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http://www.archive.org/cletails/cu31924101981235



BY THE SAME AUTHOE



INDIAN CURRENCY AND FINANCE

8vo. Pp. viii + 263. 1913.
7s. 6d. net.

THE ECONOMIC CONSEQUENCES
OF THE PEACE

8vo. Pp. vii + 279. 1919.
8s. ed. net.



A TEEATISE ON PROBABILITY



^0^m.



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

1921



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
probl^me," in the words of Poisson, "propose k un austere
jans^niste par un homme du monde, a 4td I'origine du calcul
des probabilit^s." In the intervening centuries the algebraical
exercises, in which the Chevalier de la Mer^ interested Pascal,
have so far predominated in the learned world over the pro-
founder enquiries of the philosopher into those processes of
human faculty which, by determining reasonable preference,
guide our choice, that Probability is oftener reckoned with Mathe-
matics 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
myseK 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, Gr. E. Moore, and Bertrand Eussell, 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



vi A TEEATISE ON PEOB ABILITY

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,
■ 1, 1920.



CONTENTS

PAET I

FUNDAMENTAL IDEAS



CHAPTER I

PAGE

The Meaning op Probability . . . . .3



CHAPTER II
Probability in Relation to the Theory op Knowledge . 10

CHAPTER III

The Measurement op Probabilities . . . .20

CHAPTER IV
■ — The Principle op Indipperence . . . .41

CHAPTER V

Other Methods op Determining Probabilities . . 65

CHAPTER VI

The Weight op Arguments . . . . . 71 -^-^



PAGE

79 ^



viii A TEEATISE ON PEOBABILITY

CHAPTER VII
Historical Retrospect . . • •

CHAPTER VIII

.The Freqdkncy Theory of Probability



CHAPTER IX

The Constructive Theory of Part I. summarised . .111



PART II

FUNDAMENTAL THEOREMS

CHAPTER X

Introductory . . . . . .115

CHAPTER XI

^VThe Theory of Groups, with special reference to Logical

Consistence, Inference, and Logical Priority . .123

CHAPTER XII

' ~~-^Thb Definitions and Axioms of Inference and Probability 133

CHAPTER XIII

The Fundamental Theorems of Necessary Inference. . 139

CHAPTER XIV

''■The Fundamental Theorems of Probable Inference . 144



CONTENTS ix

CHAPTER XV

PAGE

Numerical Measurement and Approximation op Proba-
bilities . . . . . . .158

CHAPTER XVI

Observations on the Theorems op Chapter XIV., and

their Developments, including Testimony . . . 1 64

CHAPTER XVII

Some Problems in Inverse Probability, including Averages 186

PART III

INDUCTION AND ANALOGY

CHAPTER XVIII
Introduction . . ■ .217

CHAPTER XIX

The Nature op Argument by Analogy . . .222

CHAPTER XX

The Value op Multiplication op Instances, or Pure Induction 233

CHAPTER XXI

The Nature op Inductive Argument continued . .242

CHAPTER XXII

The Justification op these Methods . . .251

CHAPTER XXIII

Some Historical Notes on Induction . . .265

Notes on Part III. • • • • • .274



A TEEATISE ON PEOBABILITY

PAET IV

SOME PHILOSOPHICAL APPLICATIONS OF PROBABILITY



CHAPTER XXIV

PAGE

The Meanings of Objective Chance, and of Randomness . 281



CHAPTEE XXV

Some Problems arisins odt of the Discussion of Chance . 293

CHAPTEE XXVI

'"The Application of Probability to Conduct . . 307

PAET V
THE FOUNDATIONS OF STATISTICAL INFERENCE

CHAPTEE XXVII

The Nature of Statistical Inference . . _ 337

CHAPTEE XXVIII
The Law of Great Numbers .

