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BY THE SAME AUTHOR
INDIAN CURRENCY AND FINANCE
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THE ECONOMIC CONSEQUENCES
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• 8vo. Pp. vii + 279. 1919.
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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
1921
COPYRIGHT
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
problfeme," in the words of Poisson, "propose k un austere
jansdniste par un homme du monde, a ^t^ 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
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 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 PEOBABILITY
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, Oambkidgb,
May 1, 1920.
CONTENTS
PART I
FUNDAMENTAL IDEAS
CHAPTER I
PAGE
The Meaning of Probability ..... 3
CHAPTER II
Peobabilitt in Relation to the Thboet or Knowledsb . 10
CHAPTER III
The Measurement of Probabilities . . . .20
CHAPTER IV
The Principle of Indifference . . . .41
CHAPTER V
Other Methods of Determining Probabilities . . 65
CHAPTER VI
The Weight of Arguments . . . . .71
viii A TKEATISE ON PEOBABILITY
CHAPTER VII
FADE
Historical Ebtrospect . . . . .79
QHAPTER VIII
The Frequency Theory of Probability . . .92
CHAPTER IX
The Constructive Theoby op Part I. summarised . .111
PART II
FUNDAMENTAL THEOREMS
CHAPTER X
Introductory . . . . . . .115
CHAPTER XI
The Theory of Groups, with special eeperence to Lomcal
Consistence, Inference, and Logical Priority . .123
CHAPTER XII
The 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 . r . . . . .158
CHAPTER XVI
Observations on the Theorems of Chapter XIV., and
THEIR Developments, including Testimony. . .164
CHAPTER XVII
Some Problems in Inverse Probability, including Averages 186
PART III
INDUCTION AND ANALOGY
CHAPTER XVIII
Introduction . . . . .217
CHAPTER XIX
The Nature of Argument by Analogy . . . .222
CHAPTER XX
The Value of Multiplication of Instances, or Pure Induction ,233
CHAPTER XXI
The Nature of Inductive Argument continued . .242
CHAPTER XXII
The Justification of these Methods . . .251
CHAPTER XXIII
Some Historical Notes on Induction . . . 265
Notes on Part III. ...... 274
A TEEATISE 0^ PEOBABILITY
PAET IV
SOME PHILOSOPHICAL APPLICATIONS OF PEOBABILITY
CHAPTER XXIV
PAGE
The Meanings of Objective Chance, and of Randomness . '281
CHAPTER XXV
Some Problems arising out of the Discussion of Chance . 293
CHAPTER XXVI
The Application of Probability to Conduct . . 307
PAET V
THE FOUNDATIONS OF STATISTICAL INFEEENCE
CHAPTER XXVII
The Nature of Statistical Inference . . .327
CHAPTER XXVIII
The Law of Great Numbers . . . . .332
CHAPTER XXIX
The Use of d priori Probabilities for the Prediction of
Statistical Frequency — the Theorems of Bernoulli,
poisson, and tchbbychbff .... 337
CHAPTER XXX
The Mathematical use of Statistical Frequencies for the
Determination of Probability d posteriori — the Methods
of Laplace ...... 367
CONTENTS xi
CHAPTER XXXI
PAGE
The Inversion of Bebnotjlli's Theorem . . . 384
CHAPTER XXXII
The Inductive Use of Statistical Frequencies for the
Determination of Peobabilitt d posterior'!^— tbm Methods
OF Lexis ....... 3D1
CHAPTER XXXIII
Outline of a Constructive Theory .... 406
BIBLIOGRAPHY .... .429
INDEX ........ 459
PART I
FUNDAMENTAL IDEAS
CHAPTEE I
THE MEANING OF PROBABILITY
" J'ai dit plus d'une fois qu'il faudrait une nouvelle eap^oe de logique, qui
tiaiteroit des degrSs de Probabilite." — 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 argiunent, and it treats of the different
degrees in which the results so obtained are conclusive or in-
conclusive.
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 with-
out 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 deem 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
3
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 call a proposition prob-
able unless we specify the knowledge to which we are relating it.
To this extent, therefore, probability may be caUed 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 ouj 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 out
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 probabil/ity-relation of degree a between a and h.^
In ordinary speech we often describe the conclusion as being
doubtful, uncertain, 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 afEord grounds for a
corresponding degree of rational belief.^
1- This will be written ajh = a. 2 ggg g^jgg Chapter' II. § 5.
CH. I FUNDAMENTAL IDEAS 5
4. With the term " event," wMch 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 unambiguous ; 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, jdelds 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 aUow, 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. JWhen 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 beheve 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 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
1 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 AnciUon- in Doutes sur les
bases du calcul des probabilites (1794) : " Dire qu'un fait passe, present ou k
venir est probable, c'est dire qu'ime 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, Uber den Begriff der mathe-
matischen WahrscheinluAkeit.
6 A TEEATISE 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. Itis 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 probability in the sense which is meant by the
authors of typical treatises on Wafirschemlichkeitsrechnung or
Calcul des probahilites, wHl find that I do eventually reach topics
with which they are familiar. But iu making a serious attempt
to deal with the fundamental difficulties with which aU students
of mathematical probabilities have met and which are notoriously
xmsolved, 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 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
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 prob-
abihty. From the premisses " a impUes 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 beUef 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 " 6 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
CH. I FUNDAMENTAL IDEAS 7
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 considera-
tions, implicitly or explicitly 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 intrinsically distant ; and the probabihty 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 aU other suppositions, — ^according to the
usual practice which leads to the elUptical 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
hypotheses.
Reflection will show that this accoimt 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 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
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 mUes
distant ; and the opinion still is probable in relation to the old
hypotheses, just as the destination is stiU three miles distant
from our starting-point.
8 A TREATISE ON PROBABILITY n. i
■ 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 tbe 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 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 presiunption 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 inde-
pendent. If the statement that an opinion was probable 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 terms of the other
indefinables of formal logic. The attempts at definition, which
have been made hitherto, will be criticised ia later chapters.
I do not believe that any of them 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 " 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 post-
poning it untU 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 quaUties
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
OH. I FUNDAMENTAL IDEAS 9
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 Probabihty, 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 IS RELATION TO THE THEORY OF KNOWLEDGE
1. I DO not wisli 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 wjll, there-
fore, 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, arid 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 beUef, 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
10
CH. u FUNDAMENTAL IDEAS 11
belief, and its relation to knowledge have been introduced above.
Wliat, however, is the relation to knowledge of probable degrees
of rational beUef ?
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, a) 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 a to
the set of propositions h ; and this knowledge of ours justifies
us in a rational belief of degree a 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 ration-
ally 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 beUef is entertained.
5. At this point it is desirable to colligate the three senses
ia 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. Deriva-
tive from this sense, we have the sense in which, as above, the
term probable is apphed to the degrees of rational belief arising
out of knowledge of secondary propositions which assert the
1 This classification of "primary" and "secondary" propositions was
suggested to me by Mr. W. E. Johnson.
12 A TEEATISE ON PEOBABILITY m- i
existence of probability-relations in the fundamental logical sense.
Fuitber 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 understamd,
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 behef — as opposed to the
objects of direct acquaintance which I term sensations, meanings,
and perceptions — ^I shall term propositions.
Now OUT knowledge of propositions seems to be obtained in
two ways : directly, as the result of contemplating the objects
of acquaintance ; and indirectly, hy 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 itseK 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 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 ac-
quaintance with a sensation of yellow I can pass directly to a
knowledge of the proposition " I have a sensation of yeUow."
From acquaintance with a sensation of yellow and with the
CH. II FUNDAMENTAL IDEAS 13
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 per-
ceiving a logical relation between them. With this logical rela-
tion 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 aboutj 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 per-
ception of what logical relation we have passed from the know-
ledge of one proposition to knowledge about another. But
although in some cases we seem to pass directly from one pro-
position to another, I am inclined to believe that in all legitimate
transitions of this kind some logical relation 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
14 A TREATISE ON PROBABILITY pt. i
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 know-
ing 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 know-
ledge. 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 know-
ledge. But many questions are in doubt. Of vihich 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 interpretiag 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 — vt& remember
it. We may remember it as knowledge, but forget how we origin-
ally knew it. What we once knew and now consciously re-
member, 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)— rthe last of which cannot fairly be called
knowledge, for unhke 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 behefs 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 know-
ledge, 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 vaUd and rational. When we know some-
thing by argument this must be through direct acquaintance
with some logical relation between the conclusion and the premiss.
OH. n FUKDAMENTAL IDEAS 15
In all knowledge, therefore, there is some direct element ; and
logic can never be made purely mechanical. AH it can do is
so to arrange the reasoning that the logical relations, which
have to be perceived directly, are made exphcit 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 in-
volving a degree of probabiUty lower than certainty, together
with knowledge of the premiss of the secondary proposition,
leads only to a rational belief of the appropriate degree m the
primary proposition. The knowledge present in this latter case
I have called knowledge about the primary proposition or con-
clusion 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 certairity ; 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 probabihty, 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
^ This view has often been taken, e.g., by BemoulU and, incidentally, by
Laplace ; also by Fries (see Czuber, Entwicklung, p. 12). The view, occasion-
ally 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.
16 A TREATISE ON PROBABILITY ft. i
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 impossibihty is the relation of minimum probability.^
10. We have distinguished between rational beUef 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 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 con-
ditions 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 pro-
positions h should be known. In order that we may have rational
behef in ^ 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 ^ of a degree
less than certainty except through knowing a secondary pro-
position of the prescribed tjrpe. Such belief can only arise, that
is to say, by means of the perception of some probabihty-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. AH knowledge, that is to say, which is obtained in a
manner strictly direct by contemplation of the objects of acquaint-
ance and without any admixture whatever of argument and the
contemplation of the logical bearing of any other knowledge on
this, corresponds to certmn 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
^ Necessity and Impossibility, ia 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 possibihty, corresponding to the intermediate
degrees of probabihty. 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.
OH. n FUNDAMENTAL IDEAS IT
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 know-
ledge, and partly out of a confusion between probable know-
ledge 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 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 pre-
misses 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 rela-
tive 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 ex-
ample^ — 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 do not mean to imply, however, at any rate at present, that the ultimate
premisses of an argument need always be primary propositions.
18 A TEBATISE ON PEOBABILITY pt. i
I really know, may be only a probable belief to you, or may form
no part of your rational beliefs at aU. And tliis 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. Further, the
difEerence 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 ftmdamental 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
CH. n FUNDAMENTAL IDEAS 19
by previous writers and have the advantage of being, in a general
way at least, immediately inteUigible to the reader.^
^ This question, which faces all contemporary writers on logical and philo-
sophical 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 state-
ment, 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 Ethika 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 imderstanding ; 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
comfarisons 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
probabiUty is, in the full and literal sense of the word, measurable.
I shall have to hmit, not extend, the popular doctrine. But,
keeping my own theories in the background for the moment, I
win 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 con-
ceivable 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 quota-
tion ^ puts this point of view very well :
" I do not see on what ground it can be doubted that every
1 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 country-
man. Dr. Middleton, should constitute the opposite and remote extremities,
as the former sank 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."
3 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. m FUNDAMENTAL IDEAS 21
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
probabUities 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
Probabihty has had for them. The Calculus of ProbabiHty 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 exist-
ence of which will be demonstrated at a later stage, where
algebraical representation is possible. Probabihty has become
associated, therefore, in the minds of theorists with those problems
in which we are presented with a number of exclusive and ex-
haustive alternatives of equal probabihty ; and the principles, which
are readily apphcable in such circumstances, have been supposed,
without much further enquiry, to possess general vahdity.
3. It is also the case that theories of probability have been
• " Whenever the terms greater and less can be applied, there twice, thrice,
etc., can be conceived, though not perhaps measured by us." — " Theory of Prob-
abilities," Encyclopaedia MetropoUtana, 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.
22 A TEEATISE ON PEOBABILITY pt. i
propounded and widely accepted, according to which its numerical
character is necessarily involved in the definition. It is often
said, for instance, that probabiUty is the ratio of the number of
" favourable cases " to the total number of " cases." If this
definition is accurate, it foUows 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 probabihty. These
definitions and the theories based on them wiU be discussed in
Chapter VIII. ; they are connected with fundamental difEerences
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 probabiUties can be
based on the practice of underwriters and the wilhngness 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 whoUy
rational and determinate ; or that two equally intelKgent 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
CH. m FUNDAMENTAL IDEAS 23
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 imder-
wiiting on these lines that the probabilities of these new taxes
arereaUy measurable by the figures ^, ^, and -^-fj^ ; it is sufficient
that there should be merchants wining 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 in-
solvency 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 possibiHties, the rate rises ; — the probabihty has not
changed, but the " book " is in danger of being upset. A Presi-
dential 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 pohcy. Is it possible to
maintain that these figures in any way represent reasoned
numerical estimates of probabihty ?
In some insurances the arbitrary element seems even greater.
Consider, for instance, the reinsurance rates for the Waratdh,
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 hmits 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 hmits or because it is possible to make a " book "
which covers all possibiUties, and other risks which cannot be
24 A TREATISE ON PROBABILITY pt. i
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 under-
writers 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 hmits, 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 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 plaintifE's brood
mares. In the summer of 1908 the defendant, without the con-
sent 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 com-
pensate 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 ahve and well at the time of the
intended service ; that the mare sent would be well bred and not
barren ; that she would not sUp her foal ; and that the foal would
be born ahve and healthy. In a case of this kind he could only
1 Leibniz note9 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 (SapweU v. Bass).
OH. m FUNDAMENTAL IDEAS 25
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 hne
between an estimate of damage based on probabihties, as in
" Simpson v. L. and N.W. Eailway 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 hne. Having referred to
" Mayne on Damages " (8th ed., p. 70), he pointed out that
in " Watson -y.Ambergah Railway Co." (15, Jur., 448) 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 Erie, 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 plaintifi for nominal
damages, which were all he was entitled to. They would be
assessed at Is.
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 probabiUty is calculated. This case ^ arose out of an ofEer of
a Beauty Prize ^ by the Baily 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 hving 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
1 ChapKn v. Hicks (1911).
^ The prize was to be a theatrical engagement and, according to the article,
the probability of subsequent marriage into the peerage.
