. S. Peirce, A Theory of Probable Inference, pp. 162–167; published in

Johns Hopkins Studies in Logic, 1883.

pt. iv

A TREATISE ON PROBABILITY

348
the qualities of objects may be conceived to result from variations of
a number of continuous variables; hence any lot of objects possesses
some character in common, not possessed by any other.” Hence if
the common character is not predesignate we can conclude nothing.

Cases must not be used to prove a generalisation which has only been
suggested by the cases themselves. He takes the first five poets from
a biographical dictionary with their ages at death:

Aagard . . . . . . . . . . . . 48

Abunowas . . . . . . . 48

Abeille . . . . . . . . . . . . .76

Accords . . . . . . . . . .45

Abulola . . . . . . . . . . . . 84

“These five ages have the following characters in common:

“1. The difference of the two digits composing the number,
divided by three, leaves a remainder of one.

“2. The first digit raised to the power indicated by the second,
and then divided by three, leaves a remainder of one.

“3. The sum of the prime factors of each age, including one as
a prime factor, is divisible by three.”

He compares a generalisation regarding the ages of poets based
on this evidence to Dr. Lyon Playfair’s argument about the specific
gravities of the three allotropic forms of carbon:

√

Diamond . . . . . . . . . . . . . . . . . .3.48 = 2 12

√

Graphite . . . . . . . . . . . . . . . . . . 2.29 = 3 12

√

Charcoal . . . . . . . . . . . . . . . . . . 1.88 = 4 12
approximately, the atomic weight of carbon being 12. Dr. Playfair
thinks that the above renders it probable that the specific gravities
of the allotropic forms of other elements would, if we knew them, be
found to equal the different roots of their atomic weight.

The weakness of these arguments, however, has a different
explanation. These inductions are very improbable, because they are
out of relation to the rest of our knowledge and are based on a
very small number of instances. The apparent absurdity, moreover,
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PHILOSOPHICAL APPLICATIONS

349
of the inductive law of Poets’ Ages is increased by the fact that we
take account of the knowledge we actually possess that the ages of
poets are not in fact connected by any such law. If we knew nothing
whatever about poets’ ages except what is stated above, the induction
would be as valid as any other which is based on a very weak analogy
and a very small number of instances and is unsupported by indirect
evidence.

The peculiar virtue of prediction or predesignation is altogether
imaginary.

The number of instances examined and the analogy
between them are the essential points, and the question as to whether
a particular hypothesis happens to be propounded before or after
their examination is quite irrelevant. If all our inductions had to be
thought of before we examined the cases to which we apply them,
we should, doubtless, make fewer inductions; but there is no reason
to think that the few we should make would be any better than
the many from which we should be precluded. The plausibility of
the argument is derived from a different source. If an hypothesis is
proposed à priori, this commonly means that there is some ground
for it, arising out of our previous knowledge, apart from the purely
inductive ground, and if such is the case the hypothesis is clearly
stronger than one which reposes on inductive grounds only. But if
it is a mere guess, the lucky fact of its preceding some or all of the
cases which verify it adds nothing whatever to its value. It is the
union of prior knowledge, with the inductive grounds which arise out
of the immediate instances, that lends weight to an hypothesis, and
not the occasion on which the hypothesis is first proposed. It is
sometimes said, to give another example, that the daily fulfilment of
the predictions of the Nautical Almanack constitutes the most cogent
proof of the laws of dynamics. But here the essence of the verification
lies in the variety of cases which can be brought accurately under our
notice by means of the Almanack, and in the fact that they have
all been obtained on a uniform principle, not in the fact that the
verification is preceded by a prediction.

pt. iv

A TREATISE ON PROBABILITY

350

The same point arises not uncommonly in statistical inquiries. If
a theory is first proposed and is then confirmed by the examination
of statistics, we are inclined to attach more weight to it than to a
theory which is constructed in order to suit the statistics. But the
fact that the theory which precedes the statistics is more likely than
the other to be supported by general considerations—for it has not,
presumably, been adopted for no reason at all—constitutes the only
valid ground for this preference. If it does not receive more support
than the other from general considerations, then the circumstances of
its origin are no argument in its favour. The opposite view, which the
unreliability of some statisticians has brought into existence,—that
it is a positive advantage to approach statistical evidence without
preconceptions based on general grounds, because the temptation to

‘cook’ the evidence will prove otherwise to be irresistible,—has no
logical basis and need only be considered when the impartiality of an
investigator is in doubt.

CHAPTER XXVI
the application of probability to conduct

1.

Given as our basis what knowledge we actually have, the
probable, I have said, is that which it is rational for us to believe.

This is not a definition. For it is not rational for us to believe that
the probable is true; it is only rational to have a probable belief in
it or to believe it in preference to alternative beliefs. To believe one
thing in preference to another, as distinct from believing the first
true or more probable and the second false or less probable, must
have reference to action and must be a loose way of expressing the
propriety of acting on one hypothesis rather than on another. We
might put it, therefore, that the probable is the hypothesis on which
it is rational for us to act. It is, however, not so simple as this, for
the obvious reason that of two hypotheses it may be rational to act
on the less probable if it leads to the greater good. We cannot say
more at present than that the probability of a hypothesis is one of
the things to be determined and taken account of before acting on it.

2.

I do not know of passages in the ancient philosophers which
explicitly point out the dependence of the duty of pursuing goods
on the reasonable or probable expectation of attaining them relative
to the agent’s knowledge. This means only that analysis had not
disentangled the various elements in rational action, not that common
sense neglected them. Herodotus puts the point quite plainly. “There
is nothing more profitable for a man,” he says, “than to take good
counsel with himself; for even if the event turns out contrary to one’s
hope, still one’s decision was right, even though fortune has made it
of no effect: whereas if a man acts contrary to good counsel, although
by luck he gets what he had no right to expect, his decision was not
any the less foolish.”∗

3.

The first contact of theories of probability with modern

∗Herod. vii. 10.

351
pt. iv

A TREATISE ON PROBABILITY

352
ethics appears in the Jesuit doctrine of probabilism. According to
this doctrine one is justified in doing an action for which there is any
probability, however small, of its results being the best possible. Thus,
if any priest is willing to permit an action, that fact affords some
probability in its favour, and one will not be damned for performing
it, however many other priests denounce it.∗ It may be suspected,
however, that the object of this doctrine was not so much duty as
safety. The priest who permitted you so to act assumed thereby
the responsibility. The correct application of probability to conduct
naturally escaped the authors of a juridical ethics, which was more
interested in the fixing of responsibility for definite acts, and in the
various specified means by which responsibility might be disposed of,
than in the greatest possible sum-total of resultant good.

A more correct doctrine was brought to light by the efforts of the
philosophers of the Port Royal to expose the fallacies of probabilism.

“In order to judge,” they say, “of what we ought to do in order
to obtain a good and to avoid an evil, it is necessary to consider
not only the good and evil in themselves, but also the probability
of their happening and not happening, and to regard geometrically
the proportion which all these things have, taken together.”† Locke
perceived the same point, although not so clearly.‡

By Leibniz
this theory is advanced more explicitly; in such judgments, he

∗Compare with this doctrine the following curious passage from Jeremy

Taylor:—“We being the persons that are to be persuaded, we must see that we
be persuaded reasonably. And it is unreasonable to assent to a lesser evidence
when a greater and clearer is propounded: but of that every man for himself is
to take cognisance, if he be able to judge; if he be not, he is not bound under
the tie of necessity to know anything of it.

That that is necessary shall be
certainly conveyed to him: God, that best can, will certainly take care for that;
for if he does not, it becomes to be not necessary; or if it should still remain
necessary, and he be damned for not knowing it, and yet to know it be not in
his power, then who can help it! There can be no further care in this business.”

†The Port Royal Logic (1662), Eng. Trans. p. 367.

‡Essay concerning Human Understanding, book ii. chap. xxi. § 66.

ch. xxvi

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353
says, “as in other estimates disparate and heterogeneous and, so to
speak, of more than one dimension, the greatness of that which is
discussed is in reason composed of both estimates (i.e. of goodness
and of probability), and is like a rectangle, in which there are two
considerations, viz. that of length and that of breadth. . . . Thus
we should still need the art of thinking and that of estimating
probabilities, besides the knowledge of the value of goods and evils,
in order properly to employ the art of consequences.”∗

In his preface to the Analogy Butler insists on “the absolute and
formal obligation” under which even a low probability, if it is the
greatest, may lay us: “To us probability is the very guide of life.”

4.

With the development of a utilitarian ethics largely concerned
with the summing up of consequences, the place of probability in
ethical theory has become much more explicit. But although the
general outlines of the problem are now clear, there are some elements
of confusion not yet dispersed. I will deal with some of them.

In his Principia Ethica (p. 152) Dr. Moore argues that “the first
difficulty in the way of establishing a probability that one course of
action will give a better total result than another, lies in the fact
that we have to take account of the effects of both throughout an
infinite future. . . . We can certainly only pretend to calculate the
effects of actions within what may be called an ‘immediate future.’. . .

We must, therefore, certainly have some reason to believe that no
consequences of our action in a further future will generally be such
as to reverse the balance of good that is probable in the future which
we can foresee. This large postulate must be made, if we are ever
to assert that the results of one action will be even probably better
than those of another. Our utter ignorance of the far future gives us
no justification for saying that it is even probably right to choose the
greater good within the region over which a probable forecast may
extend.”

∗Nouveaux Essais, book ii. chap. xxi.

pt. iv

A TREATISE ON PROBABILITY

354

This argument seems to me to be invalid and to depend on
a wrong philosophical interpretation of probability. Mr. Moore’s
reasoning endeavours to show that there is not even a probability by
showing that there is not a certainty. We must not, of course, have
reason to believe that remote consequences will generally be such
as to reverse the balance of immediate good. But we need not be
certain that the opposite is the case. If good is additive, if we have
reason to think that of two actions one produces more good than the
other in the near future, and if we have no means of discriminating
between their results in the distant future, then by what seems a
legitimate application of the Principle of Indifference we may suppose
that there is a probability in favour of the former action. Mr. Moore’s
argument must be derived from the empirical or frequency theory of
probability, according to which we must know for certain what will
happen generally (whatever that may mean) before we can assert a
probability.

The results of our endeavours are very uncertain, but we have a
genuine probability, even when the evidence upon which it is founded
is slight. The matter is truly stated by Bishop Butler: “From our
short views it is greatly uncertain whether this endeavour will, in
particular instances, produce an overbalance of happiness upon the
whole; since so many and distant things must come into the account.

And that which makes it our duty is that there is some appearance
that it will, and no positive appearance to balance this, on the
contrary side. . . .”∗

The difficulties which exist are not chiefly due, I think, to our
ignorance of the remote future. The possibility of our knowing
that one thing rather than another is our duty depends upon the
assumption that a greater goodness in any part makes, in the absence
of evidence to the contrary, a greater goodness in the whole more

∗This passage is from the Analogy. The Bishop adds: “. . . and also that
such benevolent endeavour is a cultivation of that most excellent of all virtuous
principles, the active principle of benevolence.”

ch. xxvi

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355
probable than would the lesser goodness of the part. We assume
that the goodness of a part is favourably relevant to the goodness of
the whole. Without this assumption we have no reason, not even a
probable one, for preferring one action to any other on the whole.

If we suppose that goodness is always organic, whether the whole
is composed of simultaneous or successive parts, such an assumption
is not easily justified. The case is parallel to the question, whether
physical law is organic or atomic, discussed in Chapter XXI. § 6.

Nevertheless we can admit that goodness is partly organic and still
allow ourselves to draw probable conclusions. For the alternatives,
that either the goodness of the whole universe throughout time is
organic or the goodness of the universe is the arithmetic sum of the
goodnesses of infinitely numerous and infinitely divided parts, are not
exhaustive. We may suppose that the goodness of conscious persons
is organic for each distinct and individual personality. Or we may
suppose that, when conscious units are in conscious relationship, then
the whole which we must treat as organic includes both units. These
are only examples. We must suppose, in general, that the units whose
goodness we must regard as organic and indivisible are not always
larger than those the goodness of which we can perceive and judge
directly.

5.

The difficulties, however, which are most fundamental from
the standpoint of the student of probability, are of a different kind.

Normal ethical theory at the present day, if there can be said to be
any such, makes two assumptions: first, that degrees of goodness
are numerically measurable and arithmetically additive, and second,
that degrees of probability also are numerically measurable. This
theory goes on to maintain that what we ought to add together,
when, in order to decide between two courses of action, we sum up
the results of each, are the ‘mathematical expectations’ of the several
results. ‘Mathematical expectation’ is a technical expression originally
derived from the scientific study of gambling and games of chance,
and stands for the product of the possible gain with the probability
pt. iv

A TREATISE ON PROBABILITY

356
of attaining it.∗ In order to obtain, therefore, a measure of what
ought to be our preference in regard to various alternative courses of
action, we must sum for each course of action a series of terms made
up of the amounts of good which may attach to each of its possible
consequences, each multiplied by its appropriate probability.

The first assumption, that quantities of goodness are duly subject
to the laws of arithmetic, appears to me to be open to a certain
amount of doubt. But it would take me too far from my proper
subject to discuss it here, and I shall allow, for the purposes of further
argument, that in some sense and to some extent this assumption
can be justified. The second assumption, however, that degrees of
probability are wholly subject to the laws of arithmetic, runs directly
counter to the view which has been advocated in Part I. of this
treatise. Lastly, if both these points be waived, the doctrine that
the ‘mathematical expectations’ of alternative courses of action are
the proper measures of our degrees of preference is open to doubt on
two grounds—first, because it ignores what I have termed in Part I.

the ‘weights’ of the arguments, namely, the amount of evidence upon
which each probability is founded; and second, because it ignores the
element of ‘risk’ and assumes that an even chance of heaven or hell is
precisely as much to be desired as the certain attainment of a state of
mediocrity. Putting on one side the first of these grounds of doubt, I
will treat each of the others in turn.

6.

In Chapter III. of Part I. I have argued that only in a
strictly limited class of cases are degrees of probability numerically
measurable. It follows from this that the ‘mathematical expectations’

∗Priority in the conception of mathematical expectation can, I think, be
claimed by Leibniz, De incerti aestimatione, 1678 (Couturat, Logigue de Leibniz,
p. 248). In a letter to Placcius, 1687 (Dutens, vi. i. 36 and Couturat, op. cit.

p. 246) Leibniz proposed an application of the same principle to jurisprudence,
by virtue of which, if two litigants lay claim to a sum of money, and if the
claim of the one is twice as probable as that of the other, the sum should be
divided between them in that proportion. The doctrine, seems sensible, but I
am not aware that it has ever been acted on.

ch. xxvi

PHILOSOPHICAL APPLICATIONS

357
of goods or advantages are not always numerically measurable; and
hence, that even if a meaning can be given to the sum of a series
of non-numerical ‘mathematical expectations,’ not every pair of such
sums are numerically comparable in respect of more and less. Thus
even if we know the degree of advantage which might be obtained
from each of a series of alternative courses of actions and know also
the probability in each case of obtaining the advantage in question, it
is not always possible by a mere process of arithmetic to determine
which of the alternatives ought to be chosen. If, therefore, the
question of right action is under all circumstances a determinate
problem, it must be in virtue of an intuitive judgment directed to the
situation as a whole, and not in virtue of an arithmetical deduction
derived from a series of separate judgments directed to the individual
alternatives each treated in isolation.

We must accept the conclusion that, if one good is greater than
another, but the probability of attaining the first less than that of
attaining the second, the question of which it is our duty to pursue
may be indeterminate, unless we suppose it to be within our power
to make direct quantitative judgments of probability and goodness
jointly.

It may be remarked, further, that the difficulty exists,
whether the numerical indeterminateness of the probability is intrinsic
or whether its numerical value is, as it is according to the Frequency

Theory and most other theories, simply unknown.

7.

The second difficulty, to which attention is called above,
is the neglect of the ‘weights’ of arguments in the conception of

‘mathematical expectation.’ In Chapter VI. of Part I. the significance of ‘weight’ has been discussed. In the present connection the question
comes to this—if two probabilities are equal in degree, ought we, in
choosing our course of action, to prefer that one which is based on a
greater body of knowledge?

The question appears to me to be highly perplexing, and it is
difficult to say much that is useful about it. But the degree of
completeness of the information upon which a probability is based
pt. iv

A TREATISE ON PROBABILITY

358
does seem to be relevant, as well as the actual magnitude of the
probability, in making practical decisions. Bernoulli’s maxim,∗ that in
reckoning a probability we must take into account all the information
which we have, even when reinforced by Locke’s maxim that we must
get all the information we can,† does not seem completely to meet the
case. If, for one alternative, the available information is necessarily
small, that does not seem to be a consideration which ought to be
left out of account altogether.

8.

The last difficulty concerns the question whether, the former
difficulties being waived, the ‘mathematical expectation’ of different
courses of action accurately measures what our preferences ought to
be—whether, that is to say, the undesirability of a given course of
action increases in direct proportion to any increase in the uncertainty
of its attaining its object, or whether some allowance ought to be
made for ‘risk,’ its undesirability increasing more than in proportion
to its uncertainty.

In fact the meaning of the judgment, that we ought to act in such
a way as to produce most probably the greatest sum of goodness, is
not perfectly plain. Does this mean that we ought so to act as to
make the sum of the goodnesses of each of the possible consequences
of our action multiplied by its probability a maximum? Those who
rely on the conception of ‘mathematical expectation’ must hold that
this is an indisputable proposition. The justifications for this view
most commonly advanced resemble that given by Condorcet in his

“Réflexions sur la règle générale, qui prescrit de prendre pour valeur
d’un événement incertain, la probabilité de cet événement multipliée

∗Ars Conjectandi,
p.

215:

“Non
sufficit
expendere
unum
alterumve
argumentum, sed conquirenda sunt omnia, quae in cognitionem nostram venire
possunt, atqne ullo modo ad probationem rei facere videntur.”

†Essay concerning Human Understanding, book ii. chap. xxi. § 67: “He that
judges without informing himself to the utmost that he is capable, cannot acquit
himself of judging amiss.”

ch. xxvi

PHILOSOPHICAL APPLICATIONS

359
par la valeur de l’événement en lui-même,”∗ where he argues from

Bernoulli’s theorem that such a rule will lead to satisfactory results
if a very large number of trials be made. As, however, it will be
shown in Chapter XXIX. of Part V. that Bernoulli’s theorem is not applicable in by any means every case, this argument is inadequate
as a general justification.

In the history of the subject, nevertheless, the theory of

‘mathematical expectation’ has been very seldom disputed.

As

D’Alembert has been almost alone in casting serious doubts upon it

(though he only brought himself into disrepute by doing so), it will
be worth while to quote the main passage in which he declares his
scepticism: “Il me sembloit” (in reading Bernoulli’s Ars Conjectandi)

“que cette matière avoit besoin d’être traitée d’une manière plus
claire; je voyois bien que l’espérance étoit plus grande, 1o que la
somme espérée étoit plus grande, 2o que la probabilité de gagner
l’étoit aussi. Mais je ne voyois pas avec la même évidence, et je ne le
vois pas encore, 1o que la probabilité soit estimée exactement par les
méthodes usitées; 2o que quand elle le seroit, l’espérance doive être
proportionnelle à cette probabilité simple, plutôt qu’à une puissance
ou même à une fonction de cette probabilité; 3o que quand il y a
plusieurs combinaisons qui donnent différens avantages ou différens
risques (qu’on regarde comme des avantages négatifs) il faille se
contenter d’ajouter simplement ensemble toutes les espérances pour
avoir l’espérance totale.”†

In extreme cases it seems difficult to deny some force to

D’Alembert’s objection; and it was with reference to extreme cases
that he himself raised it. Is it certain that a larger good, which is
extremely improbable, is precisely equivalent ethically to a smaller
good which is proportionately more probable? We may doubt whether
the moral value of speculative and cautious action respectively can be

∗Hist. de l’Acad., Paris, 1781.

†Opuscules mathématiques, vol. iv., 1768 (extraits de lettres), pp. 284, 285.

See also p. 88 of the same volume.

pt. iv

A TREATISE ON PROBABILITY

360
weighed against one another in a simple arithmetical way, just as we
have already doubted whether a good whose probability can only be
determined on a slight basis of evidence can be compared by means
merely of the magnitude of this probability with another good whose
likelihood is based on completer knowledge.

There seems, at any rate, a good deal to be said for the conclusion
that, other things being equal, that course of action is preferable
which involves least risk, and about the results of which we have
the most complete knowledge. In marginal cases, therefore, the
coefficients of weight and risk as well as that of probability are
relevant to our conclusion. It seems natural to suppose that they
should exert some influence in other cases also, the only difficulty in
this being the lack of any principle for the calculation of the degree
of their influence. A high weight and the absence of risk increase pro
tanto the desirability of the action to which they refer, but we cannot
measure the amount of the increase.

The ‘risk’ may be defined in some such way as follows. If A is the
amount of good which may result, p its probability (p + q = 1), and

E the value of the ‘mathematical expectation,’ so that E = pA, then
the ‘risk’ is R, where R = p(A − E) = p(1 − p)A = pqA = qE. This
may be put in another way: E measures the net immediate sacrifice
which should be made in the hope of obtaining A; q is the probability
that this sacrifice will be made in vain; so that qE is the ‘risk.’∗ The
ordinary theory supposes that the ethical value of an expectation is a
function of E only and is entirely independent of R.

∗The theory of Risiko is briefly dealt with by Czuber, Wahrscheinlichkeit-
srechnung, vol. i. pp. 219 et seq.

If R measures the first insurance, this
leads to a Risiko of the second order, R1 = qR = q2E.

This again may be
insured against, and by a sufficient number of such reinsurances the risk can be
completely shifted:

E

E

E + R1 + R2 + . . . = E(1 + q + q2 + . . .) =

=

= A.

1 − q
p
ch. xxvi

PHILOSOPHICAL APPLICATIONS

361

We could, if we liked, define a conventional coefficient c of weight
and risk, such as

2pw
c =

, where w measures the ‘weight,’

(1 + q)(1 + w)

which is equal to unity when p = 1 and w = 1, and to zero when p = 0
or w = 0, and has an intermediate value in other cases.∗ But if doubts
as to the sufficiency of the conception of ‘mathematical expectation’

be sustained, it is not likely that the solution will lie, as D’Alembert
suggests, and as has been exemplified above, in the discovery of some
more complicated function of the probability wherewith to compound
the proposed good. The judgment of goodness and the judgment of
probability both involve somewhere an element of direct apprehension,
and both are quantitative. We have raised a doubt as to whether
the magnitude of the ‘oughtness’ of an action can be in all cases
directly determined by simply multiplying together the magnitudes
obtained in the two direct judgments; and a new direct judgment
may be required, respecting the magnitude of the ‘oughtness’ of an
action under given circumstances, which need not bear any simple
and necessary relation to the two former.

The hope, which sustained many investigators in the course of the
nineteenth century, of gradually bringing the moral sciences under
the sway of mathematical reasoning, steadily recedes—if we mean,
as they meant, by mathematics the introduction of precise numerical
methods. The old assumptions, that all quantity is numerical and
that all quantitative characteristics are additive, can be no longer
sustained. Mathematical reasoning now appears as an aid in its
symbolic rather than in its numerical character. I, at any rate, have
not the same lively hope as Condorcet, or even as Edgeworth, “éclairer
les Sciences morales et politiques par le flambeau de l’Algèbre.” In
the present case, even if we are able to range goods in order of
magnitude, and also their probabilities in order of magnitude, yet it

∗If pA = p A , w > w , and q = q , then cA > c A ; if pA = p A , w = w , and
q < q , then cA > c A ; if pA = p A , w > w , and q < q , then cA > c A ; but if pA = p A , w = w , and q > q , we cannot in general compare cA and c A .

pt. iv

A TREATISE ON PROBABILITY

362
does not follow that we can range the products composed of each
good and its corresponding probability in this order.

9.

Discussions of the doctrine of Mathematical Expectation,
apart from its directly ethical bearing, have chiefly centred round the
classic Petersburg Paradox,∗ which has been treated by almost all
the more notable writers, and has been explained by them in a great
variety of ways. The Petersburg Paradox arises out of a game in
which Peter engages to pay Paul one shilling if a head appears at the
first toss of a coin, two shillings if it does not appear until the second,
and, in general, 2r−1 shillings if no head appears until the rth toss.

What is the value of Paul’s expectation, and what sum must he hand
over to Peter before the game commences, if the conditions are to be
fair?

n

The mathematical answer is

( 1 )r2r−1, if the number of tosses is

2

1

∞
not in any case to exceed n in all, and

( 1 )r2r−1 if this restriction

2

1
is removed. That is to say, Paul should pay n shillings in the first

2
case, and an infinite sum in the second. Nothing, it is said, could
be more paradoxical, and no sane Paul would engage on these terms
even with an honest Peter.

Many of the solutions which have been offered will occur at once
to the reader. The conditions of the game imply contradiction, say

Poisson and Condorcet; Peter has undertaken engagements which
he cannot fulfil; if the appearance of heads is deferred even to the

100th toss, he will owe a mass of silver greater in bulk than the sun.

But this is no answer. Peter has promised much and a belief in his
solvency will strain our imagination; but it is imaginable. And in any
case, as Bertrand points out, we may suppose the stakes to be, not
shillings, but grains of sand or molecules of hydrogen.

∗For the history of this paradox see Todhunter. The name is due, he says,
to its having first appeared in a memoir by Daniel Bernoulli in the Commentarii
of the Petersburg Academy.

ch. xxvi

PHILOSOPHICAL APPLICATIONS

363

D’Alembert’s principal explanations are, first, that true expectation
is not necessarily the product of probability and profit (a view which
has been discussed above), and second, that very long runs are not
only very improbable, but do not occur at all.

The next type of solution is due, in the first instance, to Daniel

Bernoulli, and turns on the fact that no one but a miser regards the
desirability of different sums of money as directly proportional to their
amount; as Buffon says, “L’avare est comme le mathématicien: tous
deux estiment l’argent par sa quantité numérique.” Daniel Bernoulli
deduced a formula from the assumption that the importance of an
increment is inversely proportional to the size of the fortune to which
it is added. Thus, if x is the ‘physical’ fortune and y the ‘moral’

fortune,
dx
dy = k

,
x
or
x
y = k log

, where k and a are constants.

a

On the basis of this formula of Bernoulli’s a considerable theory
has been built up both by Bernoulli∗ himself and by Laplace.† It
leads easily to the further formula—
x = (a + x1)p1(a + x2)p2 . . . ,
where a is the initial ‘physical’ fortune, p1, etc., the probabilities
of obtaining increments x1, etc., to a, and x the ‘physical’ fortune
whose present possession would yield the same ‘moral’ fortune as
does the expectation of the various increments x1, etc. By means of
this formula Bernoulli shows that a man whose fortune is £1000 may
reasonably pay a £6 stake in order to play the Petersburg game with

£1 units. Bernoulli also mentions two solutions proposed by Cramer.

In the first all sums greater than 224 (16, 777, 116) are regarded as

∗“Specimen Theoriae Novae de Mensura Sortis,” Comm. Acad. Petrop. vol. v.

for 1730 and 1731, pp. 175–192 (published 1738). See Todhunter, pp. 213 et seq.

†Théorie analytique, chap. x. “De l’espérance morale,” pp. 432–445.

pt. iv

A TREATISE ON PROBABILITY

364

‘morally’ equal; this leads to £13 as the fair stake. According to the
other formula the pleasure derivable from a sum of money varies as
the square root of the sum; this leads to £2 : 9s. as the fair stake.

But little object is served by following out these arbitrary hypotheses.

As a solution of the Petersburg problem this line of thought is
only partially successful: if increases of ‘physical’ fortune beyond a
certain finite limit can be regarded as ‘morally’ negligible, Peter’s
claim for an infinite initial stake from Paul is, it is true, no longer
equitable, but with any reasonable law of diminution for successive
increments Paul’s stake will still remain paradoxically large. Daniel

Bernoulli’s suggestion is, however, of considerable historical interest
as being the first explicit attempt to take account of the important
conception known to modern economists as the diminishing marginal
utility of money,—a conception on which many important arguments
are founded relating to taxation and the ideal distribution of wealth.

Each of the above solutions probably contains a part of the
psychological explanation.

We are unwilling to be Paul, partly
because we do not believe Peter will pay us if we have good fortune
in the tossing, partly because we do not know what we should do
with so much money or sand or hydrogen if we won it, partly because
we do not believe we ever should win it, and partly because we do
not think it would be a rational act to risk an infinite sum or even a
very large finite sum for an infinitely larger one, whose attainment is
infinitely unlikely.

When we have made the proper hypotheses and have eliminated
these elements of psychological doubt, the theoretic dispersal of what
element of paradox remains must be brought about, I think, by a
development of the theory of risk. It is primarily the great risk of
the wager which deters us. Even in the case where the number of
tosses is in no case to exceed a finite number, the risk R, as already
defined, may be very great, and the relative risk R will be almost

E
unity. Where there is no limit to the number of tosses, the risk is
ch. xxvi

PHILOSOPHICAL APPLICATIONS

365
infinite. A relative risk, which approaches unity, may, it has been
already suggested, be a factor which must be taken into account in
ethical calculation.

10.

In establishing the doctrine, that all private gambling must
be with certainty a losing game, precisely contrary arguments are
employed to those which do service in the Petersburg problem. The
argument that “you must lose if only you go on long enough” is well
known. It is succinctly put by Laurent:∗ Two players A and B have
a and b francs respectively. f (a) is the chance that A will be ruined.

Thus
b
f (a) =

,† so that the poorer a gambler is, relatively to his
a + b
opponent, the more likely he is to be ruined. But further, if b = ∞,
f (a) = 1, i.e. ruin is certain. The infinitely rich gambler is the public.

It is against the public that the professional gambler plays, and his
ruin is therefore certain.

Might not Poisson and Condorcet reply, The conditions of the
game imply contradiction, for no gambler plays, as this argument
supposes, for ever?‡ At the end of any finite quantity of play, the
player, even if he is not the public, may finish with winnings of
any finite size. The gambler is in a worse position if his capital is
smaller than his opponents’—at poker, for instance, or on the Stock

Exchange. This is clear. But our desire for moral improvement
outstrips our logic if we tell him that he must lose. Besides it is
paradoxical to say that everybody individually must lose and that
everybody collectively must win. For every individual gambler who
loses there is an individual gambler or syndicate of gamblers who win.

The true moral is this, that poor men should not gamble and that
millionaires should do nothing else. But millionaires gain nothing by
gambling with one another, and until the poor man departs from the

∗Calcul des probabilités, p. 129.

†This would possibly follow from the theorem of Daniel Bernoulli.

The
reasoning by which Laurent obtains it seems to be the result of a mistake.

‡Cf. also Mr. Bradley, Logic, p. 217.

pt. iv

A TREATISE ON PROBABILITY

366
path of prudence the millionaire does not find his opportunity. If it
be replied that in fact most millionaires are men originally poor who
departed from the path of prudence, it must be admitted that the
poor man is not doomed with certainty. Thus the philosopher must
draw what comfort he can from the conclusion with which his theory
furnishes him, that millionaires are often fortunate fools who have
thriven on unfortunate ones.∗

11.

In conclusion we may discuss a little further the conception
of ‘moral’ risk, raised in § 8 and at the end of § 9. Bernoulli’s formula
crystallises the undoubted truth that the value of a sum of money to
a man varies according to the amount he already possesses. But does
the value of an amount of goodness also vary in this way? May it not
be true that the addition of a given good to a man who already enjoys
much good is less good than its bestowal on a man who has little? If
this is the case, it follows that a smaller but relatively certain good is
better than a greater but proportionately more uncertain good.

In order to assert this, we have only to accept a particular theory
of organic goodness, applications of which are common enough in the
mouths of political philosophers. It is at the root of all principles of
equality, which do not arise out of an assumed diminishing marginal
utility of money. It is behind the numerous arguments that an equal
distribution of benefits is better than a very unequal distribution. If
this is the case, it follows that, the sum of the goods of all parts
of a community taken together being fixed, the organic good of the
whole is greater the more equally the benefits are divided amongst
the individuals. If the doctrine is to be accepted, moral risks, like
financial risks, must not be undertaken unless they promise a profit
actuarially.

∗From the social point of view, however, this moral against gambling may
be drawn—that those who start with the largest initial fortunes are most likely
to win, and that a given increment to the wealth of these benefits them, on the
assumption of a diminishing marginal utility of money, less than it injures those
from whom it is taken.

ch. xxvi

PHILOSOPHICAL APPLICATIONS

367

There is a great deal which could be said concerning such a
doctrine, but it would lead too far from what is relevant to the
study of Probability. One or two instances of its use, however,
may be taken from the literature of Probability. In his essay, “Sur
l’application du calcul des probabilités à l’inoculation de la petite
vérole,”∗ D’Alembert points out that the community would gain on
the average if, by sacrificing the lives of one in five of its citizens, it
could ensure the health of the rest, but he argues that no legislator
could have the right to order such a sacrifice.