CHAPTEE XXIX

The Use of a priori Probabilities for the Prediction of
Statistical Frequency— the Theorems of Bernoulli

POISSON, AND TcHEBYCHBFF . ' qo-7

001

CHAPTEE XXX

The Mathematical use of Statistical Frequencies for the
Determination of Probability d ^osfe„-o„_THE Methods
of Laplace

. 367



\ \



332



CONTENTS xi

CHAPTER XXXI

PAQE

The Inversion of Bernoulli's Theorem . . . 384



CHAPTER XXXII

The Inductive Use of Statistical Frbquenoies for the
Determination of Probability a posteriori — the Methods
OF Lexis .... . . 391



CHAPTER XXXIII

Outline of a Constructive Theory .... 406

BIBLIOGBAPHY . .... 429

INDEX ...... .459



PART I
FUNDAMENTAL IDEAS



CHAPTEE I



THE MEANING OP PROBABILITY



" J ai dit plus d'une fois qu'il faudrait une nouvelle esp^ce de logique, qui
traiteroit des degr^s de Probability." — Leibniz.

1. Part of our knowledge we obtain direct ; and part by
argument. Tbe 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 in-
conclusive. T^

In most branches of academic logic, such as the theory of the
syllogism or the geometry of ideal space, all the argmnents aim
at demonstrative certainty. They claim to be conclusive. But
many other arguments are rational and claim some weight with-
out pretending to be certain. In Metaphysics, in Science, and in
Conduct, most of the arguments, upon which we habitually base
our rational behefs, 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 ia the actual exercise of reason we do not wait
on certainty, or deem it irrational to depend on a doubtful
argument. If logic investigates the general principles of vahd
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 describejhe various degrees
of rational beliel about a proposition which different amounts of
Imowledgeauthorise us to entertain. AU-Pi^P08itions,>re Jitue
or false,-Jbut the knowledge.JEe, ha,ye of jfchem- depends on our
circumstances I and while it is often convenient to speak of



4 A TREATISE ON PROBABILITY pt. i

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 of the propositions in them-
selves. 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 caU a proposition prob-
able unless we specify the knowledge to which we are relating it.

To this extent, therefore, probability may be called sub-
jective. 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 particidar 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 rela-
tions between the propositions which embody our direct know-
ledge and the propositions about which we seek indirect know-
ledge. 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 a, we say
that there is a probability-relation of degree a between a and h.^

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

1 This wiU be written o/^ =a. 2 gee also Chapter II. § 5.



OH. I FUNDAMENTAL IDEAS 5

4. With the term " event," which has taken hitherto so im-
portant a place in the phraseology of the subject, I shall dis-
pense 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 ^nanabiguous ; and it will be
more than a verbal improvement to mscuss 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
cril4cism. In .the ..ordinary course of thought and argumentj..
we are constantly assuming that knowledge of one statement,
while_n6t"~p7Twmy-'tEe^truth"~of'a second, yields nevertheless
some gTOwM^^^F^lieving^ it. We assert that we ought on the
evidence to prefeTsucE and such a belief. We claim rational
grounds for assertions which are not conclusively demonstrate(L-
We allow, in fact, that statements may be unproved, without, for
that reason, being unfoimded. And it does not seem on reflection
that the information we convey by these expressions is wholly
subjective. When we argue that Darwin gives vahd grounds
for our accepting his theory of natural selection, we do not simply
mean that we are psychologically inchned to agree with him ;
it is certain that we also intend to convey our belief that
we are acting rationally in regarding his theory as prob-
able. We beUeve that there is some real objective relation
between Darwin's evidence and his conclusions, which is inde-
pendent of the mere fact of our behef, and which is just as real
and objective, though of a difierent 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

^ Except in those chapters (Chap. XVII., for example) where I am deahng
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 probabilites (1794) : " Dire qu'un fait pass6, present ou a
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,
Wahrscheinlichkeiisrechnung, vol. i. p. 5, and Stumpf, Vber den Begriff der mathe-
matischen Wahracheinlichkeit. ,



6 A TREATISE ON PROBABILITY pt. i

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 probabihty. 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 probabihty in the sense which is meant by the
authors of typical treatises on Wahrscheinlichkeitsrechnung or
Calcul des jyrobabilites, wiU 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
imsolved, we must begin at the beginning (or almost at the
beginning) and treat our subject widely. As soon as mathe-
matical probability ceases to be the merest algebra or pretends
to guide our decisions, it itnmediately 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 wiU
be well to know first what it means in the widest.