26 A TREATISE ON PROBABILITY pt. i
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 capable of assessment,
at £100, the question of the possibihty of assessment being post-
poned. This was argued before Mr. Justice Pickford, and sub-
sequently in the Court of Appeal before Lord Justices Vaughan
WiUiams, Fletcher Moulton, and Earwell. Two questions arose
— relative to what evidence ought the probabiUty to be cal-
culated, 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 begin-
ning 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 in-
calculable." 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
WiUiams 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 imassessable that the
doctrine of averages was not possible of apphcation because the
figures necessary to be apphed were not forthcoming. Several
CH. m FUNDAMENTAL IDEAS 27
cases were to be found in the reports where it had been so held,
but he denied the proposition that because precision 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 cer-
tainty 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 vohtion 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 them-
selves 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/ew^er 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 appUcable, 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 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
^ The jury in aasessing 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.
28 A TKEATISE ON PEOBABILITY pt. i
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 prob-
abihties are in every case even theoretically capable of comparison
in terms of numbers, has not, however, received serious considera-
tion. 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 generahsation. It is usually
held that each additional instance increases the generaUsation's
probabihty. A conclusion, which is based on three experiments
in which the unessential conditions are varied, is more trust-
worthy 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 pecuhar marks in a
picture increases the probabihty 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 Hke a third ; but there will very seldom be
any meaning in saying that it is twice as Uke. Probabihty is, so
far as measurement is concerned, closely analogous to similarity.^
^ 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 an
71+1
induction based on n instances is i. Those who have been convinced bv
m + 2 •'
the reasoning employed to estabUsh this rule must be asked to postpone judg-
ment 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
baaed on a, single instance — a conclusion which this formula would seem to
justify.
^ There are very few writers on probability who have explicitly admitted
that probabilities, though in some sense quantitative, may be incapable of
OH. in FUNDAMENTAL IDEAS 29
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 thunder-
storm bursts upon us, the probabiUty 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 probabihties
numerically comparable ; it might be argued that a knowledge
of the statistics of death by hghtning would make such a com-
parison possible. But if such information is not included within
the knowledge to which the probabiUty is referred, this fact is
not relevant to the probabiUty actually in question and cannot
affect its value. In some cases, moreover, where general statistics
are available, the numerical probabiUty which might be derived
from them is inappUcable because of the presence of additional
knowledge with regard to the particular case. Gibbon cal-
culated his prospects of Ufe 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 probabihties
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 probabihties in an order of magni-
tude, or to say that one is the greater and the other less ?
8. Consider three sets of experiments, each directed towards
estabUshing a generaUsation. The first set is more numerous ;
numerical comparison. Edgeworth, " Philosophy of Chance " (Mind, 1884, p.
225), admitted that " there may well be important quantitative, although not
numerical, estimates " of probabilities. Goldsohmidt {Wahrscheinlichkeitsrech-
nung, 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.
30 A TREATISE ON PROBABILITY pt. i
in the second set the irrelevant conditions have been more
carefuUy varied ; in the third case the generaUsation in view
is wider in scope than in the others. Which of these generaKsa-
tions is on such evidence the most probable ? There is, surely,
no answer ; there is neither equality nor inequahty between
them. We cannot always weigh the analogy against the induc-
tion, or the scope of the generahsation 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
probabiUty 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 mare
likely than not, or less Ukely than not, or as likely as not ? I am
prepared to argue that on some occasions none of these alternatives
hold, and that it wUl be an arbitrary matter to decide for or
against the umbrella. If the 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 primn facie indications of
the existence of a common unit to which the magnitudes of all
probabiKties are naturally referrible. A degree of probabihty
is not composed of some homogeneous material, and is not
apparently divisible into parts of, Hke character with one
another. An assertion, that the magnitude of a given prob-
ability is in a numerical ratio to the magnitude of every
other, seems, therefore, unless it is based on one of the current
definitions of probabihty, with which I shall deal separately
in later chapters, to be altogether devoid of the kind of support,
which can usually be suppKed in the case of quantities of which
CH. m FXJKDAMENTAL IDEAS 31
the mensurability is not open to denial. It will be worth,
while, however, to pursue the argument a Uttle further.
10. There appear to be four alternatives. Either in some
cases there is no probabiUty at all ; or probabihties 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 renmin, unknown ; or probabihties 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 alter-
natives. They argued that every conclusion has its place in
the numerical range of probabihties from to 1, if only we knew
it, and they developed their theory of unknown probabihties.
In dealing with this contention, we must be clear as to what
we mean by saying that a probabihty is unknown. Do we mean
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 probabihty,
not a fuller knowledge of the old one ; we have not discovered
the probabihty of a statement on given evidence, by determining
its probabihty in relation to quite different evidence. We must
not allow the theory of imknown probabihties to gain plausibihty
from the second sense. A relation of probabihty does not yield
us, as a rule, information of much value, unless it invests the
conclusion with a probabihty which hes between narrow numerical
limits. In ordinary practice, therefore, we do not always regard
ourselves as knowing the probabihty 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 probabihty is
unknown.' We might say, for example, that we do not know,
when we go on a railway journey, the probabihty 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 probabihty does not
exist, and not that it is unknown. For it is not this probabihty
32 A TREATISE ON PROBABILITY pt. i
that we have discovered, when the accession of new evidence
makes it possible to frame a numerical estimate.
Possibly this theory of unknown probabiUties 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 probabihty 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 since the probabihty
lies between two numerical measures, we come to beheve that it
must also, if only we knew it, possess such a measure itself.
12. To say, then, that a probabihty 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 probabihty the premisses invest
the conclusion. That probabihties 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 probabihty at all, just as we may through the
same cause estimate a probabihty wrongly. As soon as we
distinguish between the degree of beUef which it is rational to
entertain and the degree of behef actually entertained, we have
in effect admitted that the true probabihty is not known to
everybody.
But this admission must not be allowed to carry us too far.
Probabihty 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 presimie 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 probabihty 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 probabihty, if we do not limit it
CH. m FUNDAMENTAL IDEAS 33
in this way and make it, to tMs 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
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 probabiHty 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 earUer
paragraphs of this chapter, it is theoretically possible on that
evidence to assign a numerical value to the probabiKty, 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 char-
acteristics, 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 probabihties (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 tbink, 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
1 Not altogether ; for it would be natural to select the set to which the
relation of certainty belongs.
D
34 A TREATISE ON PROBABILITY pt. i
widest sense, and am averse to confining its scope to a limited
type of argument. If the opinion that not all probabiUties can
be measured 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 in-
capable of it. The confused attempts, which have been made,
to deal with numerically indeterminate probabilities imder the
title of unknown probabilities, show how difficult it is to
confine the discussion within the intended Hmits, 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 probabihty 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 earher 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 proba-
bilities 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 quanti-
tative properties of probabihty, 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 oxa theory must apply to all probabiUties
and not to a Hmited class only, and that, as we do not adopt a
definition of probabihty which presupposes its numerical men-
surability, we cannot directly argue from differences in degree
to a numerical measurement of these differences. The problem
is subtle and difficult, and the following solution is, therefore,
OH. m FUNDAMENTAL IDEAS 35
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 probabihty, which has an inter-
mediate 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 probabihty 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 behef 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 proba-
bilities, only when they and certainty all lie on the same ordered
series. But if more than one distinct series of probabihties
exist, then it is clear that only those, which belong to the same
series, can be compared. If the attribute ' greater ' as apphed
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 probabihties 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 impossibiUty ; and neither of two proba-
bihties, which he 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 relevant, in such a case a relation seems
clearly to exist between the two in virtue of which one hes
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 probabihties
of every pair of arguments.
36 A TREATISE ON PEOBABILITY pt. i
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 afiorded 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 quan-
titatively 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.
Por 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 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 probabihty 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 land 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
CH. m FUNDAMENTAL IDEAS 37
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 measure-
ment of probabihties 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 wiU be shown that a process
of compounding probabihties 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
probabihty C is equal to the sum of two other probabihties 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 probabihty, and w is a number. The relation of
certainty has been commonly taken as the unit of such con-
ventional measurements. Hence if P represents certainty,
we should say, in ordinary language, that the magnitude of the
probabihty A is i. It will be shown also that we can define a
process, apphcable to probabihties, which has the properties of
arithmetical multiphcation. 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 hmited class of numerical probabilities, seems to be
a part explanation of the behef, which it is the principal object
of this chapter to prove erroneous, that all probabihties must
belong to it.
18. We must look, then, at the quantitative characteristics of
probabihty in the following way. Some sets of probabihties
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
38 A TKEATISE ON PKOBABILITY pt. i
we can only build up these ordered series in special cases. If we
are given two distinct arguments, there is no general presump-
tion that their two probabilities and certainty can be placed
in an order. The burden of estabhshing 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 estabUshed. 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
probabihties are as follows :
(i.) Every probabihty 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
impossibihty or with certainty, that it Ues between
them. Thus certainty, impossibihty 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 inter-
mediate position.
(ii.) A path or series, composed of degrees of probabihty,
is not in general compact. It is not necessarily true,
that is to say, that any pair of probabihties in the
same series have a probabihty between them.
(iii.) The same degree of probabihty can he on more than
one path {i.e. can belong to more than one series).
Hence, if B hes between A and C, and also hes between
A' and C, it does not follow that of A and A' either hes
between the other and certainty. The fact, that the
same probabihty 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 difEerent series of probabihties and their mutual rela-
tions can be most easily pictured by means of a diagram. Let us
represent an ordered series by points lying upon a path, all the
OH. m FUNDAMENTAL IDEAS 39
points on a given path, belonging to the same series. It follows
from (i.) that the points 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 (iv.) that the
same point can lie on more than one path. It is possible, there-
fore, 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 Mth a motion constantly towards the point of
impossibiUty, 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.
represents impossibility, I certainty, and A a numerically
measurable probability inter-
mediate between and I ; U,
V, W, X, Y, Z are non-mmaerical
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 wiU 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 probabiUty or rational
behef 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 and I between
40 A TEEATISE ON PEOBABILITY m. i
whicli all other probabilities lie. If, that is to say, A
is a probability, OAI form a series. represents im-
possibility and I certainty. ^
(iii.) If A lies between 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 probabihty
P to the premiss h, or if, in other words, the hypothesis
h invests the conclusion a with probabihty P, this may
be written aPh. It may also be written ajli='?.
This latter expression, which proves to be the more useful of the
two for most purposes, is of fundamental importance. If aVh
and a'PA', i.e. if the probabihty of a relative to h is the
same as the probabihty of a' relative to h', this may be written
alh^=a' jh'. The value of the symbol ajh, which represents
what is called by other writers ' the probabihty of a,' hes in
the fact that it contains exphcit 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.
CHAPTBE IV
THE PEINCIPLE OP INDIPPERENCE
ABSOiiTJTE. ' Sure, Sir, this is not very reasonable, to summon my afEeotion
for a lady I know nothing of.'
Sib ANTEONr. ' 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 assmned that in some cases the
probabilities of two arguments may be equal. It was also argued
that there are other cases in which one probabihty is, in some
sense, greater than another. But so far there has been nothing
to show how we are to know when two probabiUties are equal or
unequal. The recognition of equahty, 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 iu Chapter VII.
2. The determination of equality between probabilities has
received hitherto much more attention than the determination
of iaequality. 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, there-
fore, 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.
1 Quoted by Mr. Bosanquet with reference to the Principle of Non-Sufficient
Reason. ' See also Chap. VII.
41
42 A TREATISE ON PROBABILITY pt. i
The Principle of Indifierence 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 Wahrschem-
Uchkeit.^
3. If every probability was necessarily either greater than,
equal to, or less than any other, the Principle of Indifference
wotdd 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 prob-
abilities, 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 — ^whUe 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 we have no outside
knowledge, it is inferred that the probability of each is ^.^ In
^ Published in 1886. A briel account of Von Kries's principal conclusions
will be given on p. 87. A useful summary of his book will be found in a review
by Meinong, pubUshed in the Gottingische gelehrte Anifeigen, for 1890 (pp. 66-75).
' Cf. (e.g.) the well-known passage in Jevons's Principles of Science, voL i.
p. 243, in which he assigns the probability } to the proposition " A Platythliptio
Coefficient is positive." Jevons points out, by way of proof, that no other
CH. IV FUNDAMENTAL IDEAS 43
the same way the probabilities of two other propositions, b and c,
having the same subject as a, may be each ^. But without
having any evidence bearing on the subject of these propositions
we may know that the predicates are contraries amongst them-
selves, 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 ^ is the proba-
bility of ' This book is red,' we could conclude equally that the
probability of each of the propositions ' This book is black ' and
' This book is blue ' is also J. So that we are faced with the
impossible case of three exclusive alternatives all as likely as not.
A defender of the Principle of IndifEerence might rejoin that we
are assuming knowledge of the proposition : ' Two difEerent
colours cannot be predicated of the same subject at the same
time ' ; and that, if we know this, it constitutes relevant out-
side evidence. But such evidence is about the predicate, not
about the subject. 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 ; 56 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=^ and ab/h=^, if A is
probability could reasonably be given. This, of course, involves the assumption
that every proposition must have some numerical probability. Such a con-
tention 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 pub-
lished work on probabiUty : " 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 J, but by the indefinite form %." Jevons's
particular example, 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 J to the proposition
' A Platythliptic Coefficient is a perfect cube ' ? What about the proposition
' A Platythliptio Coefficient is allogeneous ' ?
44 A TREATISE ON PROBABILITY m. i
irrelevant both to a and to b, in the sense required by the crude
Principle of IndifEerence.^ It follows from this that, if a is true,
b must be true also. If it foUows from the absence of positive
data that 'A is a red book' has a probability of ^, and that the
probability of ' A is red ' is also ^, then we may deduce 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 ^. "Whether or not this conclusion discredits the
Principle 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 certaiu
alternative predicates. If, to take an example, we have no
information whatever as to the area or population of the
coimtries of the world, a man is as likely to be an inhabitant
of Great Britain as of France, there beiag no reason to prefer
one alternative to the other.^ He is also as likely to be an
inhabitant of Ireland as of France. And on the same principle
he is as Ukely 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 sub-
divisions which are within our knowledge, is, iu the absence of
any evidence as to their size, a piece of relevant evidence.
At any rate, many other similar examples could be invented,
^ ajh 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 plausibility than against Von Kries's.
OH. IV FUKDAMENTAL IDEAS 45
wMch would reqtiire 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 some-
what different side.
6. Consider the specffic volume of a given substance.^ Let us
suppose that we know the specific volume to he 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 he 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 ^.