Galton, in his

Probability, the Foundation of Eugenics, employed an argument which
depends essentially on the same point. Suppose that the members of
a certain class cause an average detriment M to society, and that the
mischiefs done by the several individuals differ more or less from M by
amounts whose average is D, so that D is the average amount of the
individual deviations, all regarded as positive, from M; then, Galton
argued, the smaller D is, the stronger is the justification for taking
such drastic measures against the propagation of the class as would
be consonant to the feelings, if it were known that each individual
member caused a detriment M. The use of such arguments seems
to involve a qualification of the simple ethical doctrine that right
action should make the sum of the benefits of the several individual
consequences, each multiplied by its probability, a maximum.

On the other hand, the opposite view is taken in the Port Royal

Logic and by Butler, when they argue that everything ought to be
sacrificed for the hope of heaven, even if its attainment be thought
infinitely improbable, since “the smallest degree of facility for the
attainment of salvation is of higher value than all the blessings
of the world put together.”† The argument is, that we ought to

∗Opuscules mathématiques, vol. ii.

†Port Royal Logic (Eng. trans.), p. 369:

“It belongs to infinite things
alone, as eternity and salvation, that they cannot be equalled by any temporal
advantage; and thus we ought never to place them in the balance with any of
the things of the world.

This is why the smallest degree of facility for the
pt. iv

A TREATISE ON PROBABILITY

368
follow a course of conduct which may with the slightest probability
lead to an infinite good, until it is logically disproved that such a
result of our action is impossible. The Emperor who embraced the

Roman Catholic religion, not because he believed it, but because it
offered insurance against a disaster whose future occurrence, however
improbable, he could not certainly disprove, may not have considered,
however, whether the product of an infinitesimal probability and an
infinite good might not lead to a finite or infinitesimal result. In
any case the argument does not enable us to choose between different
courses of conduct, unless we have reason to suppose that one path is
more likely than another to lead to infinite good.

12.

In estimating the risk, ‘moral’ or ‘physical,’ it must be
remembered that we cannot necessarily apply to individual cases
results drawn from the observation of a long series resembling them
in some particular. I am thinking of such arguments as Buffon’s when
he names

1
as the limit, beyond which probability is negligible,

10,000
on the ground that, being the chance that a man of fifty-six taken at
random will die within a day, it is practically disregarded by a man of
fifty-six who knows his health to be good. “If a public lottery,” Gibbon
truly pointed out, “were drawn for the choice of an immediate victim,
and if our name were inscribed on one of the ten thousand tickets,
should we be perfectly easy?”

Bernoulli’s second axiom,∗ that in reckoning a probability we must
take everything into account, is easily forgotten in these cases of
statistical probabilities. The statistical result is so attractive in its
definiteness that it leads us to forget the more vague though more
important considerations which may be, in a given particular case,
within our knowledge. To a stranger the probability that I shall send
a letter to the post unstamped may be derived from the statistics
of the Post Office; for me those figures would have but the slightest
attainment of salvation is of higher value than all the blessings of the world put
together. . . .”

∗See p. 84.

ch. xxvi

PHILOSOPHICAL APPLICATIONS

369
bearing upon the question.

13.

It has been pointed out already that no knowledge of
probabilities, less in degree than certainty, helps us to know what
conclusions are true, and that there is no direct relation between the
truth of a proposition and its probability. Probability begins and ends
with probability. That a scientific investigation pursued on account
of its probability will generally lead to truth, rather than falsehood,
is at the best only probable. The proposition that a course of action
guided by the most probable considerations will generally lead to
success, is not certainly true and has nothing to recommend it but its
probability.

The importance of probability can only be derived from the
judgment that it is rational to be guided by it in action; and a
practical dependence on it can only be justified by a judgment that in
action we ought to act to take some account of it. It is for this reason
that probability is to us the “guide of life,” since to us, as Locke says,

“in the greatest part of our concernment, God has afforded only the

Twilight, as I may so say, of Probability, suitable, I presume, to that
state of Mediocrity and Probationership He has been pleased to place
us in here.”

PART V
the foundations of statistical
inference

CHAPTER XXVII
the nature of statistical inference

1.

The Theory of Statistics, as it is now understood,∗ can
be divided into two parts which are for many purposes better kept
distinct. The first function of the theory is purely descriptive. It
devises numerical and diagrammatic methods by which certain salient
characteristics of large groups of phenomena can be briefly described;
and it provides formulae by the aid of which we can measure or
summarise the variations in some particular character which we have
observed over a long series of events or instances. The second function
of the theory is inductive. It seeks to extend its description of certain
characteristics of observed events to the corresponding characteristics
of other events which have not been observed. This part of the
subject may be called the Theory of Statistical Inference; and it is
this which is closely bound up with the theory of probability.

2.

The union of these two distinct theories in a single science
is natural. If, as is generally the case, the development of some
inductive conclusion which shall go beyond the actually observed
instances is our ultimate object, we naturally choose those modes of
description, while we are engaged in our preliminary investigation,
which are most capable of extension beyond the particular instances
which they primarily describe. But this union is also the occasion of
a great deal of confusion. The statistician, who is mainly interested in
the technical methods of his science, is less concerned to discover the
precise conditions in which a description can be legitimately extended
by induction. He slips somewhat easily from one to the other, and
having found a complete and satisfactory mode of description he may
take less pains over the transitional argument, which is to permit him
to use this description for the purposes of generalisation.

∗See Yule, Introduction to Statistics, pp. 1–5, for a very interesting account
of the evolution of the meaning of the term statistics.

371
pt. v

A TREATISE ON PROBABILITY

372

One or two examples will show how easy it is to slip from
description into generalisation.

Suppose that we have a series
of similar objects one of the characteristics of which is under
observation;—a number of persons, for example, whose age at death
has been recorded. We note the proportion who die at each age,
and plot a diagram which displays these facts graphically. We then
determine by some method of curve fitting a mathematical frequency
curve which passes with close approximation through the points of
our diagram. If we are given the equation to this curve, the number of
persons who are comprised in the statistical series, and the degree of
approximation (whether to the nearest year or month) with which the
actual age has been recorded, we have a very complete and succinct
account of one particular characteristic of what may constitute a very
large mass of individual records. In providing this comprehensive
description the statistician has fulfilled his first function. But in
determining the accuracy with which this frequency curve can be
employed to determine the probability of death at a given age in
the population at large, he must pay attention to a new class of
considerations and must display a different kind of capacity. He must
take account of whatever extraneous knowledge may be available
regarding the sample of the population which came under observation,
and of the mode and conditions of the observations themselves. Much
of this may be of a vague kind, and most of it will be necessarily
incapable of exact, numerical, or statistical treatment. He is faced, in
fact, with the normal problems of inductive science, one of the data,
which must be taken into account, being given in a convenient and
manageable form by the methods of descriptive statistics.

Or suppose, again, that we are given, over a series of years, the
marriage rate and the output of the harvest in a certain area of
population. We wish to determine whether there is any apparent
degree of correspondence between the variations of the two within this
field of observation. It is technically difficult to measure such degree
of correspondence as may appear to exist between the variations in
ch. xxvii

STATISTICAL INFERENCE

373
two series, the terms of which are in some manner associated in
couples,—by coincidence, in this case, of time and place. By the
method of correlation tables and correlation coefficients the descriptive
statistician is able to effect this object, and to present the inductive
scientist with a highly significant part of his data in a compact and
instructive form. But the statistician has not, in calculating these
coefficients of observed correlation, covered the whole ground of which
the inductive scientist must take cognisance. He has recorded the
results of the observations in circumstances where they cannot be
recorded so clearly without the aid of technical methods; but the
precise nature of the conditions in which the observations took place
and the numerous other considerations of one sort or another, of
which we must take account when we wish to generalise, are not
usually susceptible of numerical or statistical expression.

The truth of this is obvious; yet, not unnaturally, the more
complicated and technical the preliminary statistical investigations
become, the more prone inquirers are to mistake the statistical
description for an inductive generalisation.∗ This tendency, which has
existed in some degree, as, I think, the whole history of the subject
shows, from the eighteenth century down to the present time, has been
further encouraged by the terminology in ordinary use. For several
statistical coefficients are given the same name when they are used
for purely descriptive purposes, as when corresponding coefficients
are used to measure the force or the precision of an induction. The
term ‘probable error,’ for example, is used both for the purpose of
supplementing and improving a statistical description, and for the
purpose of indicating the precision of some generalisation. The term

‘correlation’ itself is used both to describe an observed characteristic
of particular phenomena and in the enunciation of an inductive law

∗Cf. Whitehead, Introduction to Mathematics, p. 27: “There is no more
common error than to assume that, because prolonged and accurate mathematical
calculations have been made, the application of the result to some fact of nature
is absolutely certain.”

pt. v

A TREATISE ON PROBABILITY

374
which relates to phenomena in general.

3.

I have been at pains to enforce this contrast between
statistical description and statistical induction, because the chapters
which follow are to be entirely about the latter, whereas nearly all
statistical treatises are mainly concerned with the former. My object
will be to analyse, so far as I can, the logical basis of statistical
modes of argument. This involves a double task. To mark down those
which are invalid amongst arguments having the support of authority
is relatively easy. The other branch of our investigation, namely, to
analyse the ground of validity in the case of those arguments the
force of which all of us do in fact admit, presents the same kind of
fundamental difficulties as we met with in the case of Induction.

4.

The arguments with which we have to deal fall into three
main classes:

(i.) Given the probability relative to certain evidence of each
of a series of events, what are the probabilities, relative to the
same evidence, of various proportionate frequencies of occurrence for
the events over the whole series? Or more briefly, how often may
we expect an event to happen over a series of occasions, given its
probability on each occasion?

(ii.) Given the frequency with which an event has occurred on
a series of occasions, with what probability may we expect it on a
further occasion?

(iii.) Given the frequency with which an event has occurred on a
series of occasions, with what frequency may we probably expect it
on a further series of occasions?

In the first type of argument we seek to infer an unknown
statistical frequency from an à priori probability. In the second
type we are engaged on the inverse operation, and seek to base the
calculation of a probability on an observed statistical frequency. In
the third type we seek to pass from an observed statistical frequency,
not merely to the probability of an individual occurrence, but to the
probable values of other unknown statistical frequencies.

ch. xxvii

STATISTICAL INFERENCE

375

Each of these types of argument can be further complicated
by being applied not simply to the occurrence of a simple event
but to the concurrence under given conditions of two or more
events. When this two or more dimensional classification replaces
the one dimensional, the theory becomes what is sometimes termed

Correlation, as distinguished from simple Statistical Frequency.

5.

In Chapter XXVIII. I touch briefly on the observed phenomena
which have given rise to the so-called Law of Great Numbers, and the
discovery of which first set statistical investigation going. In Chapter

XXIX. the first type of argument, as classified above, is analysed,
and the conditions which are required for its validity are stated. The
crucial problem of attacking the second and third types of argument
is the subject of my concluding chapters.

CHAPTER XXVIII
the law of great numbers

Natura quidem suas habet consuetudines, natas ex reditu causarum, sed non
nisi ὡς ὲπὶ τὸ πολύ. Novi morbi inundant subiande humanum genus, quodsi ergo
de mortibus quotcunque experimenta feceris, non ideo naturae rerum limites
posuisti, ut pro futuro variare non possit.—Leibniz in a letter to Bernoulli,

December 3, 1703.

1.

It has always been known that, while some sets of events
invariably happen together, other sets generally happen together.

That experience shows one thing, while not always a sign of another,
to be a usual or probable sign of it, must have been one of the
earliest and most primitive forms of knowledge. If a dog is generally
given scraps at table, that is sufficient for him to judge it reasonable
to be there. But this kind of knowledge was slow to be made precise.

Numerous experiments must be carefully recorded before we can know
at all accurately how usual the association is. It would take a dog a
long time to find out that he was given scraps except on fast days,
and that there was the same number of these in every year.

The necessary kind of knowledge began to be accumulated during
the seventeenth and eighteenth centuries by the early statisticians.

Halley and others began to construct mortality tables; the proportion
of the births of each sex were tabulated; and so forth. These
investigations brought to light a new fact which had not been
suspected previously—namely, that in certain cases of partial
association the degree of association, i.e. the proportion of instances
in which it existed, shows a very surprising regularity, and that
this regularity becomes more marked the greater the number of the
instances under consideration. It was found, for example, not merely
that boys and girls are born on the whole in about equal proportions,
but that the proportion, which is not one of complete equality, tends
everywhere, when the number of recorded instances becomes large, to

376
ch. xxviii

STATISTICAL INFERENCE

377
approximate towards a certain definite figure.

During the eighteenth century matters were not pushed much
further than this, that in certain cases, of which comparatively
few were known, there was this surprising regularity, increasing in
degree as the instances became more numerous. Bernoulli, however,
took the first step towards giving it a theoretical basis by showing
that, if the à priori probability is known throughout, then (subject
to certain conditions which he himself did not make clear) in the
long run a certain determinate frequency of occurrence is to be
expected. Süssmilch (Die göttliche Ordnung in den Veränderungen
des menschlichen Geschlechts, 1741) discovered a theological interest
in these regularities. Such ideas had become sufficiently familiar
for Gibbon to characterise the results of probability as “so true in
general, so fallacious in particular.” Kant found in them (as many
later writers have done) some bearing on the problem of Free Will.∗

But with the nineteenth century came bolder theoretical methods
and a wider knowledge of facts. After proving his extension of

Bernoulli’s Theorem,† Poisson applied it to the observed facts, and
gave to the principle underlying these regularities the title of the

Law of Great Numbers. “Les choses de toutes natures,” he wrote,‡

∗In Idee zu einer allgemeinen Geschichte in weltbürgerlicher Absicht, 1784.

For a discussion of this passage and for the connection between Kant and

Süssmilch, see Lottin’s Quetelet, pp. 367, 368.

†See p. 391.

‡Recherches, pp. 7–12. Von Bortkiewicz (Kritische Betrachtungen, 1st part,
pp. 655–660) has maintained that Poisson intended to state his principle in a
less general way than that in which it has been generally taken, and that he
was misunderstood by Quetelet and others.

If we attend only to Poisson’s
contributions to Comptes Rendus in 1835 and 1836 and to the examples he
gives there, it is possible to make out a good case for thinking that he intended
his law to extend only to cases where certain strict conditions were fulfilled.

But this is not the spirit of his more popular writings or of the passage
quoted above.

At any rate, it is the fashion, in which Poisson influenced
his contemporaries, that is historically interesting; and this is certainly not
represented by Von Bortkiewicz’s interpretation.

pt. v

A TREATISE ON PROBABILITY

378

“sont soumises à une loi universelle qu’on peut appeler la loi des
grands nombres. . . . De ces exemples de toutes natures, il résulte
que la loi universelle des grands nombres est déjà pour nous un fait
général et incontestable, résultant d’expériences qui ne se démentent
jamais.” This is the language of exaggeration; it is also extremely
vague. But it is exciting; it seems to open up a whole new field to
scientific investigation; and it has had a great influence on subsequent
thought. Poisson seems to claim that, in the whole field of chance and
variable occurrence, there really exists, amidst the apparent disorder,
a discoverable system. Constant causes are always at work and
assert themselves in the long run, so that each class of event does
eventually occur in a definite proportion of cases. It is not clear how
far Poisson’s result is due to à priori reasoning, and how far it is a
natural law based on experience; but it is represented as displaying a
certain harmony between natural law and the à priori reasoning of
probabilities.

Poisson’s conception was mainly popularised through the writings
of Quetelet. In 1823 Quetelet visited Paris on an astronomical errand,
where he was introduced to Laplace and came into touch with “la
grande école française.” “Ma jeunesse et mon zèle,” he wrote in later
years, “ne tardèrent pas à me mettre en rapport avec les hommes
les plus distingués de cette époque; qu’on me permette de citer

Fourier, Poisson, Lacroix, spécialement connus, comme Laplace, par
leurs excellents écrits sux la théorie mathématique des probabilités. . . .

C’est donc au milieu des savants, statisticiens, et économistes de ce
temps que j’ai commencé mes travaux.”∗ Shortly afterwards began his
long series of papers, extending down to 1873, on the application of

Probability to social statistics. He wrote a text-book on Probability
in the form of letters for the instruction of the Prince Consort.

Before accepting in 1815 at the age of nineteen (with a view to a

∗For the details of the life of Quetelet and for a very full discussion of his
writings with special reference to Probability, see Lottin’s Quetelet, statisticien
et sociologue.

ch. xxviii

STATISTICAL INFERENCE

379
livelihood) a professorship of mathematics, Quetelet had studied as
an art student and written poetry; a year later an opera, of which
he was part-author, was produced at Ghent. The character of his
scientific work is in keeping with these beginnings. There is scarcely
any permanent, accurate contribution to knowledge which can be
associated with his name. But suggestions, projects, far-reaching
ideas he could both conceive and express, and he has a very fair
claim, I think, to be regarded as the parent of modern statistical
method.

Quetelet very much increased the number of instances of the

Law of Great Numbers, and also brought into prominence a slightly
variant type of it, of which a characteristic example is the law of
height, according to which the heights of any considerable sample
taken from any population tend to group themselves according to a
certain well-known curve. His instances were chiefly drawn from social
statistics, and many of them were of a kind well calculated to strike
the imagination—the regularity of the number of suicides, “l’effrayante
exactitude avec laquelle les crimes se reproduisent,” and so forth.

Quetelet writes with an almost religious awe of these mysterious laws,
and certainly makes the mistake of treating them as being as adequate
and complete in themselves as the laws of physics, and as little needing
any further analysis or explanation.∗ Quetelet’s sensational language
may have given a considerable impetus to the collection of social
statistics, but it also involved statistics in a slight element of suspicion
in the minds of some who, like Comte, regarded the application of the
mathematical calculus of probability to social science as “purement

∗Compare, for instance, the following passage from Recherches sur le penchant
au crime: “Il me semble que ce qui se rattache à l’espèce humaine, considérée
en masse, est de l’ordre des faits physiques; plus le nombre des individus est
grand, plus la volonté individuelle s’efface et laisse prédominer la série des faits
généraux qui dépendent des causes générales. . . . Ce sont ces causes qu’il s’agit
de saisir, et dès qu’on les connaîtra, on en déterminera les effets pour la société
comme on détermine les effets par les causes dans les sciences physiques.”

pt. v

A TREATISE ON PROBABILITY

380
chimérique et, par conséquent, tout à fait vicieuse.” The suspicion
of quackery has not yet disappeared. Quetelet belongs, it must be
admitted, to the long line of brilliant writers, not yet extinct, who
have prevented Probability from becoming, in the scientific salon,
perfectly respectable. There is still about it for scientists a smack of
astrology, of alchemy.

The progress of the conception since the time of Quetelet has
been steady and uneventful; and long strides towards this perfect
respectability have been taken. Instances have been multiplied and
the conditions necessary for the existence of statistical stability have
been to some extent analysed. While the most fruitful applications
of these methods have still been perhaps, as at first, in social
statistics and in errors of observation, a number of uses for them
have been discovered in quite recent times in the other sciences; and
the principles of Mendelism have opened out for them a great field of
application throughout biology.

2.

The existence of numerous instances of the Law of Great

Numbers, or of something of the kind, is absolutely essential for
the importance of Statistical Induction. Apart from this the more
precise parts of statistics, the collection of facts for the prediction of
future frequencies and associations, would be nearly useless. But the

‘Law of Great Numbers’ is not at all a good name for the principle
which underlies Statistical Induction. The ‘Stability of Statistical

Frequencies’ would be a much better name for it. The former suggests,
as perhaps Poisson intended to suggest, but what is certainly false,
that every class of event shows statistical regularity of occurrence if
only one takes a sufficient number of instances of it. It also encourages
the method of procedure, by which it is thought legitimate to take
any observed degree of frequency or association, which is shown in
a fairly numerous set of statistics, and to assume with insufficient
investigation that, because the statistics are numerous, the observed
degree of frequency is therefore stable. Observation shows that some
statistical frequencies are, within narrower or wider limits, stable.

ch. xxviii

STATISTICAL INFERENCE

381

But stable frequencies are not very common, and cannot be assumed
lightly.

The gradual discovery, that there are certain classes of phenomena,
in which, though it is impossible to predict what will happen in each
individual case, there is nevertheless a regularity of occurrence if the
phenomena be considered together in successive sets, gives the clue to
the abstract inquiry upon which we are about to embark.

CHAPTER XXIX
the use of à priori probabilities for the
prediction of statistical frequency—the
theorems of bernoulli, poisson, and tchebycheff

Hoc igitur est illud Problema, quod evulgandum hoc loco proposui, postquam
jam per vicennium pressi, et cujus tum novitas, tum summa utilitas cum
pari conjuncta difficultate omnibus reliquis hujus doctrinae capitibus pondus et
pretium superaddere potest.—Bernoulli.∗

1.

Bernoulli’s Theorem is generally regarded as the central
theorem of statistical probability. It embodies the first attempt to
deduce the measures of statistical frequencies from the measures of
individual probabilities, and it is a sufficient fruit of the twenty
years which Bernoulli alleges that he spent in reaching his result, if
out of it the conception first arose of general laws amongst masses
of phenomena, in spite of the uncertainty of each particular case.

But, as we shall see, the theorem is only valid subject to stricter
qualifications, than have always been remembered, and in conditions
which are the exception, not the rule.

The problem, to be discussed in this chapter, is as follows: Given
a series of occasions, the probability† of the occurrence of a certain
event at each of which is known relative to certain initial data h,
on what proportion of these occasions may we reasonably anticipate
the occurrence of the event? Given, that is to say, the individual
probability of each of a series of events à priori, what statistical
frequency of occurrence of these events is to be anticipated over the
whole series? Beginning with Bernoulli’s Theorem, we will consider
the various solutions of this problem which have been propounded,

∗Ars Conjectandi, p. 227.

†In the simplest cases, dealt with by Bernoulli, these probabilities are all
supposed equal.

382
ch. xxix

STATISTICAL INFERENCE

383
and endeavour to determine the proper limits within which each
method has validity.

2.

Bernoulli’s Theorem in its simplest form is as follows:

If the probability of an event’s occurrence under certain conditions
is p, then, if these conditions are present on m occasions, the most
probable number of the event’s occurrences is mp (or the nearest
integer to this), i.e. the most probable proportion of its occurrences
to the total number of occasions is p: further, the probability that
the proportion of the event’s occurrences will diverge from the most
probable proportion p by less than a given amount b, increases as
m increases, the value of this probability being calculable by a process
of approximation.

The probability of the event’s occurring n times and failing
m − n times out of the m occasions is (subject to certain conditions
to be elucidated later) pnqm−n multiplied by the coefficient of this
expression in the expansion of (p + q)m, where p + q = 1. If we
write n = mp − h, this term is
m!

pnqm−n. It is easily

(mp − h)!(mq + h)!

shown that this is a maximum when h = 0, i.e. when n = mp (or
the nearest integer to this, where mp is not integral). This result
constitutes the first part of Bernoulli’s Theorem.

For the second part of the theorem some method of approximation
is required. Provided that m is large, we can simplify the expression
m!

pnqm−n by means of Stirling’s Theorem, and

(mp − h)!(mq + h)!

obtain as its approximate value

1

√
e− h2

2mpq

2πmpq

As before, this is a maximum when h = 0, i.e. when n = mp.

It is possible, of course, by more complicated formulae to obtain
closer approximations than this.∗ But there is an objection, which

∗See, e.g., Bowley, Elements of Statistics, p. 298. The objection about to be
pt. v

A TREATISE ON PROBABILITY

384
can be raised to this approximation, quite distinct from the fact that
it does not furnish a result correct to as many places of decimals as
it might. This is, that the approximation is independent of the sign
of h, whereas the original expression is not thus independent. That
is to say, the approximation implies a symmetrical distribution for
different values of h about the value for h = 0; while the expression
under approximation is unsymmetrical. It is easily seen that this
want of symmetry is appreciable unless mpq is large. We ought,
therefore, to have laid it down as a condition of our approximation,
not only that m must be large, but also that mpq must be large.

Unlike most of my criticisms, this is a mathematical, rather than a
logical point. I recur to it in § 15.

“Par une fiction qui rendra les calculs plus faciles” (to quote

Bertrand), we now replace the integer h by a continuous variable z
and argue that the probability that the amount of the divergence
from the most probable value mp will lie between z and z + dz, is

1

√
e− z2

2mpq dz.

2πmpq

This ‘fiction’ will do no harm so long as it is remembered that we are
now dealing with a particular kind of approximation. The probability
that the divergence h from the most probable value mp will be less
than some given quantity a is, therefore,

1

+a

√
e− z2

2mpq dz.

2πmpq −a

If we put
z

√

= t, this is equal to

2mpq
a

2

√2mpq

√
e−t2dt.

π 0
raised does not apply to these closer approximations.

ch. xxix

STATISTICAL INFERENCE

385

√

Thus, if we write a = 2mpqγ, the probability∗ that the number of
occurrences will lie between
mp +

2mpqγ and mp −

2mpqγ
γ
is measured by† 2

√
e−t2dt. This same expression measures the
π 0
probability that the proportion of occurrences will lie between

2pq

2pq
p +
γ and p −
γ.

m
m
t

The different values of the integral 2

√
e−t2dt = Θ(t) are given in
π 0
tables.‡

The probability that the proportion of occurrences will lie between
given limits varies with the magnitude of

2pq , and this expression
m
is sometimes used, therefore, to measure the ‘precision’ of the
series. Given the à priori probabilities, the precision varies inversely
with the square root of the number of instances.

Thus, while
the probability that the absolute divergence will be less than a

∗The replacement of the integer h by the continuous variable z may render
the formula rather deceptive. It is certain, for example, that the error does not
lie between h and h + 1.

†The above proof follows the general lines of Bertrand’s (Calcul des
probabilités, chap. iv.). Some writers, using rather more precision, give the result
as

2
γ
e−γ2

√
e−t2 dt + √
π 0

2πmpq

(e.g. Laplace, by the use of Euler’s Theorem, and more recently Czuber,

Wahrscheinlichkeitsrechnung, vol. i. p. 121). As the whole formula is approximate,
the simpler expression given in the text is probably not less satisfactory in
practice. See also Czuber, Entwicklung, pp. 76, 77, and Eggenberger, Beiträge
zur Darstellung des Bernoullischen Theorems.

‡A list of the principal tables is given by Czuber, loc. cit. vol. i. p. 122.

pt. v

A TREATISE ON PROBABILITY

386
given amount a decreases, the probability that the corresponding
proportionate divergence (i.e. the absolute divergence divided by the
number of instances) will be less than a given amount b, increases, as
the number of instances increases. This completes the second part of

Bernoulli’s Theorem.

3.

Bernoulli himself was not acquainted with Stirling’s theorem,
and his proof differs a good deal from the proof outlined in § 2. His
final enunciation of the theorem is as follows: If in each of a given
series of experiments there are r contingencies favourable to a given
event out of a total number of contingencies t, so that r is the
t
probability of the event at each experiment, then, given any degree
of probability c, it is possible to make such a number of experiments
that the probability, that the proportionate number of the event’s
occurrences will lie between r + 1 and r − 1, is greater than c.∗
t
t

4.

We seem, therefore, to have proved that, if the à priori
probability of an event under certain conditions is p, the proportion
of times most probable à priori for the event’s occurrence on a series
of occasions where the conditions are satisfied is also p, and that if
the series is a long one the proportion is very unlikely to differ widely
from p. This amounts to the principle which Ellis† and Venn have
employed as the defining axiom of probability, save that if the series
is ‘long enough’ the proportion, according to them, will certainly

∗Ars Conjectandi, p. 236 (I have translated freely). There is a brief account
of Bernoulli’s proof in Todhunter’s History, pp. 71, 72. The problem is dealt
with by Laplace, Théorie analytique, livre ii. chap. iii.

For an account of

Laplace’s proof see Todhunter’s History, pp. 548–553.

†On the Foundation of the Theory of Probabilities: “If the probability of a
given event be correctly determined, the event will on a long run of trials tend
to recur with frequency proportional to this probability. This is generally proved
mathematically. It seems to me to be true à priori . . . . I have been unable to
sever the judgment that one event is more likely to happen than another from
the belief that in the long run it will occur more frequently.”

ch. xxix

STATISTICAL INFERENCE

387
be p. Laplace∗ believed that the theorem afforded a demonstration
of a general law of nature, and in his second edition published
in 1814 he replaces† the eloquent dedication, A Napoléon-le-Grand,
which prefaces the edition of 1812, by an explanation that Bernoulli’s

Theorem must always bring about the eventual downfall of a great
power which, drunk with the love of conquest, aspires to a universal
domination,—“c’est encore un résultat du calcul des probabilités,
confirmé par de nombreuses et funestes expériences.”

5.

Such is the famous Theorem of Bernoulli which some
have believed‡ to have a universal validity and to be applicable
to all ‘properly calculated’ probabilities. Yet the theorem exhibits
algebraical rather than logical insight. And, for reasons about to be
given, it will have to be conceded that it is only true of a special
class of cases and requires conditions, before it can be legitimately
applied, of which the fulfilment is rather the exception than the rule.

For consider the case of a coin of which it is given that the two faces
are either both heads or both tails: at every toss, provided that the
results of the other tosses are unknown, the probability of heads is 12
and the probability of tails is 1; yet the probability of m heads and

2
m tails in 2m tosses is zero, and it is certain à priori that there
will be either 2m heads or none. Clearly Bernoulli’s Theorem is
inapplicable to such a case. And this is but an extreme case of a
normal condition.

For the first stage in the proof of the theorem assumes that, if p is
the probability of one occurrence, pr is the probability of r occurrences

∗Essai philosophique, p. 53: “On peut tirer du théorème précédent cette
conséquence qui doit être regardée comme une loi générale, savoir, que les
rapports des effets de la nature, sont à fort peu près constans, quand ces effets
sont considérés en grand nombre.”

†Introduction, pp. liii, liv.

‡Even by Mr. Bradley, Principles of Logic, p. 214. After criticising Venn’s
view he adds: “It is false that the chances must be realised in a series.

It
is, however, true that they most probably will be, and true again that this
probability is increased, the greater the length we give to our series.”

pt. v

A TREATISE ON PROBABILITY

388
running. Our discussion of the theorems of multiplication will have
shown how considerable an assumption this involves. It assumes
that a knowledge of the fact that the event has occurred on every
one of the first r − 1 occasions does not in any degree affect the
probability of its occurrence on the rth. Thus Bernoulli’s Theorem is
only valid if our initial data are of such a character that additional
knowledge, as to the proportion of failures and successes in one part
of a series of cases is altogether irrelevant to our expectation as to
the proportion in another part. If, for example, the initial probability
of the occurrence of an event under certain circumstances is one in
a million, we may only apply Bernoulli’s Theorem to evaluate our
expectation over a million trials, if our original data are of such a
character that, even after the occurrence of the event in every one of
the first million trials, the probability in the light of this additional
knowledge that the event will occur on the next occasion is still no
more than one in a million.

Such a condition is very seldom fulfilled. If our initial probability
is partly founded upon experience, it is clear that it is liable to
modification in the light of further experience. It is, in fact, difficult
to give a concrete instance of a case in which the conditions for the
application of Bernoulli’s Theorem are completely fulfilled. At the
best we are dealing in practice with a good approximation, and can
assert that no realised series of moderate length can much affect our
γ
initial probability. If we wish to employ the expression 2

√
e−t2dt
π 0
we are in a worse position. For this is an approximate formula
which requires for its validity that the series should be long; whilst
it is precisely in this event, as we have seen above, that the use of

Bernoulli’s Theorem is more than usually likely to be illegitimate.

6.

The conditions, which have been described above, can be
expressed precisely as follows:

Let mxn represent the statement that the event has occurred
on m out of n occasions and has not occurred on the others; and
ch. xxix

STATISTICAL INFERENCE

389
let 1x1/h = p, where h represents our à priori data, so that p
is the à priori probability of the event in question. Bernoulli’s

Theorem then requires a series of conditions, of which the following
is typical: m+1xn+1/mxn h =1 x1/h, i.e. the probability of the event
on the n + 1th occasion must be unaffected by our knowledge of its
proportionate frequency on the first n occasions, and must be exactly
equal to its à priori probability before the first occasion.

Let us select one of these conditions for closer consideration. If
yr represents the statement that the event has occurred on each of
r successive occasions, yr/h = yr/yr−1h yr−1/h and so on, so that
s=r
yr/h =
ys/ys−1h. Hence if we are to have yr/h = pr, we must
s=1
have ys/ys−1h = p for all values of s from 1 to r. But in many
particular examples ys/ys−1h increases with s, so that yr/h > pr.

Bernoulli’s Theorem, that is to say, tends, if it is carelessly applied,
to exaggerate the rate at which the probability of a given divergence
from the most probable decreases as the divergence increases. If
we are given a penny of which we have no reason to doubt the
regularity, the probability of heads at the first toss is 1; but if heads

2
fall at every one of the first 999 tosses, it becomes reasonable to
estimate the probability of heads at the thousandth toss at much
more than 1. For the à priori probability of its being a conjurer’s

2
penny, or otherwise biassed so as to fall heads almost invariably, is
not usually so infinitesimally small as (1)1000. We can only apply

2

Bernoulli’s Theorem with rigour for a prediction as to the penny’s
behaviour over a series of a thousand tosses, if we have à priori such
exhaustive knowledge of the penny’s constitution and of the other
conditions of the problem that 999 heads running would not cause us
to modify in any respect our prediction à priori.

7.

It seldom happens, therefore, that we can apply Bernoulli’s

Theorem with reference to a long series of natural events. For in such
cases we seldom possess the exhaustive knowledge which is necessary.