7. Between two sets of propositions, therefore, there exists
a relation, in virtue of which, if we know the first, we can attach
to the latter some degree of rational beUef. 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 prob-
abihty. From the premisses " a imphes 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, there-
fore, to say " b is probable " as it would be to say " 6 is equal,"
or " b is greater than," and as unwarranted to conclude that,
because a makes b probable, therefore a and c together make 6



CH. I FUNDAMENTAL IDEAS 7

probable, as to argue that because a is less than h, therefore a
and c together are less than h.

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 considera-
tions, implicitly or expHcitly present to our minds at the moment,
are taken into account. We use the word for the sake of short-
ness, 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 intrinsicaUy 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 elUptical form of conunon
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 corjpus of
knowledge of ourselves or others, or which are merely
hypotheses.

Reflection will show that this account harmonises with
familiar experieace. 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 fijst to hand, but on further informa-
tion 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
stiU 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 tie 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 TEEATISE ON PROBABILITY pt. i

8. A definition of probability is not possible, unless it contents
us to define degrees of the probability-relation by reference to
I degrees of rational beKef. We cannot analyse the probability-
relation in terms of simpler ideas. As soon as we have passed
from the logic of imphcation and the categories of truth and
falsehood to the logic of probabiHty and the categories of know-
ledge, 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 posi-
tive proof. The presumption in its favour must arise partly
out of our failure to fimd a defiboition, and partly because the
notion presents itself to the mind as something uew and inde-
pendent. If the statement that an opinion was ptrobable on the
evidence at first to hand, but became untenable on further in-
formation, is not solely concerned with 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 ternis 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 tiiem accurately represent that par-
ticular 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 " proUble " seems to
be used consistently by different persons to d^cribe 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 post-
poning it until our enquiry into the object of definition is far
advanced. In the case of " probability " the o|)ject before the
mind is so familiar that the danger of misdescriljing its qualities
through lack of a definition is less than if it were a highly abstract
entity far removed from the normal channels of jhought.

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



CH. I FUNDAMENTAL IDEAS 9

of a most frmdamental 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 Probabihty, the principal properties of this
relation. Eirst, however, we must digress in order to consider
briefly what we mean by knowledge, rational belief, and argument.



CHAPTEE 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, there-
fore, expand some part of what has been outUned or assumed
in the first chapter.

2. There is, first of all, the distinction between that part of
our beUef 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 beUeves it and it is in fact true. On the
other hand, a man may rationally bSlieve a proposition to be
jrrobable, when it is in fact false. The distinction between
rational belief and mere behef, therefore, is not the same as the
distinction between true beUefs and false behefs. 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

10



OH. u FUNDAMENTAL IDEAS 11

belief, and its relation to knowledge have been introduced above.
What, however, is the 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, f) in which we have a probable
degree {say, a) of rational belief. If the evidence upon which
we base our belief is h, then what we hnow, namely q, is that
the proposition f bears the probability-relation of degree a to
the set of propositions li ; and this knowledge of ours justifies
us in a rational belief of degree a in the proposition f. It will
be convenient to call propositions such as p,. which do not contain
assertions about probability-relatipns, " 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 ia the proposition itself. We cannot know a proposition
unless it is in fact true. A probable degree of rational beUef
in a proposition, on the other hand, arises out of knowledge of
some corresponding secondary proposition. A man may ration-
ally beUeve 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 beUef 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



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