Our data remaining as before, we know that the specific density
must lie between 1 and ^, and, by the same use of the Principle
of Indifference as before, that it is as Hkely to he between
1 and f as between f and ^. But the specific volume being
a determinate function of the specific density, if the latter hes
between 1 and |, the former Hes between 1 and 1|, and if the
latter lies between | and ^, the former Hes between 1| and 3.
It follows, therefore, that the specific volume is as Hkely to He
between 1 and IJ as between If and 3 ; whereas we have already
proved, relatively to precisely the same data, that it is as Hkely
to He 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 jdelds sub-intervals of equal probabiHty.
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
appHcation of the Principle of Indifference a different probabiHty
for a given objective variation in the quantity.^
1 This example is taken from Von Kries, op. cit. p. 24. Von Kjies does
not seem to me to explain correctly how the contradiction arises.
' A. Nitsche ("Die Dimensionen der WahrsoheinUohkeit nnd die Evidenz der
UngewiBsheit," VierteJjah-sschr. f. maaenach. Philoa. vol. xvi. p. 29, 1892), in
46 A TREATISE ON PROBABILITY pt. i
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 neces-
sary 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 contimMus, 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 quantitative-
ness 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 character-
istics, 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
oritioising Von Kries, argues that the alternatives to which the principle must
be applied are the smallest physically distinguishable intervals, and that the
probabiUty 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 Indifierence. 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 Ues 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 conven-
tional basis as the methods of specific volume and specific density, and is no
more the one correct measure than these are ?
OH. IV FUNDAMENTAL IDEAS 47
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 difEerences depend upon which convention of measure-
ment 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 posi-
tion of a point or infinitesimal area or volume within a con-
tinuimi.^ 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 volvmies of the continmmi 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 Ijdng 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 probabiUty that it will be less
than the side of the inscribed equilateral triangle. One can
argue : —
(a) It is indifferent at what point one end of the chord lies.
If we suppose this end fixed, the direction is then
^ The best accounts of this subject aie to be found in Czuber, Oeometrische
Wahrscheinlichkeiten und Mittelwerte ; Czuber, Wahrscheinlichkeitsrechnung,
voL i. pp. 75-109; Crofton, Encyd. Brit. (9th edit.), article 'Probability';
Borel, Elements de la theorie des probabilitis, chapa. 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 {Wahrscheinlichkeitsre<Anung, 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 Lammel, Vntersuchungen.
48 A TEEATISE ON PEOBABILITY pt. i
chosen at random. In this case the answer is easily
shown to be |.
(6) It is indifEerent in what direction we suppose the chord
to he. Beginning with this apparently not less justifi-
able assumption, we find that the answer is ^.
(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 | of the whole. Hence
our answer is |. ^
In general, if x and /(a;) are both continuous variables, varying
always in the same or in the opposite sense, and x must he
between a and b, then the probability that x lies between c
d — c
and d, where a<c<d<b, seems to be ^ 'and the probabihty
that f(x) hes between /(c) and f{d) to be -^tjt — ^^-r. These
expressions, which represent the probabihties 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
apphed to examples of geometrical probabihty from those in
which it cannot. M. Borel argues that the mathematician can
define the geometrical probabihty that a point M hes 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 d 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 others.
Whether or not this is so, the proposed procedure amounts to
an abandonment of the Principle of IndijEEerence as a vahd
criterion, and leaves our choice undetermined when further
evidence is not forthcoming.
M. PoLQcare, who also held that judgments of equiprobabihty
in such cases depend upon a ' convention,' endeavoured to mini-
^ Beitrand, Calcul dea probabilitea, p. 5.
' See {e.g.) Borel, MUmerUs de la theorie des probabilites, p. 85.
OH. IV FUNDAMENTAL IDEAS 49
mise the importance of the arbitrary element by showing that,
under certain conditions, the result is independent of the particu-
lar convention which is chosen. Instead of assuming that the
point is equally hkely to he in every infinitesimal interval dx
we may represent the probabihty of its Ijnng in this interval by
the function ^(x)dx. M. Poincare showed that, in the game of
rouge et novr, for instance, where we have a number of compart-
ments arranged in a circle coloured alternately black and white,
if we can assume that <^{x) is a regular function, continuous and
with continuous differential coefficients, then, whatever the
particular form of the function, the probabihty of black is
approximately equal to that of white.^
Whether or not investigations on these hues prove to have
a practical value, they have not, I think, any theoretical import-
ance. If, as I maintain, the probabihty ^{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 probabihty is very nearly equal to that
of the white. Naturally in such a case we can get an approxi-
mately equal probabihty for the whites as a whole and for the
blacks as a whole, without assuming equal probabihty 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
geometrical probability that he wishes to exclude aU examples
in which the number of alternatives is infinite? It will be argued
in the sequel that something resembUng this is true. The dis-
cussion 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 baUs in an
^ Poinoar6, Calcul des probabilites, pp. 126 et seq.
^ Bertrand, Calcul des probabiUtea, p. 4: "L'infini n'eat pas un nombre;
on ne doit pas, sans explication, I'introduire dans les raisonnements. La
precision Ulusoire des mots pourrait faire naStre des contradictions. Choisir
au hasard, entie un nombre infini de cas possibles, n'est pas mie indication
sufSsante."
E
50 A TEEATISE ON PEOBABILITY pt- i
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 aU possible constiivAions ^
of the system of balls are equally probable, so that each individual
ball is assumed equally likely to be black or white. It wotild
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 proba-
bihties 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 hypo-
theses seems to satisfy the Principle of Indifference, and a believer
ia 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 Kkely
to be black as red. If we prefer this assumption, we must relin-
^ The diffloulty 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 invol-riug 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 Stump^
Vber den Begrijf 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 oases are indistinguishable, and count as one only.
^ 0. S, Peiroe in hia 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 h3rpotheses, only this one is consistent with itself.
I agree with his conclusion, and shall give at the close of the chapter the funda-
mental considerations which lead to the rejection of the ' ratio ' hypothesis.
Stumpf points out that the probabihty of drawing a white ball is, in any
case, ^. 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.
OH. IV FUM)AMENTAL IDEAS 51
quish. 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 i£ 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 |.^ The Principle of Indifference
has nothing to say against either solution. Until some further
criterion has been proposed we seem compelled to agree with
Poincar6 that a preference for either h3rpothesis 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
origiaal 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 ride 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 applica-
tion of the Calculus. Many candid persons, when confronted
with the results of Probability, feel a strong sense of the un-
certainty of the logical basis upon which it seems to rest. It is
difficult to find an inteUigible account of the meaning of ' proba-
bility,' 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
1 This is Foisson's solution, Becherches, p. 96.
52 A TREATISE ON PROBABILITY w. i
to be mucli 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."
Yet such scepticism goes too far. The judgments of proba-
bility, 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
" hvely 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 dis-
cussion 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 probabiHty, 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 /oJfows /row 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
CH. IV FUNDAMENTAL IDEAS 53
of probability must be, in the fixst instance, directly recognised
as such, and cannot be evolved by rule out of data which them-
selves contain no statements of probability.
On the other hand, although we cannot exclude every element
of direct judgment, these judgments may be limited and con-
trolled, perhaps, by logical rules and principles which possess a
general application. While we may possess a faculty of direct
recognition of many relations of probabihty, 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 probabihty 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 probabihty
of drawing a black ball from an urn containing balls which are
1 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, -yhich
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.
54 A TREATISE ON PROBABILITY pt. i
black and white, we assume that the difference of colour be-
tween 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 baUs 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 un-
affected 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, imless 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 re-
quired, 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 likeUhood of a given conclusion. In symboUc
language we may wish to compare xjh with yjh, or x/h with
xlhjh. 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 xjh and x/hji, of irrelevance. In the first
we consider whether or not a; is to be preferred to y on evidence h ;
in the second we consider whether the addition of h^ 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,
OH. IV FUKDAMENTAL IDEAS 55
involves, if it is to be a guiding rule at all, and not a fetitio
'princifii, an appeal to judgments of irrelevance.
Th& simplest definition of Irrelevance is as follows: h^ is
irrelevant to x on evidence h, if the probability of x on evidence h\
is the same as its probabihty on evidence h,}- But for a reason
which will appear in Chapter "VT., a stricter and more complicated
definition, as foUows, is theoretically preferable : \ is irrelevant
to X on evidence h, if there is no proposition, inferrible from hji
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 avoid-
ing this proves in the sequel more convenient. If the condition
xlhjt=xlh alone is satisfied, we may say that the evidence h^
is ' irrelevant as a whole.' ^
It will be convenient to define also two other phrases. Aj
and ^2 S'le 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 h^ and h^ are independent
and complementary parts of the evidence, then h^ is relevant if
the addition of it to h^ affects the probability of a;.*
Some propositions regarding irrelevance wiU be proved in
Part II. If K^ is the contradictory of h^ and xlhyh=xjh, then
x/JiJi^xjh. Thus the contradictory of irrelevant evidence is
also irrelevant. Also, if xjyh=x/h, it foUows 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
"■ That is to say, h^ is irrelevant to x/h if xlhji,=xjh.
^ That is to say, h^ is irrelevant to x/h, if there is no proposition h\ such that
h'Jhjh = l, h\lh=¥l, and xjh'-JidFXJh.
^ Where no misunderstanding can arise, the qualification ' as a whole ' will
be sometimes omitted.
* /.e (in symbolism) \ and 1i^ are independent and complementary parts of
h if ^1^2 =fe, ^1/^2=1=1, and ^2/^14=1. Also h^ is relevant if xjli^xlh^.
56 A TREATISE ON PROBABILITY ft. i
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 rele-
vance, 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 symbohc
language. Let us assume, to begin with, that the alternative
conclusions are expressible in the forms ^(a) and ^(6), where
^{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
<^{a) and </)(6), when the evidence h is so constituted that, if f{a)
is an independent part of h (see § 14) which is relevant to <^{a),
and does not contain any independent parts which are irrelevant
to (^(os), then h includes /(6) also.
The rule can be extended by successive steps to cases in
which we have more than one variable. We can, if the necessary
conditions are fulfilled, successively compare the probabihties
of ^{a^a^ and <\>{\a^, and of <}>{b^a.^) and ^{b^b^), and establish
equaUty between ^{cij^a,^) and ^{bjb.^).
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 appHed, 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 equi-
probability of each ' constitution ' is alone legitimate, and the
^ If 0(o), 0(6), etc., are propositions, and a; is a variable, capable of taking
the values a, b, etc., then ^(a;) is a propositional function.
OH. IV FUNDAMENTAL IDEAS 57
equiprobability of each numerical ratio erroneous. Let us write
the alternative ' The proportion of black balls is x '=(j){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 indifEerence, <f){x)lh =
^{y)lh. In order that this may be vahd, it must be possible to
state the relevant evidence in the form J{x) f{y). But this is
not the case. If a; = ^ and y = ^, we have relevant knowledge
about the 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 of irrelevance can we
ignore them. On the ' constitution ' hypothesis, however,
where A, B black and A, C black are treated as distinct alter-
natives, 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 probabiUties
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 meaniag 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
58 A TEEATISE ON PEOBABILITY ft. i
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 wJiat particular proposition it stands for ; for, as
we have seen, the proposition itseK 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 probabihty but
with truth, what is true of a variable must be equally true of all
instances of the variable. In Probability, on the other hand,
we must be on oux guard wherever a variable occurs. In Im-
plication we may conclude that a^ is true of anything of which
(j) is true. In Probability we may conclude no more than that
■yfr is probable of anything of which we only know that <ji is true of
it. If X stands for anything of which (ji(x) is true, as soon as
we substitute in probabihty any particular value, whose meaning
we know, for x, the value of the probabihty may be affected ;
for knowledge, which was irrelevant before, may now become
relevant. Take the following example : Does t^{a)l-^{a) =
<j}{b)/tjr(b) 1 That is to say, is the probabihty of ^'s being true
of a, given only that i/r is true of a, equal to the probabihty of
<^'s being true of b, given only that yfr is true of 6 ? If this simply
means that the probabihty of an object's satisfjdng (f> about
which nothing is known except that it satisfies yjr is equal to
ditto ditto, the equation is an identity. For in this case <j){a)l'\fr{a)
means the same as <j)(b)l-\lr{b), i.e. we know nothing about x and y
except that they satisfy t/t, and there is nothing whatever by
which we can distinguish a from 6. But if a and b represent
specific entities, which we can distinguish, then the equalijby
does not necessarily hold. If, for instance, <f){x) stands for ' x is
Socrates,' then it is plainly false that <ji{a)/-\lr{a) = <l>{b)j'\]r{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
accoimt of so far as it is relevant ; and the Principle is only satis-
fied if we have corresponding knowledge about the meaning of b.
Thus <f){a)/h = (j)(b)jh may be true for one pair of values a, b, and
not true for another pair of values a', b'.
OH. IV FUNDAMENTAL IDEAS 59
This makes it possible to explain in part the contradiction
discussed in § 4. Even if it were true that the probability of a is
\, when we know nothing except that a is a proposition, it does
not follow that the probabihty of ' This book is red ' is |, when
we know the meanings of ' book ' and ' red,' even if we Imow no
more than this. Knowledge arising directly out of acquaititance
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 wiU be discussed in §§ 20, 21.
But the contradictions are not yet really solved ; for some
of the difficulties discussed iu § 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 Priaciple 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 iudefinite 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 Kkely as its contradictory, its con-
tradictory is not an ultimate and indivisible alternative (in the
sense to be explained in § 21 below), even if the proposition itseH
satisfies this condition. For its contradictory can be disjunct-
ively 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, there-
fore, an ultimate alternative.
In the same way the contradiction of § 5 arose out of the possi-
biUty of sphtting the alternatives ' He inhabits the British
60 A TEEATISE ON PKOBABILITY pt. i
Isles ' into the sub-alternatives ' He inhabits Ireland or he
inhabits Great Britain.' 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 Ues in the interval 1 to If or 1 J to 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 t}>{aj),
^(fflg) . . . ^(ffly),^ and let the evidence be h. Then it is a neces-
sary condition for the appUcation of the principle, that these
should be, relatively to the evidence, indivisible alternatives of
the form <j>(x). We may define a divisible alternative in the
following manner :
An alternative <^(a^) is divisible if
(i.) [</,(«,) ^<^(a,,) + «^K")]A=l.