Even where the series is short, the perfectly rigorous application
pt. v

A TREATISE ON PROBABILITY

390
of the Theorem is not likely to be legitimate, and some degree of
approximation will be involved in utilising its results.

Not so infrequently, however, artificial series can be devised
in which the assumptions of Bernoulli’s Theorem are relatively
legitimate.∗ Given, that is to say, a proposition a1, some series
a1a2 . . . can be found, which satisfies the conditions:

(i.) a1/h = a2/h . . . = ar/h.

(ii.) ar/as . . . ¯at . . . h = ar/h.

Adherents of the Frequency Theory of Probability, who use the
principal conclusion of Bernoulli’s Theorem as the defining property
of all probabilities, sometimes seem to mean no more than that,
relative to given evidence, every proposition belongs to some series,
to the members of which Bernoulli’s Theorem is rigorously applicable.

But the natural series, the series, for example, in which we are most
often interested, where the a’s are alike in being accompanied by
certain specified conditions c, is not, as a rule, rigorously subject to
the Theorem. Thus ‘the probability of a in certain conditions c is 1’

2
is not in general equivalent, as has sometimes been supposed, to ‘It
is 500 to 1 that in 40, 000 occurrences of c, a will not occur more
than 20, 200 times, and 500 to 1 that it will not occur less than

19, 800 times.’

8.

Bernoulli’s Theorem supplies the simplest formula by which
we can attempt to pass from the à priori probabilities of each of a
series of events to a prediction of the statistical frequency of their
occurrence over the whole series. We have seen that Bernoulli’s

Theorem involves two assumptions, one (in the form in which it is
usually enunciated) tacit and the other explicit. It is assumed, first,
that a knowledge of what has occurred at some of the trials would

∗In the discussion in Chapter XVI., p. 195, of the probability of a divergence from an equality of heads and tails in coin-tossing, an example has been given
of the construction of an artificial series in which the application of Bernoulli’s

Theorem is more legitimate than in the natural series.

ch. xxix

STATISTICAL INFERENCE

391
not affect the probability of what may occur at any of the others; and
it is assumed, secondly, that these probabilities are all equal à priori.

It is assumed, that is to say, that the probability of the event’s
occurrence at the rth trial is equal à priori to its probability at the
nth trial, and, further, that it is unaffected by a knowledge of what
may actually have occurred at the nth trial.

A formula, which dispenses with the explicit assumption of equal
à priori probabilities at every trial, was proposed by Poisson,∗ and
is usually known by his name. It does not dispense, however, with
the other inexplicit assumption. The difference between Poisson’s

Theorem and Bernoulli’s is best shown by reference to the ideal
case of balls drawn from an urn. The typical example for the valid
application of Bernoulli’s Theorem is that of balls drawn from a
single urn, containing black and white balls in a known proportion,
and replaced after each drawing, or of balls drawn from a series
of urns, each containing black and white balls in the same known
proportion. The typical example for Poisson’s Theorem is that of
balls drawn from a series of urns, each containing black and white
balls in different known proportions.

Poisson’s Theorem may be enunciated as follows:† Let s trials be
made, and at the λth trial (λ = 1, 2 . . . s) let the probabilities for the
occurrence and non-occurrence of the event be pλ, qλ respectively.

Then, if
pλ = p, the probability that the number of occurrences m
s
of the event in the s trials will lie between the limits sp ± l is given by

2
l
e− l2
k2s

P =

√
e− x2
k2s dx +

√
k πs 0
k πs
where

2
p
k =
λqλ .

s

∗Recherches, pp. 246 et seq.

†For the proof see Poisson, Recherches, loc. cit., or Czuber, Wahrschein-
lichkeitsrechnung, vol. i. pp. 153–159.

pt. v

A TREATISE ON PROBABILITY

392

By substituting x

√ = t and
l

√ = γ, this may be written in a
k s
k s
form corresponding to that of Bernoulli’s Theorem,∗ namely:

The probability that the number of occurrences of the event will

√
lie between sp ± γk s is given by

2
γ
e−γ2

P = √
e−t2 dt + √
π 0
k πs

9.

This is a highly ingenious theorem and extends the application
of Bernoulli’s results to some important types of cases. It embraces,
for example, the case in which the successive terms of a series are
drawn from distinct populations known to be characterised by differing
statistical frequencies; no further complication being necessary beyond
the calculation of two simple functions of these frequencies and of the
number of terms in the series. But it is important not to exaggerate
the degree to which Poisson’s method has extended the application
of Bernoulli’s results. Poisson’s Theorem leaves untouched all those
cases in which the probabilities of some of the terms in the series of
events can be influenced by a knowledge of how some of the other
terms in the series have turned out.

Amongst these cases two types can be distinguished. In the
first type such knowledge would lead us to discriminate between
the conditions to which the different instances are subject. If, for
example, balls are drawn from a bag, containing black and white
balls in known proportions, and not replaced, the knowledge whether
or not the first ball drawn was black affects the probability of the
second ball’s being black because it tells us how the conditions in
which the second ball is drawn differ from those in which the first ball
was drawn. In the second type such knowledge does not lead us to
discriminate between the conditions to which the different instances
are subject, but it leads us to modify our opinion as to the nature

∗For the analogous form of Bernoulli’s Theorem see p. 385 (footnote).

ch. xxix

STATISTICAL INFERENCE

393
of the conditions which apply to all the terms alike. If, for instance,
balls are drawn from a bag, which is one, but it is not certainly
known which, out of a number of bags containing black and white
balls in differing proportions, the knowledge of the colour of the first
ball drawn affects the probabilities at the second drawing, because it
throws some light upon the question as to which bag is being drawn
from.

This last type is that to which most instances conform which are
drawn from the real world. A knowledge of the characteristics of some
members of a population may give us a clue to the general character
of the population in question. Yet it is this type, where there is a
change in knowledge but no change in the material conditions from
one instance to the next, which is most frequently overlooked.∗ It
will be worth while to say something further about each of these two

∗Numerous instances could be quoted. To take a recent English example,
reference may be made to Yule, Introduction to the Theory of Statistics, p. 251.

Mr. Yule thinks that the condition of independence is satisfied if “the result
of any one throw or toss does not affect, and is unaffected by, the results
of the preceding and following tosses,” and does not allow for the cases in
which knowledge of the result is relevant apart from any change in the physical
conditions.

pt. v

A TREATISE ON PROBABILITY

394
types.∗

10.

For problems of the first type, where there is physical or
material dependence between the successive trials, it is not possible, I
think, to propose any general solution; since the probabilities of the
successive trials may be modified in all kinds of different ways. But
for particular problems, if the conditions are precise enough, solutions
can be devised. The problem, for instance, of an urn, containing black
and white balls in known proportions, from which balls are drawn
successively and not replaced,† is ingeniously solved by Czuber‡ with
the aid of Stirling’s Theorem. If σ is the number of balls and s the
number of drawings, he reaches the interesting conclusion (assuming
that σ, s and σ − s are all large) that the probability of the number of
black balls lying within given limits is the same as it would be if the
balls were replaced after each drawing and the number of drawings
were σ − ss instead of s.

σ

In addition to the assumptions already stated, Professor Czuber’s
solution applies only to those cases where the limits, for which we
wish to determine the probability, are narrow compared with the

∗The types which I distinguish under four heads (the Bernoullian, the

Poissonian, and the two described above) Bachelier (Calcul des probabilités,
p. 155) classifies as follows:

(i.) When the conditions are identical throughout, the problem has uniformité;

(ii.) When they vary from stage to stage, but according to a law given from
the beginning and in a manner which does not depend upon what has happened
at the earlier stages, it has indépendance;

(iii.) When they vary in a manner which depends upon what has happened
at the earlier stages, it has connexité.

Bachelier gives solutions for each type on the assumption that the number
of trials is very great, and that the number of successes or failures can be
regarded as a continuous variable. This is the same kind of assumption as that
made in the proof of Bernoulli’s Theorem given in § 2, and is open to the same
objections,—or rather the value of the results is limited in the same way.

†It is of no consequence whether the balls are drawn successively and not
replaced, or are drawn simultaneously.

‡Loc. cit. vol. i. pp. 163, 164.

ch. xxix

STATISTICAL INFERENCE

395
total number of black balls pσ. Professor Pearson∗ has worked out
the same problem in a much more general manner, so as to deal
with the whole range, i.e. the frequency or probability of all possible
ratios of black balls, even where s > pσ. The various forms of curve,
which result, according to the different relations existing between p, s,
and σ, supply examples of each of the different types of frequency
curve which arise out of a classification according to (i.) skewness or
symmetry, (ii.) limitation of range in one, both or neither direction;
and he designates, therefore, the curves which are thus obtained as
generalised probability curves. His discussion of the properties of these
curves is interesting, however, to the student of descriptive statistics
rather than to the student of probability. The most generalised and,
mathematically, by far the most elegant treatment of this problem,
with which I am acquainted, is due to Professor Tschuprow.†

Poisson, in attempting a somewhat similar problem,‡ arrives at a
result, which seems obviously contrary to good sense, by a curious,
but characteristic, misapprehension of the meaning of ‘independence’

in probability. His problem is as follows: If l balls be taken out from
an urn, containing c black and white balls in known proportions, and
not replaced, and if a further number of balls µ be then taken out,
the probability that a given proportion
m
of these µ balls will
m + n
be black is independent of the number and the colour of the l balls
originally drawn out. For, he argues, if l + µ balls are drawn out,
the probability of a combination, which is made up of l black and
white balls in given proportions followed by µ balls, of which m are
white and n black, must be the same as that of a similar combination
in which the µ balls precede the l balls. Hence the probability of
m white balls in µ drawings, given that the l balls have already been
drawn out, must be equal to the probability of the same result, when

∗“Skew Variation in Homogeneous Material,” Phil. Trans. (1895), p. 360.

†“Zur Theorie der Stabilität statistischer Reihen,” p. 216, published in the

Skandinavisk Aktuarietidskrift for 1919.

‡Loc. cit. pp. 231, 232.

pt. v

A TREATISE ON PROBABILITY

396
no balls have been previously drawn out. The reader will perceive
that Poisson, thinking only of physical dependence, has been led to
his paradoxical conclusion by a failure to distinguish between the
cases where the proportion of black and white balls amongst the
l balls originally drawn is known and where it is not. The fact of
their having been drawn in certain proportions, provided that only
the total number drawn is known and the proportions are unknown,
does not influence the probability. Poisson states in his conclusion
that the probability is independent of the number and colour of the
l balls originally drawn. If he had added—as he ought—‘provided the
number of each colour is unknown,’ the air of paradox disappears.

This is an exceedingly good example of the failure to perceive that
a probability cannot be influenced by the occurrence of a material
event but only by such knowledge, as we may have, respecting the
occurrence of the event.∗

11.

For problems of the second type, where knowledge of the
result of one trial is capable of influencing the probability at the next
apart from any change in the material conditions, there is, likewise,
no general solution. The following artificial example, however, will
illustrate the sort of considerations which are involved.

In the cases where Bernoulli’s Theorem is applied to practical
questions, the à priori probability is generally obtained empirically
by reference to the statistical frequency of each alternative in past
experience under apparently similar conditions. Thus the à priori

∗For an attempt to solve other problems of this type see Bachelier, Calcul
des probabilités, chap. ix. (Probabilités connexes).

I think, however, that the
solutions of this chapter are vitiated by his assuming in the course of them
both that certain quantities are very large, and also, at a later stage, that the
same quantities are infinitesimal. On this account, for example, his solution of
the following difficult problem breaks down: Given an urn A with m white and
n black balls and an urn B with m white and n black balls, if at each move
a ball is taken from A and put into B, and at the same time a ball is taken
from B and put into A, what is the probability after x moves that the urns

A and B shall have a given composition?

ch. xxix

STATISTICAL INFERENCE

397
probability of a male birth is estimated by reference to the recorded
proportion of male births in the past.∗ The validity of estimating
probabilities in this manner will be discussed later. But for the
purposes of this example let us assume that the à priori probability
has been calculated on this basis. Thus the à priori probability
r
p =
of an event is based on the observation of its occurrence
s
r times out of s occasions on which the given conditions were
present. Now, according to Bernoulli’s Theorem directly applied, the
n
probability of the event’s occurring n times running is pn or r .

s

But, if the event occurs at the first trial, the probability at the
second becomes r + 1, and so on. Hence the probability P, properly
s + 1
calculated, of n successive occurrences is
r
r + 1
r + 2
r + n − 1

. . .

.

s
s + 1
s + 2
s + n − 1

∗Cf. Yule, Theory of Statistics, p. 258: “We are not able to assign an à priori
value to the chance p (i.e. of a male birth) as in the case of dice-throwing, but
it is quite sufficiently accurate for practical purposes to use the proportion of
male births actually observed if that proportion be based on a moderately large
number of observations.”

pt. v

A TREATISE ON PROBABILITY

398

Hence

(r + n − 1)! (s − 1)!

P = (s + n − 1)! (r − 1)!

(r + n − 1)r+n− 12 e−(r+n−1)ss− 12 e−(s−1)

=
by Stirling’s

(s + n − 1)s+n− 12 e−(s+n−1)rr− 12 e−(r−1)

Theorem, provided that r and s are large;
n − 1 r+n− 12

1 +
r n
r

=
s
n − 1 s+n− 12

1 +
s
r− 1

2
n − 1

+1.

n

1 +
r

= pnQn, where Q =
s− 1

2
n − 1

+1.

n

1 +
s

Thus, in this case, the assumption of Bernoulli’s Theorem is
approximately correct, only if Q is nearly unity. This condition is
not satisfied unless n is small both compared with r and compared
with s. It is very important to notice that two conditions are involved.

Not only must the experience, upon which the à priori probability
is based, be extensive in comparison with the number of instances
to which we apply our prediction; but also the number of previous
instances multiplied by the probability based upon them, i.e. sp (= r),
must be large in comparison with the number of new instances. Thus,
even where the prior experience, upon which we found the initial
probability P, is very extensive, we must not, if P is very small, say
that the probability of n successive occurrences is approximately pn,
unless n is also small. Similarly if we wish to determine, by the
methods of Bernoulli, the probability of n occurrences and m failures
on m + n occasions, it is necessary that we should have m and n small
ch. xxix

STATISTICAL INFERENCE

399
compared with s, n small compared with r, and m small compared
with s − r.∗

The case solved above is the simplest possible. The general
problem is as follows: If an event has occurred x times in the first
y trials, its probability at the y + 1th is r + x; determine the à priori
s + y
probability of the event’s occurring p times in q trials. If the à priori
probability in question is represented by φ(p, q), we have
r + p − 1
s + q − 1 − r − p
φ(p, q) =
φ(p − 1, q − 1) +
φ(p, q − 1).

s + q − 1
s + q − 1

I know of no solution of this, even approximate. But we may say that
the conditions are those of supernormal dispersion as compared with

Bernoulli’s conditions. That is to say, the probability of a proportion
differing widely from r is greater than in Bernoullian conditions;
s
for when the proportion begins to diverge it becomes more probable
that it will continue to diverge in the same direction. If, on the
other hand, the conditions of the problem had been such, that when
the proportion begins to diverge it becomes more probable that it
will recover itself and tend back towards r (as when we draw balls
s
without replacing them from a bag of known composition), we should
have subnormal dispersion.†

12.

The condition elucidated in the preceding paragraph is
frequently overlooked by statisticians. The following example from

Czuber‡ will be sufficient for the purpose of illustration. Czuber’s

∗This paragraph is concerned with a different point from that dealt with
in Professor Pearson’s article “On the Influence of Past Experience on Future

Expectation,” to which it bears a superficial resemblance. Professor Pearson’s
article which deals, not with Bernoulli’s Theorem, but with Laplace’s “Rule of

Succession,” will be referred to in § 16 of this chapter and in § 12 of the next.

†Bachelier (Calcul des probabilités, p. 201) classifies these two kinds of
conditions as conditions accélératrices and conditions retardatrices.

‡Loc. cit. vol. ii. p. 15. I choose my example from Professor Czuber because
he is usually so careful an exponent of theoretical statistics.

pt. v

A TREATISE ON PROBABILITY

400
argument is as follows:

In the period 1866–1877 there were registered in Austria
m = 4, 311, 076 male births
n = 4, 052, 193 female births
s = 8, 363, 269 ;
for the succeeding period, 1877–1899, we are given only
m = 6, 533, 961 male births;
what conclusion can we draw as to the number n of female births?

We can conclude, according to Czuber, that the most probable value
nm
n =

= 6, 141, 587,

0
m
and that there is a probability P = .9999779 that n will lie between
the limits 6, 118, 361 and 6, 164, 813.

It seems in plain opposition to good sense that on such evidence we
should be able with practical certainty

1

P = .9999779 = 1 − 45250
to estimate the number of female births within such narrow limits.

And we see that the conditions laid down in § 11 have been flagrantly
neglected. The number of cases, over which the prediction based
on Bernoulli’s Theorem is to extend, actually exceeds the number of
cases upon which the à priori probability has been based. It may be
added that for the period, 1877–1894, the actual value of n did lie
between the estimated limits, but that for the period, 1895–1905, it
lay outside limits to which the same method had attributed practical
certainty.

That Professor Czuber should have thought his own argument
plausible, is to be explained, I think, by his tacitly taking account in
his own mind of evidence not stated in the problem. He was relying
upon the fact that there is a great mass of evidence for believing that
ch. xxix

STATISTICAL INFERENCE

401
the ratio of male to female births is peculiarly stable. But he has not
brought this into the argument, and he has not used as his à priori
probability and as his coefficient of dispersion the values which the
whole mass of this evidence would have led him to adopt. Would
not the argument have seemed very preposterous if m had been the
number of males called George, and n the number of females called

Mary? Would it not have seemed rather preposterous if m had been
the number of legitimate births and n the number of illegitimate
births? Clearly we must take account of other considerations than
the mere numerical values of m and n in estimating our à priori
probability. But this question belongs to the subject-matter of later
chapters, and, quite apart from the manner of calculation of the
à priori probability, the argument is invalidated by the fact than
an à priori probability founded on 8, 363, 269 instances, without
corroborative evidence of a non-statistical character, cannot be
assumed stable through a calculation which extends over 12, 700, 000
instances.

13.

Before we leave the theorems of Bernoulli and Poisson,
it is necessary to call attention to a very remarkable theorem by

Tchebycheff, from which both of the above theorems can be derived
as special cases.

This result is reached rigorously and without
approximation, by means of simple algebra and without the aid of
the differential calculus. Apart from the beauty and simplicity of the
proof, the theorem is so valuable and so little known that it will be
worth while to quote it in full:∗

∗From Journ. Liouville (2), xii., 1867, “Des valeurs moyennes,” an article
translated from the Russian of Tchebycheff.

This proof is also quoted by

Czuber, loc. cit. p. 212, through whom I first became acquainted with it. Most
of Tchebycheff’s work was published previous to 1870 and appeared originally
in Russian.

It was not easily accessible, therefore, until the publication at

Petrograd in 1907 of the collected edition of his works in French. His theorems
are, consequently, not nearly so well known as they deserve to be, although his
most important theorems were reproduced from time to time in the Journals of

Euler and Liouville. For full references see the Bibliography.

pt. v

A TREATISE ON PROBABILITY

402

Let x, y, z . . . represent certain magnitudes, of which x can take
the values x1x2 . . . xk with probabilities p1p2 . . . pk respectively, y the
values y1y2 . . . yl with probabilities q1q2 . . . ql, z the values z1z2 . . . zm
with probabilities r1r2 . . . rm and so on, so that
k
l
m
p = 1,
q = 1,
r = 1, etc.

1

1

1
k
l
m

Write
pκxκ = a,
qλyλ = b,
rµzµ = c, etc.,

1

1

1
k
l
m
and
pκx2 = a
q

= b
r

= c
κ

1,
λy2
λ

1,
µz2
µ

1, etc.,

1

1

1
so that we can describe a as the mathematical expectation or average
value of x and a1 as the mathematical expectation or average value
of x2, etc.

Consider the expression:

(xκ + yλ + zµ + . . . − a − b − c − . . .)2pκqλrµ . . .

k

Now

(x2 − 2ax

− 2a
p
κ
κ + a2)pκ =
pκx2κ
κxκ + a2
pκ

1

= a1 − 2a2 + a2 = a1 − a2.

Also
qλrµ . . . = 1 summed for all values of λ, µ . . ., and
k
k

2(xκ − a)(yλ − b)pκ =

2(xκyλ − bxκ − ayλ + ab)pκ

1

1

= 2 (yλ
pκxκ − b
pκxκ − ayλ
pκ + ab
pκ)

= 2(ayλ − ab − ayλ + ab) = 0.

Therefore

(xκ + yλ + zµ + . . . − a − b − c . . .)2pκqλrµ . . .

= a1 + b1 + c1 + . . . − a2 − b2 − c2 − . . . ,
whence

(xκ + yλ + zµ + . . . − a − b − c . . .)2pκqλrµ . . .

1

=

,
α2(a1 + b1 + c1 + . . . − a2 − b2 − c2 − . . .)

α2
ch. xxix

STATISTICAL INFERENCE

403
where the summation extends over all values of κ, λ, µ . . . and α is
some arbitrary number greater than unity.

If we omit those terms of the sum on the left-hand side of the
above equation for which

(xκ + yλ + zµ + . . . − a − b − c . . .)2

< 1,
α2(a1 + b1 + c1 + . . . − a2 − b2 − c2 − . . .)

and write unity for this expression in the remaining terms, both
these processes diminish the magnitude of the left-hand side. Hence

1
pκqλrµ . . . <

, where the summation covers those sets of values
α2
only for which

(xκ + yλ + zµ + . . . − a − b − c . . .)2

≥ 1.

α2(a1 + b1 + c1 + . . . − a2 − b2 − c2 . . .)

If P is the probability that

(xκ + yλ + zµ + . . . − a − b − c . . .)2
α2(a1 + b1 + c1 + . . . − a2 − b2 − c2 − . . .)

is equal to or less than unity, it follows that

1

1 − P < α2

1
i.e.

P > 1 − α2

Hence the probability that the sum
xκ + yλ + zµ . . .

lies between the limits
a + b + c + . . . − α
a1 + b1 + c1 + . . . − a2 − b2 − c2 − . . .

and a + b + c + . . . + α a1 + b1 + c1 + . . . − a2 − b2 − c2 − . . .

is greater than

1

1 −

, where α is some number greater than unity.

α2
pt. v

A TREATISE ON PROBABILITY

404

This result constitutes Tchebycheff’s Theorem. It may also be
written in the following form:

Let n be the number of the magnitudes x, y, z . . ., and write α =

√n; then the probability that the arithmetic mean xκ +yλ +zµ +...

t
n
lies between the limits
a + b + c + . . .

1
a
a2 + b2 + c2 + . . .

±

1 + b1 + c1 + . . . −
n
t
n
n
is greater than
t2

1 −

.

n

It is also easy to show∗ as a deduction from Tchebycheff’s Theorem
that, if an amount A is won when an event of probability p [p = 1 − q]

occurs and an amount B lost when it fails, then in s trials the
probability that the total winnings (or losses) will lie between the
limits

√
s(pA − qB) ± α(A + B) spq
is greater than

1

1 −

.

α2

14.

From this very general result for the probable limits of
a sum composed of a number of independently varying magnitudes,

Bernoulli’s Theorem is easily derived. For let there be s observations
or trials, and s magnitudes x1x2 . . . xs corresponding, such that x = 1
when the event under consideration occurs, and x = 0 when it fails.

If the probability of the events occurrence is p, we have a = p, b = p,
etc., and a1 = p, b1 = p, etc. Hence the probability P that the number
of the event’s occurrences will lie between the limits sp ± α sp − sp2,

√

1
i.e. between the limits sp ± α spq where q = 1 − p, is > 1 −

. If
α2
we compare this formula with the formula for Bernoulli’s Theorem
already given, we find that, where this formula gives

1

P > 1 −

,
α2

∗For a proof see Czuber, loc. cit. vol. i. p. 216.

ch. xxix

STATISTICAL INFERENCE

405

Bernoulli’s Theorem with greater precision gives
α

P = Θ

√

. The

2
degree of superiority in the matter of precision supplied by the latter
can be illustrated by the following table:

α2.

Θ
α

√

.

1 − 1 .

2
α2

1.5

.7788

.333

2

.8427

.5

4.5

.9661

.7778

8

.9953

.875

12.5

.9996

.92

18

.99998

.9445

Thus when the limits are narrow and α is small, Bernoulli’s formula
gives a value of

1

P very much in excess of 1 −

. But Bernoulli’s
α2
formula involves a process of approximation which is only valid when
s is large.

Tchebycheff’s formula involves no such process and is
equally valid for all values of s. We have seen in § 11 that there
are numerous cases in which for a different reason Bernoulli’s formula
exaggerates the results, and, therefore, Tchebycheff’s more cautious
limits may sometimes prove useful.

The deduction of a corresponding form of Poisson’s Theorem from

Tchebycheff’s general formula obviously follows on similar lines. For
we put∗ a = p1, b = p2, etc., and a1 = p1, b1 = p2, etc., and find that
the probability that the number of the event’s occurrences will lie
between the limits
λ
λ
λ
pλ ± α
pλ −
p2 ,
λ

1

1

1

∗I am using the same notation as that used for Poisson’s Theorem in § 8.

pt. v

A TREATISE ON PROBABILITY

406
λ
i.e. between the limits
sp ± α
pλqλ,

√

√
i.e. between the limits
sp ±

2 αk s,
is greater than

1
t −

.

α2

In Crelle’s Journal∗ Tchebycheff proves Poisson’s Theorem directly
by a method similar to his general method, and also obtains several
supplementary results such as the following:

I. If the chances of an event E in µ consecutive trials are p1p2 . . . pµ
respectively, and their sum is s, the probability that E will occur at
least m times is less than

1
m(µ − m)

s
m
µ − s
µ−m+1

2(m − s)

µ
µ
µ − m
provided that m > s + 1;

II. and the probability that E will not occur more than n times is
less than

1
µ(µ − n)

µ − s
µ−n
s n+1

2(s − n)

µ
µ − n
n
provided that n < s − 1.

III. Hence the probability that E will occur less than m times and
more than n is greater than

1
m(µ − m)

s
µ−m+1
m
µ − s

1 − 2(m − s)

µ
m
µ − m

1
n(µ − n)

s
n−µ
n+1
µ − s

− 2(s − n)

µ
n
µ − n

∗Vol. 33 (1846), Démonstration élémentaire d’une proposition générale de la
théorie des probabilités.

ch. xxix

STATISTICAL INFERENCE

407
provided m > s + 1, n < s − 1.

15.

Tchebycheff’s methods have been set out and his results
admirably extended by A. A. Markoff.∗ And some developments along
the same lines by Tschuprow (“Zur Theorie der Stabilität statistischer

Reihen,” Skandinavisk Aktuarietidskrift, 1919) have convinced me that

Tchebycheff’s discovery is far more than a technical device for solving
a special problem, and points the way to the fundamental method for
attacking these questions on the mathematical side. The Laplacian
mathematics, although it still holds the field in most text-books, is
really obsolete, and ought to be replaced by the very beautiful work
which we owe to these three Russians.

16.

There is one other investigation relating to Bernoulli’s

Theorem which deserves remark. I have already pointed out, in

§ 2, that the dispersion about the most probable value, even when
the conditions for the applicability of Bernoulli’s Theorem in its
non-approximate form are strictly fulfilled, is unsymmetrical. The
fact, that the usual approximation for the probability of a divergence h
from the most probable number of occurrences (the notation is that of

§ 2 above) takes the form

1

√
e− h2

2mpq , which is the same for +h

2πmpq
as for −h, has led to this want of symmetry being very generally
overlooked; and it is not uncommon to assume that the probability
of a given divergence less than pm is equal to that of the same
divergence in excess of pm, and, in general, that the probability of
the frequency’s exceeding pm in a set of m trials is equal to that of
its falling short of pm.

That this is not strictly the case is obvious. If a die is cast

60 times, the most probable number of appearances of the ace is 10;
but the ace is more likely to appear 9 times than 11 times; and

∗The reader is referred to Markoff’s Wahrscheinlichkeitsrechnung, and
particularly to p. 67, for a striking development, along mathematical lines, of

Tchebycheff’s leading idea. Further references to later memôirs, which, being in
the Russian language, are inaccessible to me, will be found in the Bibliography.

pt. v

A TREATISE ON PROBABILITY

408
much more likely (about 5 times as likely) not to appear at all than
to appear exactly 20 times. That this must be so will be clear to
the reader (without his requiring to trouble himself with the algebra),
when he reflects that the ace cannot appear less often than not at all,
whereas it may well appear more than 20 times, so that the smallness
of the possible divergence in defect from the most probable value 10,
as compared with the possible divergence in excess, must be made
up for by the greater frequency of any given defection as compared
with the corresponding excess. Thus the actual frequency in a series
of trials of an event, of which the probability at each trial is less
than 1, is likely to fall short of its most probable value more often

2
than it exceeds it. What is in fact true is that the mathematical
expectation of deficiency is equal to the mathematical expectation of
excess, i.e. that the sum of the possible deficiencies each multiplied
by its probability is equal to the sum of the possible excesses each
multiplied by its probability.

The actual measurement of this want of symmetry and the
determination of the conditions, in which it can be safely neglected,
involves laborious mathematics, of which I am only acquainted with
one direct investigation, that published in the Proceedings of the

London Mathematical Society by Mr. T. C. Simmons.∗

For the details of the proof I must refer the reader to

∗“A New Theorem in Probability.”

Mr. Simmons claimed novelty for
his investigation, and so far as I know this claim is justified; but recent
investigations obtaining closer approximations to Bernoulli’s Theorem by means
of the Method of Moments are essentially directed towards the same problem.

A somewhat analogous point has, however, been raised by Professor Pearson
in his article (Phil. Mag., 1907) on “The Influence of Past Experience on

Future Expectation.”

He brings out an exactly similar want of symmetry in
the probabilities of the various possible frequencies about the most probable
frequency, when the calculation is based, not on Bernoulli’s Theorem as in

Mr. Simmons’s investigation, but on Laplace’s rule of succession (see next
chapter). The want of symmetry has also been pointed out by Professor Lexis

(Abhandlungen, p. 120).

ch. xxix

STATISTICAL INFERENCE

409

Mr. Simmons’s article. His principal theorem∗ is as follows: If

1
is the probability of the event at each trial and n(a + 1) the
a + 1
number of trials, n and a being integers,† the probability that the
frequency of occurrence will fall short of n is always greater than
the probability that it will exceed n; the difference between the two
probabilities being a maximum when n = 1, constantly diminishing
as
a − 1
n increases, lying always between 1
times the greatest term

3 a + 1
n(a+1)

in
a

1
a − 1

+
and 1
times the greatest term in
a + 1
a + 1

3 a + 1
a

1

(n+1)(a+1)

+

, and being approximately equal, when n is
a + 1
a + 1
very large, to 1
a − 1

.

3

2πna(a + 1)

The following table gives the value of the excess ∆ of the
probability of a frequency less than pm over the probability of a
frequency greater than pm for various values of p the probability and

1
m the number of trials p =

, m = n(a + 1) , as calculated by
a + 1

∗I am not giving his own enunciation of it.

†Mr. Simmons does not seem to have been able to remove this restriction
on the generality of his theorem, but there does not seem much reason to doubt
that it can be removed.

pt. v

A TREATISE ON PROBABILITY

410

Mr. Simmons:

p.

m.

∆.

1

3

.037037

3

1

15

.02243662

3

1

24

.0182706

3

1

4

.054687

4

1

20

.03201413

4

1

10

.084777

10

1

20

.068673713

10

1

100

.101813

100

1

200

.081324387

100

1

1000

.103454

1000

Thus unless not only m but mp also is large the want of symmetry
is likely to be appreciable. Thus it is easily found that in 100 sets of

1

4 trials each, where p = , the actual frequency is likely to exceed

4
the most probable 26 times and to fall short of it 31 times; and in

1

100 sets of 10 trials each, where p =

, to exceed 26 times and to

10
fall short 34 times.

Mr. Simmons was first directed to this investigation through
noticing in the examination of sets of random digits that “each digit
presented itself, with unexpected frequency, less than 1 of the

10
number of times. For instance, in 100 sets of 150 digits each, I
found that a digit presented itself in a set more frequently under

15 times than over 15 times; similarly in the case of 80 sets each of

250 digits, and also in other aggregations.” Its possible bearing on
such experiments with dice and roulette, as are described at the end
of this chapter, is clear. But apart from these artificial experiments,
it is sometimes worth the statistician’s while to bear in mind this
ch. xxix

STATISTICAL INFERENCE

411
appreciable want of symmetry in the distribution about the mode or
most probable value in many even of those cases in which Bernoullian
conditions are strictly fulfilled.

17.

I will conclude this chapter by an account of some of the
attempts which have been made to verify à posteriori the conclusions
of Bernoulli’s Theorem. These attempts are nearly useless, first,
because we can seldom be certain à priori that the conditions
assumed in Bernoulli’s Theorem are fulfilled, and, secondly, because
the theorem predicts not what will happen but only what is, on
certain evidence, likely to happen. Thus even where our results do not
verify Bernoulli’s Theorem, the theorem is not thereby discredited.

The results have bearing on the conditions in which the experiments
took place, rather than upon the truth of the theorem. In spite,
therefore, of the not unimportant place which these attempts have in
the history of probability, their scientific value is very small. I record
them, because they have a good deal of historical and psychological
interest, and because they satisfy a certain idle curiosity from which
few students of probability are altogether free.∗

18.

The data for these investigations have been principally
drawn from four sources—coin-tossing, the throw of dice, lotteries,
and roulette; for in such cases as these the conditions for Bernoulli’s

Theorem seem to be fulfilled most nearly. The earliest recorded
experiment was carried out by Buffon,† who, assisted by a child
tossing a coin into the air, played 2048 partis of the Petersburg game,
in which a coin is thrown successively until the parti is brought
to an end by the appearance of heads. The same experiment was
repeated by a young pupil of De Morgan’s ‘for his own satisfaction.’‡

∗Mr. Yule (Introduction to Statistics, p. 254) recommends its indulgence:

“The student is strongly recommended to carry out a few series of such
experiments personally, in order to acquire confidence in the use of the theory.”

Mr. Yule himself has indulged moderately.

†Essai d’arithmètique morale (see Bibliography), published 1777, said to have
been composed about 1760.

‡Formal Logic, p. 185, published 1847. De Morgan gives Buffon’s results, as
pt. v

A TREATISE ON PROBABILITY

412

In Buffon’s trials there were 1992 tails to 2048 heads; in Mr. H.’s

(De Morgan’s pupil) 2044 tails to 2048 heads. A further experiment,
due to Buffon’s example, was carried out by Quetelet∗ in 1837. He
drew 4096 balls from an urn, replacing them each time, and recorded
the result at different stages, in order to show that the precision of
the result tended to increase with the number of the experiments.

He drew altogether 2066 white balls and 2030 black balls. Following
in this same tradition is the experiment of Jevons,† who made 2048
throws of ten coins at a time, recording the proportion of heads at
each throw and the proportion of heads altogether. In the whole
number of 20, 480 single throws, he obtained heads 10, 353 times;

More recently Weldon‡ threw twelve dice 4096 times, recording the
proportion of dice at each throw which showed a number greater than
three.

All these experiments, however, are thrown completely into
the shade by the enormously extensive investigations of the Swiss
astronomer Wolf, the earliest of which were published in 1850 and the
latest in 1893.§ In his first set of experiments Wolf completed 1000
sets of tosses with two dice, each set continuing until every one of the

21 possible combinations had occurred at least once. This involved
altogether 97, 899 tosses, and he then completed a total of 100, 000.

These data enabled him to work out a great number of calculations,
of which Czuber quotes the following, namely a proportion of .83533
well as his pupil’s, in full.

Buffon’s results are also investigated by Poisson,

Recherches, pp. 132–135.

∗Letters on the Theory of Probabilities (Eng. trans.), p. 37.

†Principles of Science (2nd ed.), p. 208.

‡Quoted by Edgeworth, “Law of Error” (Ency. Brit. 10th ed.), and by Yule,

Introduction to Statistics, p. 254.

§See Bibliography.

Of the earliest of these investigations I have no
first-hand knowledge and have relied upon the account given by Czuber, loc.

cit. vol. i. p. 149. For a general account of empirical verifications of Bernoulli’s

Theorem reference may be made to Czuber, Wahrscheinlichkeitsrechnung, vol. i.

pp. 139–152, and Czuber, Entwicklung der Wahrscheinlichkeitstheorie, pp. 88–91.

ch. xxix

STATISTICAL INFERENCE

413
of unlike pairs, as against the theoretical value

5

.83333, i.e.

. In

6
his second set of experiments Wolf used two dice, one white and one
red (in the first set the dice were indistinguishable), and completed

20, 000 tosses, the details of each result being recorded in the

Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich. He
studied particularly the number of sequences with each die, and the
relative frequency of each of the 36 possible combinations of the two
dice. The sequences were somewhat fewer than they ought to have
been, and the relative frequency of the different combinations very
different indeed from what theory would predict.∗ The explanation of
this is easily found; for the records of the relative frequency of each
face show that the dice must have been very irregular, the six face
of the white die, for example, falling 38 per cent more often than
the four face of the same die. This, then, is the sole conclusion of
these immensely laborious experiments,—that Wolf’s dice were very ill
made. Indeed the experiments could have had no bearing except upon
the accuracy of his dice. But ten years later Wolf embarked upon one
more series of experiments, using four distinguishable dice,—white,
yellow, red, and blue,—and tossing this set of four 10, 000 times.

Wolf recorded altogether, therefore, in the course of his life 280, 000
results of tossing individual dice. It is not clear that Wolf had
any well-defined object in view in making these records, which are
published in curious conjunction with various astronomical results,
and they afford a wonderful example of the pure love of experiment
and observation.†

19.

Another series of calculations have been based upon the

∗Czuber quotes the principal results (loc. cit. vol. i. pp. 149–151).

The
frequencies of only 4, instead of 18, out of the 36 combinations lay within the
probable limits, and the standard deviation was 76.8 instead of 23.2.

†The latest experiment of the kind, of which I am aware, is that of Otto

Meissner (“Würfelversuche,” Zeitschrift für Math. und Phys. vol. 62 (1913),
pp. 149–156), who recorded 24 series of 180 throws each with four distinguishable
dice.

pt. v

A TREATISE ON PROBABILITY

414
ready-made data provided by the published results of lotteries and
roulette.∗

Czuber† has made calculations based on the lotteries of Prague

(2854 drawings) and Brünn (2703 drawings) between the years

1754 and 1886, in which the actual results agree very well with
theoretical predictions. Fechner‡ employed the lists of the ten State
lotteries of Saxony between the years 1843 and 1852. Of a rather

∗For the publication of such returns there has always been a sufficient
demand on the part of gamblers.

An Almanach romain sur la loterie royale
de France was published at Paris in 1830, which contained all the drawings
of the French lottery (two or three a month) from 1758 to 1830.

Players at

Monte Carlo are provided with cards and pins with which to record the results
of successive coups, and the results at the tables are regularly published in Le

Monaco.

Gamblers study these returns on account of the belief, which they
usually hold, that as the number of cases is increased the absolute deviation
from the most probable proportion becomes less, whereas at the best Bernoulli’s

Theorem shows that the proportionate deviation decreases while the absolute
deviation increases.

Cf. Houdin’s Les Tricheries des Grecs dévoilées:

“In a
game of chance, the oftener the same combination has occurred in succession,
the nearer we are to the certainty that it will not recur at the next cast
or turn-up.

This is the most elementary of the theories on probabilities; it
is termed the maturity of the chances.”

Laplace (Essai philosophique, p. 142)

quotes an amusing instance of the same belief not drawn from the annals of
gambling: “J’ai vu des hommes désirant ardemment d’avoir un fils, n’apprendre
qu’avec peine les naissances des garçons dans le mois où ils allaient devenir
pères. S’imaginant que le rapport de ces naissances à celles des filles devait être
le même à la fin de chaque mois, ils jugaient que les garçons déjà nés rendaient
plus probables les naissances prochaines des filles.”

The literature of gambling is very extensive, but, so far as I am acquainted
with it, excessively lacking in variety, the maturity of the chances and the
martingale continually recurring in one form or another. The curious reader will
find tolerable accounts of such topics in Proctor’s Chance and Luck, and Sir

Hiram Maxim’s Monte Carlo Facts and Fallacies.

†Zum Gesetz der grossen Zahlen.

The results are summarised in his

Wahrscheinlichkeitsrechnung, vol. i. p. 139.

‡Kollektivmasslehre, p. 229. These results also are summarised by Czuber,
loc. cit.

ch. xxix

STATISTICAL INFERENCE

415
more interesting character are Professor Karl Pearson’s investigations∗
into the results of Monte Carlo Roulette as recorded in Le Monaco
in the course of eight weeks. Applying Bernoulli’s Theorem, on the
hypothesis of the equiprobability of all the compartments throughout
the investigation, he found that the actually recorded proportions
of red and black were not unexpected, but that alternations and
long runs were so much in excess that, on the assumption of the
exact accuracy of the tables, the à priori odds were at least a
thousand millions to one against some of the recorded deviations.

Professor Pearson concluded, therefore, that Monte Carlo Roulette
is not objectively a game of chance in the sense that the tables on
which it is played are absolutely devoid of bias. Here also, as in the
case of Wolf’s dice, the conclusion is solely relevant, not to the theory
or philosophy of Chance, but to the material shapes of the tools of
the experiment.

Professor Pearson’s investigations into Roulette, which dealt with

33, 000 Monte Carlo coups, have been overshadowed, just as all other
tosses of coins and dice have been outdone by Wolf, by Dr. Karl

Marbe,† who has examined 80, 000 coups from Monte Carlo and
elsewhere. Dr. Marbe arrived at exactly opposite conclusions; for he
claims to have shown that long runs, so far from being in excess, were
greatly in defect. Dr. Marbe introduces this experimental result in
support of his thesis that the world is so constituted that long runs
do not as a matter of fact occur in it.‡ Not merely are long runs very

∗The Chances of Death, vol. i.

†Naturphilosophische Untersuchungen zur Wahrscheinlichkeitstheorie.

‡Dr. Marbe’s monograph has given rise in Germany to a good deal of
discussion, not directed towards showing what a preposterous method this is for
demonstrating a natural law, but because the experimental result itself does not
really follow from the data and is due to a somewhat subtle error in Marbe’s
reasoning, by which he has been led into an incorrect calculation of the probable
proportions à priori of the various sequences. The problem is discussed by Von

Bortkiewicz, Brömse, Bruns, Grimsehl, and Grünbaum (for exact references to
these see the Bibliography), and by Lexis (Abhandlungen, pp. 222–226) and
pt. v

A TREATISE ON PROBABILITY

416
improbable. They do not, according to him, occur at all. But we
may doubt whether roulette can tell us very much either of the laws
of logic or of the constitution of the universe.

Dr. Marbe’s main thesis is identical, as he himself recognises, with
one of the heterodox contentions of D’Alembert.∗ But this principle
of variety, precisely opposite to the usual principle of Induction, can
have no claim to be accepted à priori and, as a general principle,
there is no adequate evidence to support it from experience. Its
origin is to be found, perhaps, in the fact that in a certain class
of cases, especially where conscious human agency comes in, it may
contain some element of truth. The fact of an act’s having been done
in a particular way once may be a special reason for thinking that
it will not be performed on the next occasion in precisely the same
manner. Thus in many so-called random events some slight degree

Czuber (Wahrscheinlichkeitsrechnung, vol. i. pp. 144–149). Largely as a result
of this controversy, Von Bortkiewicz has lately devoted a complete treatise (Die

Iterationen) to the mathematics of ‘runs.’ Dr. Marbe has been given far more
attention by his colleagues in Germany than he conceivably deserves.

∗D’Alembert’s principal contributions to Probability are most accessible in
the volumes of his Opuscules mathématiques (1761).

Works on Probability
usually contain some reference to D’Alembert, but his sceptical opinions, rejected
rather than answered by the orthodox school of Laplace, have not always
received full justice.

D’Alembert has three main contentions to which in his
various papers he constantly recurs:

(1) That a probability very small mathematically is really zero;

(2) That the probabilities of two successive throws with a die are not
independent;

(3) That ‘mathematical expectation’ is not properly measured by the product
of the probability and the prize.

The first and third of these were partly advanced in explanation of the

Petersburg paradox (see p. 362). The second is connected with the first, and was also used to support his incorrect evaluation of the probability of heads
twice running; but D’Alembert, in spite of many of his results being wrong, does
not altogether deserve the ridicule which he has suffered at the hands of writers,
who accepted without sceptical doubts the hardly less incorrect conclusions of
the orthodox theory of that time.

ch. xxix

STATISTICAL INFERENCE

417
of causal and material dependence between successive occurrences
may, nevertheless, exist. In these cases ‘runs’ may be fewer and
shorter than those which we should predict, if a complete absence
of such dependence is assumed. If, for example, a pack of cards
be dealt, collected, and shuffled, to the extent that card-players do
as a rule shuffle, there may be a greater presumption against the
second hand’s being identical with the first than against any other
particular distribution. In the case of croupiers long experience
might possibly suggest some psychological generalisation,—that they
are very mechanical, giving an excess of numbers belonging to a
particular section of the wheel, or, on the other hand, that when a
croupier sees a run beginning, he tends to vary his spin more than
usual, thus bringing runs to an end sooner than he ought.∗ At any
rate, it is worth emphasising once more that from such experiments
as these this is the only kind of knowledge which we can hope to
obtain,—knowledge of the material construction of a die or of the
psychology of a croupier.

∗A good roulette table is, however, so delicate an instrument that no
probable degree of regularity of habit on the part of the spinner could be
sufficient to produce regularity in the result.

CHAPTER XXX
the mathematical use of statistical frequencies
for the determination of probability à
posteriori —the methods of laplace

Utilissima est aestimatio probabilitatum, quanquam in exemplis juridicis
politicisque plerumque non tam subtili calculo opus est, quam accurata omnium
circumstantiarum enumeratione.—Leibniz.

1.

In the preceding chapter we have assumed that the
probability of an event at each of a series of trials is given, and
have considered how to infer from this the probabilities of the various
possible frequencies of the event over the whole series, without
discussing in detail by what method the initial probability had been
determined. In statistical inquiries it is generally the case that this
initial probability is based, not upon the Principle of Indifference,
but upon the statistical frequencies of similar events which have been
observed previously. In this chapter, therefore, we must commence
the complementary part of our inquiry,—namely, into the method
of deriving a measure of probability from an observed statistical
frequency.

I do not myself believe that there is any direct and simple method
by which we can make the transition from an observed numerical
frequency to a numerical measure of probability. The problem, as

I view it, is part of the general problem of founding judgments
of probability upon experience, and can only be dealt with by the
general methods of induction expounded in Part III. The nature of the problem precludes any other method, and direct mathematical
devices can all be shown to depend upon insupportable assumptions.

In the next chapters we will consider the applicability of general
inductive methods to this problem, and in this we will endeavour to
discredit the mathematical charlatanry by which, for a hundred years

418
ch. xxx

STATISTICAL INFERENCE

419
past, the basis of theoretical statistics has been greatly undermined.

2.

Two direct methods have been commonly employed,
theoretically inconsistent with one another, though not in every case
noticeably discrepant in practice. The first and simplest of these
may be termed the Inversion of Bernoulli’s Theorem, and the other

Laplace’s Rule of Succession.

The earliest discussion of this problem is to be found in the

Correspondence of Leibniz and Jac. Bernoulli,∗ and its true nature
cannot be better indicated than by some account of the manner in
which it presented itself to these very illustrious philosophers. The
problem is tentatively proposed by Bernoulli in a letter addressed
to Leibniz in the year 1703.

We can determine from à priori
considerations, he points out, by how much it is more probable
that we shall throw 7 rather than 8 with two dice, but we cannot
determine by such means the probability that a young man of twenty
will outlive an old man of sixty. Yet is it not possible that we
might obtain this knowledge à posteriori from the observation of a
great number of similar couples, each consisting of an old man and
a young man? Suppose that the young man was the survivor in

1000 cases and the old man in 500 cases, might we not conclude that
the young man is twice as likely as the old man to be the survivor?

For the most ignorant persons seem to reason in this way by a sort
of natural instinct, and feel that the risk of error is diminished as
the number of observations is increased. Might not the solution tend
asymptotically to some determinate degree of probability with the
increase of observations? Nescio, Vir Amplissime, an speculationibus
istis soliditatis aliquid inesse Tibi videatur.

Leibniz’s reply goes to the root of the difficulty. The calculation of
probabilities is of the utmost value, he says, but in statistical inquiries
there is need not so much of mathematical subtlety as of a precise
statement of all the circumstances. The possible contingencies are

∗For the exact references see Bibliography.

pt. v

A TREATISE ON PROBABILITY

420
too numerous to be covered by a finite number of experiments, and
exact calculation is, therefore, out of the question. Although nature
has her habits, due to the recurrence of causes, they are general, not
invariable. Yet empirical calculation, although it is inexact, may be
adequate in affairs of practice.∗

Bernoulli in his answer fell back upon the analogy of balls drawn
from an urn, and maintained that without estimating each separate
contingency we might determine within narrow limits the proportion
favouring each alternative. If the true proportion were 2 : 1, we might
estimate it with moral certainty à posteriori as lying between 201 : 100
and 199 : 100. “Certus sum,” he concluded the controversy, “Tibi
placituram demonstrationem, cum publicavero.” But whether he was
impressed by the just caution of Leibniz, or whether death intercepted
him, he advances matters no further in the Ars Conjectandi. After
dealing with some of Leibniz’s objections† and seeming to promise
some mode of estimating probabilities à posteriori by an inversion
of his theorem, he proves the direct theorem only and the book is
suddenly at an end.

3.

In dealing with the correspondence of Leibniz and Bernoulli,

I have not been mainly influenced by the historical interest of it. The
view of Leibniz, dwelling mainly on considerations of analogy, and

∗Leibniz’s actual expressions (in a letter to Bernoulli, December 3, 1703)

are as follows: Utilissima est aestimatio probabilitatum, quanquam in exemplis
juridicis
politicisque
plerumque
non
tam
subtili
calculo
opus
est,
quam
accurata omnium circumstantiarum enumeratione.

Cum empirice aestimamus
probabilitates per experimenta successuum, quaeris an ea via tandem aestimatio
perfecte obtineri possit. Idque a Te repertum scribis. Difficultas in eo mihi inesse
videtur, quod contingentia seu quae infinitis pendent circumstantiis, per finita
experimenta determinari non possunt; natura quidem suas habet consuetudines,
natas ex reditu causarum, sed non nisi ὡς ὲπὶ τὸ πολύ. Novi morbi inundant
subinde humanum genus, quodsi ergo de mortibus quotcunque experimenta
feceris, non ideo naturae rerum limites posuisti, ut pro futuro variare non possit.

Etsi autem empirice non posset haberi perfecta aestimatio, non ideo minus
empirica aestimatio in praxi utilis et sufficiens foret.

†The relevant passages are on pp. 224–227 of the Ars Conjectandi.

ch. xxx

STATISTICAL INFERENCE

421
demanding “not so much mathematical subtlety as a precise statement
of all the circumstances,” is, substantially, the view which will be
supported in the following chapters. The desire of Bernoulli for an
exact formula, which would derive from the numerical frequency of
the experimental results a numerical measure of their probability,
preludes the exact formulas of later and less cautious mathematicians,
which will be examined immediately.

4.

During the greater part of the eighteenth century there is
no trace, I think, of the explicit use of the Inversion of Bernoulli’s

Theorem. The investigations carried out by D’Alembert, Daniel

Bernoulli, and others relied upon the type of argument examined in

Chapter XXV. They showed, that is to say, that certain observed
series of events would have been very improbable, if we had supposed
independence between some two factors or if some occurrence had
been assumed to be as likely as not, and they inferred from this that
there was in fact a measure of dependence or that the occurrence
had probability in its favour. But they did not endeavour to pass
from the observed frequency of occurrence to an exact measure of
the probability. With the advent of Laplace more ambitious methods
took the field.

Laplace began by assuming without proof a direct inversion of

Bernoulli’s Theorem. Bernoulli’s Theorem, in the form in which

Laplace proved it, states that, if p is the probability à priori, there
is a probability P that the proportion of times
m
of the event’s
m + n
occurrence in

2pq
µ (= m + n) trials will lie between p ± γ

, where
µ

2
γ

1

P = √
e−t2 dt + √
e−γ2.

The inversion of the theorem,
π 0

2πµpq
which he assumes without proof, states that, if the event is observed
to happen m times in µ trials, there is a probability P that the
probability of the event

2mn
p will lie between m ± γ

, where
µ
µ3
pt. v

A TREATISE ON PROBABILITY

422

2
γ

1

P = √
e−t2 dt +
e−γ2. The same result is also given by
π
m

0

2πµ µ2

Poisson.∗ Thus, given the frequency of occurrence in µ trials, these
writers infer the probability of occurrence at subsequent trials within
certain limits, just as, given the à priori probability, Bernoulli’s

Theorem would enable them to predict the frequency of occurrence in
µ trials within corresponding limits.

If the number of trials is at all numerous, these limits are narrow
and the purport of the inversion of Bernoulli’s Theorem may therefore
be put briefly as follows. By the direct theorem, if p measures the
probability, p also measures the most probable value of the frequency;
by the inversion of the theorem, if
m
measures the frequency,
m + n
m
also measures the most probable value of the probability.

m + n

The simplicity of the process has recommended it, since the time of

Laplace, to a great number of writers. Czuber’s argument, criticised
on p. 399, with reference to the proportions of male and female births in Austria, is based upon an unqualified use of it. But examples
abound throughout the literature of the subject, in which the theorem
is employed in circumstances of greater or less validity.

The theorem was originally given without proof, and is indeed

∗For an account of the treatments of this topic both by Laplace and by

Poisson, see Todhunter’s History, pp. 554–557.

Both of them also obtain a
formula slightly different from that given above by a method analogous to the
first part of the proof of Laplace’s Rule of Succession; i.e. by an application
of the inverse principle of probability to the assumption that the probability
of the probability’s lying within any interval is proportional to the length
of the interval.

This discrepancy has given rise to some discussion.

See

Todhunter, loc. cit.; De Morgan, On a Question in the Theory of Probabilities;

Monro, On the Inversion of Bernoulli’s Theorem in Probabilities; and Czuber,

Entwicklung, pp. 83, 84. But this is not the important distinction between the
two mathematical methods by which this question has been approached, and
this minor point, which is of historical interest mainly, I forbear to enter into.

ch. xxx

STATISTICAL INFERENCE

423
incapable of it, unless some illegitimate assumption has been
introduced. But, apart from this, there are some obvious objections.

We have seen in the preceding chapter that Bernoulli’s Theorem itself
cannot be applied to all kinds of data indiscriminately, but only
when certain rather stringent conditions are fulfilled. Corresponding
conditions are required equally for the inversion of the theorem, and
it cannot possibly be inferred from a statement of the number of
trials and the frequency of occurrence merely, that these have been
satisfied. We must know, for instance, that the examined instances
are similar in the main relevant particulars, both to one another and
to the unexamined instances to which we intend our conclusion to be
applicable. An unanalysed statement of frequency cannot tell us this.

This method of passing from statistical frequencies to probabilities
is not, however, like the method to be discussed in a moment,
radically false. With due qualifications it has its place in the solution
of this problem. The conditions in which an inversion of Bernoulli’s

Theorem is legitimate will be elucidated in Chapter XXXI. In the
meantime we will pass on to Laplace’s second method, which is more
powerful than the first and has obtained a wider currency. The more
extreme applications of it are no longer ventured upon, but the theory
which underlies it is still widely adopted, especially by French writers
upon probability, and seldom repudiated.

5.

The formula in question, which Venn∗ has called the Rule of

Succession, declares that, if we know no more than that an event has
occurred m times and failed n times under given conditions, then the
probability of its occurrence when those conditions are next fulfilled
is
m + 1

. It is necessary, however, before we examine the proof
m + n + 2
of this formula, to discuss in detail the reasoning which leads up to it.

This preliminary reasoning involves the Laplacian theory of

‘unknown probabilities.’

The postulate, upon which it depends,
is introduced to supplement the Principle of Indifference, and is

∗Logic of Chance, p 190.

pt. v

A TREATISE ON PROBABILITY

424
in fact the extension of this principle from the probabilities of
arguments, when we know nothing about the arguments, to the
probabilities that the probabilities of arguments have certain values,
when we know nothing about the probabilities. Laplace’s enunciation
is as follows: “Quand la probabilité d’un événement simple est
inconnue, on peut lui supposer également toutes les valeurs depuis
zéro jusqu’à l’unité. La probabilité de chacune de ces hypothèses
tirée de l’événement observé est . . . une fraction dont le numérateur
est la probabilité de l’événement dans cette hypothèse, et dont le
dénominateur est la somme des probabilités semblables relatives à
toutes les hypothèses. . . .”∗

Thus when the probability of an event is unknown, we may
suppose all possible values of the probability between 0 and 1 to
be equally likely à priori. The probability, after the event has
occurred, that the probability à priori was 1 (say), is measured by
r
a fraction of which 1 is the numerator and the sums of all the
r
possible à priori values the denominator. The origin of this rule
is evident. If we consider the problem in which a ball is drawn
from a bag containing an infinite number of black and white balls in
unknown proportions, we have hypotheses, corresponding to each of
the possible constitutions of the bag, the assumption of which yields
in turn every value between 0 and 1 as the à priori probability of
drawing a white ball. If we could assume that these constitutions are
equally probable à priori, we should obtain probabilities for each of
them à posteriori according to Laplace’s rule.

On the analogy of this Laplace assumes in general that, where
everything is unknown, we may suppose an infinite number of
possibilities, each of which is equally likely, and each of which leads to
the event in question with a different degree of probability, so that for
every value between 0 and 1 there is one and only one hypothetical

∗Essai philosophique, p. 16.

ch. xxx

STATISTICAL INFERENCE

425
constitution of things, the assumption of which invests the event with
a probability of that value.

6.

It might be an almost sufficient criticism of the above to point
out that these assumptions are entirely baseless. But the theory has
taken so important a place in the development of probability that it
deserves a detailed treatment.

What, in the first place, does Laplace mean by an unknown
probability? He does not mean a probability, whose value is in fact
unknown to us, because we are unable to draw conclusions which
could be drawn from the data; and he seems to apply the term to any
probability whose value, according to the argument of Chapter III.,
is numerically indeterminate. Thus he assumes that every probability
has a numerical value and that, in those cases where there seems to
be no numerical value, this value is not non-existent but unknown;
and he proceeds to argue that where the numerical value is unknown,
or as I should say where there is no such value, every value between

0 and 1 is equally probable. With the possible interpretations of
the term ‘unknown probability,’ and with the theory that every
probability can be measured by one of the real numbers between

0 and 1, I have dealt, as carefully as I can, in Chapter III. If the view
taken there is correct, Laplace’s theory breaks down immediately.

But even if we were to answer these questions, not as they have been
answered in Chapter III., but in a manner favourable to Laplace’s
theory, it remains doubtful whether we could legitimately attribute
a value to the probability of an unknown probability’s having such
and such a value. If a probability is unknown, surely the probability,
relative to the same evidence, that this probability has a given value,
is also unknown; and we are involved in an infinite regress.

7.

This point leads on to the second objection; Laplace’s theory
requires the employment of both of two inconsistent methods. Let
us consider a number of alternatives a1, a2, etc., having probabilities
p1, p2, etc.; if we do not know anything about a1, we do not know
the value of its probability p1, and we must consider the various
pt. v

A TREATISE ON PROBABILITY

426
possible values of p1, namely b1, b2, etc., the probabilities of these
possible values being q1, q2, etc. respectively. There is no reason
why this process should ever stop. For as we do not know anything
about b1, we do not know the value of its probability q1, and we
must consider the various possible values of q1 namely c1, c2, etc., the
probabilities of these possible values being r1, r2, etc., respectively;
and so on. This method consists in supposing that, when we do not
know anything about an alternative, we must consider all the possible
values of the probability of the alternative; these possible values can
form in their turn a set of alternatives, and so on. But this method
by itself can lead to no final conclusion. Laplace superimposes on
it, therefore, his other method of determining the probabilities of
alternatives about which we know nothing,—namely, the Principle of

Indifference. According to this method, when we know nothing about
a set of alternatives, we suppose the probabilities of each of them to
be equal. In some parts of his writings—and this is true also of most
of his followers—he applies this method from the beginning. If, that
is to say, we know nothing about a1, since a1 and its contradictory
form a pair of exhaustive alternatives two in number, the probability
of these alternatives is equal and each is 1. But in the reasoning

2
which leads up to the Law of Succession he chooses to apply this
method at the second stage, having used the other method at the first
stage. If, that is to say, we know nothing about a1, its probability p1
may have any of the values b1, b2, etc. where b1 is any fraction
between 0 and 1; and, as we know nothing about the probabilities
q1, q2, etc. of these alternatives b1, b2, etc., we may by the Principle of

Indifference suppose them to be equal. This account may seem rather
confused; but it is not easy to give a lucid account of so confused a
doctrine.

8.

Turning aside from these considerations, let us examine the
theory, for a moment, from another side. When we reach the Rule of

Succession, it will be seen that the hypothetical à priori probabilities
ch. xxx

STATISTICAL INFERENCE

427
are treated as if they were possible causes of the event. It is assumed,
that is to say, that the number of possible sets of antecedent conditions
is proportional to the number of real numbers between 0 and 1; and
that these fall into equal groups, each group corresponding to one of
the real numbers between 0 and 1, this number measuring the degree
of probability with which we could predict the event, if we knew
that an antecedent condition belonging to that group was fulfilled.

It is then assumed that all of these possible antecedent conditions
are à priori equally likely. The argument has arisen by false analogy
from the problem in which a ball is drawn from an urn containing
an infinite number of black and white balls. But for the assumption
that we have in general the kind of knowledge which is necessary
about the possible antecedents, no reasonable foundation has been
suggested.

De Morgan endeavoured to deal with the difficulty in much the
same way in the following passage:∗ “In determining the chance which
exists (under known circumstances) for the happening of an event a
number of times which lies between certain limits, we are involved
in a consideration of some difficulty, namely, the probability of a
probability, or, as we have called it, the presumption of a probability.

To make this idea more clear, remember that any state of probability
may be immediately made the expression of the result of a set of
circumstances, which being introduced into the question, the difficulty
disappears. The word presumption refers distinctly to an act of
the mind, or a state of the mind, while in the word probability we
feel disposed rather to think of the external arrangements on the
knowledge of which the strength of our presumption ought to depend,
than of the presumption itself.” The point of this explanation lies
in the assumption that “any state of probability may be immediately
made the expression of the result of a set of circumstances.” It cannot
be allowed that this is generally true;† and even in those cases in

∗Cabinet Encyclopaedia, p. 87.

†For instance, it is not true even in the standard instance of balls drawn
pt. v

A TREATISE ON PROBABILITY

428
which it is true we are thrown back on the à priori probabilities of
the various sets of circumstances which need not be, as De Morgan
assumes, either equal or exhaustive alternatives.

9.

The proof of the Rule of Succession, which is based upon this
theory of unknown probabilities, is, briefly, as follows:

If x stands for the à priori probability of an event in given
conditions, then the probability that the event will occur m times
and fail n times in these conditions is xm(1 − x)n. If, however,
x is unknown, all values of it between 0 and 1 are à priori equally
probable. It follows from these two sets of considerations that, if the
event has been observed to occur m times out of m+n, the probability
à posteriori that x lies between x and x + dx is proportional to
xm(1 − x)ndx, and is equal, therefore, to Axm(1 − x)ndx where A is a
constant. Since the event has in fact occurred, and since x must have
one of its possible values, A is determined by the equation

Γ(m + n + 2)

Axm(1 − x)ndx = 1

∴ A =

.

Γ(m + 1)Γ(n + 1)

0

Hence the probability that the event will occur at the (m + n + 1)th
trial, when we know that it has occurred m times in m + n trials, is

1

A
xm+1(1 − x)ndx.

0

If we substitute the value of A found above, this is equal to
m + 1

.∗
m + n + 2
from an urn containing black and white in unknown proportions, unless the
number of balls is infinite.

∗The theorem is sometimes enunciated by contemporary writers in a much
more guarded form, e.g. by Czuber, Wahrscheinlichkeitsrechnung, vol. i. p. 197,
and by Bachelier, Calcul des probabilités, p. 487. Bachelier, instead of assuming
that the à priori probabilities of all possible values of the probability of the
event are equal, writes ˆ
ω(y)dy as the à priori probability that the probability
ch. xxx

STATISTICAL INFERENCE

429

The class of problem to which the theorem is supposed to apply is
the following: There are certain conditions such that we are ignorant
à priori as to whether they do or do not lead to the occurrence of
a particular event; on m out of m + n occasions, however, on which
these conditions have been observed, the event has occurred; what is
the probability in the light of this experience that the event will occur
on the next occasion? The answer to all such problems is
m + 1

.

m + n + 2

In the cases where n = 0, i.e. when the event has invariably occurred,
the formula yields the result m + 1. In the case where the conditions
m + 2
have been observed once only and the event has occurred on that
occasion, the result is 2. If the conditions have never been met with

3
at all, the probability of the event is 1. And even in the case where

2
on the only occasion on which the conditions were observed, the event
did not occur, the probability is 1.

3

Some of the flaws in this proof have been already explained. One
minor objection may be pointed out in addition. It is assumed that,
if x is the à priori probability of the event’s happening once, then
xn is the à priori probability of its happening n times in succession,
whereas by the theorem’s own showing the knowledge that the event
has happened once modifies the probability of its happening a second
time; its successive occurrences are not, therefore, independent. If
is y, so that after m occurrences in m + n trials the probability that the
ym(1 − y)n ˆ
ω(y)dy
probability lies between y and y + dy is

.

If one has no
ym(1 − y)n ˆ
ω(y)dy
idea of ˆ
ω à priori, he suggests that the simplest hypothesis is to put ˆ
ω = 1,
which leads, as above, to Laplace’s Law of Succession.