(ii.) <p{a^) . (f)(a^.)/h = o,
(iii.) <f){a^')lh4=o and ^{a^.^jhJrO
The condition that the sub-alternatives must be of the same,
form as the original alternatives, i.e. expressible by means of the
same prepositional function ^(a;), deserves attention. It might
be the case that the original alternatives had nothing substantial
in common ; i.e. (j}{x) = a; is the only propositional function
common to aU of them, the alternatives being %, a^, . . ., a^. In
these circumstances the condition in question cannot be satisfied.
For the proposition a^ can always be resolved into the disjunction
ajb +a^E, where b is any proposition and 5 its contradictory. If,
on the other hand, the alternatives which we are comparing can
be expressed in the forms <f>(a^) and ^{a^, where the function
<f)(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
^ 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 61
alternative </)(a^) into a disjunction of two sub-alternatives, (i.)
which are themselves expressible in the same form (p, (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 funda-
mental 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 impUcit 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 con-
tradictions 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 pro-
position and its contradictory constitute the alternatives. For
if b is the proposition and B its contradictory, we cannot find
a propositional function ^(a;) which will satisfy the necessary
conditions. It deals also with the type of contradiction which
arose in considering the probabiUty 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 <j>(x), where <j>(x) = ' the individual is an in-
habitant of coimtry x,' satisfies the conditions. And it deals
with the example of ranges of specific volume and specific density,
62 A TREATISE ON PROBABILITY pt. i
because there is no range wMch does not contain vsdthin itself two
similar ranges. As there are in this case no definite units by
which we can define eqvul ranges, the device, which will be referred
to in § 25 for dealing with geometrical probabilities, is not avail-
able.
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 alter-
natives which satisfy the conditions. For if the original alterna-
tives 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 sub-
division 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 tf>(x)=x, or where a; is a part of un-
specified 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 ra 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 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 Ues
on a line of length m.l., we may write the alternative ' the interval
of length I on which the point lies is the art;h interval of that
length as we move along the line from left to right ' =<f>{x) ; and
the Principle of Indifference can then be applied safely to the m
alternatives ^(1), <^(2) . . . ^(m), the number m increasing as the
length I of the intervals is diminished. There is no reason why
I should not be of any definite length however small.
If we deal with the problems of geometrical probability in
CH. IV FUNDAMENTAL IDEAS 63
tHs 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 h3rpotheses lead inevitably to different resiilts. If
we were deahng with a strictly linear chord, the Priaciple of
Indifference would 3deld 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 alter-
natives in a form to which the Principle of Indifference can be
applied unambiguously, we shall be prevented from confusing
together distinct problems, and shall be able to reach conclusions
in geometrical probability which are imambiguously valid.
The substance of this explanation can be put in a sHghtly
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 imambiguously. 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 hmit. Mathemati-
cians will recognise an analogy between this case and the deter-
mination 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 deter-
mination of valid judgments of equiprobabihty 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 bringiug 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
64 A TKEATISE ON PROBABILITY pt. i
direct judgments of relevance and irrelevance can lead on to
judgments of preference and indifference. In the second place,
some tjrpes of consideration, which are in fact relevant, but which
are in danger of being overlooked, have been brought into promi-
nence. 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
niunerical, 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 iu 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 ia 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 incHned 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 abjh and ajh,
or (ii.) of the type a/hh^ and ajh,
provided the additional evidence h^ contains only one inde-
pendent piece of relevant information.
* Parts of Chap. XV. are closely connected with the topics of the follow-
ing paragraphs, and the discussion which is commenced here is concluded there.
65 P
66 A TEEATISE ON PROBABILITY pt. i
(i.) The propositions of Part II. will enable us to prove that
ahjh < 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 probabihty a/hh^ is always greater than, equal to,
or less than the probabiHty a/h, if h^ contains no pair of comple-
mentary and independent parts ^ both relevant to a/h. If h^
is favourable, ajhh^ > a/h. Similarly, ii h^is favourable to a/hhi,
a/hhji^ > a/hh^. The reverse holds if h^ and h^ 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 favour-
able and part xmf avoiurable, 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 un-
affected 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 unfavour-
able, 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.
Other comparisons are possible by a combination of these
two principles with the Principle of Indifference. We may
find, for instancCp that a/hh^>a/h, that a/A =6/A, that b/h>b/hh^,
and that, therefore, a/h\>b/h\. We have thus obtained a
comparison between a pair of probabilities, which are not
of the types discussed above, but without the introduction
'^ See Chap. IV. § 14 for the meaning of these terms.
CH. V FimDAMENTAL IDEAS 67
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 imdoubtedly 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 Mnd 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 behef in Romulus
mthovt 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
T|r^(x); *the historical tradition x has been handed down from
the time of Caesar' by 1/^2(05) ; ' the historical tradition x has
extra-traditional support ' by 1/^3(3;) ; and the two traditions,
the Romulus tradition and the Caesar tradition respectively,
by a and b. Then if our relevant evidence h were of the form
■>^j(a)i/r2(6)i|r3(6), it is easily seen that the comparison alh<b/h
could be justified on the lines laid down above.^ A further judg-
ment, that our actual evidence presented no relevant divergence
from this schematic form, would then estabUsh 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
1 This phrase is used by Von Eiies, op. cit. p. 179, in a somewhat similar
connection.
2 For alf^{a,) = bH^^{b); o/^('i(o) < o/f j(o) ; blf^{b)<cblMb)Mb);
a/i'i{a) = alh ; and b/xl'^{b)^,{b) = b/h.
68 A TEEATISB ON PEOBABILITY pt. i
doubting the possibility oi comparing the probabilities of argu-
ments 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 argu-
ment 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 hj^ to a doubtful ^ argument
a/h is favourably relevant, if either of the following conditions
is fulfilled : — (a) if ajh\=(i ; (6) if ajhh^ = l. Divested of sym-
bolism, 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
favourable to our conclusion which is favourable to favourable
CAddence — i.e. that, if A^ is favourable to x/h and x is favourable to
a/h, hy is favourable to ajh. 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 iavolved in the above supposition, Mr. Johnson has
suggested to me the name of the Fallacy of the Middle Term. The
general question — ^If h^ is favourable to x/h and x is favourable to
alh, m what conditions is h^ favourable to ajh 1 — will be examined
ia Chapter XIV. §§ 4 and 8 (49.1). In the meantime, the iutui-
tion 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 h^ and a be concordamt in regard to each of
1 Aj is new evidence so long as Aj/A + 1.
^ The argument is doubtful so long as ajTi is neither certain nor impossible.
CH. V FUNDAMENTAL IDEAS 69
tkese alternatives : i.e. any hypothesis whicli is strengthened by
h^ wUl strengthen a, and any hypothesis which is weakened by
h^ will weaken a. It is obvious that, if h^ 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 A^
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 shaU show) is
obviously satisfied iu 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
h^ would not weaken the probability of a.
These two conditions are sufficient to ensure that h^ 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 h^ 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 analogous to the requirement of distribution of the middle
term in ordinary syllogism. Thus, from premises " All P is a;,
all S is X," the conclusion that " S's are P " does not formally
follow ; but given " all P is a; 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 : aU a; S is
P ; aU a;' S is P ; aU a;" S is P ; . . . therefore all S is P.
Fi/rst Typical Problem. — ^An urn contains an unknown pro-
portion of differently coloured balls. A ball is drawn and replaced.
Then x, x' , x" . . . stand for the various possible proportions.
Let h^ 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 h^ wiU strengthen a; and any hypothesis
which is weakened by h^ wUl weaken a. Moreover, were any
one of these hypotheses known to be true, the knowledge of h^
would not weaken the probabiUty of a. Hence, in the absence
of definite knowledge as regards x, x', x" . . ., the knowledge
of Aj would strengthen the probability of a.
Second Typical Problem. — ^Let a certaiti event have taken
place ; which may have been x, x', x" or . . . Let A^ mean that
A reports so and so ; and let a mean that B reports similarly or
70 A TREATISE ON PROBABILITY pt. i
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 natxire 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 h^ would strengthen
the probability of a.
5. Before leaving this part of the argument we must emphasise
the part played by direct judgment ia the theory here presented.
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 argu-
ment 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 magni-
tude of the probability of the argument may either decrease or
increase, according as the new knowledge strengthens the un-
favourable 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 argu-
ment. 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 in-
cludes and exceeds the evidence in the other. If the new evidence
71
72 A TREATISE ON PROBABILITY pt. i
is ' irrelevant,' in the more precise of the two senses defined in § 14
of Chapter IV., the weight is 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 defini-
tion, 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 a priori probability. In
this case the weight of the argument is at its lowest. Start-
ing, therefore, with minimum weight, corresponding to d 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 wiU 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 probabihty 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 ^a/-\jra.\ = <f)il'\}rb,h^,
where h^ and h^ are irrelevant to the arguments,
(h.) ajhh^^ajh, where h^ is a single unit of information,
containing no independent parts which are relevant.
(iii.) ablh^ajh.
Let us represent the evidential weight of the argument,
whose probability is ajh, by Y{alh). Then, corresponding to
OH. VI FUNDAMENTAL IDEAS 73
the above, we find that the following comparisons of weight are
possible : —
(i.) Y{<f)a/'\lra.h^) =Y{(f>b/-\lrb.h2), where h^ and h^ are irrelevant
in the strict sense. Arguments, that is to say, to which the
Principle of IndifEerence is applicable, have equal evidential
weights.
(ii.) Y{a/hhj)>Y{ajh), unless h^ is irrelevant, in which case
V(a/AAj)=V(a/A). The restriction on the composition of \,
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 corre-
sponding to (iii.) above. It might be thought that Y{abjh) < V(a/A),
on the ground that the more comphcated an argument is, relative
to given premisses, the less is its evidential weight. But this
is invalid. The argument abjh is further off proof than was the
argimient a/h ; but it is nearer disproof. For example, if ab/h =
and ajh>0, then V(a6/A)>V(a/A). In fact it would seem to
be the case that the weight of the argimient a/h is always
equal to that of d/h, where a is the contradictory of a ; i.e.,
Y(a/h)=Y{d/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 ajh = bjh, it does not neces-
sarily follow that Y{a/h)=Y{blh). It has been asserted already
that if the first equaUty f oUows directly from a single appHcation 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 ajh = b/h ; but
these argimients 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/A) = Y{bjh). For in proving or dis-
proving one, 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 con-
fidence. Suppose, for instance, that we have three exclusive
and exhaustive alternatives, a, b, and c, and that ajh = bjh
in virtue of the Principle of Indifference, then we have
Y{a/h) = Y{b/h) and Y{a/h) = Y{d/h), so that Y{b/h) = Y{djh). It is
74 A TEEATISE ON PROBABILITY m. i
also true, since dj{b + c)h = l and {b + c)lah = l, that V(a/A) =
V((6 + c)lh). Hence V(6/A) = V({6 + c)/A).
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 xjhjiz^^ and xj\=^, it has sometimes
been supposed that it is more probable that xjhji^ really is f than
that x/hi really is |. According to this view, an increase in the
amount of evidence strengthens the probability of the proba-
bility, or, as De Morgan would say, the presumption of the
probability. A Uttle reflection will show that such a theory is
untenable. For the probability of x on hypothesis hy is inde-
pendent 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 h^ the probabiUty of x is f . Simi-
larly the fact that xjh^^ is f does not impugn the conclusion that
xjh^ is I, and unless we have made a mistake in our judgment or
our calculation on the evidence, the two probabilities are f and f
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 ques-
tion. ' Probable error,' it should be explained, is the name
given, rather inconveniently perhaps, to an expression which
arises when we consider the probabiUty 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 niagnitude 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 hkely as not to exceed,
is the ' probable error.' In many practical questions the exist-
ence of a small probable error is of the greatest importance,
if our conclusions are to prove valuable. The probability that
CH. VI FUNDAMENTAL IDEAS 75
the quantity has any particular magnitude may be very small ;
but this may matter very Uttle, 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 difEerent 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 magni-
tudes 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 probabiUty of a; = 5 is J, of a; = 6 is \,
of a; = 7 is i, of a; = 8 is ^, and of a; = 9 is xir. Additional evidence
might show that x must either be 5 or 8 or 9, the probabilities of
each of these conclusions being iV> Aj tt- The evidential weight
of the 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 ^radical 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 propor-
tions ; in the second case the proportion of each colour is unknown,
and each ball is as Hkely to be black as white. It is evident that
in either case the probability of drawing a white ball is \, but
that the weight 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, it after
76 A TEEATISE ON PKOBABILITY pt. i
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 hjrpothesis 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
hkely, 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 accom-
panied 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 Ukely to be right ' than
one of low weight ; for the probabihties of these arguments only
state relations between premiss and conclusion, and these re-
lations are stated with equal accuracy in either case. Nor is an
argument of 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 e\ddence 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 apphcation of probabUity
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 mijch
practical significance.
BernoulH'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
I See also Chapter XXVI. § 7.
* Cf. Locke, Essay concerning HvMMn Understanding, hook ii. chap. xxi. § 67:
" He that judges without informing himself to the utmost that he is capable,
cannot acquit himself of judging amiss."
OH. VI FUKDAMENTAL IDEAS 77
difficult to see, however, to what point the strengthening of an
argument's weight by increasing the evidence ought to b^ pushed.
We may argue that, when our knowledge is slight but capable of
increase, the course of action, which will, relative to such know-
ledge, probably produce the greatest amount of good, will often
consist La 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 hm far we ought
to carry our maxim of strengthening the weight of our argument.
A little reflection will probably con-vince 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 proposi-
tions that one set embodies Tnore 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 sym of the favourable and unfavour-
able 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
probabiUty 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 possibiUty is contracted.
The same distinction may be explained in the language of the
frequency theory.^ We should then say that the weight is in-
creased 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
1 See Chap. VIII.
78 A TEEATISE ON PROBABILITY n. i
by writers on probability, and I know of only two by whom the
question, has been expHcitly raised : ^ Meinong, who threw out , a
suggestion at the conclusion of his review of Von Kries' "Princi-
pien," published in the Gottingische gelehrte Anzeigen for 1890
(see especially pp. 70-74), and A. Nitsche, who took up Meinong's
suggestion in an article in the Vierteljahrsschrifi fiir wissenschaft-
liche Philosophie, 1892, vol. xvi. pp. 20-35, entitled "Die Dimen-
sionender Wahrscheinlichkeit und die Evidenz der Ungewissheit."
Meinong, who does not develop the point in any detail, dis-
tinguishes probabihty and weight as ' Intensitat ' and ' Qualitat,'
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 reliabihty (Sicherheit) of the probability, and
holds that the probabihty 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.