He also proposes the
hypothesis ˆ
ω(y) = a + a1y + a2y2 + . . ., in which case the denominator is a series
of Eulerian integrals. There is a discussion of the Law of Succession, and of the
contradictions and paradoxes to which it leads, by E. T. Whittaker and others
in Part VI. vol. viii. (1920) of the Transactions of the Faculty of Actuaries in

Scotland.

pt. v

A TREATISE ON PROBABILITY

430
the à priori probability of the event is 1, and if, after it has been

2
observed once, the probability that it will occur a second time is 2,

3
then it follows that the à priori probability of its occurring twice is
not 1

1

2

1

× , but 1 × , i.e. ; and in general the à priori probability

2

2

2

3

3
n
of its happening n times in succession is not 1
but

1

.

2
n + 1

10.

But refinements of disproof are hardly needed. The
principle’s conclusion is inconsistent with its premisses. We begin
with the assumption that the à priori probability of an event, about
which we have no information and no experience, is unknown, and
that all values between 0 and 1 are equally probable. We end with
the conclusion that the à priori probability of such an event is 1. It

2
has been pointed out in § 7 that this contradiction was latent, as soon
as the Principle of Indifference was superimposed on the principle of
unknown probabilities.

The theorem’s conclusions, moreover, are a reductio ad absurdum
of the reasoning upon which it is based. Who could suppose that the
probability of a purely hypothetical event, of whatever complexity,
in favour of which no positive argument exists, the like of which
has never been observed, and which has failed to occur on the one
occasion on which the hypothetical conditions were fulfilled, is no less
than 1? Or if we do suppose it, we are involved in contradictions,—for

3
it is easy to imagine more than three incompatible events which
satisfy these conditions.

11.

The theorem was first suggested by the problem of the
urn which contains black and white balls in unknown proportions: m
white and n black balls have been successively drawn and replaced;
what is the probability that the next draw will yield a white ball? It
is supposed that all compositions of the urn are equally probable, and
the proof then proceeds precisely as in the case of the more general
ch. xxx

STATISTICAL INFERENCE

431
rule of succession. The rule of succession has been, sometimes, directly
deduced from the case of the urn, by assimilating the occurrence of
the event to the drawing of a white ball and its non-occurrence to the
drawing of a black ball.

On the hypothesis that all compositions of the urn are equally
probable, an hypothesis to which in general there is nothing
corresponding, and on the further hypothesis that the number of balls
is infinite, this solution is correct.∗ But the rule of succession does
not apply, as it is easy to demonstrate, even to the case of balls
drawn from an urn, if the number of balls is finite.†

12.

If the Rule of Succession is to be adopted by adherents of
the Frequency Theory of Probability,‡ it is necessary that they should
make some modification in the preliminary reasoning on which it is
based. By Dr. Venn, however, the rule has been explicitly rejected on
the ground that it does not accord with experience.§ But Professor

∗This second condition is often omitted (e.g. Bertrand, Calcul des probabilités,
p. 172).

†The correct solution for the case of a finite number of balls, on the
hypothesis that each possible ratio is equally likely, is as follows: The probability
of a black ball at a further trial, after black balls have been successively

1 sp+1
withdrawn and replaced p times, is
where there are n balls and
n
sp
sr represents the sum of the rth powers of the first n natural numbers.

p + 1

This reduces to

—the solution usually given,—when n is infinite.

More
p + 2
generally, if p black balls and q white balls have been drawn and replaced, the
chance that the next ball will be black is
r=n
rp+1(n − r)q

1 r=0

.

n
r=n
rp(n − r)q
r=0

‡See Chapter VIII.

§Logic of Chance, p. 197.

pt. v

A TREATISE ON PROBABILITY

432

Karl Pearson, who accepts it, has made the necessary restatement,∗
and it will be worth while to examine the reasoning when it is put in
this form. Professor Pearson’s proof of the Rule of Succession is as
follows:

“I start, as most mathematical writers have done, with ‘the equal
distribution of ignorance,’ or I assume the truth of Bayes’ Theorem.

I hold this theorem not as rigidly demonstrated, but I think with

Edgeworth† that the hypothesis of the equal distribution of ignorance
is, within the limits of practical life, justified by our experience of
statistical ratios, which à priori are unknown, i.e. such ratios do not
tend to cluster markedly round any particular value. ‘Chances’ lie
between 0 and 1, but our experience does not indicate any tendency
of actual chances to cluster round any particular value in this range.

The ultimate basis of the theory of statistics is thus not mathematical
but observational. Those who do not accept the hypothesis of the
equal distribution of ignorance and its justification in observation are
compelled to produce definite evidence of the clustering of chances, or
to drop all application of past experience to the judgment of probable
future statistical ratios. . . .

“Let the chance of a given event occurring be supposed to lie
between x and x + dx, then if on n = p + q trials an event has been
observed to occur p times and fail q times, the probability that the
true chance lies between x and x + dx is, on the equal distribution of
our ignorance,
xp(1 − x)q dx

Pz =

.

1 xp(1 − x)q dx

0

∗“On the Influence of Past Experience on Future Expectation,” Phil. Mag.

1907, pp. 365–378. The quotations given below are taken from this article.

†This reference is, no doubt, to Edgeworth’s “Philosophy of Chance” (Mind,

1884, p. 230), when he wrote: “The assumption that any probability-constant
about which we know nothing in particular is as likely to have one value as
another is grounded upon the rough but solid experience that such constants
do, as a matter of fact, as often have one value as another.” See also Chapter

VII. § 6, above.

ch. xxx

STATISTICAL INFERENCE

433

“This is Bayes’ Theorem. . . .∗

“Now suppose that a second trial of m = r + s instances be made,
then the probability that the given event will occur r times and fail s,
is on the à priori chance being between x and x + dx

Γm

= Px
nr(1 − x)s,

ΓrΓs
and accordingly the total chance Cr, whatever x may be of the event
occurring r times in the second series, is

1

Γm
xp+r(1 − x)q+s dx

C

0
r =

.

ΓrΓs

1 xp(1 − x)q dx

0

This is, with a slight correction, Laplace’s extension of Bayes’

Theorem.”†

13.

This argument can be restated as follows. Of all the
objects which satisfy φ(x), let us suppose that a proportion p also
satisfy f(x). In this case p measures the probability that any object,
of which we know only that it is φ, is in fact also f. Now if we
do not know the value of p and have no relevant information which
bears upon it, we can assume à priori that all values of p between

0 and 1 are equally likely. This assumption, which is termed the

∗Professor Pearson’s use of this title for the above formula is not, I think,
historically correct. Bayes’ Theorem is the Inverse Principle of Probability itself,
and not this extension of it.

†The rest of the article is concerned with the determination of the probable
error when Laplace’s Rule of Succession is used not simply to yield the
probability of a single additional occurrence, but to predict the probable limits
within which the frequency will lie in a considerable series of additional trials.

Professor Pearson’s method applies more rigorous methods of approximation to
the fundamental formulae given above than have been sometimes used. As my
main purpose in this chapter is to dispute the general validity of the fundamental
formulae, it is not worth while to consider these further developments here. If
the validity of the fundamental formula were to be granted, Professor Pearson’s
methods of approximation would, I think, be satisfactory.

pt. v

A TREATISE ON PROBABILITY

434

‘equal distribution of ignorance,’ is justified by our experience of
statistical ratios. Our experience, that is to say, leads us to suppose
that of all the theories, which could be propounded, there are just as
many which are always true as there are which are always false, just
as many which are true once in fifty times as there are which are true
once in three times, and so on. Professor Pearson challenges those
who do not accept this assumption to produce definite evidence to
the contrary.

The challenge is easily met. It would not be difficult to produce

10, 000 positive theories which are always false corresponding to
every one which is always true, and 10, 000 correlations of positive
qualities which hold less often than once in three times for every
one we can name which holds more often than once in three times.

And the converse is the case for negative theories and correlations
between negative qualities; for corresponding to every positive theory
which is true there is a negative theory which is false, and so
on. Thus experience, if it shows anything, shows that there is a
very marked clustering of statistical ratios in the neighbourhoods of
zero and unity,—of those for positive theories and for correlations
between positive qualities in the neighbourhood of zero, and of those
for negative theories and for correlations between negative qualities
in the neighbourhood of unity. Moreover, we are seldom in so
complete a state of ignorance regarding the nature of the theory or
correlation under investigation as not to know whether or not it is a
positive theory or a correlation between positive qualities. In general,
therefore, whenever our investigation is a practical one, experience,
if it tells us anything, tells us not only that the statistical ratios
cluster in the neighbourhood of zero and unity, but in which of these
two neighbourhoods the ratio in this particular case is most likely
à priori to be found. If we seek to discover what proportion of the
population suffer from a certain disease, or have red hair, or are
called Jones, it is preposterous to suppose that the proportion is as
likely à priori to exceed as to fall short of (say) fifty per cent. As
ch. xxx

STATISTICAL INFERENCE

435

Professor Pearson applies this method to investigations where it is
plain that the qualities involved are positive, he seems to maintain
that experience shows that there are as many positive attributes
which are shared by more than half of any population as there are
which are shared by less than half.

It is also worthwhile to point out that it is formally impossible
that it should be true of all characters, simple and complex, that
they are as likely to have any one frequency as any other. For let
us take a character c which is compound of two characters a and b,
between which there is no association, and let us suppose that a
has a frequency x in the population in question and that b has a
frequency y, so that, in the absence of association, the frequency z
of c is equal to xy. Then it is easy to show that, if all values of
x and y between 0 and 1 are equally probable, all values of z between

0 and 1 are not equally probable. For the value 1 is more probable

2
than any other, and the possible values of z become increasingly
improbable as they differ more widely from 1.

2

It may be added that the conclusions, which Professor Pearson
himself derives from this method, provide a reductio ad absurdum of
the arguments upon which they rest. He considers, for example, the
following problem: A sample of 100 of a population shows 10 per cent
affected with a certain disease. What percentage may be reasonably
expected in a second sample of 100? By approximation he reaches
the conclusion that the percentage of the character in the second
sample is as likely to fall inside as outside the limits, 7.85 and 13.71.

Apart from the preceding criticisms of the reasoning upon which
this depends, it does not seem reasonable upon general grounds
that we should be able on so little evidence to reach so certain a
conclusion. The argument does not require, for example, that we
have any knowledge of the manner in which the samples are chosen,
of the positive and negative analogies between the individuals, or
indeed anything at all beyond what is given in the above statement.

pt. v

A TREATISE ON PROBABILITY

436

The method is, in fact, much too powerful. It invests any positive
conclusion, which it is employed to support, with far too high a degree
of probability. Indeed this is so foolish a theorem that to entertain it
is discreditable.

14.

The Rule of Succession has played a very important part
in the development of the theory of probability. It is true that it
has been rejected by Boole∗ on the ground that the hypotheses on
which it is based are arbitrary, by Venn† on the ground that it does
not accord with experience, by Bertrand‡ because it is ridiculous, and
doubtless by others also. But it has been very widely accepted,—by

De Morgan,§ by Jevons,¶ by Lotze,‖ by Czuber,∗∗ and by Professor

Pearson,††—to name some representative writers of successive schools
and periods. And, in any case, it is of interest as being one of the most
characteristic results of a way of thinking in probability introduced by

Laplace, and never thoroughly discarded to this day. Even amongst
those writers who have rejected or avoided it, this rejection has been
due more to a distrust of the particular applications of which the law
is susceptible than to fundamental objections against almost every
step and every presumption upon which its proof depends.

Some of these particular applications have certainly been surprising.

The law, as is evident, provides a numerical measure of the probability
of any simple induction, provided only that our ignorance of its

∗Laws of Thought, p. 369.

†Logic of Chance, p. 197.

‡Calcul des probabilités, p. 174.

§Article in Cabinet Encyclopaedia, p. 64.

¶Principles of Science, p. 297.

‖Logic, pp. 373, 374; Lotze propounds a “simple deduction” “as convincing”

to him “as the more obscure analysis, by which it is usually obtained.”

The
proof is among the worst ever conceived, and may be commended to those who
seek instances of the profound credulity of even considerable thinkers.

∗∗Wahrscheinlichkeitsrechnung, vol. i. p. 199,—though much more guardedly
and with more qualifications than in the form discussed above.

††Loc. cit.

ch. xxx

STATISTICAL INFERENCE

437
conditions is sufficiently complete, and, although, when the number
of cases dealt with is small, its results are incredible, there is, when
the number dealt with is large, a certain plausibility in the results
it gives. But even in these cases paradoxical conclusions are not
far out of sight. When Laplace proves that, account being taken
of the experience of the human race, the probability of the sun’s
rising to-morrow is 1, 826, 214 to 1, this large number may seem in a
kind of way to represent our state of mind about the matter. But
an ingenious German, Professor Bobek,∗ has pushed the argument a
degree further, and proves by means of these same principles that the
probability of the sun’s rising every day for the next 4000 years, is
not more, approximately, than two-thirds,—a result less dear to our
natural prejudices.

∗Lehrbuch der Wahrscheinlichkeitsrechnung, p. 208.

CHAPTER XXXI
the inversion of bernoulli’s theorem

1.

I conclude, then, that the application of the mathematical
methods, discussed in the preceding chapter, to the general problem
of statistical inference is invalid. Our state of knowledge about our
material must be positive, not negative, before we can proceed to
such definite conclusions as they purport to justify. To apply these
methods to material, unanalysed in respect of the circumstances of
its origin, and without reference to our general body of knowledge,
merely on the basis of arithmetic and of those of the characteristics
of our material with which the methods of descriptive statistics are
competent to deal, can only lead to error and to delusion.

But I go further than this in my opposition to them. Not only are
they the children of loose thinking, and the parents of charlatanry.

Even when they are employed by wise and competent hands, I doubt
whether they represent the most fruitful form in which to apply
technical and mathematical methods to statistical problems, except
in a limited class of special cases. The methods associated with the
names of Lexis, Von Bortkiewicz, and Tschuprow (of whom the last
named forms a link, to some extent, between the two schools), which
will be briefly described in the next chapter, seem to me to be much
more clearly consonant with the principles of sound induction.

2.

Nevertheless it is natural to suppose that the fundamental
ideas, from which these methods have sprung, are not wholly égarés.

It is reasonable to presume that, subject to suitable conditions and
qualifications, an inversion of Bernoulli’s Theorem must have validity.

If we knew that our material could be likened to a game of chance,
we might expect to infer chances from frequencies, with the same sort
of confidence as that with which we infer frequencies from chances.

This part of our inquiry will not be complete, therefore, until we
have endeavoured to elucidate the conditions for the validity of an

438
ch. xxxi

STATISTICAL INFERENCE

439

Inversion of Bernoulli’s Theorem.

3.

The problem is usually discussed in terms of the happening of
an event under certain conditions, that is to say, of the coexistence
of the conditions, as affecting a particular event, with that event.

The same problem can be dealt with more generally and more
conveniently as an investigation of the correlation between two
characters A(x) and B(x), which, as in Part III., are propositional functions which may be said to concur or coexist when they are both
true of the same argument x. Given that, within the field of our
knowledge, B(x) is true for a certain proportion of the values of x for
which A(x) is true, what is the probability for a further value a of x
that, if A(a) holds, B(a) will hold also?

Let us suppose that the occurrence of an instance of A(x) is a
sign of one of the events e1(x), e2(x) . . . or em(x), and that these are
exhaustive, exclusive, and ultimate alternatives. By exhaustive it is
meant that, whenever there is an instance of A(x), one of the e’s is
present; by exclusive, that the presence of one of the e’s is not a sign
of the presence of any other, but not that the concurrence of two or
more of the e’s is in fact impossible; by ultimate, that no one of the e’s
is a disjunction of two or more alternatives which might themselves be
members of the e’s. Let us assume that these alternatives are initially
and throughout the argument equally probable, which, subject to the
above conditions, is justified by the Principle of Indifference. We have
no reason, that is to say, and no part of our evidence ever gives us
one, for thinking that A(a) is more likely to be a sign of one of the e’s
than of any other, or even for thinking that some e’s, although we do
not know which, are more likely to occur than others. Let us also
assume that, out of e1(x), e2(x) . . . em(x), the set e1(x), e2(x) . . . el(x),
and these only, are signs or occasions of B(x); and further that we
have no evidence bearing on the actual magnitude of the integers
l and m, so that the ratio l/m is the only factor of which the
probability varies as the evidence accumulates. Let us assume, lastly,
that our knowledge of the several instances of B(x) is adequate
pt. v

A TREATISE ON PROBABILITY

440
to establish a perfect analogy between them; the instances a, etc.,
of B(x), that is to say, must not have anything in common except B,
unless we have reason to know that the additional resemblances are
immaterial. Even by these considerable simplifications not every
difficulty has been avoided. But a development along the usual lines
with the assistance of Bernoulli’s Theorem is now possible.

Let l/m = q. If the value of q were known, the problem would
be solved. For this numerical ratio would represent the probability
that A is, in any random instance, a sign of B; and no further
evidence, which satisfies the conditions of the preceding hypothesis,
can possibly modify it. But in the inverse problem q is not known;
and our problem is to determine whether evidence can be forthcoming
of such a kind, that, as this evidence is increased in quantity, the
probability that A will be in any instance a sign of B, tends to a limit
which lies between two determinate ratios, just as the probability
of an inductive generalisation may tend towards certainty, when the
evidence is increased in a manner satisfying given conditions.

Let f(q) represent the proposition that q is the true value of l/m.

Let q represent the ratio of the number of instances actually before
us in which A has been accompanied by B to that of the instances
in which A has not been accompanied by B; and let f (q ) be the
proposition which asserts this. Now if the ratio q is known, then,
subject to the assumptions already stated, the number q must also
represent the à priori probability in any instance, both before and
after the results of other instances are known, that A, if it occurs, will
be accompanied by B. We have, in fact, the conditions as set forth in

Chapter XXIX., in which Bernoulli’s Theorem can be validly applied,
so that this theorem enables us to give a numerical value, for all
numerical values of q and q , to the probability f (q )/h f(q),—which
expression represents the likelihood à priori of the frequency q ,
given q.

An application of the inverse formula allows us to infer from the
ch. xxxi

STATISTICAL INFERENCE

441
above the à posteriori probability of q, given q , namely:

f (q)/h f (q )/h f (q)

f (q)/h f (q )/h f (q)

where the summation in the denominator covers all possible values
of q. In rough applications of this inverse of Bernoulli’s Theorem
it has been usual to suppose that f(q)/h is constant for all values
of q,—that, in other words, all possible values of the ratio q are
à priori equally likely. If this supposition were legitimate, the formula
could be reduced to the algebraical expression
f (q )/h f (q) ,
f (q )/h f (q)

all the terms of which can be determined numerically by Bernoulli’s

Theorem. It is easy to show that it is a maximum when q = q ,
i.e. that q is the most probable value of l/m, and that, when the
instances are very numerous, it is very improbable that l/m differs
from q widely. If, therefore, the number of instances is increased in
such a manner that the ratio continues in the neighbourhood of q , the
probability that the true value of l/m is nearly q tends to certainty;
and, consequently, the probability, that A is in any instance a sign
of B, also tends to a magnitude which is measured by q .

I see, however, no justification for the assumption that all possible
values of the ratio q are à priori equally likely. It is not even
equivalent to the assumptions that all integral values of l and m
respectively are equally probable. I am not satisfied either that
different values of q, or that different values of m, satisfy the
conditions which have been laid down in Part I. for alternatives which are equal before the Principle of Indifference. There seem, for
instance, to be relevant differences between the statement that A can
arise in exactly two ways and the statement that it can arise in
exactly a thousand ways. We must, therefore, be content with some
pt. v

A TREATISE ON PROBABILITY

442
lesser assumption and with a less precise form for our final conclusion.

4.

Since, in accordance with our hypothesis, m cannot exceed
some finite number, and since l must necessarily be less than m, the
possible values of m, and therefore of q, are finite in number. Perhaps
we can assume, therefore, as one of our fundamental assumptions,
that there is à priori a finite probability in favour of each of these
possible values. Let µ be the finite number which m cannot exceed.

Then there is a finite probability for each of the intervals∗

1 to 2 2
µ − 1

,
to 3 , . . .

to 1
µ
µ
µ
µ
µ
that q lies in this interval; but we cannot assume that there is an
equal probability for each interval.

We must now return to the formula
f q/h f (q )/hf (q)

,
f (q)/h f (q )/hf (q)

which represents the à posteriori probability of q, given q . Since
by sufficiently increasing the number of instances, the sum of terms
f (q )/hf (q) for possible values of q within a certain finite interval in
the neighbourhood of q can be made to exceed the other terms by
any required amount, and since the sum of the values of f(q)/h for
possible values of q within this interval is finite, it clearly follows that
a finite number of instances can make the probability, that q lies in
an interval of magnitude 1/µ in the neighbourhood of q , to differ
from certainty by less than any finite amount however small.

5.

We have, therefore, reached the main part of the conclusion
after which we set out—namely, that as the number of instances
is increased the probability, that q is in the neighbourhood of q ,
tends towards certainty; and hence that, subject to certain specified
conditions, if the frequency with which B accompanies A is found

∗The intervals are supposed to include their lower but not their upper limit.

ch. xxxi

STATISTICAL INFERENCE

443
to be q in a great number of instances, then the probability
that A will be accompanied by B in any further instance is also
approximately q . But we are left with the same vagueness, as in the
case of generalisation, respecting the value of µ and the number of
instances that we require. We know that we can get as near certainty
as we choose by a finite number of instances, but what this number
is we do not know. This is not very satisfactory, but it accords very
well, I think, with what common sense tells us. It would be very
surprising, in fact, if logic could tell us exactly how many instances we
want, to yield us a given degree of certainty in empirical arguments.

Nobody supposes that we can measure exactly the probability of
an induction. Yet many persons seem to believe that in the weaker
and much more difficult type of argument, where the association
under examination has been in our experience, not invariable, but
merely in a certain proportion, we can attribute a definite measure
to our future expectations and can claim practical certainty for
the results of predictions which lie within relatively narrow limits.

Coolly considered, this is a preposterous claim, which would have
been universally rejected long ago, if those who made it had not so
successfully concealed themselves from the eyes of common sense in a
maze of mathematics.

6.

Meantime we are in danger of forgetting that, in order
to reach even our modified conclusion, material assumptions have
been introduced. In the first place, we are faced with exactly the
same difficulties as in the case of universal induction dealt with in

Part III., and our original starting-point must be the same. We have the same difficulty as to how our initial probability is to be
obtained; and I have no better suggestion to offer in this than
in the former case—namely, the supposed principle of a limitation
of independent variety in experience. We have to suppose that if

A and B occur together (i.e. are true of the same object), this is some
just appreciable reason for supposing that in this instance they have a
common cause; and that, if A occurs again, this is a just appreciable
pt. v

A TREATISE ON PROBABILITY

444
reason for supposing that it is due to the same cause as on the
former occasion. But in addition to the usual inductive hypothesis,
the argument has rested on two particularly important assumptions,
first, that we have no reason for supposing that some of the events of
which A may be a sign are more likely to be exemplified in some of
the particular instances than in others, and secondly, that the analogy
amongst the examined B’s is perfect. The first assumption amounts,
in the language of statisticians, to an assumption of random sampling
from amongst the A’s. The second assumption corresponds precisely
to the similar condition which we discussed fully in connection with
inductive generalisation. The instances of A(x) may be the result
of random sampling, and yet it may still be the case that there
are material circumstances, common to all the examined instances
of B(x), yet not covered by the statement A(x)B(x). In so far as
these two assumptions are not justified, an element of doubt and
vagueness, which is not easily measured, assails the argument. It is
an element of doubt precisely similar to that which exists in the case
of generalisation. But we are more likely to forget it. For having
overcome the difficulties peculiar to correlation,∗ it is, possibly, not
unnatural for a statistician to feel as if he had overcome all the
difficulties.

In practice, however, our knowledge, in cases of correlation just as
in cases of generalisation, will seldom justify the assumption of perfect
analogy between the B’s; and we shall be faced by precisely the same
problems of analysing and improving our knowledge of the instances,
as in the general case of induction already examined. If B has
invariably accompanied A in 100 cases, we have all kinds of difficulties
about the exact character of our evidence before we can found on

∗I am here using this term in distinction to generalisation; that is to say, I
call the statement that A(x) is always accompanied by B(x) a generalisation,
and the statement that A(x) is accompanied by B(x) in a certain proportion
of cases a correlation.

This is not quite identical with its use by modern
statisticians.

ch. xxxi

STATISTICAL INFERENCE

445
this experience a valid generalisation. If B has accompanied A, not
invariably, but only 50 times in the 100 cases, clearly we have just
the same kind of difficulties to face, and more too, before we can
announce a valid correlation. Out of the mere unanalysed statement
that B has accompanied A as often as not in 100 cases, without
precise particulars of the cases, or even if there were 1, 000, 000 cases
instead of 100, we can conclude very little indeed.

CHAPTER XXXII
the inductive use of statistical frequencies for
the determination of probability
à posteriori —the methods of lexis

1.

No one supposes that a good induction can be arrived at
merely by counting cases. The business of strengthening the argument
chiefly consists in determining whether the alleged association is
stable, when the accompanying conditions are varied. This process
of improving the Analogy, as I have called it in Part III., is, both logically and practically, of the essence of the argument.

Now in statistical reasoning (or inductive correlation) that part of
the argument, which corresponds to counting the cases in inductive
generalisation, may present considerable technical difficulty. This
is especially so in the particularly complex cases of what in the
next chapter (§ 9) I shall term Quantitative Correlation, which have
greatly occupied the attention of English statisticians in recent
years. But clearly it would be an error to suppose that, when
we have successfully overcome the mathematical or other technical
difficulties, we have made any greater progress towards establishing
our conclusion than when, in the case of inductive generalisation, we
have counted the cases but have not yet analysed or compared the
descriptive and non-numerical differences and resemblances. In order
to get a good scientific argument we still have to pursue precisely
the same scientific methods of experiment, analysis, comparison,
and differentiation as are recognised to be necessary to establish
any scientific generalisation. These methods are not reducible to
a precise mathematical form for the reasons examined in Part III.

of this treatise. But that is no reason for ignoring them, or for
pretending that the calculation of a probability, which takes into
account nothing whatever except the numbers of the instances, is
a rational proceeding. The passage already quoted from Leibniz

446
ch. xxxii

STATISTICAL INFERENCE

447

(In exemplis juridicis politicisque plerumque non tamen subtili calculo
opus est, quam accurata omnium circumstantiarum enumeratione) is
as applicable to scientific as to political inquiries.

Generally speaking, therefore, I think that the business of
statistical technique ought to be regarded as strictly limited to
preparing the numerical aspects of our material in an intelligible form,
so as to be ready for the application of the usual inductive methods.

Statistical technique tells us how to ‘count the cases’ when we are
presented with complex material. It must not proceed also, except in
the exceptional case where our evidence furnishes us from the outset
with data of a particular kind, to turn its results into probabilities;
not, at any rate, if we mean by probability a measure of rational
belief.

2.

There is, however, one type of technical, statistical
investigation not yet discussed, which seems to me to be a valuable
aid to inductive correlation. This method consists in breaking up a
statistical series, according to appropriate principles, into a number
of sub-series, with a view to analysing and measuring, not merely
the frequency of a given character over the aggregate series, but the
stability of this frequency amongst the sub-series; that is to say, the
series as a whole is divided up by some principle of classification into
a set of sub-series, and the fluctuation of the statistical frequency
under examination between the various sub-series is then examined.

It is, in fact, a technical method of increasing the Analogy between
the instances, in the sense given to this process in Part III.

3.

The method of analysing statistical series, as opposed
to the Laplacian or mathematical method, one might designate the
inductive method. Independently of the investigations of Bernoulli
or Laplace, practical statisticians began at least as early as the
end of the seventeenth century∗ to pay attention to the stability of

∗Graunt in his Natural and Political Observations upon the Bills of Mortality
has been quoted as one of the earliest statisticians to pay attention to these
considerations.

pt. v

A TREATISE ON PROBABILITY

448
statistical series when analysed in this manner. Throughout the
eighteenth century, students of mortality statistics, and of the ratio
of male to female births (including Laplace himself), paid attention
to the degree of constancy of the ratios over different parts of their
series of instances as well as to their average value over the whole
series. And in the early part of the nineteenth century, Quetelet,
as we have already noticed, widely popularised the notion of the
stability of various social statistics from year to year. Quetelet,
however, sometimes asserted the existence of stability on insufficient
evidence, and involved himself in theoretical errors through imitating
the methods of Laplace too closely; and it was not until the last
quarter of the nineteenth century that a school of statistical theory
was founded, which gave to this way of approaching the problem
the system and technique which it had hitherto lacked, and at the
same time made explicit the contrast between this analytical or
inductive method and the prevailing mathematical theory. The sole
founder of this school was the German economist, Wilhelm Lexis,
whose theories were expounded in a series of articles and monographs
published between the years 1875 and 1879. For some years Lexis’s
fundamental ideas did not attract much notice, and he himself seems
to have turned his attention in other directions. But more recently
a considerable literature has grown up round them in Germany, and
their full purport has been expressed with more clearness than by

Lexis himself—although no one, with the exception of Ladislaus von

Bortkiewicz, has been able to make additions to them of any great
significance.∗ Lexis devised his theory with an immediate view to

∗A list of Lexis’s principal writings on these topics will be found in the

Bibliography.

There is little of first-rate importance which is not contained
either in the volume, Zur Theorie der Massenerscheinungen in der menschlichen

Gesellschaft,
or in the Abhandlungen zur Theorie der Bevölkerungs- und

Moral-Statistik.

In this latter volume the two important articles on “Die

Theorie der Stabilität statistischer Reihen” and on “Das Geschlechtsverhältnis
der Geborenen und die Wahrscheinlichkeitsrechnung,” originally published in

Conrad’s Jahrbüche, are reprinted.

ch. xxxii

STATISTICAL INFERENCE

449
its practical application to the problems of sex ratio and mortality.

The fact that his general theory is so closely intermingled with these
particular applications of it is, probably, a part explanation of the
long interval which elapsed before the general theoretical importance
of his ideas was widely realised. I cannot help doubting how fully

Lexis himself realised it in the first instance. It would certainly
be easy to read his earlier contributions to the question without
appreciating their generalised significance. After 1879 Lexis added
nothing substantial to his earlier work, and later developments are
mainly due to Von Bortkiewicz. Those of the latter’s writings, which
have an important bearing on the relation between probability and
statistics, are given in the Bibliography.∗

On the logic and philosophy of Probability writers of the school
of Lexis are in general agreement with Von Kries; but this seems
to be due rather to the reaction which is common both to him and
to them against the Laplacian tradition, than to any very intimate
theoretical connection between Von Kries’s main contributions to

Probability and those of Lexis, though it is true that both show
a tendency to find the ultimate basis of Probability in physical
rather than in logical considerations. I am not acquainted with
much work, which has been appreciably influenced by Lexis, written
in other languages than German (including with Germans, that is

∗The reader may be specially referred to the Kritische Betrachtungen zur
theoretischen Statistik (first instalment—the later instalments being of less interest
to the student of Probability), the Anwendungen der Wahrscheinlichkeitsrechnung
auf Statistik, and Homogeneität und Stabilität in der Statistik. Of other German
and Russian writers it will be sufficient to mention here Tschuprow, who in

“Die Aufgaben der Theorie der Statistik” (Schmoller’s Jahrbuch, 1905) and

“Zur Theorie der Stabilität statistischer Reihen” (Skandinavisk Aktuarietidskrift )

gives by far the best and most lucid general accounts that are available of
the doctrines of the school, he alone amongst these authors writing in a
style from which the foreign reader can derive pleasure, and Czuber, who in
his Wahrscheinlichtkeitsrechnung (vol. ii. part iv. section 1) supplies a useful
mathematical commentary.

pt. v

A TREATISE ON PROBABILITY

450
to say, those Russians, Austrians, and Dutch who usually write in

German, and are in habitual connection with the German scientific
world). In France Dormoy∗ published independently and at about
the same time as Lexis some not dissimilar theories, but subsequent

French writers have paid little attention to the work of either. Such
typical French treatises as that of Bertrand, or, more recently, that
of Borel, contain no reference to them.† In Italy there has been
some discussion recently on the work of Von Bortkiewicz. Among

Englishmen Professor Edgeworth has shown a close acquaintance with
the work of the German school,‡ he providing for nearly forty years
past, on this as on other matters where the realms of Statistics and

Probability overlap, almost the only connecting link between English
and continental thought.

Nevertheless, an account in English of the main doctrines of this
school is still lacking. It would be outside the plan of the present
treatise to attempt such an account here. But it may be useful
to give a short summary of Lexis’s fundamental ideas. After giving
this account I shall find it convenient, in proceeding to my own
incomplete observations on the matter, to approach it from a rather
different standpoint from that of Lexis or of Von Bortkewicz, though
not for that reason the less influenced or illuminated by their eminent
contributions to this problem.

4.

It will be clearer to begin with some analysis due to

Von Bortkiewicz,§ and then to proceed to the method of Lexis

∗Journal des actuaires français, 1874, and Théorie mathématique des
assurances sur la vie, 1878; on the question of priority see Lexis, Abhandlungen,
p. 130.

†Though both these writers touch on closely cognate matters, where Lexis’s
investigations would be highly relevant—Bertrand, Calcul, pp. 312–314; Borel,

Éléments, p. 160.

‡See especially his “Methods of Statistics” in the Jubilee Volume of the Stat.

Journ., 1885, and “Application of the Calculus of Probabilities to Statistics,”

International Statistical Institute Bulletin, 1910.

§What follows is a free rendering of some passages in his Kritiscke
ch. xxxii

STATISTICAL INFERENCE

451
himself, although the latter came first in point of time.

A group of observations may be made up of a number of
subgroups, to which different frequencies for the character under
investigation are properly applicable. That is to say, a proportion z1z
of the observations may belong to a group, for which, given the
frequency, the à priori probability of the character under observation
in a particular instance would be p1, a proportion z2 may belong to
z
a second group for which p2 is the probability, and so on. In this
case, given the frequencies for the subgroups, the probability p for
the group as a whole would be made up as follows:

z
z
p = 1 p

2

1 +
p2 + . . . .

z
z

We may call p a general probability, and p1, etc., special
probabilities.

But the special probabilities may in their turn be
general probabilities, so that there may be more than one way of
resolving a general probability into special probabilities.

If p1 = p2 = . . . = p, then p, for that particular way of resolving
the total group into partial groups, is, in Bortkiewicz’s terminology,
indifferent.

If p is indifferent for all conceivable resolutions into
partial groups,∗ then, borrowing a phrase from Von Kries, Bortkiewicz
says of it that it has a definitive interpretation. In dealing with
à priori probabilities, we can resolve a total probability until we
reach the special probabilities of each individual case; and if we find
that all these special probabilities are equal, then, clearly, the general
probability satisfies the condition for definitive interpretation.

So far we have been dealing with à priori probabilities. But the
object of the analysis has been to throw light on the inverse problem.

We want to discover in what conditions we can regard an observed
frequency as being an adequate approximation to a definitive general
probability.

Betrachungen.

∗This is clearly a very loose statement of what Bortkiewicz really means.

pt. v

A TREATISE ON PROBABILITY

452

If p is the empirical value of p (or, as I should prefer to call it,
the frequency) given by a series of n observations, we may have
n
n
p =

1 p + 2 p + . . . .

n

1
n

2

Even if this particular way of resolving the series of observations
is indifferent, the actually observed frequencies p , p , etc., may

1

2
nevertheless be unequal, since they may fluctuate round the norm p
through the operation of ‘chance’ influences. If, however, n1, n2, etc.,
are large, we can apply the usual Bernoullian formula to discover
whether, if there was a norm p , the divergences of p , p , etc.,

1

2
from it are within the limits reasonably attributable on Bernoullian
hypotheses to ‘chance’ influences. We can, however, only base a sound
argument in favour of the existence of a ‘definitive’ probability p by
resolving our aggregate of instances into sub-series in a great variety
of ways, and applying the above calculations each time. Even so,
some measure of doubt must remain, just as in the case of other
inductive arguments.

Bortkiewicz goes on to say that probabilities having definitive
interpretation (definitive Bedeutung) may be designated elementary
probabilities (Elementarwahrscheinlichkeiten). But the probabilities
which usually arise in statistical inquiries are not of this type, and may
be termed average probabilities (Durchschnittswahrscheinlichkeiten).

That is to say, a series of observed frequencies (or, as he calls them,
empirical probabilities) does not, as a rule, group itself as it would if
the series was in fact subject to an elementary probability.

5.

This exposition is based on a philosophy of Probability
different from mine; but the underlying ideas are capable of
translation.

Suppose that one is endeavouring to establish an
inductive correlation, e.g. that the chance of a male birth is m. The
conclusion, which we are seeking to establish, takes no account of
the place or date of birth or the race of the parents, and assumes
that these influences are irrelevant. Now, if we had statistics of
ch. xxxii

STATISTICAL INFERENCE

453
birth ratios for all parts of the world throughout the nineteenth
century, and added them all up and found that the average frequency
of male births was m, we should not be justified in arguing from
this that the frequency of male births in England next year is
very unlikely to diverge widely from m. For this would involve
the unwarranted assumption, in Bortkiewicz’s terminology, that the
empirical probability m is elementary for any resolution dependent on
time or place, and is not an average probability compounded out of a
series of groups, relating to different times or places, to each of which
a distinct special probability is applicable. And, in my terminology,
it would assume that variations of time and place were irrelevant to
the correlation, without any attempt having been made to employ
the methods of positive and negative Analogy to establish this.

We must, therefore, break up our statistical material into groups
by date, place, and any other characteristic which our generalisation
proposes to treat as irrelevant. By this means we shall obtain a
number of frequencies m , m , m , . . . m , m , m , . . . etc., which are

1

2

3

1

2

3
distributed round the average frequency m. For simplicity let us
consider the series of frequencies m , m , m , . . . obtained by breaking

1

2

3
up our material according to the date of the birth. If the observed
divergences of these frequencies from their mean are not significant,
we have the beginnings of an inductive argument for regarding date
as being in this connection irrelevant.

6.

At this point Lexis’s fundamental contribution to the problem
must be introduced. He concentrated his attention on the nature
of the dispersion of the frequencies m , m , m . . . round their mean

1

2

3
value m; and he sought to devise a technical method for measuring
the degree of stability displayed by the series of sub-frequencies, which
are yielded by the various possible criteria for resolving the aggregate
statistical material into a number of constituent groups.

For this purpose he classified the various types of dispersion which
could occur. It may be the case that some of the sub-frequencies
show such wide and discordant variations from the mean as to suggest
pt. v

A TREATISE ON PROBABILITY

454
that some significant Analogy has been overlooked. In this event
the lack of symmetry, which characterises the oscillations, may be
taken to indicate that some of the subgroups are subject to a relevant
influence, of which we must take account in our generalisation, to
which some of the other subgroups are not subject.

But amongst the various types of dispersion Lexis found one class
clearly distinguishable from all the others, the peculiarity of which
is that the individual values fluctuate in a ‘purely chance’ manner
about a constant fundamental value. This type he called typical

(typische) dispersion. He meant by this that the dispersion conformed
approximately to the distribution which would be given by some
normal law of error.

The next stage of Lexis’s argument∗ was to point out that series
of frequencies which are typical in character may have as their
foundation either a constant probability,† or one which is itself subject
to chance variations about a mean. The first case is typified by the
example of a series of sets of drawings of balls, each set being drawn
from a similar urn; the second case by the example of a series of sets
of drawings, the urns from which each set is drawn being not similar,
but with constitutions which vary in a chance manner about a mean.

As his measure of dispersion Lexis introduces a formula, which
is evidently in part conventional (as is the case with so many other
statistical formulae, the particular shape of which is often determined
by mathematical convenience rather than by any more fundamental
criterion). He expresses himself as follows. Where the underlying
probability is constant, the probable error in a particular frequency

2v(1 − v)

à priori is r = ρ

, where ρ = .4769, v is the underlying
g

∗I am here following fairly closely his paper, “Über die Theorie der

Stabilität statisticher Reihen,” reprinted in his Abhandlungen zur Theorie der

Bevölkerungsund Moral-Statistik, pp. 170–212.

†This mode of expression, which is not in accurate conformity with my
philosophy of Probability, is Lexis’s, not mine. His meaning is intelligible.

ch. xxxii

STATISTICAL INFERENCE

455
probability, and g is the number of instances to which the frequency
refers. This follows from the usual Bernoullian assumptions. Now let

R be the corresponding expression derived à posteriori by reference
to the actual deviations of a series of observed frequencies from their
mean, so that

2[δ2]

R = ρ

, where [δ2] is the sum of the squares
n − 1
of the deviations of the individual frequencies from their mean and
n is their number. Now, if the observed facts are due to merely
chance variations about a constant v, we must have approximately

R = r, though, if g is small, comparatively wide deviations between

R and r will not be significant.

If, on the other hand, v itself
is not constant but is subject to chance variations, the case stands
differently. For the fluctuations of the observed frequencies are now
due to two components. The one which would be present, even if
the underlying probability were constant, Lexis terms the ordinary or
unessential component; the other he terms the physical component.

If p is the probable deviation of the various values of v from their
mean, then, on the same assumptions and as a deduction from the
same theory as before, R will tend to equal not r but
r2 + p2. In
this event R cannot be less than r. If, therefore, R < r, one must
suppose that the individual instances of each several series on which
each frequency is based are not independent of one another. Such a
series Lexis terms an organic or dependent (gebundene) series, and
explains that it cannot be handled by purely statistical methods.

Since, therefore, we have three types of series, differing fundamen-
tally from one another according as R = r, > r, or < r, Lexis puts

R = Q, and takes Q as his measure of dispersion.∗ If Q = 1, we have
r
normal dispersion; if Q > 1, we have supernormal dispersion; and if

∗In Tschuprow’s notation (Die Aufgaben der Theorie der Statistik, p. 45),
k=n

2

(pk − p)2

Q = P/C, where P (the Physical modulus) =
k=1
and C (the
n
pt. v

A TREATISE ON PROBABILITY

456

Q < 1, we have subnormal dispersion, which is an indication that the
series is ‘organic.’

If the number of instances on which the frequencies are based is
very great, r becomes negligible in comparison with p (the physical
component), and, therefore, R =
r2 + p2 becomes approximately

R = p. On the other hand, if p is not very large and the base number
of instances is small, p becomes negligible in comparison with r, and
we have a delusive appearance of normal dispersion.∗ Lexis well
illustrates the former point by the example that the statistics of the
ratio of male to female births for the forty-five registration districts of

England over the years 1859–1871 approximately satisfy the relation

R = r. But if we take the figures for all England over those thirteen
years, although the extreme limits of the fluctuation of the ratio
about its mean 1.042 are 1.035 and 1.047, nevertheless R = 2.6 and
r = 1.6, so that Q = 1.625; the explanation being that the base
number of instances, namely 730, 000, is so large that r is very small,
with the result that it is swamped by the physical component p.

And he illustrates the latter point by the assertion that, if in 20 or

30 series each of 100 draws from an urn containing black and white
balls equally, the number of black balls drawn each time were only to
vary between 49 and 51, he would have confidence that the game was
in some way falsified and that the draws were not independent. That
is to say, undue regularity is as fatal to the assumption of Bernoullian
conditions as is undue dispersion.

7.

In a characteristic passage† Professor Edgeworth has applied
these theories to the frequency of dactyls in successive extracts from

2p(1 − p)

Combinatorial modulus) =

, M being the number of instances in

M
each set, n the number of sets, pk the frequency for set k, and p the mean of
the n frequencies.

∗This is part of the explanation of Bortkiewicz’s Law of Small Numbers. See
also p. 458.

†“On Methods of Statistics,” Jubilee Volume of the Royal Statistical Society,
p. 211.

ch. xxxii

STATISTICAL INFERENCE

457
the Aeneid. The mean for the line is 1.6, exclusive of the fifth foot,
thus sharply distinguishing the Virgilian line from the Ovidian, for
which the corresponding figure is 2.2. But there is also a marked
stability. “That the Mean of any five lines should differ from the
general Mean by a whole dactyl is proved to be an exceptional
phenomenon, about as rare as an Englishman measuring 5 feet, or

6 feet 3 inches.

An excess of two dactyls in the Mean of five
lines would be as exceptional as an Englishman measuring 6 feet

10 inches.”

But not only so—the stability is excessive, and the
fluctuation is less “than that which is obtained upon the hypothesis
of pure sortition. If we could imagine dactyls and spondees to be
mixed up in the poet’s brain in the proportion of 16 to 24 and
shaken out at random, the modulus in the number of dactyls would
be 1.38, whereas we have constantly obtained a smaller number, on
an average (the square root of the average fluctuation) 1.2.” On

Lexian principles these statistical results would support the hypothesis
that the series under investigation is ‘organic’ and not subject to

Bernoullian conditions, an hypothesis in accordance with our ideas
of poetry. That Edgeworth should have put forward this example in
criticism of Lexis’s conclusions, and that Lexis∗ should have retorted
that the explanation was to be found in Edgeworth’s series’ not
consisting of an adequate number of separate observations, indicates,
if I do not misapprehend them, that these authorities are at fault in
the principles, if not of Probability, of Poetry.

The dactyls of the Virgilian hexameter are, in fact, a very good
example of what has been termed connexité, leading to subnormal
dispersion. The quantities of the successive feet are not independent,
and the appearance of a dactyl in one foot diminishes the probability
of another dactyl in that line. It is like the case of drawing black and
white balls out of an urn, where the balls are not replaced. But Lexis
is wrong if he supposes that a supernormal dispersion cannot also

∗“Über die Wahrscheinlichkeitsrechnung,” p. 444 (see Bibliography).

pt. v

A TREATISE ON PROBABILITY

458
arise out of connexité, or organic connection between the successive
terms. It might have been the case that the appearance of a dactyl
in one foot increased the probability of another dactyl in that line.

He should, I think, have contemplated the result R > r as possibly
indicating a non-typical, organic series, and should not have assumed
that, where R is greater than r, it is of the form
r2 + p2.

In short, Lexis has not pushed his analysis far enough, and he has
not fully comprehended the character of the underlying conditions.

But this does not affect the fact that it was he who made the
vital advance of taking as the unit, not the single observation, but
the frequency in given conditions, and of conceiving the nature of
statistical induction as consisting in the examination, and if possible
the measurement, of the stability of the frequency when the conditions
are varied.

8.

There is one special piece of work illustrative of the above
methods, due to Von Bortkiewicz, which must not be overlooked, and
which it is convenient to introduce in this place—the so-called Law of

Small Numbers.∗

Quetelet, as we have seen in Chapter XXVIII., called attention
to the remarkable regularity of comparatively rare events.

Von

Bortkiewicz has enlarged Quetelet’s catalogue with modern instances
out of the statistical records of bureaucratic Germany. The classic
instance, perhaps, is the number of Prussian cavalrymen killed each
year by the kick of a horse. The table is worth giving as a statistical curiosity. (The period is from 1875 to 1894; G stands for the Corps
of Guards, and I.–XV. for the 15 Army Corps.)

The agreement of this table with the theoretical results of a
random distribution of the total number of casualties is remarkably
close:†

∗There are numerous references to this phenomenon in periodical literature;
but it is sufficient to refer the reader to Von Bortkiewicz’s Das Gesetz der
kleinen Zahlen.

†Bortkiewicz, op. cit. p. 24.

ch. xxxii

STATISTICAL INFERENCE

459

94

1

..

..

..

..

..

..

..

1

..

1

1

..

..

93

..

1

..

..

..

1

..

2

..

..

1

3

..

..

92

1

3

2

..

1

1

3

..

1

1

..

1

1

..

91

..

..

..

1

1

1

..

1

1

..

3

3

1

..

90

1

2

..

2

..

1

1

2

..

2

1

1

2

2

89

..

..

1

1

..

1

1

..

..

1

2

2

..

2

88

..

1

1

..

..

1

1

..

..

..

..

1

1

..

87

1

1

2

1

..

..

3

2

1

1

..

1

2

..

86

2

1

..

..

1

1

1

..

..

1

..

1

3

..

85

..

..

..

..

..

..

1

..

..

2

..

1

..

1

84

3

..

1

..

..

..

..

1

..

..

2

..

1

1

83

..

..

1

2

..

1

2

1

..

1

..

3

..

..

82

1

2

..

..

..

..

1

..

1

1

2

1

4

1

81

1

..

..

2

1

..

..

1

..

1

..

..

..

..

80

..

3

2

1

1

1

..

..

..

2

1

4

3

..

79

..

..

..

1

1

2

2

..

1

..

..

2

1

..

78

1

2

2

1

1

..

..

..

..

..

1

..

1

..

77

2

..

..

..

..

..

1

1

..

..

1

..

2

..

76

2

..

..

..

1

..

..

..

..

..

..

..

1

1

75

..

..

..

..

..

..

..

1

1

..

..

..

1

..

G.

I.

II.

I.

III.

IV.

V.

VI.

II.

X.

VI

IX.

XI.

VI

XIV.

XV.

pt. v

A TREATISE ON PROBABILITY

460

Casualties in a Number of Occasions on which the Annual

Year.

Casualties in a Corps reach the Figure
in Column 1.

Actual.

Theoretical.

0

144

143.1

1

91

92.1

2

32

33.3

3

11

8.9

4

2

2.0

5 and more

..

0.6

Other instances are furnished by the numbers of child suicides in

Prussia, and the like.

It is Von Bortkiewicz’s thesis that these observed regularities have
a good theoretical explanation behind them, which he dignifies with
the name of the Law of Small Numbers.

The reader will recall that, according to the theory of Lexis, his
measure of stability Q is, in the more general case, made up of
two components r and p, combined in the expression
r2 + p2, of
which one is due to fluctuations from the average of the conditions
governing all the members of a series, which furnishes us with one of
our observed frequencies, and of which the other is due to fluctuations
in the individual members of the series about the true norm of the
series. Bortkiewicz carries the same analysis a little further, and
shows that Lexis’s Q is of the form

1 + (n − 1)c2, where n is the
number of times that the event occurs in each series.∗ That is to
say, Q increases with n, and, when n is small, Q is likely to exceed
unity to a less extent than when n is large. To postulate that n is
small, is, when we are dealing with observations drawn from a wide

∗I refer the reader to the original, op. cit. pp. 29–31, for the interpretation
of c (which is a function of the mean square errors arising in the course of the
investigation) and for the mathematical argument by which the above result is
justified.

ch. xxxii

STATISTICAL INFERENCE

461
field, the same thing as to say that the event we are looking for is a
comparatively rare one. This, in brief, is the mathematical basis of
the Law of Small Numbers.

In his latest published work on these topics,∗ Von Bortkiewicz
builds his mathematical structure considerably higher, without,
however, any further underpinning of the logical foundations of it. He
has there worked out further statistical constants, arising out of the
conceptions on which Lexis’s Q is based (the precise bearing of which
is not made any clearer by his calling them coefficients of syndromy),
which are explicitly dependent on the value of n; and he elaborately
compares the theoretical value of the coefficients with the observed
value in certain actual statistical material. He concludes with the
thesis, that Homogeneity and Stability (defined as he defines them)

are opposed conceptions, and that it is not correct to premise, that
the larger statistical mass is as a rule more stable than the smaller,
unless we also assume that the larger mass is less homogeneous.

At this point, it would have helped, if Von Bortkiewicz, excluding
from his vocabulary homogeneity, paradromy, γ , and the like, had

M
stopped to tell in plain language where his mathematics had led him,
and also whence they had started. But like many other students of

Probability he is eccentric, preferring algebra to earth.

9.

Where, then, though an admirer, do I criticise all this? I think
that the argument has proceeded so far from the premisses, that it
has lost sight of them. If the limitations prescribed by the premisses
are kept in mind, I do not contest the mathematical accuracy of the
results. But many technical terms have been introduced, the precise
signification and true limitations of which will be misunderstood if

∗“Homogeneität und Stabilität in der Statistik,” published in the Skandinavisk

Aktuarielidskrift, 1918. Those readers, who look up my references, will, I think,
agree with me that Von Bortkiewicz does not get any less obscure as he goes
on. The mathematical argument is right enough, and often brilliant. But what
it is all really about, what it all really amounts to, and what the premisses are,
it becomes increasingly perplexing to decide.

pt. v

A TREATISE ON PROBABILITY

462
the conclusion of the argument is allowed to detach itself from the
premisses and to stand by itself. I will illustrate what I mean by two
examples from the work of Von Bortkiewicz described above.

Von Bortkiewicz enunciates the seeming paradox that the larger
statistical mass is only, as a rule, more stable if it is less homogeneous.

But an illustration which he himself gives shows how misleading his
aphorism is. The opposition between stability and homogeneity is
borne out, he says, by the judgment of practical men. For actuaries
have always maintained that their results average out better, if their
cases are drawn from a wide field subject to variable conditions of
risk, whilst they are chary of accepting too much insurance drawn
from a single homogeneous area which means a concentration of risk.

But this is really an instance of Von Bortkiewicz’s own distinction
between a general probability p and special probabilities p1 etc., where
z
z
p = 1 p

2

1 +
p2 + . . . .

z
z

If we are basing our calculations on p and do not know p1, p2, etc.,
then these calculations are more likely to be borne out by the result
if the instances are selected by a method which spreads them over
all the groups 1, 2, etc., than if they are selected by a method which
concentrates them on group 1. In other words, the actuary does
not like an undue proportion of his cases to be drawn from a group
which may be subject to a common relevant influence for which
he has not allowed. If the à priori calculations are based on the
average over a field which is not homogeneous in all its parts, greater
stability of result will be obtained if the instances are drawn from
all parts of the non-homogeneous total field, than if they are drawn
now from one homogeneous subfield and now from another. This is
not at all paradoxical. Yet I believe, though with hesitation, that
this is all that Von Bortkiewicz’s elaborately supported mathematical
conclusion really amounts to.

My second example is that of the Law of Small Numbers. Here
also we are presented with an apparent paradox in the statement that
ch. xxxii

STATISTICAL INFERENCE

463
the regularity of occurrence of rare events is more stable than that
of commoner events. Here, I suspect, the paradoxical result is really
latent in the particular measure of stability which has been selected.

If we look back at the figures, which I have quoted above, of Prussian
cavalrymen killed by the kick of a horse, it is evident that a measure
of stability could be chosen according to which exceptional instability
would be displayed by this particular material; for the frequency
varies from 0 to 4 round a mean somewhat less than unity, which is a
very great percentage fluctuation. In fact, the particular measure of
stability which Von Bortkiewicz has adopted from Lexis has about it,
however useful and convenient it may be, especially for mathematical
manipulation, a great deal that is arbitrary and conventional. It
is only one out of a great many possible formulae which might
be employed for the numerical measurement of the conception of
stability, which, quantitatively at least, is not a perfectly precise
one. The so-called Law of Small Numbers is, therefore, little more
than a demonstration that, where rare events are concerned, the

Lexian measure of stability does not lead to satisfactory results. Like
some other formulae which involve a use of Bernoullian methods in
an approximative form, it does not lead to reliable results in all
circumstances, I should add that there is one other element which
may contribute to the total psychological reaction of the reader’s
mind to the Law of Small Numbers, namely, the surprising and
piquant examples which are cited in support of it. It is startling and
even amusing to be told that horses kick cavalrymen with the same
sort of regularity as characterises the rainfall. But our surprise at this
particular example’s fulfilling the Law of Great Numbers has little or
nothing to do with the exceptional stability about which the Law of

Small Numbers purports to concern itself.

CHAPTER XXXIII
outline of a constructive theory

1.

There is a great difference between the proposition. “It is
probable that every instance of this generalisation is true” and the
proposition “It is probable of any instance of this generalisation taken
at random that it is true.” The latter proposition may remain valid,
even if it is certain that some instances of the generalisation are false.

It is more likely than not, for example, that any number will be
divisible either by two or by three, but it is not more likely than not
that all numbers are divisible either by two or by three.

The first type of proposition has been discussed in Part III. under the name of Universal Induction. The latter belongs to Inductive

Correlation or Statistical Induction, an attempt at the logical analysis
of which must be my final task.

2.

What advocates of the Frequency Theory of Probability
wrongly believe to be characteristic of all probabilities, namely, that
they are essentially concerned not with single instances but with series
of instances, is, I think, a true characteristic of statistical induction.

A statistical induction either asserts the probability of an instance
selected at random from a series of propositions, or else it assigns the
probability of the assertion, that the truth frequency of a series of
propositions (i.e. the proportion of true propositions in the series) is
in the neighbourhood of a given value. In either case it is asserting a
characteristic of a series of propositions, rather than of a particular
proposition.

Whilst, therefore, our unit in the case of Universal Induction is a
single instance which satisfies both the condition and the conclusion of
our generalisation, our unit in the case of Statistical Induction is not a
single instance, but a set or series of instances, all of which satisfy the
condition of our generalisation but which satisfy the conclusion only
in a certain proportion of cases. And whilst in Universal Induction we

464
ch. xxxiii

STATISTICAL INFERENCE

465
build up our argument by examining the known positive and negative

Analogy shown in a series of single instances, the corresponding task
in Statistical Induction consists in examining the Analogy shown in a
series of series of instances.

3.

We are presented, in problems of Statistical Induction,
with a set of instances all of which satisfy the conditions of our
generalisation, and a proportion f of which satisfy its conclusion; and
we seek to generalise as to the probable proportion in which further
instances will satisfy the conclusion.

Now it is useless merely to pay attention to the proportion (or
frequency) f discovered in the aggregate of the instances. For any
collection whatever, comprising a definite number of objects, must, if
the objects be classified with reference to the presence or absence of
any specified characteristic whatever, show some definite proportion
or statistical frequency of occurrence; so that a mere knowledge of
what this frequency is can have no appreciable bearing on what the
corresponding frequency will be for some other collection of objects,
or on the probability of finding the characteristic in an object which
does not belong to the original collection. We should be arguing in
the same sort of way as if we were to base a universal induction
as to the concurrence of two characteristics on a single observation
of this concurrence, and without any analysis of the accompanying
circumstances.

Let the reader be clear about this. To argue from the mere fact that
a given event has occurred invariably in a thousand instances under
observation, without any analysis of the circumstances accompanying
the individual instances, that it is likely to occur invariably in future
instances, is a feeble inductive argument, because it takes no account
of the Analogy. Nevertheless an argument of this kind is not entirely
worthless, as we have seen in Part III. But to argue, without analysis of the instances, from the mere fact that a given event has a frequency
of 10 per cent in the thousand instances under observation, or even in
a million instances, that its probability is 1/10 for the next instance,
pt. v

A TREATISE ON PROBABILITY

466
or that it is likely to have a frequency near to 1/10 in a further
set of observations, is a far feebler argument; indeed it is hardly an
argument at all. Yet a good deal of statistical argument is not free
from this reproach;—though persons of common sense often conclude
better than they argue, that is to say, they select for credence, from
amongst arguments similar in form, those in favour of which there is
in fact other evidence tacitly known to them though not explicit in
the premisses as stated.

4.

The analysis of statistical induction is not fundamentally
different from that of universal induction already attempted in Part

III. But it is much more intricate; and I have experienced exceptional difficulty, as the reader may discover for himself in the following
pages, both in clearing up my own mind about it and in expounding
my conclusions precisely and intelligibly. I propose to begin with a
few examples of what commonly impresses us as good arguments in
this field, and also of the attendant circumstances which, if they were
known to exist, might be held to justify such a mode of reasoning;
and, having thus attempted to bring before the reader’s mind the
character of the subject-matter, to proceed to an abstract analysis.

Example One.—Let us investigate the generalisation that the
proportion of male to female births is m.

The fact that the
aggregate statistics for England during the nineteenth century yield
the proportion m would go no way at all towards justifying the
statement that the proportion of male births in Cambridge next year
is likely to approximate to m. Our argument would be no better if
our statistics, instead of relating to England during the nineteenth
century, covered all the descendants of Adam. But if we were able to
break up our aggregate series of instances into a series of sub-series,
classified according to a great variety of principles, as for example
by date, by season, by locality, by the class of the parents, by the
sex of previous children, and so forth, and if the proportion of male
births throughout these sub-series showed a significant stability in
the neighbourhood of m, then indeed we have an argument worth
ch. xxxiii

STATISTICAL INFERENCE

467
something. Otherwise we must either abandon our generalisation,
amplify its conditions, or modify its conclusion.

Example Two.—Let us take a series of objects s all alike in some
specified respect, this resemblance constituting membership of the
class F; let us determine of how many members of the series a certain
property φ is true, the frequency of which is to be the subject of
our generalisation; and if a proportion f of the series s have the
property φ we may say that the series s has a frequency f for the
property φ.

Now if the whole field F has a finite number of constituents,
it must have some determinate frequency p, and if, therefore, we
increase the comprehensiveness of s until eventually it includes the
whole field, f must come in the end to be equal to p. This is obvious
and without interest and not what we mean by the law of great
numbers and the stability of statistical frequency.

Let us now divide up the field F, according to some determinate
principle of division D, into subfields F1, F2, etc.; and let the series s1
be taken from F1, s2 from F2, and so on. Where F1, F2, etc., have
a finite number of constituents, s1, s2 etc., may possibly coincide
with them; if s1, s2 etc., do not coincide with F1, F2, etc., but are
chosen from them, let us suppose that they are chosen according
to some principle of random or unbiassed selection—s1, that is to
say, will be a random sample from F1. Now it may happen that
the frequencies f1, f2, etc., of the series s1, s2, etc., thus selected
cluster round some mean frequency f. If the frequencies show this
characteristic (the measurement and precise determination of which I
am not now considering), then the series of series s1, s2, etc., has a
stable frequency for the classification D. ‘Great numbers’ only come
in because it is difficult to ascertain the existence of stable frequency
unless the series s1, s2, etc., are themselves numerous and unless each
of these comprises numerous individual instances.

Let us then apply a different principle of division D , leading
to series s , s , etc., and to frequencies f , f , etc.; and then again

1

2

1

2
pt. v

A TREATISE ON PROBABILITY

468
a third principle of division D leading to frequencies f , f , etc.;

1

2
and so on, to the full extent that our knowledge of the differences
between the individual instances permits us.

If the frequencies
f1, f2, etc., f , f , etc., f , f , etc., and so on are all stable about f ,

1

2

1

2
we have an inductive ground of some weight for asserting a statistical
generalisation.

Let the field F, for example, comprise all Englishmen in their
sixtieth year, and let the property φ, about the frequency of which
we are generalising, be their death in that year of their age. Now
the field F can be divided into subfields F1, F2, etc., on innumerable
different principles. F1 might represent Englishmen in their sixtieth
year in 1901, F2 in 1902, and so on; or we might classify them
according to the districts in which they live; or according to the
amount of income tax they pay; or according as they are in
workhouses, in hospitals, in asylums, in prisons, or at large. Let
us take the second of these classifications and let the subfields

F1, F2, etc., be constituted by the districts in which they live. If we
take large random selections s1, s2, etc., from F1, F2, etc., respectively,
and find that the frequencies, f1, f2, etc., fluctuate closely round a
mean value f, this can be expressed by the statement that there
is a stable frequency f for death in the sixtieth year in different

English districts. We might also find a similar stability for all the
other classifications. On the other hand, for the third and fourth
classifications we might find no stability at all, and for the first a
greater or less degree of stability than for the second. In the latter
case the form of our statistical generalisation must be modified or the
argument in its favour weakened.

Example Three.—Let us return to the example given in Chapter

XXVII. of the dog which is fed sometimes by scraps at table and so
judges it reasonable to be there. From one year to another, let us
assume, the dog gets scraps on a proportion of days more or less
stable. What sorts of explanation might there be of this? First,
it might be the case that he was fed on the movable feasts of the
ch. xxxiii

STATISTICAL INFERENCE

469

Church; there would be the same number of these in each year, but it
would not be easy for any one who had not the clue to discover any
regularity in the occasions of their individual occurrence. Second, it
might be the case that he was given scraps whenever he looked thin,
and that the scraps were withheld whenever he looked fat, so that if
he was given scraps on one day, this diminished the likelihood of his
getting scraps on the next day, whilst if they were withheld this would
increase the likelihood; the dog’s constitution remaining constant, the
number of days for scraps would tend to fluctuate from year to year
about a stable value. Third, it might be the case that the company
at table varied greatly from day to day, and that some days people
were there of the kind who give dogs scraps and other days not; if
the set of people from whom the company was drawn remained more
or less the same from year to year, and it was a matter of chance (in
the objective sense defined in § 8 of Chapter XXIV. above) which of
them were there from day to day, the proportion of days for scraps
might again show some degree of stability from year to year. Lastly,
a combination between the first and third type of circumstance gives
rise to a variant deserving separate mention. It might be the case
that the dog was only given scraps by his master, that his master
generally went away for Saturday and Sunday, and was at home the
rest of the week unless something happened to the contrary, and
that “chance” causes would sometimes intervene to keep him at home
for the week-end and away in the week; in this case the frequency
of days for scraps would probably fluctuate in the neighbourhood of
five-sevenths. In circumstances of this third type, however, the degree
of stability would probably be less than in circumstances of the first
two types; and in order to get a really stable frequency it might be
necessary to take a longer period than a year as the basis for each
series of observations, or even to take the average for a number of
dogs placed in like circumstances instead of one dog only.

It has been assumed so far that we have an opportunity of
observing what happens on every day of the year. If this is not the
pt. v

A TREATISE ON PROBABILITY

470
case and we have knowledge only of a random sample from the days
of each year, then the stability, though it will be less in degree, may
be nevertheless observable, and will increase as the number of days
included in each sample is increased. This applies equally to each of
the three types.

5.

What is the correct logical analysis of this sort of reasoning?

If an inductive generalisation is a true one, the conclusion which
it asserts about the instance under inquiry is, so far as it goes,
definite and final, and cannot be modified by the acquisition of more
detailed knowledge about the particular instance. But a statistical
induction, when applied to a particular instance, is not like this;
for the acquisition of further knowledge might render the statistical
induction, though not in itself less probable than before, inapplicable
to that particular instance.

This is due to the fact that a statistical induction is not really
about the particular instance at all, but has its subject, about
which it generalises, a series; and it is only applicable to the
particular instance, in so far as the instance is relative to our
knowledge, a random member of the series. If the acquisition of
new knowledge affords us additional relevant information about the
particular instance, so that it ceases to be a random member of the
series, then the statistical induction ceases to be applicable; but the
statistical induction does not for that reason become any less probable
than it was—it is simply no longer indicated by our data as being the
statistical generalisation appropriate to the instance under inquiry.

The point is illustrated by the familiar example that the probability
of an unknown individual posting a letter unaddressed can be based
on the statistics of the Post Office, but my expectation that I shall
act thus, cannot be so determined.