^ There are also some remarks by Czuber {Wahrscheinlichheitsrechnung,
voL i. p. 202) on the Erkenntnisswert of probabiUties obtained by different
methods, which may have been intended to have some bearing on it.
CHAPTEE VII
HISTORICAL BETEOSPEOT
1. The characteristic features of our Philosophy of Probability-
must be determined by the solutions which we offer to the
problems attacked m Chapters III. and IV. Whilst a great part
of the logical calculus, which will be developed in Part II., would
be applicable with sKght modification to several distinct theories
of the subject, the ultimate problems of establishing the premisses
of the calculus bring into the Ught 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
1 " 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 fuU 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."
79
80 A TEEATISE ON PROBABILITY pt. i
and of the Uniformity of Nature. As Aristotle says, " the
probable is that which usually happens." Events do not always
occur in accordance with the expectations of experience ; but
the laws of experience afEord 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 modem manner : "In order for me to
judge of the truth of an event, and to be determined to believe
it or not beheve 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 ij;, 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 beheve it. This being done, if all the circumstances are
such that it never or rarely happens that the hke circumstances
are the concomitants of falsehood, our mind is led, naturally,
to beheve that it is true."^ Locke follows the Port Royal
Logicians very closely : " Probability is hkeUness 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 hkeness to
some other event, which we have observed has come to pass.
And this observation forms, in numberless instances, a pre-
sumption, opinion, or full conviction that such event has or will
come to pass." ^
Against this view of the subject the criticisms of Hume were
directed : " The idea of cause and efEect is derived from experi-
ence, which informs us, that such particular objects, in all past
1 Eng. Trans., p. 353.
^ An Essay concerning Human Understanding, book iv. " Of Knowledge and
Opinion."
' Introduction to the Analogy.
OH. vn FUKDAMENTAL IDEAS 81
instances, have been constantly conjoined with each other. . . .
According to this account of things . . . probabihty 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 probabihty." ^ "When we are accustomed to see two impres-
sions conjoined together, the appearance or idea of the one im-
mediately carries us to the idea of the other. . . . Thus aU prob-
able reasoning is nothing but a species of sensation. 'Tis not
solely in poetry and music, we must follow our taste and senti-
ment, but hkewise 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 feehng concerning the superi-
ority of their influence." ^ Hume, in fact, points out that, while
it is true that past experience gives rise to a psychological anticipa-
tion of some events rather than of others, no ground has been
given for the vahdity 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 probabihty 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 approxi-
mately 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 them-
selves to precise numerical appraisement. Accordingly James
Bernoulh,' the real founder of the classical school of mathematical
probabihty, 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 aU the possible conclusions to a number
of ' equi-probable cases.' And, according to the Principle of
^ Treatise of Human Nature, p. 391 (Green's edition).
» Op. cit. p. 403.
^ See especially Ars Oonjectandi, p. 224. Cf. Laplace, Theorie analytique,
p. 178.
G
82 A TREATISE ON PROBABILITY pt. i
IndifEerence, ' cases ' are held to be equi-probable 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 ns to assimie 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
probabiUties 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 IndifEerence, Laplace
established his Law of Succession by which the influence of any
experience, however Umited, 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 wiU again accompany A on A's
next appearance. No other formula iu the alchemy of logic
has exerted more astonishing powers. For it has established
the existence of God from the premiss of total ignorance ; and it
has measured with nimierical precision the probability that the
sun win rise to-morrow.
Yet the new principles did not win acceptance without
opposition. D'Alembert,^ Hume, and AnciUon ^ stand out as
the sceptical critics of probability, against the credulity of
^ Wah/rscheinlichkeitsreclmwng, p. 9.
' D'Alembert's scepticism was directed towaids the cuireat mathematical
theory only, and was not, like Hume's, fundamental and far-reaching. Hia
opposition to the received opinions was, perhaps, more splendid than dis-
criminating.
' AnciUon's communication to the Berlin Academy in 1794, entitled Doutes
sur les bases du calcul des probabilites, 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,
AnciUon was, at one time, the Prussian Minister of Foreign Affairs.
OH. vn FUNDAMENTAL IDEAS 83
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 Eeason.^
. 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. " Prob-
abihty," 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 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,
estabhshed 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
vaUdity.
4. The reaction against the traditional teaching during the
past hundred years has not possessed sufficient force to displace
^ French philosophy of the latter half of the eighteenth century was pro-
foundly 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 Condoroet evolved his doctrine
of the perfectibility of the human race. The continuity and oneness of
modem European thought may be illustrated, if such things amuse the
reader, by the reflection that Condoroet derived from BernouUi, that Godwin
was inspired by Condoroet, that Malthus was stimulated by Godwin's foUy
into stating hia 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).
3 Op. cit. p. 425.
84 A TREATISE ON PROBABILITY pt. i
the established doctrine, and the Principle of IndifEerence is
still very widely accepted in an unqualified form. Criticism
has proceeded along two distinct lines ; the one, originated by
LesHe 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
ia Germany, where its ablest exponent has been Von Eaies.
France has remained uninfluenced by either, and faithful, on
the whole, to the tradition of Laplace. Even Henri Poincar6,
who had his doubts, and described the Principle of IndifEerence
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," but, upon which calculation in
probability invariably rests.^
5. Before following up in detail these two Unes of develop-
ment, I will summarise again the earlier doctrine with which the
leaders of the new schools found themselves confronted.
The earher philosophers had in mind in dealing with prob-
ability the apphcation to the future of the inductions of experience,
to the almost complete exclusion of other problems. Eor the
dcAa of probabihty, 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
1 Poincar^'s opiniona on Probability are to be found in his Oaleul des Prob-
abilites and in bis Science et Hypothise. ITeither 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 probabUit^s ofEre une
contradiction dans les termes m§mes qui servent a le designer, et, si je ne crai-
gnais de rappeler ici un mot trop souvent r^p^t6, 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 practi-
cal application, and he speaks of ' metaphysics ' legitimising particular conven-
tions. 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. •
CH. vn FUKDAMENTAL IDEAS 85
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 expUcit introduction of the Principle of Indifference,
or, as it was then termed, the Principle of Non-Sufficient Reason.
The great achievement of the eighteenth century was, ia 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 satis-
fied was the theory. What was required was not a wide experi-
ence 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 quahfication 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 ElUs in 1842.^
At the conclusion of his Remarks on an alleged froof 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 ElUs'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 concep-
tions, we must appeal to the things themselves to obtain it, for
we cannot find much help elsewhere." Professor Edgeworth *
was an early disciple of the same school : " The probability," he
^ On the Foundations of the Theory of Probabilities.
' Republished in Miseellaneous Writings.
^ Logic of Chance, p. 74.
* Metretike, p. 4.
86 A TEEATISE ON PROBABILITY ft. i
says, " of head occurring n times if the coin is of the ordinary
make is approximately at least (J)". This value is rigidly deducible
from positive experience, the observations made by gamesters,
the experiments recorded by Jevons and De Morgan."
The doctrines of the empirical school wiU 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 remainiag relics of Realism,"
as well as the later Laplacian Law of Succession, thus destroying
the link between the empirical and the A '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 hable. 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 hnown to
diSer 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 preten-
tious 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 d priori principle. Experience, as opposed to in-
tuition, cannot possibly afEord us a criterion by which to 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 ProbabiHty
^ WahracheirUichlceitsreehnung, p. 11.
CH. VII FUNDAMENTAL IDEAS 87
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 appre-
hend, 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 equahty 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 dis-
cover a proper basis for the numerical measurement of probabiU-
ties, and he is thus led to examine with care the grounds of vahd
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 qualify-
ing conditions are expressed is ofteU; to an EngUsh 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 sunmiary, and the reader must be
^ Die Principien der Wahrscheinlichkeitsrechnung, Eine logische Unter-
auchung. Freiburg, 1886.
88 A TEEATISE ON PEOBABILITY pt. i
referred to the original for a proper exposition of the Doctrine of
Spiekdume. Briefly, but not very inteUigibly perhaps, he may
be said to hold that the hj^otheses for the probabihties of which
we wish to obtain a numerical comparison, must refer to 'fields'
(Spielrmime) which are ' indifferent,' ' comparable ' in magnitude,
and ' original ' {urspriinglich). 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 exceed-
ingly 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 correla-
tion between these objects and the objects of the fields. The
qualification of comparabihty is intended to deal with difficulties
such as that connected with the population of different areas of
unknown extent ; and the quahfication of originaUty 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 formulation of his conditions.
The condition of UrsprilngUcJikeit, for instance, seems to depend
upon physical rather than logical criteria, and is, as a result,
much more restricted in its apphcabUity 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 TV.
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 treat-
ment Hes primarily in the fact that in the disjunctive judgment
* Sigwart, Logic (Eng. edition), vol. ii. p. 220.
CH. vn FUNDAMENTAL IDEAS 89
the number of terms in the disjunction plays a decisive part.
Inasmuch as a limited number of mutually exclusive possi-
bilities 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 deaUng with equal
parts of a spatial area, or equal parts of a period of time. . . .
But even where this obvious quahty is not forthcoming, we may
ground our expectations upon a hypothetical equivalence, where
we see no reason for considering the extent of one possibihty 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 speciahsation of
a general concept." But in the last sentence quoted he surrenders
the advantages he has gained in the earher part of his explana-
tion, and, instead of insisting on a knowledge of an equal degree
of speciahsation, he is satisfied with an absence of any knowledge
to the contrary. Hence, in spite of his initial quaUfications, he
ends unrestrainedly in the arms of Non-Sufficient Eeason.^
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 coordinated 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 con-
dition B is fulfilled, one of the kinds f-^f^f^ ... 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 h-p^^ ... in which that universal
condition is fulfilled. ... A coordinated case is a case which
answers to one and only one of the mutually exclusive values
bp^ ... of the condition B, and these rival values may occur in
* 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.
2 Lotze, Logic (Eng. edition), pp. 364, 365.
90 A TREATISE ON PEOBABILITY pt. i
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 h^^. . . ."
This certainly meets some of the difficulties, and its resem-
blance to the conditions formulated in Chapter IV. wiU 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 measure-
ment of continuous quantity. By combining the suggestions of
Von EJies, Sigwart, and Lotze, we might, perhaps, patch up a
fairly comprehensive rule. We might say, for instance, that if
6j and 6, are classes, their members must be finite in number and
enumerable or they must compose 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 6^ and h^ are concepts, they must represent
concepts of an equal degree of speciahsation. But quahfications
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 speciahsation ?
9. That probability is a relation has often received incidental
recognition from logicians, in spite of the general faUure to place
proper emphasis on it. The earliest writer, with whom I am
acquainted, explicitly to notice this, is Kahle in his Elementa
logicae ProbahiUum methodo mathematica in icsum Scientiarum
et Vitae adornata published at HaUe 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
'- 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 pio-
babilis, si desunt quaedam requisita ad reritatem demonstrativam (p. 15).
Propositio probabiUs esse potest falsa, et improbabilis esse potest vera ; ergo
cognitio hodie possibilis, crastina luce mutari potest improbabilem, si accedunt
leUqua requisita omnia, in certitudinem (p. 26). . . . Certitudo est terminus
relatives : considerare potest ratione representationum in intelleotu nostro.
. . . Incerta nobis dependent a defectu cognitionis (p. 35). . . . Actionem
imprudeuter et contra regulas probabiUtatis susceptam eventus felix sequi
potest. Ergo prudentia actionum ex successu solo non est aestimanda (p. 62).
. . . Logica probabiUum est scientia dijudicandi gradum certitudinis eorum,
quibuB 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
OH. vn FUNDAMENTAL IDEAS 91
state of infonnation and varies with 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 vraisem-
blables, la difi6rence des donnees que chaque homme a sur eUes,
est une des causes principales de la diversity des opinions que
Ton voit regner sur les mSmes objets . . . c'est ainsi que le
mSme fait, recite devant une nombreuse assemblee, obtient divers
degres de croyance, suivant I'etendue 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 difiicult problem.
to the Question of the Combinatioii of Testimonies or Judgments " (Edin. Phil.
Trans, xxi. p. 600) : " Our estimate of the probability of an event varies not
absolutely with the ciroumstanoes which actually affect its occurrence, but with
our knowledge of those circumstances."
^ T}\x. Principles of Logic, p. 208.
* Essai philosophique, p. 7.
CHAPTEE VIII
THE FREQUENCY THEORY OP PROBABILITY
1. The theory of probability, outlined in the preceding chapters,
has serious difficulties to overcome. There is a theoretical, as
weU as a practical, difficulty in measuring or comparing degrees
of probability, and a further difficulty in determining them
d 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 unUke this was entertained by those philoso-
phers of the seventeenth and eighteenth centuries who approached
the problems of probability uninfluenced by the work of mathe-
maticians. But the underlying conception of earlier writers
received at the hands of some Enghsh 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 Probabihty, first appears ^ as the basis of a proposed
logical scheme in a brief essay by Leslie Ellis On the Foundations
of the Theory of Probdbilities, and is somewhat further developed
in his Remarks on the Fundamental Principles of the Theory of
1 Shet. 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 Coumot's
Exposition, also published in 1843 : " La theorie des probabilit^s a pour objet
certains rapports numeriques qui prendraient des valeurs fixes et oompWtement
d^termin^es, si Ton pouvait rlp6ter k I'infini les 6preuves des mSmes hasards,
et qui, pour un nombre flni d'^preuves, osoillent entre des Umites d'autant plus
resserrSes, d'autant plus voisines des valeMTB finales, que le nombre des ^preuves
est plus grand."
92
CH. vin FUNDAMENTAL IDEAS 93
Probabilities.^ " If the probability of a given event be correctly
deternained," he says, "the event will on a long run of trials tend
to recur with frequency proportional to their probabihty. This
is generally proved mathematically. It seems to me to be true
d priori. ... I have been unable to sever the judgment that
one even,t is more likely to happen than another from the beHef
that in the long run it will occur more frequently." Ellis ex-
phcitly introduces the conception that probabihty is essentially
concerned with a group or series.
Although the priority of invention must be allowed to Leslie
EUis, the theory is commonly associated with the name of Venn.
In his Logic of Chcmce ^ 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
impUes, 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 con-
ception 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 probabihty is this, — the occa-
sional 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
1 These essays were published in the Transactions of the Camb. Phil. Soo., 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 fuU of
spirit and originality, and are not now so well known as they deserve to be.
2 The first edition appeared in 1866. Revised editions were issued in 1876
and 1888. References are given to the third edition of 1888.
94 A TREATISE ON PROBABILITY pt. i
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 beUeve that it
is in such things only, as distinguished from things artificial, that
the property ia question is to be found (p. 65). How, in any
particular case, are we to estabhsh the existence of a probabiUty
series ? Experience is our sole guide. If we want to discover
what is in reaUty a series of things, not a series of our own con-
ceptions, we must appeal to the things themselves to obtain it,
for we cannot find much help elsewhere (p. 174). When proba-
bihty is divorced from direct reference to objects, as it substanti-
ally is by not being founded upon experience, it simply resolves
itseK 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, aU 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 charac-
teristic is -, is to mean that the event is one of a number of events,
a proportion - of which have the characteristic in question ; and
the 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 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 indefimables, no appeals to intuition.
Measurement leads to no difficulties ; our probabilities or fre-
quencies are ordinary numbers, upon which the arithmetical
apparatus can be safely brought to bear. And at the same time it
OH. vm FUNDAMENTAL IDEAS 95
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 probabihty 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 probabihty gives us. The importance
of probabihty depends solely upon the actual existence of such
groups or real kinds in the world of experience, and a judgment
of probabihty must necessarily depend for its vahdity upon our
empirical knowledge of them.
4. It is the obvious, as well ais the correct, criticism of such a
theory, that the identification of probabihty with statistical
frequency is a very grave departure from the estabhshed use of
words ; for it clearly excludes a great number of judgments
which are generally beUeved to deal with probabihty. Venn
himself was well aware of this, and cannot be accused of supposing
that all behefs, which are commonly called probable, are really
concerned with statistical frequency. But some of his followers,
to judge from their pubUshed 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,
96 A TREATISE ON PROBABILITY pt. i
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 plausibihty of Venn's theory is derived, I
think, from a failure to recognise the narrow limits of its ap-
pHcability, 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 com-
plicated 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 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 vaia to endeavour to
force them into one science." All these cases, therefore, in which
we are ' not certain,' Venn expHcitly excludes from what he
chooses to call the science of probability, and he pays no further
attention to them. The science of probabihty 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 behef might be observed.
Probability only considers the case in which this variation is
brought about in a certain definite statistical way."^ He points
* 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
Galeulus 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
OH. ym FUNDAMENTAL IDEAS 97
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 manu-
facture : " 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 the
diminution of its value by the processes of attainment."
It must be recognised, therefore, that Venn has deUberately
excluded from Ms 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
probabiHty 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 behef is occasioned by purely
psychological causes, and that his argument is valid against those
who have defined probability as measuring the degree of sub-
jective belief. But this has not been the usual way of
looking at the subject. ProbabiUty is the study of the
grounds which lead us to entertain a rational preference for
one behef 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 valiie ' of our conclusion is influenced by many other con-
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 foUow it consistently. But, like Venn's,
it leaves aside the most important questions. The Calculus of Probabili-
ties, thus interpreted, is no guide by itself as to which opinion we ought
to foUow, and is not a measure of the weight we should attach to conflicting
arguments.
H
98 A TREATISE ON PROBABILITY pt. i
siderations than that of statistical frequency. Venn's theory,
therefore, cannot be faixly propounded by his disciples as alterna-
tive 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, dehberately, not treated
in the Logic of Chance.
7. Apart from two circumstances, it would scarcely be neces-
sary to say anything further ; but in the first place some writers
have believed that Venn has propounded a complete theory
of probabihty, failing to realise that he is not at all concerned
with the sense in which we may saythat one induction or analogy,
or testimony, or memory, or train of argxmient is more probable
than another ; and in the second place he himiself 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 modahty 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 prefer-
ence for one alternative over another from a knowledge of statis-
tical frequencies, a statistical frequency by itseK is insufficient
to determine us. We may call a statistical frequency a prob-
abihty, 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
OH. vm FUKDAMENTAL IDEAS 99
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 probabiUty 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 indiiction, 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. But we
cannot assert on this ground, as Venn wishes to do, that the prob-
abiUty of the death of an infant in its first four years is i^ths.
We can say no more than that it is probable (in my sense) that
there is such a probabihty (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 probabiUty
(in my sense) that the statistical frequency really is T^rths.
The cases in which we can determine 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 foimd in his belief that probabiUties which are
based on statistical frequencies are alone capable of accurate
measurement. The term ' probabiUties,' 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 estabUsh 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 practipe, d priori
determination is often easy, whilst d posteriori appeal to experi-
ence would be not merely tedious but utterly impracticable.''
But these cases which are usually based on tlie Principle of
Indifference can, he maintains, be justified on statistic^ grounds.
In the case of coin tossing there is a considerable experience o^
the equaUy frequent occurrence of heads and 'tails ; the experi-
ence gaiaed in this simple case is to be extended, to the coi^iplex
cases by "Induction and Analogy." In one Simple /case the
100 A TREATISE ON PEOBABILITY pt. i
result to which the Principle of IndifEerence 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 IndifEerence. This is to assert that, because in one case,
where there is no known reason to the contrary, there actually
is 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 care-
ful 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 d posteriori arguments ia 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 foUow 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 at-
tempted 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 pre-
slippositioAs wbuld greatly add to the completeness of his work.
In the(mea]itime it is only possible to raise general objections to
1 Let the' reader, yrho is acquainted with this chapter, consider what precise
assumption Venn's rpasoning requires on p. 187 in the example which seeks to
show the efficacy of Lord Lister's antiseptic treatment d posteriori. What is
thn 'inevitable assumption about the bags ' when it is translated into the
language of this example ?
CH. vra FUNDAMENTAL IDEAS 101
any theory of probability which seeks to found itself upon the
conception of statistical frequency.
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 beHeve 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 probabihty of a proposition always depends
upon referring it to some class whose truth-frequency is known
within wide or narrow limits.
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 prob-
^ 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.
102 A TREATISE ON PROBABILITY w. i
abilities whicli are numerical. In some cases we may know the
exact nmnber whicli 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 iu cases where the real •
probabilities were partially unknown. On the frequency theory,
therefore, there is an important sense ia which probabilities can
be unknown, and the relative vagueness of the probabilities
employed in ordinary reasoning is explained as belonging not
to the probabUities 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 probabihties which is imphcit
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 corre-
sponding 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 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
CH. vm FUNDAMENTAL IDEAS 103
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 sometlung of this kind, what principle is there for
uniquely determining the class, the truth-frequency of which is
to measure the probabihty 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 iasurmoimtable.
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 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 generahsation,
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 prob-
ability we seek to determine. And, indeed, in those examples
104 A TREATISE ON PROBABILITY pt. i
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
oharacteristicSj 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 considera-
tion. 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 probabiUty, and goes rather to show that
the theory of this Treatise is the generalised theory, compre-
hending within it such appUcations of the idea of statistical truth-
frequency as have vahdity.
* Relevance ' is an important term in probabiLtty, of which
the meaning is readily inteUigible. I have given my own defini-
tion of it already. But I do not know how it is to be explained
ia 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 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 earher chapters. With this admissionj it might be argued, the
theory would still stand, divested, it is true, of some of its original
simphcity, 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
estabhshed on a theory of frequency. This wiU 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.'
OH. vm FUKDAMEiNTAL IDEAS 105
If ajh denotes the probability of a on hypothesis h, this theorem
may be written {a + b)/h=a/h+bjh-ablh, and may be read
' On hypothesis h the probability of " a or 6 " is equal to the
probability of a + the probability of 6 - the probability of
" both a and 6." ' 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 wiU assist the argument : Let ay represent the truth-
frequency of any class a, and let aJh stand for ' the probability
of a on hypothesis h, a being the class of reference determined
by this hypothesis.' ^ We then have aJh = ay, and we require to
prove a proposition, for values of y and S not yet determined,
which wiU be of the form :
{a + i)Jh = aJh + h^jh - abjh.
Now if S' is the class of propositions {a + b) such that a is an
a and 6 a ;8, it is easily shown by the ordinary arithmetic of classes
that Sy = ay,+ /Sy-aySy where a/8 is the class of propositions which
are members of both a and /3. In the case, therefore, where
S = S' and j = a^, an addition theorem of the required kind has
been established.
But it does not follow by any reasonable rule that, if h deter-
mines a and /3 as the appropriate classes of reference for a and 6,
h must necessarily determine S' and a/3 as the appropriate classes
of reference for (a+b) and ab ; it may, for iastance, be the case
that h, while it renders a and /3 determinate, yields no informa-
tion whatever regarding a^, and points to some quite different
class fi, 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 8 = 8' and y = a^.
The following is a good example : We are given
that the proportion of black-haired men in the population
V V
is — and the proportion of colour-blind men — , and there is no
known connection between black - hair and colour - blindness :
what is the probabiUty that a man, about whom nothing special
^ The question, previously at issue, as to how the class of reference is deter-
mined by the hypothesis, is now ignored.
106 A TEEATISE ON PROBABILITY ft. i
is known, is ^ either black-haired or colour-blind ? If we represent
the hypotheses by h and the alternatives by a and 6, it would
usually be held that, colour-blindness and black hair being
p p
independent for knowledge ^ relative to the given data, al/h = -^j
p
and that, therefore, by the addition theorem, (a + &)/A = - +
p P V
— - -Tg^- But, on the frequency theory, this result might be
invaUd; for a^j= -^, only if this is the actual proportion in fact
of persons who are both colour-blind and black-haired, and that
this is the actual proportion cannot possibly be inferred from
the independence for knowledge of the characters in question.^
Precisely the same difficulty arises in connection with the
multiphcation theorem ab/h^ajbh.b/h.* In the frequency nota-
tion, which is proposed above, the corresponding theorem wiU
be of the form ahjh = a Jbh . b^/h. For this equation to be satisfied
it is easily seen that S must be the class of propositions xy such
that a; is a member of a and y of y8, and 7 the class of propositions
xb such that a; is a member of a ; and, as in the case of the addition
theorem, we have no guarantee that these classes 7 and S will be
those which the hypotheses bh and h wiU respectively determine
as the appropriate classes of reference for a and ah.
In the case of the theorem of inverse probability ^
b/ah ajbh b/h
c/ah a/ch cjh
the same difficulty again arises, with an additional one when
practical apphcations are considered. For the relative proba-
bihties of our d 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
^ In the course of the present discussion the disjunctive a + 6 is never inter-
preted so as to exclude the conjunctive db.
' For a discussion of this term see Chapter XVI. § 2.
8 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.
5 Vide Chapter XIV. § 5.
OH. vm FUNDAMENTAL IDEAS 107
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 proba-
bility, 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, inde-
pendent for knowledge. 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 a and /S 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 estab-
lished 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 wiH 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 probabihty, it is itself
dependent on them. In any case, it is generally agreed that
an iaductive 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 behef 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 possibihty that a majority are
false is an actuality. Yet what possible reason can the frequency
theory ofEer, which does not beg the question, for supposing that
a majority are true ? And failing this, what groimd have we
for believing the inductive process to be reasonable ? Yet we
108 A TEEATISE ON PROBABILITY pt. i
invaoriably assume that with our existing knowledge it is logically
reasonable to attach some weight to the inductive method, even
it future experience shows that 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 know-
ledge 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 maimer. It is not stifficient,
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 iato a shape in
which they would be seen to rest upon statistical frequency !
Many individual arguments, of course, are exphcitly 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 beiag 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 intelhgible poiat 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 e^lain why some species have changed and others
OH. vm FUNDAMENTAL IDEAS 109
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 know-
ledge 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 probabiHty 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 probabiHty 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 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
1 Letter to G. Bentham, Life and Letters, vol. iii. p. 25.
' E.g. in the discussion on the relative efEeot of disuse and selection in
reducing unnecessary organs to a rudimentary condition.
110 A TREATISE ON PROBABILITY pt. i
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 incon-
sistent 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 OP PART I. SUMMARISED
1. That part of our knowledge which we obtain directly,
suppKes 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 perceiv-
ing 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 ProbabiUty is to
systematise such processes of inference. In particular it aims
at elucidating rules by means of which the probabihties 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. ia
addition to the symmetrical evidence) for one of the conclusions,
and this evidence is favourably relevant, then that conclusion is
111
112 A TEEATISE ON PROBABILITY ft. i
the more probable. Certain rules have been given by which to
judge whether or not evidence is favourably relevant. And by
combinations of these judgments of preference with the judg-
ments of indifference warranted by the Principle of Indifference
more compUcated 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 probabihties 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 hke a third object, so there are
different orders of probabiUty, 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
OUT expectation, to say not only that one probabiUty 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
ntimerical 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 symboUc 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 com-
monly known as statistical.
PART II
FUNDAMENTAL THEOKEMS
113
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 probabUity.
The reader wiU readily perceive that this Part would never have
been written except under the influence of Mr. Russell's Princijna
Maihematica. 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 rehability 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 Probabihty
as a branch of Formal Logic first gave rise.
115
116 A TREATISE ON PROBABILITY m. n
A fnrtlier occasion of diffidence and apology in introducing
this Part of my Treatise arises out of the extent of my debt to
Mx. 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 Uttle likelihood of the appearance of any
work on ProbabiUty 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 cmy 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 seK-contradictory and formally
inconsistent with themselves.
For the purpose of this limitation we have to make a distinc-
tion between a set of propositions which is merely false in fact
and a set which is formally inconsistent with itseK.^ This leads
'- Spinoza had in mind, I think, the distinction between Truth and Prob-
ability in his treatment of Necessity, Contingenee, and Possibility. Res
enim omnes ex data Dei natura necessario sequutae sunt, et ex necessitate naturae
Dei determinatae sunt ad cerio modo ezistendum et operandv/m {Mhiees i. 33).
Xliat is to say, everything is, without qualification, true or false. At res
OH. X FUNDAMENTAL THEOEEMS 117
us to the conception of a growp 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-^ or q ' both belong to it, then the proposition
q also belongs to it ; (iii.) if any proposition f belongs to it, then
the contradictory of f is occluded 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 relative to a specific body of knowledge, or,
as it has been termed in the traditional logiCj to the Universe of
Reference.
It also enables us to reach a definition of inference distinct from
implication, as defined by Mr. Kussell. This is a matter of very
great importance. Readers who are acquaiuted with thej 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
aliqua nulla alia de causa contingens dicitur, nisi respectu defectus noslrae
cognitionis (Etliices 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 d priori or formal probability exceeding zero,
and Possibility ; that is to say, into formal possibility and empirical possibility.