Thus a statistical generalisation is always of the form: ‘The
probability, that an instance taken at random from the series S will
have the characteristic φ, is p;’ or, more precisely, if a is a random
member of S(x), the probability of φ(a) is p.

ch. xxxiii

STATISTICAL INFERENCE

471

It will be convenient to recapitulate from Chapter XXIV. § 11
the definition of ‘an instance taken at random’: Let φ(x) stand for

‘x has the characteristic φ,’ and S(x) for ‘x is a member of the
class S’; then, on evidence h, a is a random member of the class S
for characteristic φ, if ‘x is a’ is irrelevant to φ(x)/S(x) h,∗ i.e. if we
have no information about a relevant to φ(a) except S(a).

Or alternatively we might express our definition as follows:

Consider a particular instance a, where the object of our inquiry is
the probability of φ(a) relative to evidence h. Let us discard that
part of our knowledge h(a) which is irrelevant to φ(a), leaving us
with relevant knowledge h (a). Let the class of instances a1, a2, etc.,
which satisfy h (x) be designated by S. Then, relative to evidence h,
a is a random member of the class or series S for the characteristic φ.

Let us denote the proposition ‘x is, on evidence h, a random
member of S for characteristic φ’ by R(x, S, φ, h); then our statistical
generalisation is of the form φ(x)/R(x, S, φ, h) h = p.

If R(a, S, φ, h) holds, then, on evidence h, S is the appropriate
statistical series to which to refer a for the purposes of the
characteristic φ.

It is not always the case that the evidence indicates any series at
all as ‘appropriate’ in the above sense. In particular, if evidence h
indicates S as the appropriate series, and evidence h indicates S as
the appropriate series, then relative to evidence hh (assuming these
to be not incompatible), it may be the case that no determinate series
is indicated as appropriate. In this case the method of statistical
induction fails us as a means of determining the probability under
inquiry.

6.

We can now remove our attention from the individual
instance a to the properties of the series S. What sort of evidence is

∗The use of variables in probability, as has been pointed out on p. 63, is very dangerous.

It might therefore be better to enunciate the above: a is a
random member of S for characteristic φ, if φ(a)/S(a) h = φ(b)/S(b) h where

S(b) h contains no information about b, except that b is a member of S.

pt. v

A TREATISE ON PROBABILITY

472
capable of justifying the conclusion that p is the probability that a
random member of the series S will have the characteristic φ?

In the simplest case, S is a finite series of which we know the
truth frequency for the characteristic φ, namely f.∗ Then by a
straightforward application of the Principle of Indifference we have
p = f , so that φ(x)/R(x, S, φ, h) h = f .

In another important type S is a series, with an indefinite number
of members which, however, group themselves in such a way that for
every member of which φ(x) is true, there corresponds a determinate
number of members of which φ(x) is false. The series, that is to say,
contains an indefinite number of atoms, but each atom is made up of
a set of molecules of which φ(x) is true and false respectively in fixed
and determinate proportions. If this determinate proportion is known
to be f, we have, as before, p = f. The typical instance of this type
is afforded by games of chance. Every possible state of affairs which
might lead to a divergence in one direction is balanced by another
probability leading in the opposite direction; and these alternative
possibilities are of a kind to which the Principle of Indifference is
applicable. Thus for every poise of the dice box which leads to the
fall of the six-face, there is a corresponding poise which leads to the
fall of each of the other faces; so that if S is the series of possible
poises, we may equate p to 1 where φ is the fall of the six-face. It is

6
not necessary, in order to obtain this result, to assert that S is a finite
series with an actual determinate frequency f for the fall of each face.

So far no inductive element enters in. But in general we do
not know the constitution of S for certain, and can only infer it
inductively from its resemblance to other series of which we know
the constitution. This presents a normal inductive problem—the
determination by an analysis of the positive and negative analogies
as to whether the respects in which S differs or may differ from the
other series is or is not relevant in the particular context φ; and it

∗I.e. if f is the proportion of the members of the series for which φ(x) is
true.

ch. xxxiii

STATISTICAL INFERENCE

473
involves the same sort of considerations as those discussed in Part III.

There is, however, a further difficulty to be introduced before we
have reached the typical statistical problem. In the case now to
be considered our actual data do not consist of positive knowledge
of the constitutions either of S itself or of other series more or
less resembling S, but only of the frequency of the characteristic in
actually observed sets of selections, great or small, either from S itself
or from other series more or less resembling S.

Thus in the most general case our inquiry falls into two parts.

We are given the observed frequency in statistical sets selected from

S1, S2, etc., respectively. The first part of our inquiry is the problem of
arguing from these observed frequencies to the probable constitutions
of S1, S2, etc., i.e. of determining the values of φ(x)/R(x, S1, φ, h) h,
etc.; we may call this part the statistical problem. The second part of
our inquiry is the problem of arguing from the probable constitutions
of S1, S2, etc., to the probable constitution of S, where S, S1, S2
resemble one another more or less, and we have to determine whether
the differences are or are not relevant to our inquiry; we may call this
part the inductive problem.

Now if the observed statistical sets are made up of random
instances of S1, S2, etc., we can argue in certain conditions from
the observed frequencies to the probable constitutions of the series,
out of which the random selections have been made, by an inverse
application of Bernoulli’s Theorem on the lines explained in Chapter

XXXI. Moreover, if the series S1, S2, etc., are finite series and
the observed selections cover a great part of their members, we
can reach an at least approximate conclusion without raising all
the theoretical difficulties or satisfying all the conditions of Chapter

XXXI. The commonly received opinions as to the bearing of the
observed frequencies in a random sample on the constitution of the
universe out of which the sample is drawn, though generally stated
too precisely and without sufficient insistence on the assumptions they
involve, our actual evidence not warranting in general more than an
pt. v

A TREATISE ON PROBABILITY

474
approximate result, are not, I think, fundamentally erroneous. The
most usual error in modern method consists in treating too lightly
what I have termed above the inductive problem, i.e. the problem
of passing from the series S1, S2, etc., of which we have observed
samples, to the series S of which we have not observed samples.

Let us, then, assume that we have ascertained p1, p2, etc.,
with more or less exactness, by examining either all the instances
of the series S1, S2, etc., or random selections from them, i.e.

φ(x)/R(x, S1, φ, h) h = p1, etc. This can be expressed for short by
saying that the series S1, S2, etc., are subject to probable-frequencies
p1, p2, etc., for the characteristic φ. Our problem is to infer from
this the probable-frequency p of the unexamined series S. The class
characteristics of the series S1, S2, etc., will be partly the same and
partly different. Using the terminology of Part III. we may term the class characteristics which are common to all of them the Positive

Analogy, and the class characteristics which are not common to all of
them the Negative Analogy.

Now, if the observed or inferred probable-frequencies of the series

S1, S2, are to form the basis of a statistical induction, they must show
a stable value; that is to say, either we must have p1 = p2 = etc.,
or at least p1, p2, etc., must be stably grouped about their mean
value. Our next task, therefore, must be to discover whether the
probable-frequencies p1, p2, etc., display a significant stability. It is
the great merit of Lexis that he was the first to investigate the
problem of stability and to attempt its measurement. For, until a
primâ facie case has been established for the existence of a stable
probable-frequency, we have but a flimsy basis for any statistical
induction at all; indeed we are limited to the class of case where the
instance under inquiry is a member of identically the same series as
that from which our samples were drawn, i.e. where S = S1, which in
social and scientific inquiries is seldom the case.

What is the meaning of the assertion that p1, p2, etc., are stably
grouped about their mean value? The answer is not simple and
ch. xxxiii

STATISTICAL INFERENCE

475
not perfectly precise.

We could propound various formulae for
the measurement of stability and dispersion, respectively, and the
problem of translating the conception of stability, which is not
quantitatively precise, into a numerical formula involves an arbitrary
or approximative element. For practical purposes, however, I doubt
if it is possible to improve on Lexis’s measure of stability Q, the
mathematical definition of which has been given above on p. 455.

Lexis describes the stability as subnormal, normal, or supernormal
according as Q is less than, equal to, or greater than 1. This is too
precise, and it is better perhaps to say that the stability about the
mean is normal if the dispersion is such as would not be improbable
à priori, if we had assumed that the members of S1, S2, etc., were
obtained by random selection out of a single universe U, that it is
subnormal if the dispersion is less than one would have expected on
the same hypothesis, and that it is supernormal if the dispersion is
greater than one would have expected.

Let us suppose that we find that on this definition p1, p2, etc.,
are stable about p, and let us postpone consideration of the cases of
subnormal or supernormal dispersion. This is equivalent to saying
that the frequencies of S1, S2, etc., are within limits which we should
expect à priori, if we had knowledge relative to which their members
were chosen at random from a universe U of which the frequency
was p for the characteristic under inquiry. We next seek to extend
this result to the unexamined series S and to justify anticipations
about it on the basis of the members of S also being chosen at
random from the universe U. This leads us to the strictly inductive
part of our inquiry.

The class characteristics of the several series S1, S2, etc., will
be partly the same and partly different, those that are the same
constituting the positive analogy and those that are different
constituting the negative analogy, as stated above. The series S will
share part of the positive analogy. The argument for assimilating
the properties of S, in relation to the characteristic under inquiry, to
pt. v

A TREATISE ON PROBABILITY

476
the properties of S1, S2, etc., in relation to this characteristic depends
on the differences between S, S1, S2, etc., being irrelevant in this
particular connection. The method of strengthening this argument
seems to me to be the same as the general inductive method discussed
in Part III. and to present the same, but not greater, difficulties.

In general this inductive part of our inquiry will be best advanced
by classifying the aggregate series of instances with which we are
presented in such a way as to analyse most clearly the significant
positive and negative analogies, to group them, that is to say, into
sub-series S1, S2, etc., which show the most marked and definite
class characteristics. Our knowledge of the differences between the
particular observed instances which constitute our original data will
suggest to us one or more principles of classification, such that the
members of each sub-series all have in common some set of positive
or negative characteristics, not all of which are shared in common by
all the members of any of the other sub-series. That is to say, we
classify our whole set of instances into a series of series S1, S2, etc.,
which have frequencies f1, f2, etc., for the characteristic under inquiry;
and then again we classify them by another principle or criterion of
classification into a second series of series S , S , etc., with frequencies

1

2
f , f , etc.; and so on, so far as our knowledge of the possible relevant

1

2
differences between the instances extends; the whole result being then
summed up in a statement of the positive and negative analogies of
the series of series. If we then find that all the frequencies f1, f2, etc.,
f , f
etc., are stable about a value p, and if, on the basis of the

1

2
above positive and negative analogies, we have a normal inductive
argument for assimilating the unexamined series S to the examined
series S1, S2, etc., S , S , etc., in respect of the characteristic under

1

2
inquiry, in this case we have, not conclusive grounds, but grounds of
some weight for asserting the probability p, that an instance taken at
random from S will have the characteristic in question.

Let me recapitulate the two essential stages of the argument.

We first find that the observed frequencies in a set of series are
ch. xxxiii

STATISTICAL INFERENCE

477
such as would have been not improbable à priori if, relative to our
knowledge, these series had all been made up of random members
of the same universe U; and we next argue that the positive and
negative analogies of this set of series furnish an inductive argument
of some weight for supposing that a further unexamined series S
resembles the former series in having a frequency for the characteristic
under inquiry such as would have been not improbable à priori if,
relative to our knowledge, S was also made up of random members of
the hypothetical universe U.

7.

It is very perplexing to decide how far an argument of this
character involves any new and theoretically distinct difficulties or
assumptions, beyond those already admitted as inherent in Universal

Induction. I believe that the foregoing analysis is along the right lines
and that it carries the inquiry a good deal further than it has been
carried hitherto. But it is not conclusive, and I must leave to others
its more exact elucidation.

There is, however, a little more to be said about the half-felt
reasons which, in my judgment, recommend to common sense some at
least of the scientific (or semi-scientific) arguments which run along
the above lines. In expressing these reasons I shall be content to use
language which is not always as precise as it ought to be.

I gave in Chapter XXIV. §§ 7–9 an interpretation of what is
meant by an ‘objectively chance’ occurrence, in the sense in which
the results of a game, such as roulette, may be said to be governed
by ‘objective chance.’ This interpretation was as follows: “An event
is due to objective chance if in order to predict it, or to prefer
it to alternatives, at present equiprobable, with any high degree of
probability, it would be necessary to know a great many more facts
of existence about it than we actually do know, and if the addition of
a wide knowledge of general principles would be little use.” The ideal
instance of this is the game of chance; but there are other examples
afforded by science in which these conditions are fulfilled with more
or less perfection. Now the field of statistical induction is the class of
pt. v

A TREATISE ON PROBABILITY

478
phenomena which are due to the combination of two sets of influences,
one of them constant and the other liable to vary in accordance with
the expectations of objective chance,—Quetelet’s ‘permanent causes’

modified by ‘accidental causes.’ In social and physical statistics
the ultimate alternatives are not as a rule so perfectly fixed, nor
the selection from them so purely random, as in the ideal game of
chance. But where, for example, we find stability in the statistics
of crime, we could explain this by supposing that the population
itself is stably constituted, that persons of different temperaments are
alive in proportions more or less the same from year to year, that
the motives for crime are similar, and that those who come to be
influenced by these motives are selected from the population at large
in the same kind of way. Thus we have stable causes at work leading
to the several alternatives in fixed proportions, and these are modified
by random influences. Generally speaking, for large classes of social
statistics we have a more or less stable population including different
kinds of persons in certain proportions and on the other hand sets of
environments; the proportions of the different kinds of persons, the
proportions of the different kinds of environments, and the manner of
allotting the environments to the persons vary in a random manner
from year to year (or, it may be, from district to district). In all such
cases as these, however, prediction beyond what has been observed is
clearly open to sources of error which can be neglected in considering,
for example, games of chance;—our so-called ‘permanent’ causes are
always changing a little and are liable at any moment to radical
alteration.

Thus the more closely that we find the conditions in scientific
examples assimilated to those in games of chance, the more confidently
does common sense recommend this method. The rather surprising
frequency with which we find apparent stability in human statistics
may possibly be explained, therefore, if the biological theory of

Mendelism can be established. According to this theory the qualities
apparent in any generation of a given race appear in proportions
ch. xxxiii

STATISTICAL INFERENCE

479
which are determined by methods very closely analogous to those of
a game of chance. To take a specific example (I am giving not the
correct theory of sex but an artificially simplified form of it), suppose
there are two kinds of spermatozoa and two kinds of ova and of
the four possible kinds of union two produce males and two females,
then if the kinds of spermatozoa and ova exist in equal numbers and
their union is determined by random considerations in precisely the
same sense in which a game of chance such as roulette depends upon
random considerations, we should expect the observed proportions
to vary from equality, as indeed they do, in the same manner as
variations from equality of red and black occur at roulette.∗ If
the sphere of influence of Mendelian considerations is wide, we have
both an explanation in part of what we observe and also a large
opportunity in future of using with profit the methods of statistical
analysis.

This is all familiar. This is the way in which in fact we do think
and argue. The inquiry as to how far it is covered by the abstract
analysis of the preceding paragraphs, and by what logical principle
the use of this analysis can be justified as rational, I have pushed as
far as I can. It deserves a profounder study than logicians have given
it in the past.

8.

Two subsidiary questions remain to be mentioned. The first
of these relates to the character of series which, in the terminology of

Lexis, show a subnormal or supernormal stability; for I have pressed
on to the conclusion of the argument on the assumption that the
stabilities are normal. Subnormal stability conceals two types: the
one in which there is really no stability at all and the results are
in fact chaotic; and the other in which there is mutual dependence

∗The fluctuations in the proportion of the sexes which, as is well known, is
not in fact one of equality, correspond, as Lexis has shown, to what one would
expect in a game of chance with an astonishing exactitude. But it is difficult
to find any other example, amongst natural or social phenomena, in which his
criteria of stability are by any means as equally well satisfied.

pt. v

A TREATISE ON PROBABILITY

480
between the successive instances of such a kind that they tend to
resemble one another so that any divergence from the normal tends
to accentuate itself. Super-*normal stability corresponds in the other
direction to the second of these two types; that is to say, there
is mutual dependence of a regulative kind between the successive
instances which tends to prevent the frequency from swinging away
from its mean value. The case, where the dog was fed with scraps
when he looked thin and not fed when he looked fat, illustrated
this. The typical example of this type is where balls are drawn from
urns, containing black and white balls in certain proportions and not
replaced; so that every time a black ball is drawn the next ball is
more likely than before to be white, and there is a tendency to redress
any excess of either colour beyond the proper proportions. Possibly
the aggregate annual rainfall may afford a further illustration.

Where there is no stability at all and the frequencies are chaotic,
the resulting series can be described as ‘non-statistical.’ Amongst

‘statistical series,’ we may term ‘independent series’ those of which
the instances are independent and the stability normal, and ‘organic
series,’ those of which the instances are mutually dependent and the
stability abnormal, whether in excess or in defect. ‘Organic series’

have been incidentally discussed elsewhere in this volume. I shall
not pursue them further now, because I do not think that they
introduce any new theoretical difficulty into the general problem of
statistical inference; although the problem of fitting them into the
ch. xxxiii

STATISTICAL INFERENCE

481
general theoretical scheme is not easy.∗

9.

The second question is concerned with the relation between
the Inductive Correlation, which has been the subject-matter of this
chapter, and the Correlation Coefficient or, as I should prefer to call
it, the Quantitative Correlation, with which recent English statistical
theory has chiefly occupied itself. I do not propose to discuss this
theory in detail, because I suspect that it is much more concerned,
at any rate in its present form, with statistical description than with
statistical induction. The transition from defining the ‘correlation
coefficient’ as an algebraical expression to its employment for purposes
of inference is very far from clear even in the work of the best
and most systematic writers on the subject, such as Mr. Yule and

Professor Bowley.

∗The following more precise definitions bring these ideas into line with what
has gone before: consider the terms a1, a2, . . . an of a series s(x); let ‘ar is g’

≡ gr and let gr/h = pr, where h is our data. Then, if gr/gs . . . gt . . . h = pr
for all values of r, s, . . . , t . . ., the terms of the series are independent relative to h.

If p1 = p2 = . . . = p the terms are uniform.

If the terms are both
independent and uniform, the series may be called an independent Bernoullian
series, subject to a Bernoullian probability p. If the terms are independent but
not uniform, the series may be called an independent compound series, subject
to a compounded probability 1/n
pr. If the terms are not independent, the
series is an organic series.

The same terminology can then be applied to the series S1, S2, . . . Sn, regarded
as members of the series of series S(x).

Let the frequencies of the series
for the characteristic under inquiry be x1, x2, . . . xn, and let x1/h = θ1(x1),
i.e. θ1(x1) is the probability of a frequency x1 in the first series.

Then if
xr/xs . . . h = θr(xr) for all values of r, s, etc., the frequencies are independent;
and if θ1(x) = θ2(x2) = . . . θ(x), the frequencies are stable. If the frequencies
are stable and independent, the series of series may be called Gaussian. If the
frequencies are stable and independent, and if in addition each individual series
is subject to a Bernoullian probability, the probable dispersion of the frequency
is normal and symmetrical. If the individual series are organic, the dispersion
of the frequencies may be normal, subnormal, or supernormal. If the series of
series is Gaussian, and the individual series Bernoullian, we have the type of
the perfect statistical series.

pt. v

A TREATISE ON PROBABILITY

482

In the notation employed in the earlier part of this chapter

I have classified each examined instance a according as it did or
did not possess the characteristic φ, i.e. satisfy the propositional
function φ(x), or, in other words, according as φ(a) was true or false.

Thus only two possible alternatives were contemplated, and φ was not
considered as a quantitative characteristic which the instance could
satisfy in greater or less degree. Equally the common element in all
the instances, required to constitute them as instances for the purpose
of our statistical generalisation (or, as I have sometimes put it,
required to satisfy the condition of the generalisation), was regarded
as definite and unique and not capable of quantitative variation. That
is to say, all the instances satisfied a function ψ(x), and the question
was, what proportion of them also satisfied the function φ(x). A
typical example was that of sex-ratio,—ψ(x) being the birth of a
child and φ(x) its sex, where there is no question of degree in either
ψ(x) or φ(x).

It might be the case, however; that the characteristics under
examination were capable of degree or quantitative variation; for
example ψ(x) might be the age of the mother and φ(x) the
weight of the child at birth, in this case we should have a series
ψ1(x), ψ2(x), etc., corresponding to the various age-periods of the
mothers, and a series φ1(x), φ2(x), etc., corresponding to the various
weights of the children. Now if we concentrated our attention on
ψ1(x) and φ1(x) alone, i.e. on mothers of a particular age and the
proportions of their children which had a particular weight at birth,
we have a one-dimensional problem of the same kind as before; out
of all the instances which satisfy ψ1(x) a certain proportion satisfy
ψ1(x) also. But clearly we can push our observations further and we
can take note what proportion of the instances which satisfy ψ1(x)

satisfy φ2(x), φ3(x), and so on, respectively; and then we can do the
same as regards the instances which satisfy ψ2(x), ψ3(x), etc. The
total results of this two-dimensional set of observations can then be
tabulated in what is called a twofold correlation table. Thus if frs is
ch. xxxiii

STATISTICAL INFERENCE

483
the proportion of instances satisfying ψs(x) which also satisfy φr(x)

we have a table as follows:

ψ1(x)

ψ2(x)

ψ3(x)

. . .

φ1(x)

f11
f12
f13
φ2(x)

f21
f22
f23
φ3(x)

f31
f32
f33

..

.

.

.

.

..

..

..

We could, further, increase the complexity and completeness of
our observations to any required degree. For example we might
take account also of θ(x), the age of the father, and construct a
threefold table where frst is the proportion of instances satisfying
φr(x), ψs(x), θt(x); and so on up to an n-fold table.

Clearly it is not necessary for the construction of tables of this kind
that φ(x) and ψ(x) should stand for degrees of the same quantitative
characteristic; they might be any set of exclusive alternatives; for
example, ψ(x) might be the colour of the baby’s eyes, and φ(x) its

Christian name.

But in order that the correlation table may be of any practical
interest for the purposes of inference, it is necessary—and this,

I think, is one of the critical assumptions of correlation—that
ψ1(x), ψ2(x), . . . and also φ1(x), φ2(x), . . . should be arranged in an
order that is significant, i.e. such that we have some à priori reason
for expecting some connection to exist between the order of the ψ’s
and the order of the φ’s. The point of this will be illustrated
by concentrating our attention on the simplest type of case where
ψ(x) and φ(x) are quantitative characteristics arranged in order
of magnitude. Now suppose it were the case that the younger
mothers tended to bear heavier babies, then, if ψ1(x) ψ2(x) are the
ages increasing upwards and φ1(x) φ2(x) the weights diminishing
pt. v

A TREATISE ON PROBABILITY

484
downwards, f11 would probably be the greatest of the fr1’s and,
generally speaking, fr1 would be greater than fr+1,1; also f22 might be
the greatest of the fr2’s, and so on; so that the frequencies lying on
the diagonal of the table would be the greatest and the frequencies
would tend to be less the farther they lay from the diagonal. If we
had some reason à priori (i.e. based on our pre-existing knowledge), if
only a slight one, for supposing that there might be some connection
between the age of the mother and the weight of the baby, then, if in
a particular set of instances the frequencies were grouped about the
diagonal as suggested above, this might be taken as affording some
inductive support for the hypothesis.

Now the theory of correlation, as it is expounded in the text-books,
is almost entirely concerned with measuring how nearly the observed
frequencies are grouped about the diagonal of the table (though
the complete theory is not, of course, so restricted as this). The

‘coefficient of correlation’ is an algebraical formula which may be
regarded as measuring this phenomenon in a way that is sufficiently
satisfactory for all ordinary purposes. If it is defined thus, it is simply
a statistical description of a particular set of observations arranged in
a particular order. How can we make use of this coefficient for the
purposes of inference?

Dr. Bowley faces this problem a little more definitely than do
most statistical writers.

Mr. Yule warns the student that the
problem exists,∗ but he does not himself attack it systematically or
do more than apply common sense to particular problems. So much
greater emphasis, however, has been laid hitherto on the mathematical
complications, that many statistical students hazily float from defining
the correlation coefficient as a statistical description to employing

∗Introduction to the Theory of Statistics, p. 191:

“The coefficient of
correlation, like an average or a measure of dispersion, only exhibits in a
summary and comprehensible form one particular aspect of the facts on which
it is based, and the real difficulties arise in the interpretation of the coefficient
when obtained.”

ch. xxxiii

STATISTICAL INFERENCE

485
it as a measure of the probability of a statistical generalisation as
to the association between quantitative variations of φ(x) and ψ(x)

respectively. If, for example, it is found in a particular set of
observations of mothers’ ages and babies’ weights that the frequencies
are closely ranged about the diagonal, this is considered a sufficiently
good reason for attributing probability to a generalisation as to the

‘correlation’ (i.e. tendency to quantitative correspondence) between
the age of the mother and the weight of the baby.

Dr. Bowley’s line of thought is as follows. He begins by defining the
correlation coefficient r merely as a statistical description (Elements
of Statistics, p. 354). He then shows (p. 355), as an illustration of the
nature of r, that if x and y are two variable quantities which depend

(more strictly, are known to depend) on other variables U, V, W in
such a way that

Xt = 1Ut + 2Ut + . . . + pUt + 1Vt + 2Vt + . . . + qVt

Yt = 1Ut + 2Ut + . . . + pUt + 1Wt + 2Wt + . . . + qWt
where 1Ut, 2Ut . . . 1Vt, 2Vt . . . 1Wt, 2Wt . . . are selected at random
each from an independent group of quantities (more strictly, are
relative to our data, random members of independent groups); then,
if we know à priori certain statistical coefficients descriptive of the
constitution of these groups, the value of r will probably tend towards
a certain value. So far we are on fairly safe, but not very fruitful,
ground. We have no basis for arguing backwards from the observed
value of r; but, provided we have rather extensive and peculiar
knowledge à priori as to how Xt and Yt are constituted, then we have
calculable expectations as to the limits within which the value of r,
namely the correlation coefficient between X and Y, will probably
turn out to lie, when we have observed it.

Dr. Bowley’s next move is more dubious. If the constitutions of the
independent groups are similar in a certain, statistical respect (i.e. if
they have the same standard deviations), then, Dr. Bowley concludes,
pt. v

A TREATISE ON PROBABILITY

486
p
r =

, which “expressed in words shows that the correlation

(p + q)

coefficient tends to be the ratio of the number of causes common
in the genesis of two variables to the whole number of independent
causes on which each depends.” By this time the student’s mind,
unless anchored by a more than ordinary scepticism, will have been
well launched into a vague, fallacious sea.

Neglecting, however, the dictum just quoted, we find that the
second stage of the argument consists in showing that, if we have
a certain sort of knowledge à priori as to how our variables are
constituted, then the various possible values for the coefficients of
correlation, which would be yielded by actual sets of observations
made in prescribed conditions, will have, à priori, and before the
observations have been made, calculable probabilities, certain ranges
of values being probable and others improbable.

As a rule, however, we are not arguing from knowledge about
the variables to anticipations about their correlation coefficient; but
the other way round, that is from observations of their correlation
coefficients to theories about the nature of the variables. Dr. Bowley
perceives that this involves a third stage of the argument, and appeals
accordingly (p. 409) to “the difficult and elusive theory of inverse
probability.” He apprehends the difficulty but he does not pursue it;
and, like Mr. Yule, he really falls back for practical purposes on the
criteria of common sense, an expedient well enough in his case, but
not a universal safeguard.

The general argument from inverse probability to which Dr. Bowley
makes his vague appeal is doubtless on the following lines: If there
is no causal connection between the two sets of quantities, then
a close grouping of the frequencies about the diagonal would be
à priori improbable (and the greater the number of the individual
observations, the greater the improbability since, if the quantities are
independent, there is, then, all the more opportunity for ‘averaging
out’); therefore, inversely, if the frequencies do group themselves
ch. xxxiii

STATISTICAL INFERENCE

487
about the diagonal, we have a presumption in favour of a causal
connection between the two sets of quantities.

But if the reader recalls our discussion of the principle of inverse
probability, he will remember that this conclusion cannot be reached
unless à priori, and quite apart from the observations in question,
we have some reason for thinking that there may be such a causal
connection between the quantities. The argument can only strengthen
a pre-existing presumption; it cannot create one. And in the absence
of reasons peculiar to the particular inquiry, we have no choice but
to fall back on the general methods and the general presumptions of
induction.

It is apparent that, where the correlation argument seems plausible,
some tacit assumption must have slipped in, if we return to the
case where our correlation table relates to the weights of the babies
and their Christian names. Either by accident or because we had
arranged the order of the Christian names to suit, it might happen
with a particular set of observations, even a fairly numerous set, that
the correlation coefficient was large. Yet on that evidence alone we
should hardly assert a generalisation connecting the weights of babies
with their Christian names.

The truth is that sensible investigators only employ the correlation
coefficient to test or confirm conclusions at which they have arrived
on other grounds. But that does not validate the crude way in which
the argument is sometimes presented, or prevent it from misleading
the unwary,—since not all investigators are sensible.

If we abandon the method of inverse probability in favour of the
less precise but better founded processes of induction, ‘quantitative
correlation,’ as I should like to term this particular branch of
statistical induction, is more complicated than, but not theoretically
distinct from, the kind of arguments which have occupied the
earlier paragraphs of this chapter. The character of the additional
complication can be described by saying that we are presented with a
two-dimensional problem instead of a one-dimensional problem. The
pt. v

A TREATISE ON PROBABILITY

488
mere existence of a particular correlation coefficient as descriptive
of a group of observations, even of a large group, is not in itself a
more conclusive or significant argument than the mere existence of
a particular frequency coefficient would be. Of course if we have a
considerable body of pre-existing knowledge relevant to the particular
inquiry, the calculation of a small number of correlation coefficients
may be crucial. But otherwise we must proceed as in the case of
frequency coefficients; that is to say we must have before us, in order
to found a satisfactory argument, many sets of observations, of which
the correlation coefficients display a significant stability in the midst
of variation in the non-essential class characteristics (i.e. those class
characteristics which our generalisation proposes to neglect) of the
different sets of observations.

10.

I am now at the conclusion of an inquiry in which,
beginning with fundamental questions of logic, I have endeavoured to
push forward to the analysis of some of the actual arguments which
impress us as rational in the progress of knowledge and the practice
of empirical science. In writing a book of this kind the author must,
if he is to put his point of view clearly, pretend sometimes to a little
more conviction than he feels. He must give his own argument a
chance, so to speak, nor be too ready to depress its vitality with a
wet cloud of doubt. It is a heavy task to write on these problems;
and the reader will perhaps excuse me if I have sometimes pressed on
a little faster than the difficulties were overcome, and with decidedly
more confidence than I have always felt.

In laying the foundations of the subject of Probability, I have
departed a good deal from the conception of it which governed
the minds of Laplace and Quetelet and has dominated through
their influence the thought of the past century,—though I believe
that Leibniz and Hume might have read what I have written with
sympathy. But in taking leave of Probability, I should like to say
that, in my judgment, the practical usefulness of those modes of
inference, here termed Universal and Statistical Induction, on the
ch. xxxiii

STATISTICAL INFERENCE

489
validity of which the boasted knowledge of modern science depends,
can only exist—and I do not now pause to inquire again whether such
an argument must be circular—if the universe of phenomena does
in fact present those peculiar characteristics of atomism and limited
variety which appear more and more clearly as the ultimate result to
which material science is tending:

fateare necessest
materiem quoque finitis differre figuris.

The physicists of the nineteenth century have reduced matter to the
collisions and arrangements of particles, between which the ultimate
qualitative differences are very few; and the Mendelian biologists
are deriving the various qualities of men from the collisions and
arrangements of chromosomes. In both cases the analogy with the
perfect game of chance is really present; and the validity of some
current modes of inference may depend on the assumption that it is to
material of this kind that we are applying them. Here, though I have
complained sometimes at their want of logic, I am in fundamental
sympathy with the deep underlying conceptions of the statistical
theory of the day. If the contemporary doctrines of Biology and

Physics remain tenable, we may have a remarkable, if undeserved,
justification of some of the methods of the traditional Calculus of

Probabilities. Professors of probability have been often and justly
derided for arguing as if nature were an urn containing black and
white balls in fixed proportions. Quetelet once declared in so many
words—“l’urne que nous interrogeons, c’est la nature.” But again in
the history of science the methods of astrology may prove useful to
the astronomer; and it may turn out to be true—reversing Quetelet’s
expression—that “La nature que nous interrogeons, c’est une urne.”

BIBLIOGRAPHY

490

BIBLIOGRAPHY

INTRODUCTION

There is no opinion, however absurd or incredible, which has not been
maintained by some one of our philosophers.—Descartes.

The following Bibliography does not pretend to be complete,
but it contains a much longer list of what has been written about

Probability than can be found elsewhere. I have hesitated a little
before burdening this volume with the titles of many works, so few of
which are still valuable. But I was myself much hampered, when first

I embarked on the study of this subject, by the absence of guide-posts
to the scattered but extensive literature of the subject; and a list
which I drew up for my own convenience, without much attention to
bibliographical nicety or to exact uniformity in the style of entry, may
be useful to others.