Res singulares voco contingentes, quaienus, dum ad earum solam essentiam
attendimus, nihil invenimus, quod earum existentiam necessario ponat, vel
quod ipsam necessario seeludat. Easdem res singulares voco possibiles, quatenus,
dum ad causae, ex quibus produci detent, attendimus, nescimus, an ipsae
determinatae sint ad easdem producendum (EtMces iv. Def 3, 4).
118 A TREATISE ON PEOBABILITY w. n
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 foUowiag it. As a method of setting
forth the system of formal truth, which shall possess beauty,
iater-dependence, 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 princi-
pally interested but which he ignores, and the relation of implica-
tion on which his scheme depends.
' p implies q ' is, according to his definition, exactly equivalent
to the disjunction ' q is true or f is false.' If q is true, ' p itnpUes
q ' holds for aU values of p ; and similarly if f is false, the im-
plication holds for all values of q. This is not what we mean
when we say that q can be inferred or follows from f. For what-
ever 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 etoety 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 implica-
tions 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 growp. Inference
and Probability depend for their importance upon the fact that
in actual reasoning the limitation of our knowledge presents us
OH. X FUNDAMENTAL THEOEEMS 119
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. EusseU cannot
avoid concerning himself with groups. For his aim is to discover
the smallest set of propositions which specify our formal know-
ledge, 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. Passhig on from these preliminary reflections, our first
task is to establish the axioms and definitions which are to make
operative our sjonbolical processjes. These processes are almost
entirely a development of the idea of representing a probability
by the symbol a/h, where h is the premiss of an argument and a
its conclusion. It might have been a notation more in accord-
ance with our fundamental ideas, to have employed the symbol
a/h to designate the argwm&nZ 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
F{ajh). This would correspond to the symbol Y{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/A)
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 expHcit 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.
120 A TEBATISE ON PROBAEILITY ra. n
4. Inasmucli as relations of Probability cannot be assumed
to possess the properties of numbers, tbe 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
multipUcation 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 : ajh = l.
III. The Definition of Impossibility : a/h=0.
VI. The Definition of Inconsistency : ah is inconsistent if
a/A=0.
VII. The Definition of a Group : the class of propositions a
such that a/h = 1 is the group h.
Vni. The Defimition of Equivalence : if b/ah = 1 and a/bh = 1
(amb)lh = l.
IX. The Definition of Addition: ab/h + aE/h^=a/h.
X. The Definition of Multiplication: ab/h=ajbh .blh =
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
h given h.'
XI. The Definition of Independence: if aila^h=aj^/h and
ajajh=a2]h, ajh and ajh are independent.
XII. The Definition of Irrelevance: if ar^aji=ayjh, a^ is
irrelevant to ajh.
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
^ b stands for the contradictoiy of b.
OH. X FUNDAMENTAL THEOEEMS 121
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 Eational 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 :
Addition Theorem: {a + b)/h=alh+bfh-ab/h, which reduces
to (a + b)/h = ajh + b/h, where a and b are mutually exclusive ;
and, if p^^ • ■ ■ Pn ioim, relative to h, a set of exclusive and
n
exhaustive alternatives, a/h='tpfajh.
1
Theorem of Irrelevance: If ajhji2=a/hi, then ajhji2=ajh^;
i.e. if a proposition is irrelevant, its contradictory also is irrelevant.
Theorem of Independence : li a2/ajh=a^h, aja2h=a^/h; i.e.
if «! is irrelevant to aJh, it foUows that a^ is irrelevant to a^/h
and that a^/h and a^/A are independent.
Multiplication Theorem : If aJh and aJh are independent,
aja2lh=ai/h . a^jh.
Theorem of Inverse Probability : -^ — =J—L. . -11—. Further,
a^jbh bjaji a^/h
if 0^1^= Pit <^2l^=Pz> ^/<hfi'=iv ^/<'a^ = 9'2' ^^^ ajbh + a2lbh = \,
then ajbh= — ^-^ — ; and if a^lh^aJh, aJbh= — =^, which
is equivalent to the statement that the probability of Oj when
we know b is equal to — i^, where q^ is the probability of b when
we know a^ and q^ its probability when we know a^. 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
122 A TREATISE ON PROBABILITY fp. n
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 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 hypo-
thetical.
In a symbolical treatment of the subject it is important
that we should be free to consider hypothetical premisses, and
to take accoimt of relations of probability as existing between
any pair of sets of propositions, whether or not the premiss is
actually part of knowledge. But iu acting thus we must be
careful to avoid two possible sources of error.
2. The first is that which is Hable to arise wherever va/riables
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 afEect 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 know-
ledge ? How are we to deal with hypothetical premisses which
are self-contradictory or formally inconsistent with themselves,
and which caimot be the subject of rational belief of any degree ?
1 Or more strictly, " perception of which, together with knowledge of the
first set, justifies an appropriate degree of rational belief about the second."
123
124 A TREATISE ON PROBABILITY pt. n
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 growp 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 con-
junction of them. 2 To say, therefore, that a proposition foUows
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 ' wiU 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 ' foUows from ' one
part of them, and its contradictory from another part.
5. We have stiU, however, to make precise what we mean in
this definition by one proposition /oZtowM^/rom or being logically
invohed 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 5 ' is a formal implication if ' not-j9 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
* ' a can be inferred from b,' ' a foUows from b,' ' a is certain in relation to
b,' ' a is logically involved in 6,' I regard as equivalent expressions, the precise
meaning of which will be defined in succeeding paragraphs. ' a is implied by 6,'
I use in. a different sense, namely, in Mr. Russell's sense, as the equivalent of
' b ornot-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.
OH. XI FUNDAMENTAL THEOREMS 125
tlian the arguments are logical constants, and of which all the
values are true.
We might define a group in such a way that aU logical principles
belonged to every group. In this case all formally true proposi-
tions 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, i£ we were to
assume the premisses in every argument to include aU 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. I"or 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,
aU propositions which are instances of the same formal proposi-
tional 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.
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 com-
prehending 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
126 A TREATISE ON PROBABILITY w. n
the form of our pxemisses. If, sometimes or as a rule, o\a
premisses in fact include some more powerful principle of argu-
ment, 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 pro-
positions jPjPz . . . 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 p^ belongs to the group
specified by the rest, this term is redundant).
When the fundamental set is uniquely determined, a group h'
is a sub-group to the group h, if the set fundamental to h' is
included in the set fundamental to h.
Logically there can be more than one distinct set of proposi-
tions fundamental to a given group ; and some extra-logical test
must be appHed 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 pro-
positions, includes the most remote deductions not only amongst
those known to mathematicians, but amongst those which time
OH. XI FUNDAMENTAL THEOREMS 127
and skill have not yet served to solve. If we define cert^iinty
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 unhkely, 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 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
probabiKty. If we are unable through lack of sldll 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 prob-
ability is not unknown. When Newton held that the binomial
theorem possessed for empirical reasons sufficient probabihty
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 probabihty 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 prob-
abihty 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
128 A TREATISE ON PEOBABILITY w. n
to evidence from which tiie 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, ia other woids, 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 contra-
dictory to one another, when they are iuconsistent in the sense
that, relative to our evidence, they cannot belong to the same
group. On the other hand, a proposition, which is not itseK
included in the group specified by the premiss and whose contra-
dictory is not included either, has in relation to the premiss an
intermediate degree of probability.
If a follows from h and is, therefore, included in the group
specified by h, this is denoted hy a/h = 1. The relation of certainty,
that is to say, is denoted by the symbol of imity. 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 probabUity has been explained. If we represent
the relation of certaiuty by 7 and any other probability by
a, the product a.<y=a. Similarly, if a is excluded from the
group specified by h and is impossible in relation to it, this is
denoted by ajh=0. The use of the symbol zero to denote
impossibility arises out of the fact that, if a denotes impossibihty
and a any other relation of probability, then, in the senses of
multiphcation and addition to be defined later, the product
a .a) = o), and the sum a + (o=a. Lastly, if a is not included
ia the group specified by h, this is written o/A+1 or u/JkI; .
and if it is not excluded, this is written a/h=i=0 or ajh>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 iu 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
CH. XI FUNDAMENTAL THEOEEMS 129
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
fimdamental, 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 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 = l, q/h=i=l ; hence
p requires q if there is no real group to which p belongs
and q does not belong.
p does not require q withi/n 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'jh = 1, p/h' = 1,
qjh'^l. This reduces to the proposition next but one above
if A is the Universe of Keference. In § 13 these definitions
will be generalised to cover intermediate degrees of prob-
ability.
12. Inference and logical priority can be defined m terms of
requirement and real groups. It is convenient to distiaguish
two types of inference correspondiag to hypothetical and real
1 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.
K
130 A TEEATISE ON PEOBABILITY pt. h
groups — i.e. to cases where the argument is only hypothetical,
and cases where the conclusion can be asserted :
Hypothetical Inference. — ' If j), q,' which may also be read
' q is hypothetically inferrible from p,' means that there is a
real group h such that q/ph = l, and g/A + l. In order that this
may be the case, ph must specify a group ; i.e. p/h4=0, or in
other words p must not be excluded from h. Hypothetical
inference is also equivalent to : ' p implies g,' 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' oi' q may be asserted by inference from p,' means that ' Jlp,q'
is true, and in addition " p ' belongs to a real group ; i.e. there
are proper groups h and A' such th.a,tpjh = l, q/ph' =1, qjh'^l,
and pjh' 4= 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 = a means that the proposition p has the probable relation
of degree a to the proposition h ; while it is postulated that h is
self-consistent and therefore specifies a group.
plh = l means that p follows from h and is, therefore, in-
cluded in the group specified by h.
p/h = means that p is excluded from the group specified by h.
If h specifies the Universe of Eeference, i.e. if its group com-
prehends the whole of our knowledge, p/h is called the absolute
probahility 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
CH. XI FUNDAMENTAL THEOREMS 131
or (for short) certain. Thus f 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 ' im-
possible ' 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 probabiUty, 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 A' and is included in fe(t.e. h'/h" =\,h" jh = \),pjh" =plh',
and q/h' =f= 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 con-
clusions 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 hypo-
thetical 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 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
132 A TEEATISE ON PROBABILITY pt. n
being necessary if every hypothetical group, which includes the
proposition together with the Universe of Reference, includes
the condition, and suflSlcient if every hypothetical group, which
includes the condition together with the Universe of Reference,
includes the proposition.
CHAPTER XII
THE DEFINITIONS AND AXIOMS OP 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 oui 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 prob-
ability which we shall enunciate.
The object of this and the chapters immediately following is
to show that all the usually assumed conclusions in the funda-
mental 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
133
134 A TEEATISE ON PROBABILITY pt. h
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 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 equiprobabihty 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, whoUy independent.
They deal with a different part of the subject, not so closely
comiected with epistemology.
3. In this chapter I confine myself to Definitions and Axioms.
Propositions wUl be denoted by small letters, and relations
by capital letters. In accordance with common usage, a dis-
junctive combination of propositions is represiented by the sign
of addition, and a conjunctive combination by simple juxta-
position (or, where it is necessary for clearness, by the sign of
multipUcation) : e.g. ' a or 6 or c ' is written ' a + b+c,' and ' a
and b and c ' is written ' abc' ' a + 6 ' 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/A=P Del
II. If P is the relation of certainty ^
P=l Def.
III. If P is the relation of impossibihty ^
P=0 Def.
IV. If P is a relation of probabiUty, but not the relatipn of
certainty P<1. Def.
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/A = l is the
group specified by h ox (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,iib follows from ah, and a from bh.
^ These symbols were first employed by Leibnitz. See p. 155 below.
CH. XII FUNDAMENTAL THEOEEMS 135
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 rela-
tion of probabihty 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=i)jh=l, and a; is a proposition, x/ah = x/bh. This
is the Axiom of Equivalence.
(iii.) (a + b=aE)/h = l
{aa^a)/h = 1
{&=a)/h = 1
(ab+ab=b)/h = l.
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 MuUijaUcaiion. — If we were to assume that
probabilities are numbers or ratios, these operations could be
given their usual arithmetical signification. In adding or
multiplpng probabilities we should be simply adding or multi-
plying 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 opera-
tions is not of practical importance.
We define the sum of the probable relations ab/h and aBjh
as being the probable relation a/h ; and the prodiict of the probable
relations ajbh and b/h as being the probable relation ab/h. That
is to say :
IX. ab/h +aSjh= a/h. Def.
X. ab/h = a/bh . b/h = b/ah . a/h. Def.
Before we proceed to the axioms which will make these sym-
bols 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 hypo-
136 A TEBATISE ON PEOBABILITY pi- u
thesis." X. may be read : " The probability of ' both a and 6,'
assuming A, is equal to the product of the probability of 6, assum-
ing A, and the probability of a, assuming both 6 and A." 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 probabiUty 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 multiphcation :
XI. If PQ=E,P=5 Def.
XII. If P + Q=E, P=E-Q. Def.
Thus we have to introduce as definitions what would be axioms
if the meaning of addition and multiphcation were already defined.
In this latter case we should have been able to apply the ordinary
processes of addition and multiphcation 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
commonsense, full of preconceptions as to the meaning of addition
and multiphcation. I define these processes, for the theory of
probabiUty, by reference to a comparatively unfamiUar property,
and associate the more famihar properties with this one by means
of axioms. These axioms are as follows :
(iv.) If P, Q, E are relations of probabihty such that the
products PQ, PE and the sums P + Q, P +E 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. 6) PQ<P unless Q = l or P = 0; P + Q>P unless Q = 0.
PQ=P if Q = l or P=0; P + Q=P if Q = 0.
(iv. e) If PQSPE, then Q|E unless P = 0. If P + Q|P + E,
then Q=E and conversely.
^ E.g. Bertrand, Calcul des probabilites, p. 26.
CH. xn FUNDAMENTAL THEOREMS 137
A meaning has not been given, it is important to notice, to
the signs of addition and multipKcation 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. Further-
more, if P + Q=R and Q=S + T, it does not follow that
P + S-i-T=R, since no meaning has been assigned to such an
expression as P + S + T. The ecLuation must be written P + (S + T)
=Rj and we cannot infer from the foregoing axioms that
(P-I-S)+T=R. The following axioms allow us to make 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] +[±E±S] =[±P±R] - [tQtS] =[±P±R] +
[±Q±S] = [±P±Q]-[tRtS]
in every case in which the probabilities [±P±Q], [±R±S],
[±P±R], etc., exist, i.e., in which these sums satisfy the con-
ditions 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 prob-
abilities. They enable us to prove, for instance, that if P + Q =
R+S then P-R=S-Q, provided that the differences P-R
andS-Qexist; and that (P 4- 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 re-
arrangement 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 probabiUties 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 probabiUties — ^for our object is.