It is rather an arbitrary matter to decide what to include and
what to exclude. Probability overlaps many other topics, and some
of the most important references to it are to be found in books, the
main topic of which is something else. On the other hand it would be
absurd to include every casual reference; and no useful purpose would
have been served by cataloguing the very numerous volumes dealing
with Insurance, Games of Chance, Statistics, Errors of Observation,
and Least Squares, which treat in detail these various applications of
the Theory of Probability. It has been a matter of some difficulty,
therefore, to know precisely where to draw the line. Where the main
subject of a book or paper is Probability proper, I have included it,
nearly regardless of my own view as to its importance, and have not
attempted to act as censor; but where Probability is not the main
subject or where an application of Probability is concerned, the chief
interest of which is solely in the application itself, I have only included
the entry where I think it important, intrinsically or historically or
from the celebrity of the author. In particular, the existence of

491

A TREATISE ON PROBABILITY

492

Professor Mansfield Merriman’s very extensive bibliography, published
in the Transactions of the Connecticut Academy for 1877, has made it
possible to deal very lightly (and to the extent of but few entries) with
the inordinately large literature of Least Squares. This list comprises

408 titles of writings relating to the Method of Least Squares and
the theory of accidental errors of observation, and is sufficiently
exhaustive so far as relates to memoirs on this topic published before

1877.

Of bibliographical sources for Probability proper, Todhunter’s

History of the Mathematical Theory of Probability and Laurent’s

Calcul des probabilités are alone important. Of mathematical works
published before the time of Laplace, Todhunter’s list, and also his
commentary and analysis, are complete and exact,—a work of true
learning, beyond criticism. The bibliographical catalogue at the
conclusion of Laurent’s Calcul (published in 1873) is the longest list
published hitherto of general works on Probability. But it is unduly
swollen by the inclusion of numerous items on Insurance and Errors
of Observation, the bearing of which on Probability is very slight;∗ it
is chiefly mathematical in bias; and it is now nearly fifty years old.

I have not read all these books myself, but I have read more of
them than it would be good for any one to read again. There are
here enumerated many dead treatises and ghostly memoirs. The list
is too long, and I have not always successfully resisted the impulse
to add to it in the spirit of a collector. There are not above a
hundred of these which it would be worth while to preserve,—if only
it were securely ascertained which these hundred are. At present a
bibliographer takes pride in numerous entries; but he would be a
more useful fellow, and the labours of research would be lightened,
if he could practise deletion and bring into existence an accredited

Index Expurgatorius. But this can only be accomplished by the slow
mills of the collective judgment of the learned; and I have already

∗Laurent’s list contains 310 titles, of which I have excluded 174 from my list
as being insufficiently relevant.

BIBLIOGRAPHY

493
indicated my own favourite authors in copious footnotes to the main
body of the text.

The list is long; yet there is, perhaps, no subject of equal
importance and of equal fascination to men’s minds on which so
little has been written. It is now fifty-five years since Dr. Venn,
still an accustomed figure in the streets and courts of Cambridge,
first published his Logic of Chance; yet amongst systematic works
in the English language on the logical foundations of Probability my

Treatise is next to his in chronological order.

The student will find many famous names here recorded. The
subject has preserved its mystery, and has thus attracted the notice,
profound or, more often, casual, of most speculative minds. Leibniz,

Pascal, Arnauld, Huygens, Spinoza, Jacques and Daniel Bernoulli,

Hume, D’Alembert, Condorcet, Euler, Laplace, Poisson, Cournot,

Quetelet, Gauss, Mill, Boole, Tchebychef, Lexis, and Poincaré, to
name those only who are dead, are catalogued below.

Abbott, T. K. “On the Probability of Testimony and Arguments.”

Phil.

Mag. (4), vol. 27, 1864.

Adrain, R. “Research concerning the Probabilities of the Errors which happen
in making Observations.” The Analyst or Math. Museum, vol. 1, pp. 93–109,

1808.

[This paper, which contains the first deduction of the normal law of
error, was partly reprinted by Abbé with historical notes in Amer. Journ.

Sci. vol. i. pp. 411–415, 1871.]

Ammon, O. “Some Social Applications of the Doctrine of Probability.” Journ.

Pol. Econ. vol. 7, 1899.

Ampère. Considérations sur la théorie mathématique du jeu. Pp. 63. 4to.

Lyon, 1802.

Ancillon. “Doutes sur les bases du calcul des probabilités.” Mém. Ac. Berlin,
pp. 3–32, 1794–5.

Arbuthnot, J. Of the Laws of Chance, or a Method of Calculation of the

Hazards of Game plainly Demonstrated. 16mo. London, 1692.

[Contains a translation of Huygens, De ratiociniis in ludo aleae.]

A TREATISE ON PROBABILITY

494

4th edition revised by John Hans. By whom is added a demonstration of
the gain of the banker in any circumstance of the game call’d Pharaon, etc.

Sm. 8vo. London, 1738.

[For a full account of this book and discussion of the authorship, see

Todhunter’s History, pp. 48–53.]

“An Argument for Divine Providence, taken from the constant Regularity
observ’d in the Births of both Sexes.”

Phil. Trans, vol. 27, pp. 186–190

(1710–12).

[Argues that the excess of male births is so invariable, that we may
conclude that it is not an even chance whether a male or female be born.]

Aristotle. Anal. Prior. ii. 27, 70a 3. Rhet. i. 2, 1357 a 34.

[See Zeller’s Aristotle for further references.]

Arnauld. (The Port Royal Logic.) La Logique ou l’Art de penser. 12mo.

Paris, 1662.

Another ed. C. Jourdain, Hachette, 1846.

Transl. into Eng.

with introduction by T. S. Baynes.

London, 1851.

xlvii + 430.

See
especially pp. 351–370.

Babbage, C. An Examination of some Questions connected with Games of

Chance. 4to. 25 pp. Trans. R. Soc. Edin., 1820.

Bachelier, Louis. Calcul des probabilités. Tome i. 4to. Pp. vii + 517.

Paris, 1912.

Le Jeu, la chance, et le hasard. Pp. 320. Paris, 1914.

[Bailey, Samuel. ] Essays on the pursuit of truth, on the progress of
knowledge and on the fundamental principle of all evidence and expectation.

Pp. xii + 302. London, 1829.

Baldwin. Dictionary of Philosophy. Bibliographical volumes; s.v. “Probability.”

Baniol, A. “Le Hasard.” Revue Internationale de Sociologie. Pp. 16. 1912.

Barbeyrac. Traité du jeu. 1st ed. 1709. 2nd ed. 1744.

[Todhunter states (p. 196) that Barbeyrac is said to have published a
discourse “Sur la nature du sort.”]

Bayes, Thomas. An Essay towards solving a Problem in the Doctrine of

Chances.

Phil. Trans. vol. liii. pp. 370–418, 1763.

A demonstration, etc.

Phil. Trans. vol. liv. pp. 296–325, 1764.

[Both the above were communicated by the Rev. Richard Price, and the
second is partly due to him.]

German transl.

Versuch zur Lösung eines Problems der Wahrschein-
lichkeitsrechnung. Herausgegeben von H. E. Timerding. Sm. 8vo. Leipzig,

1908. Pp. 57.

Béguelin. “Sur les suites ou séquences dans le loterie de Gênes.” Hist. de

BIBLIOGRAPHY

495
l’Acad. Pp. 231–280. Berlin, 1765.

“ Sur l’usage du principe de la raison suffisante dans le calcul des
probabilités.” Hist. de l’Acad. Pp. 382–412. Berlin, 1767. (Publ. 1769.)

Bellavitis. “Osservazioni sulla theoria delle probabilità.” Atti del Instituto

Veneto di Scienze, Lettere, ed Arti, Venice, 1857.

Benard. “Note sur une question de probabilités.”

Journal de l’École royale
politechnique. Vol. 15, Paris, 1855.

Bentham, J. Rationale of Judicial Evidence.

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Germ. transl. 1896, by A. Pringsheim:

Die Grundlage der modernen

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Part IV.

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[Not connected with the subject of

A TREATISE ON PROBABILITY

496

Probability.]

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[The most important sections, including Bernoulli’s Theorem, are in

Part IV.

For a very full account of the whole volume see Todhunter’s

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Engl. Transl. of Part II. only, vide Maseres.

Fr. transl. of Part I. only, vide Vastel.

Germ. transl.: Wahrscheinlichkeitsrechnung.

4 Teile mit dem Anhange:

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Bertrand, J. Calcul des probabilités. Pp. lvii + 332. Paris, 1889.

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A TREATISE ON PROBABILITY

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[Doutes et questions sur le calcul des probabilités, vol. v. pp. 223–246.

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(These two papers were reprinted in
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INDEX∗

Acquaintance, direct, 11

Bachelier, 394 n.

Addition, of probabilities, 39, 148
and Rule of Succession, 428 n.

definition of, 131
and statistical frequency, 396 n.,
and measurement, 179

399

Theorem of, 115, 132, 160

Bacon, 304 f.

Analogy, principle of, 74
and limited variety, 310
tables of, 309
and Bacon, 308

Bayes, and Inverse Probability, 199
and generalisation, 256

Theorem of, 431
and induction, 251, 255

Belief, rational, 2 f., 8, 15, 350
and Jevons, 313
degrees of, 9
and Leibniz, 312

Bentham, measurement of Probability,
logical foundation of, 296

19
negative, 252, 256, 269, 474

Bernoulli’s Theorem, 121, 359, 364 n., positive, 253, 256, 474

377, 381 f.

and statistics, 445, 464, 474 f.

and asymmetry, 407 f.

Ancillon, 4 n., 91
empirical verification of, 411 f.

Apprehension, direct, and ethical

Inverse of, 419 f., 439 f.

judgment, 361

Bernoulli, Daniel, and Inverse

Argument, 13

Probability, 199

Aristotle, 87, 102
and Petersburg Paradox, 361, 362
and induction, 315
and planets, 334 n., 336

Arithmetic mean (or average), 237

Bernoulli, Jac., 14 n., 43, 83, 89, 91,

95, 419, 420

Gauss on, 238
and regular frequency, 376

Laplace on, 238
second axiom of, 368
and laws of error, 227
and statistical series, 446

Astronomers and Least Squares, 243
weight of evidence, 357

Asymmetry, and Bernoulli’s Theorem,

Bertrand, 52 n., 54

407 f.

and Bernoulli’s Theorem, 384

Atomic Uniformity, 286
and chance, 324

Averages, 237 f.

and independence, 198
and discordant observations, 248
and Law of Error, 241 n.

weighting of, 244
and Maxwell, 197 n.

Axioms, 148 f.

on multiplication, 149
non-self-evident, 342
and Petersburg Paradox, 362

∗This Index does not cover the Bibliography.

INDEX

530
and Rule of Succession, 436
and Psychical Research, 345

Bicquilley and testimony, 211 n.

and Sociology, 380

Bobek and Rule of Succession, 436

‘Casual ’, 329

Bode’s Law, 347

Causality, 302, 317

Boole, 46 n., 54 n., 92, 336 n.

and independence, 187
and approximation, 184

‘Cause’, 316
and Challenge Problem, 215

Cause, final, 339
and combination of premisses, 206

Cayley, and tradition, 213
and Cournot, 324 n.

and Challenge Problem, 214
and German logicians, 96

Certainty, 9, 139, 141
and independence, 191
and Bacon, 307
and relation of Probability, 100
definition of, 131
and Rule of Succession, 436

Kahle and, 99 n.

and symbolic probability, 177
and Leibniz, 313
and testimony, 207
relation of, 147
and Whately, 206
and truth, 14

Borel, 50 n., 52

Chance, objective, 321, 326 f., 337,

Bortkiewicz, von, and great numbers,

477

376 n.

and binary stars, 337
and Law of Small Numbers, 457 f.

Condorcet on, 324
and Lexis, 448 f.

Couturat on, 323
and Marbe, 415 n.

definition of, 327
method of, 437
and planets, 334
and Quetelet, 458

Poincaré on, 324, 330

Boscovitch and Least Squares, 243

Coefficient of Credibility, 210

Bowley, 481, 484 f.

of Correlation, 481 f.

Bradley, 364 n.

Combination of premisses, 168, 205
and Bernoulli’s Theorem, 387 n.

Comte
and relativity of Probability, 101
and ‘seven’, 284

Broad, C. D., 295 n.

and statistics, 379

Brömse and Marbe, 415 n.

Condorcet, 91 n., 362

Brünn and lotteries, 414
and chance, 322, 324

Bruns and Marbe, 415 n.

and ethics, 358, 361

Buffon, 363, 368
and gambling, 365
and coin-tossing, 411
and testimony, 207

Butler, Bishop, 86, 87, 353, 354

Conduct and Probability, 350
and risk, 366

Consistence and group theory, 135

Contradiction, 159

Calculus of Probability, 91 n., 168,

Coover, J., 340 n.

187, 346, 489

Correlation, 373, 444

INDEX

531
coefficient, 481 f.

Darwin, 120

Inductive, 464
and Lyell, 183

Quantitative, 446, 487
and Mill, 305
and statistical frequency, 375

De Morgan, 21, 81, 91

Cournot, and frequency theory, 101
and combination of premisses, 205
and causality, 316
and independence, 192
and chance, 321, 323
and inference, 152
and independence, 190
and Inverse of Bernoulli’s
on testimony, 206

Theorem, 421 n.

Couturat, 312 n., 355 n.

and Inverse Probability, 204

Craig and tradition, 211
and planets, 335

Cramer and Petersburg Paradox, 363
pupil of, 411

Crofton, 50 n.

and Rule of Succession, 427, 435

Cumulative Formula, 169
and tradition, 211 n.

Johnson and, 133

De Witt and arithmetic averages, 238

Czuber, 50 n., 85, 90, 96, 384 n., Dedekind and ‘Challenge Problem’,

391 n., 394

215 n.

and Bernoulli’s Theorem, 385 n.

definition of, 255
and ‘cause’, 316 n.

from statistics, 371
and Inverse of Bernoulli’s

Definitions, 146 f.

Theorem, 422 n.

de la Placette, Jean, and chance,
and lotteries, 414

323
and Marbe, 416
summary of, 131
and risk, 360 n.

Dice-tossing, 410 f.

and Rule of Succession, 428 n., 436

Diderot on testimony, 210
and statistical frequency, 399, 449

Discordant observations, rejection of,
and symbolic probability, 178

246
and Tchebycheff’s Theorem,

Donkin, W. F., 19

401 n., 404 n.

and Inverse Probability, 201
and verification of Bernoulli, 412 n.

Dormoy, 450

D’Alembert, 91, 194 n., 367, 415 n., Edgeworth, 30 n., 92, 95, 411 n., 432,

421

456
and chance, 322
and ethics, 361
and ethics, 361
and German statisticians, 450
and Marbe, 416
and Psychical Research, 341 n.

and mathematical expectation, 359
and randomness, 332
and Petersburg Paradox, 363
use of ‘Probability’, 107 n.

and planets, 335

Eggenberger, 385 n.

Darbon, A., and Cournot, 324

Ellis, Leslie, 92, 93

INDEX

532
and Bacon, 304, 305 n., 309 n., and Rule of Succession, 431

311 n., 315 n.

Frequency, statistical, 374
and Bernoulli’s Theorem, 386

Fresnel and simplicity, 238
and frequency theory, 101

Fries, 14 n.

and Least Squares, 239 n., 242

Empirical School, 93, 95

Galton, 367

Epistemology, 345
and Fechner’s law, 241
and inductive hypothesis, 301

Gambling, 365

Equiprobability, 44, 70, 71

Gauss, and laws of error, 226 n., 229

Equivalence, definition of, 131, 146
and arithmetic mean, 238
axiom of, 148
and Least Squares, 243
principle of, 156

Generalisation, 444

Error, probable, 373

Generator properties, 292

Ethics, 350 f.

plurality of, 292, 294, 295

Euler and Least Squares, 243

Geometrical probability, 51, 68

Event, probability of, 4

German logicians, 96

Evidence, and measurement of

Gibbon, 30, 368, 377

Probability, 6, 37

Gilman, B. I., and symbolic
addition of, 71, 74
probability, 177
external, 62

Goldschmidt, 30 n.

independent and complementary,

Goodness, organic nature of, 354

60

Graunt, 446 n.

and Induction, 254

Great Numbers, Law of, 90, 374,
relevant and irrelevant, 57, 59

376 f.

weight of, 77

Grimsehl, 285 n.

Excluded Middle, Law of, 159
and Marbe, 415 n.

Experience and the Principle of

Groups, of propositions, 128, 135

Indifference, 110
definition of, 131, 137
real and hypothetical, 141

Fechner, and median, 232

Grünbaum and Marbe, 415 n.

and law of sensation, 240
and lotteries, 414

Hagen, and error, 239

Fermat, formula of, 278
and discordant observations, 247 n.

Forbes, J. D., 19 n., 21, 336 n.

Halley and mortality statistics, 375

Frazer, Sir J., 282

Herodotus, 351

Frequency curves, 231

Herschell and binary stars, 336
and statistics, 371

Houdin, 414 n.

Frequency theory, 101 f.

Hudson, W. H., and animism, 284 n.

and Bernoulli’s Theorem, 390

Hume, 56, 77, 89, 91, 276, 488
and randomness, 331
and analogy, 256, 258

INDEX

533
and chance, 322
hypothetical and assertoric, 142
and Induction, 251, 269, 305, 312
necessary, 132, 153
and testimony, 210
statistical, 371 f.

Huyghens, 90

Insurance, 22, 326, 462
and ‘six ’, 284

Intuition versus experience, 95

Hypothesis, 7
and ethical judgment, 356

Hypothetical entities, 341

Inverse Probability, 167, 200
and Bowley, 486

Implication, 137
and frequency theory, 118

Impossibility, 15
and statistics, 420, 421 n.

definition of, 131

Theorem of, 133
relation of, 147
and Venn, 111

Inconsistency, definition of, 131

Irrelevance, 294

Independence, for knowledge, 118, 188
definition of, 60, 132, 152
judgments of, 59
and averages, 245

Theorem of, 133, 164
and chance, 323
definition of, 131, 151
and discordant observations, 247

James, W., and spirits, 344
and law of error, 225

Jesuits, 351
and measurement, 236

Jevons, 281 n.

of events, 187
and analogy, 283

Theorem of, 132, 163
and coin-tossing, 412

Index numbers, 244
and equiprobability, 46 n.

Induction, 107
and index numbers, 245
and frequency theory, 108, 109,
and Induction, 255, 275, 280, 305,

119

314, 315
and Logic, 250
and Inverse Probability, 205

Principle of, 74
and Rule of Succession, 436
pure, 251

Johnson, W. E., 126
statistical, 464 f.

and added evidence, 74
and statistics, 371 f.

and cumulative formula, 132, 168,
universal, 253, 464, 477

173, 176
validity of, 254
and groups, 135

‘Induction’, 315
and propositions, 9 n.

Inductive correlation, 253, 296, 447,
and testimony, 210

452, 464

Judgments, 59

Inductive hypothesis, 299, 304
direct, 76

Inductive method, 299
disjunctive, 84

Inference, 141
of preference and relevance, 71

INDEX

534

Kahle and the Probability relation, 99
and Least Squares, 243

Kant, 377
and planets, 335 n.

and Hume, 313
and Quetelet, 378

Kapteyn, Prof. J. C.

and relation of Probability, 101
and law of error, 231
and Rule of Succession, 399 n.,

Knowledge, 9

408 n., 419
and chance, 330
school of, 47, 56, 95, 407, 416
direct and indirect, 11, 301
and statistical series, 447
homologic and ontologic, 317, 329
and testimony, 207, 210
and ignorance, 321
and unknown probabilities, 421
incomplete and proper, 12

Laurent and gambling, 364
kinds of, 2

Law, 355 n.

of logical relations, 13

Law of error, 224 f.

probable and vague, 16
and arithmetic mean, 227
relativity of, 16
and geometric mean, 229
vague and distinct, 58

Lexis and, 454

Kries, von, 44, 47–49 n., 54, 73 n., 92
and median, 231
and Cournot, 324 n.

and mode, 235
and equiprobability, 96
normal law, 230, 234, 238
and independence, 198

Least Squares and Venn, 238
and Inverse Probability, 201
method of, 234, 237, 238, 242
and knowledge, 317

Lee and tradition, 211 n.

and Principle of Indifference, 197

Legendre and Least Squares, 243
and School of Lexis, 449

Leibniz, 24 n., 352, 419, 447, 488
and arithmetic average, 238

Lacroix, 211 n.

and Induction, 313

Lambert and Least Squares, 243

Lexis, and asymmetry of statistical

Lämmel, 50 n.

frequency, 408 n.

and symbolic probability, 177
and Edgeworth, 457

Laplace, 14 n., 29 n., 32, 90, 92, 363,
and Marbe, 415 n.

488
method of, 437, 448 f., 452 f.

and arithmetic mean, 239
and statistical stability, 474, 478 n.

and Bayes’ Theorem, 433

Locke, 84, 88, 90, 91, 352, 369
and Bernoulli’s Theorem, 386, 421
on tradition, 213
and birth proportions, 414
and weight of evidence, 358
and chance, 322

Logic, academic, 2
and doctrine of averages, 234
and Induction, 250, 282
and independence, 194
and initial probability, 341
and Induction, 254, 276, 305, 314
of implication, 63
and Inverse Probability, 200, 204
of probability, 7

INDEX

535

Logical priority, 142
and psychical research, 345

Lotteries, 376 n., 411, 414 f.

Median and laws of error, 231
published results of, 414

Meinong, 85

Lotze, 99

Meissner, Otto, and dice-throwing,
and Rule of Succession, 436

413

Lucretius, 488

Memory, 13

Mendelism and statistics, 379, 478,

M’Alister, Sir Donald, and laws of

489
error, 229

Merriman, Mansfield, and Least

Macaulay and Bacon, 305

Squares, 242

Macfarlane, and independence, 193 n.

Metaphysics and certainty, 275
and ‘Challenge Problem’, 214 n.

Method of Difference, 283
and tradition, 213

Michell, 346

Maclaurin, Theorem of, 239
and binary stars, 336

Marbe, Dr. Karl, and roulette, 415
and Inverse Probability, 200

Marginal utility, 364

Middle Term, Fallacy of, 74, 176

Markoff, A. A., 203 n.

Mill, and inductive correlations, 253
and Inverse Probability, 201
and induction, 304 f.

and Tchebycheff’s Theorem, 405
and limited variety, 311

Mathematical Expectation, 356, 360,
methods of, 310

362
and plurality of causes, 307 n.

Mathematicians, and probability, 92
and probability, 308 n.

and cumulative formula, 172
and pure induction, 309
and ethics, 361

Modality and probability, 15 n.

and laws of error, 239

Venn and, 108

Maxim, Sir Hiram, 414 n.

Mode, and law of error, 235

Maxwell, 197 n.

asymmetry about, 410
and theory of gases, 197

Monte Carlo, 415

Mayer and Least Squares, 243

Moore, G. E., 18, 276 n., 353

McColl, and symbolic probability, 176

Multiplication, 148
definition of, 131
and Boole, 191 n.

of instances, 269 f.

and ‘Challenge Problem’, 216 n.

theorems of, 133, 166, 388
and Inverse Probability, 201

Munro, 421 n.

Means and laws of error, 224 f.

Measurement of Probability, 35, 179,

Necessary connection, law of, 288

355

Newton, and induction, 281
and ethics, 355
and ‘seven’, 284
and frequency theory, 104
and Bacon, 304
and induction, 298, 442

Nitsche, A., 48 n., 54 n., 85, 197 n.

INDEX

536

Occurrences, remarkable, 345
and probabilism, 352

Prediction, value of, 348

Pascal, 90

Price and Bayes, 199 n.

Pearson, Karl, 92, 399 n.

Primitive people and rational belief,
and arithmetic mean, 240

282
and asymmetry, 394, 408 n.

Principle of compelling reason, 96
and frequency theory, 110

Principle of Indifference, 45, 90, 91 f., and generalised Probability curves,

96, 115, 118, 196

394
analysis of, 57
and roulette, 414
and ethics, 354
and Rule of Succession, 431, 435
and induction, 109
and stars, 339
and Laplace, 423, 426

Peirce, 54 n., 347
and measurement, 182
and randomness, 331
modification of, 60, 64

Petersburg Paradox, 362
and Psychical Research, 345
and Buffon, 411
and Rule of Succession, 430
psychology of, 363
and statistics, 418

Peterson and tradition, 211

Principle of Non-Sufficient Reason,

Physics and initial probability, 341

44, 94

Planets, movements of, 334

Principle of superposition of small

Playfair, Dr. Lyon, 348
effects, 286

Plurality of causes and Mill, 307

Probabilism, 352

Poetry and statistics, 457

‘Probability’, 7

Poincaré, Henri, 52, 92

Edgeworth’s use of, 106 n.

and chance, 324, 330

Venn’s use of, 105
and independence, 198

Probability relation, 2, 7, 12, 146

Poisson, 55 n., 411 n.

intuition of, 56
and gambling, 365

Probability, and relevant knowledge, 2
and great numbers, 377, 380
and inverse of Bernoulli’s
and binary stars, 336

Theorem, 422
comparison of, 35, 71, 182
and least errors, 240
dependent on evidence, 6
on testimony, 207
and ethics, 350
and Petersburg Paradox, 362
and final causes, 339
and statistical frequency, 395
finite, 273
and Tchebycheff, 406
from statistics, 417 f.

Theorem of, 391

‘geometrical ’, 50, 52, 68

Poretzki, Platon S., and symbolic

Inverse, 117, 167
probability, 178
and law, 24

Port Royal logic, 76, 87, 367
mathematical, 5

INDEX

537
measurement of, 19 f., 39
and statistical stability, 448
negative, 152
objective relation of, 4, 7, 321

Randomness, 321, 331, 470
philosophical definition of, 7

Pearson’s use of, 340
and planetary orbits, 334

Relation, of probability, 5
and randomness, 332
of ‘between’, 37, 42
and rational belief, 107

Relativity, of knowledge, 16
series of, 37, 41
doctrine of, and the Law of
and similarity, 29, 38

Uniformity, 285 n.

and spirits, 342, 344
of probabilities, 113
and star drifts, 338

Relevance, judgments of, 59
and statistical frequency, 108
and frequency theory, 115
and telepathy, 342
theorems of, 165
three senses of, 9

Remarkableness, 345
and truth, 126, 368

Requirement, 142
and truth frequency, 111 f., 381 f.

Risk, 360

‘unknown,’ and Laplace, 423
and ethics, 358

Probable error, 81

‘moral ’, 366, 368

Proctor, 414 n.

and Petersburg Paradox, 364

Proposition, characterisation of, 2

‘physical ’, 368
classes of, 111 f.

Roulette, 410, 415
disjunction and conjunction of, 146
published results of, 414 n.

existential, 317

Rule of Succession, 408 n., 419, 423,
groups of, 128, 135

426
knowledge of, 11
and frequency theory, 431
primary and secondary, 9, 12
and Pearson, 433 n.

self-evident, 16
proof of, 428
subgroups of, 138, 141

Russell, Bertrand, 18, 126, 135 n., 138
synthetic, 302

Propositional function, 61
and implication, 135
and induction, 255
and inference, 128
and randomness, 333

Psychical Research, 319 f.

Schematisation, 73

Psychology and probability, 56

Schröder and symbolic probability,

Pythagoras and ‘seven’, 283

178

Selection, random, 333

Quetelet, 376 n., 378, 379, 458, 478,

Series of probabilities, 37, 41

488, 489
and frequency theory, 102
and arithmetic mean, 241

Gaussian, 481 n.

and balls, 412
independent, 323, 479

INDEX

538
organic, 455, 479

Terrot, Bishop, 46 n.

Sigwart, 98
and combination of premisses, 205
and induction, 314
and Whately, 205 n.

and inverse probability, 205

Testimony, theory of, 206

Simmons and asymmetry in

Time, 294

Bernoulli’s
irrelevance of, 345

Theorem, 408
and uniformity, 261

Simpson and Least Squares, 243

Todhunter, 336 n., 363 n., 421 n.

Small Numbers, Law of, 457 f.

and Bayes, 201

Society for Psychical Research, 340 n.

and Bernoulli’s Theorem, 386 n.

Space, 294
and Craig, 212
irrelevance of, 345
and Petersburg Paradox, 362
and uniformity, 261

Truth and probability, 126 n., 368

Spedding and Ellis and Bacon, 304,

Truth frequency, 111, 464

305 n.

Tschuprow, 407, 455 n.

Spielräume, doctrine of, 97
method of, 438

Spinoza, 126 n., 321 n.

and statistical frequency, 395,

Spirits, probability of, 342

449 n.

Star drifts, 339

Stars, binary, 336

Uniformity of Nature, Law of, 260,

Statistical frequency, theory of, 102 f.

285, 293, 302, 317
criticism of, 114
and Mill, 310
fluctuation of, 446

Universal Causation, Law of, 285
generalisation of, 111

Universal Induction and statistical
stability of, 380, 446–475
methods, 443, 463–477

Statistical inference, 371 f.

Universe of reference, 128, 142, 143
induction, 463 f.

Unknown probabilities, 423, 424, 428

Statistics, and prediction, 349
descriptive and inductive, 371

Variables in Probability, 63, 134,

Stumpf, 47 n., 54 n., 197 n.

470 n.

Sub-analogies, 257, 264

Variety, 270

Subgroups of propositions, 138, 141
and induction, 252

Succession, Law of, 90
limitation of, 297, 300, 489

Süssmilch and regular frequencies, 376

Venn, 92, 117 n., 336 n.

and Bernoulli, 95, 386

Taylor, Jeremy, 351 n.

and chance, 329

Tchebycheff
and experience, 94
and Poisson’s Theorem, 405
and frequency theory, 103 f.

Tchebycheff, Theorem of, 401, 403
and induction, 314

Telepathy, probability of, 342
and inverse probability, 111

INDEX

539
and Least Squares, 239 n.

Whittaker, E. T., and Rule of
and ‘random’, 331

Succession, 428 n.

and Rule of Succession, 423, 431

Wilbraham, H., and Boole, 191 n.

Wolf and dice, 411

Weight, of evidence, 356
and ethics, 360

Weighting of averages, 244

Yule, 396 n., 410 n.

Weldon and dice, 411
and approximation, 184

Whately and combination of misses,
and coin-tossing, 393 n., 411 n.

204
and correlation, 481, 484

Whitehead, and frequency theory
and independence, 190
and invalid inference, 373 n.

and ‘statistics’, 371

O False and treacherous Probability,

Enemy of truth, and friend to wickednesse;

With whose bleare eyes Opinion learnes to see,

Truth’s feeble party here, and barrennesse.

THE END

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Document Outline

Front Matter

- PG Boilerplate
- Transcriber's Note
- Preface
- Contents

Part I. Fundamental Ideas

Chapter I

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

Chapter II

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

Chapter III

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

- Section 15.

- Section 16.

- Section 17.

- Section 18.

- Section 19.

- Section 20.

- Section 21.

Chapter IV

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

- Section 15.

- Section 16.

- Section 17.

- Section 18.

- Section 19.

- Section 20.

- Section 21.

- Section 22.

- Section 23.

- Section 24.

- Section 25.

- Section 26.

Chapter V

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

Chapter VI

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

Chapter VII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

Chapter VIII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

- Section 15.

- Section 16.

- Section 17.

Chapter IX

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

Part II. Fundamental Theorems

Chapter X

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

Chapter XI

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

Chapter XII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

Chapter XIII

- Section 1.

- Section 2.

Chapter XIV

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

Chapter XV

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

Chapter XVI

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

- Section 15.

- Section 16.

- Section 17.

- Section 18.

- Section 19.

- Section 20.

Chapter XVII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

Part III. Induction and Analogy

- Chapter XVIII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Chapter XIX

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Chapter XX

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Chapter XXI

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Chapter XXII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

- Chapter XXIII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Notes on Part III

- Section 1.

- Section 2.

- Section 1.

- Section 2.

- Section 3.

- Section 4.

Part IV. Philosophical Applications

- Chapter XXIV

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Chapter XXV

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Chapter XXVI

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

Part V. Statistical Inference

Chapter XXVII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

Chapter XXVIII

- Section 1.

- Section 2.

Chapter XXIX

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

- Section 15.

- Section 16.

- Section 17.

- Section 18.

- Section 19.

Chapter XXX

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

- Section 11.

- Section 12.

- Section 13.

- Section 14.

Chapter XXXI

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

Chapter XXXII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

Chapter XXXIII

- Section 1.

- Section 2.

- Section 3.

- Section 4.

- Section 5.

- Section 6.

- Section 7.

- Section 8.

- Section 9.

- Section 10.

Back Matter

- Bibliography

- A
- B
- C
- D
- E
- F
- G
- H
- J
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
- Y

- Index

- Project Gutenberg License

Table of Contents

- 2
- 9
- 20
- 44
- 71
- 78
- 87
- 102
- 123
- 126
- 135
- 146
- 153
- 161
- 180
- 188
- 214
- 250
- 256
- 269
- 279
- 289
- 305
- 315
- 321
- 335
- 351
- 371
- 376
- 382
- 418
- 438
- 446
- 464
- Part II.
- p. 96.
- Part III.
- IV.,
- V.
- Part I.
- p. 207
- p. 201.
- 74
- 165.
- p. 316.
- p. 363).
- p. 84.
- p. 391.
- p. 195,
- p. 385