138 A TREATISE ON PEOBABILITY pt. n
always, to compare one probability with another — ^and we are
not incommoded, therefore, in our symbolic operations by the
circumstance that sums and products do not exist between
every pair of probabihties.
8. Independence :
XIII. If a.Ja2h = a.Jfi and aja-^=a^jh, the probabilities
a■^]h and a^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 unafEected.
Irrelevance : ^
XIV. If a^aji=ajh, ag is irrelevant on the whole, or, for
short, irrelevant to Or^jh. 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.
CHAPTEE 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 wiU be convenient to include here a few theorems dealing
with intermediate degrees of probabihty. 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 + alh =1.
For ablh+ab/h=b/h by IX.,
a/bh . bjh + ajbh . b/h = b/h by X.
Put 6/A = 1, then a/bh + ajbh = 1 by (iv. b),
since 6/A-=.l, ih=h by (iii.).
Thus ajh+d/h = l by (ii.).
(1.1) If a/h = l,dlh = 0,
a/h+djh=l by (1),
.-. afh + d/h = ajh = a/h + by (iv. &) ,
.-. djh=0 by (iv. c).
(1.2) Similarly, if d/h^l, a/h=0.
(1.3) If a/h^O, djh^l,
a/h + d/h = 1 by (1),
.-. + a/A =0 + 1 by (iv.6),
.-. a//i = l by (iv. c).
(1.4) Similarly, if a/A = 0, a/A = 1.
(2) a/A<lora/A = l by IV.
(3) a/h>0 01 a/h =0 by V.,
i.e. there are no negative probabilities.
139
140 A TREATISE ON PROBABILITY pt. n
(4) ab/h<b/h or abfh^^bjh by X. and (iv. b).
(5) 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 = by V.
.-. P + Q>0 unless Q = 0.
Hence, if P + Q=0, Q = and similarly P=0.
(6) If PQ=0, P=0 or Q=0,
Q>0 unless Q=0 by V.
Hence PQ>P . 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 = l and Q = l,
PQ<P unless P = or Q = l by (iv. &),
PQ=PiE P = or Q = l by(iv.6),
and P<1 unless P = l by IV.,
.-. PQ<1 unless P = l.
Hence P = 1 ; similarly Q = 1.
(8) If ajh=0, ah/h=0 and a/bk = if bh is not incon-
sistent.
For ab/h = b/ah . a/h = a/bh . bjh by X.,
and since a/A = 0, b/ah. a/h = by(iv. 6),
.•. ab/h = and a/bh . b/h = 0,
:. unless h/h=0, a/bh = by (5),
whence the result by VI.
Thus, if a conclusion is impossible, we may add to the con-
clusion or add consistently to the premisses without afEecting the
argument.
(9) If a/h = l, a/bh = \ if bh is not inconsistent.
Since a/h = l, a/h = by (l.i),
.-. d/bh = by (8) if bh is not inconsistent,
whence a/bh = l by (1.4).
Thus we may add to premisses, which make a conclusion
certain, any other premisses not inconsistent with them, without
afEecting the result.
(10) If a/A = l, ab/h = b/ah = b/h,
ab/h = b/ah . a/h = a/bh . b/h by X.
Since a/h = l, a/bh = \ by (9) unless b/h = 0,
.: b/ah . a/h = b/ah and a/bh . b/h = b/h by (iv. b),
whence the result, unless b/h = 0.
CH. xm FUNDAMENTAL THBOEBMS 141
If h/h=0, the result follows from (8).
(11) liallh = \, alh = l.
For ahlh = l/ah .a/h by X.,
.-. a/A = l by (7).
(12) If (a=b)lh = 'l,ajh = bjh,
b/ah . a/h — ajbh . b/h by X.
and b/ah = l, albh = l byVIIL,
.-. alh = b/h by (iv. &).
(12.1) If {amb)/h = l and hx is not inconsistent,
a/hx = b/hx.
afhx . x/h=x/ah . a/h,
and b/hx . x/h =x/bh . b/h by X.,
x/ah=x/hh by (ii.),
and a/h = b/h by (12),
.". a/hx = b/hx unless x/h=0.
TMs is the principle of equivalence. In virtue of it and of
axiom (H.), if (a=b)/h = l, we can substitute a for b and vice versa,
wherever they occur in a probabiUty 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 = l by (ii.), unless a/a = 0,
i.e. a/a = l, unless a is inconsistent by (iii.), (12), and VI.
(13.1) d/a=0, unless a is inconsistent. This follows from
(13) and (l.l).
(13.2) a/d=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/ be the group of assumptions, common to a and 6, 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 «//=*= 0, since a is not inconsistent,
.-. b/af=0,
whence 6/^=0.
Thus, if a is impossible given b, then b is impossible given a.
(15) If V^2 = 0, hjhjh = 0,
hjhjh =h-j]iji . h^/h by II.,
and since h.Jh^=0, h.jhjh = by (8), unless h/h^ = 0, whence
the result by (iv. b), unless h/h^=0.
142 A TEBATISE ON PROBABILITY pi n
If hlh^=0, hjh=0 ' by (14),
since we assume that h is not inconsistent, and hence
h^hjh=0 by (8).
Thus, if hj^ is impossible given h^,, hji^ is always impossible and is
excluded from every group.
(15.1) If hjhjh=0 and h^h is not inconsistent, hjh2h = 0.
This, which is the converse of (15), follows from X. and (6).
(16) li hjh^ = l, {\ + K^)lh = l,
hjh^=0 ■ by (1),
.-. n^hJh = Q by (15),
.-. iy_hjh = l by (1.3),
.-. {hy + h^lh = l by (12) and (iii.).
(16.1) We may write (16) :
Ji h.Jh2 = l, (^2 5^i)/^ = lj where ' d' sjonbolises ' implies.'
Thus if hi follows from h^, then it is always certain that
^2 implies A^.
(16.2) If {h-^+h^jh = 1 and h^h is not inconsistent,
lijh2h = \.
Kjhjh=0, as in (16),
.■. ^1/^2^=0 hy (15.1), since h^h is not inconsistent,
.■.hjh^h = l by (1.4).
This is the converse of (14).
(16.3) We may write (16.2) :
If (^2 'sh-^lh = l and hji is not inconsistent, -A j/A.2^ = l-
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 ^2 ^^1 ^^^ ^2 belong to a
group hji, then h-^ also belongs to this group.
(17) If {h-y D : a=i)lh = 1 and h-Ji is not inconsistent, a/h-Ji
^Ijhji. This follows from (16.3) and (12).
(18) ala = \ or a/d = l.
a/a = l, unless a is inconsistent, by (13).
If a is inconsistent, a/h = 0, where h is not inconsistent, and
therefore d/h=l by (1.3).
Thus unless a is inconsistent, a is not inconsistent, and therefore
a/d = l by (13).
(19) adfh = 0,
d/d = l or n/a = l by (18);
.-. a/d = or ffl/a = by (l.l) and (1.2).
In either case adlh=0 by (15).
OH. xm FUNDAMENTAL THBOEEMS 143
Thus it is impossible that both a and its contradictory
should be true. This is the Law, of Contradiction.
(20) {a + d)/h = l.
Since {ad=a + d)/h = l by (iii.),
a+dlh=0 by (19) and (12),
.-. (a+a)/A = l 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/Ai = l and a/h^ = 0, h^hJh^O.
For ajhji^ . h-Jh^ = hjah^ ■ ajh^,
and d/hjh^. hjhj^ = hjdh-^.d/hi by X.,
.•. a/hjh^ . h-Jh2 = and d/h^h^ ■ hjhj^=0,
since, by hypothesis and (1), d/hj^=0 and ajh2 = 0,
:. a/h-ih2 = or hjh2=0,
and alhji2 = \ or hjhj^ = 0,
.: h-^jh^^Q or hJh-^^ = 0.
In either case hjhjh = 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.
(22) If a/hj^ = and hjh = l, ajh=0,
ahjh=0 by (15), .*. hjah .a/h=0,
hjah = l by (9), unless a/h = 0.
.". in any case a/h^O.
(23) If l/a = and h/d = 0, b/h = 0.
ah/h = and dh[h = by (15),
.-. a/hh^O or h/h = 0,
and d/bh = or b/h = by II. and (iv.),
whence &/A=0 by (1.4).
CHAPTER XIV
THE FtraDAMENTAL THEOREMS OF PROBABLE mFEBENCE
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 + l)/h==a/h + b/h-ablh.
In IX. write (a + b) for a, and db for b.
Then (a + &)a6/A + {a + b)abjh = (a + &)/A,
whence dblh + {a + b){a+5)/h = {a + b)/h by (iii.),
d/bh . b/h + a/h = (a + b)/h by (iii.) and IX.
That is to say, {a + b)/h = ajh + (1 - ajbh) . b/h,
= a/h + blh-ablh.
In accordance Asrith the principles of Chapter XII. § 6, this
should be written, strictly, in the form a/h + (b/h - abjh), or in
the form h/h + {a/h-ah/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 6 ' 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 6 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.
(24.2) ab/h+db/h = b/h,
since ab+db^b by (iii.),
and adb/h=0 by (19) and (8).
(24.3) (a + b)/h = a/h + bd/h. This follows from (24) and
(24.2).
144
OH. XIV FUNDAMENTAL THEOREMS 145
(24.4) {a + i + c)/h = {a + h)jh + cjh - {ac + bc)lh
= a/h + h/h + cjh - abjh - bc/h - cajh + abcjh.
(24.5) And in general
(24.6) If pj>tjh = for all pairs of values of s and t, it follows
by repeated application of X. that
n ,
(?)i+^2 + ...+pJ/A = 2p,/A.
1
(24.7) If i3,?)t/A,=0, etc., and (p^ +^2 + • • • +1^ JM = 1' *•«•
if PxPi, ■ ■ -Pn form, relatively to h, a set of exclusive and
exhaustive alternatives, then
1
(25) If j)jP2 • • -Vn form, relative to h, a set of exclusive
and exhaustive alternatives,
71
1
Since (:Pi+P2 + - • •+FJM = 1 ^7 hypothesis,
.-. {p-y +P2 + • • • +i'7i)/«^ = 1 Ijy (9) if ah is not inconsistent ;
and since Pspjl^ = by hypothesis,
■'• PsPtl'^^^^ ^y (9). if f^h is not inconsistent.
ji
Hence "Zpr/ah = (p^ +P2 + ... +Pn)la'h by (24.6)
1
= 1.
Also p^ajh =Pr/a,h . a/h.
Summing Sp^ajh = a/h . "Zpjah,
1 1
n
:. a/h = Xp^a/h, if ah is not inconsistent.
1
If ah is inconsistent, i.e. if a/h = (for A is by hypothesis con-
sistent), the result follows at once by (8).
(25.1) If p^a/h=X„ the above may be written
/ J, ^r
2X,
1
L
146 A TREATISE ON PEOBABILITY n. u
(26) alh = {a+K)/h.
For (a + K)/h = a/h + K/h - aK/h by (24),
= ajh by (13.1) and (8).
(26.1) This may be ■written
a/h = {hz>a)/h.
(27) li{a + b)/h = 0, ajh=0.
a/h + [h/h-al/h] =0, by (24) and hypothesis
.-. a/h = by (v.).
(27.1) If a/A=0 and 6/A=0, {a + b)/h=0. This follows
from (24).
(28) If a/A = 1, (a + h)/h = l,
(a + B)/h = a/h + Eajh by (24.3),
whence {a + d)/h = a/h = l by (l.l) and (8), together with the
hypothesis. That is to say, a certaia proposition is imphed by
every proposition.
(28.1) If a/h=0, (d + b)/h=l by substituting a for a and b
for 5 in (28). That is to say, a certainly false proposition
impUes every proposition.
(29) If a/(Ai + A2) = l, a/hi = l,
dl(h^ + hz)=0,
and .-. a{li-j^ + h^l\ = by (15).
Hence ah^h^ = Q by (27),
whence the result.
(29.1) If a/Ai = l and alh^ = \, al{h^ + li^)'^l.
As in (20) ah-J{Ji-^ + h^=Q and ahj(hj^ + h^=0.
Hence a{hj^ + hz)/{hj^ + h^)=0 by (27.1),
whence the result.
(29.2) If a/{hj^ + hz)=0, a/Ai=0. This follows from (29).
(29.3) If a/Ai = and a/Aa = 0, a/{hj^ + h2)=0. This follows
from (29.1).
3. Irrelevance and Independence :
(30) If a/h-ih^^a/h-i, then aj'h-Ji2 = ajhy^, if h-Ji^ is not incon-
sistent.
alh-y = ah^h-^^ + ah^h-^ by (24.2),
= a/hjJi^ ■ h^hj^+a/hJiQ ■ Hjh-y, •
= ajh-^. . hjhj^ + a/hji^ ■ ^Jh^,
.: a/hj^ . hjh-i = ajh-Ji^ . K^/h-^,
whence a/hi = a/hiK2, unless Jijh-y = 0, i.e. if hji^ is not in-
consistent.
CH. XIV FUNDAMENTAL THEOREMS 147
Thus, if a proposition is irrelevant to an argument, then the
contradictory of the proposition is also irrelevant.
(31) If a^aji = a^h and aji is, not inconsistent, a-jaji=ailh.
This follows by (iv. c), since a^jaji .ajh =ajaji.a2lh by X.
If, that is to say, a^ is irrelevant to the argument ajh (see
XIV.), and aa is not inconsistent with h : then a^ is irrelevant
to the argument a-Jh ; and ajh and ajh are independent
(see XIII.).
4. Theorems of Relevance :
(32) If a/Mj >a/h, hjah >'h.Jh.
ah is consistent since, otherwise, a/M^ = ajh = 0.
Therefore ajh . h^ah^ajhh^ . h^jh by X.,
>ajh . hjh by hypothesis ;
so that hi/ah>hjh.
Thus if Aj is favourably relevant to the argument a/h, a is
favourably relevant to the argument hjh.
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 a/h, than q is to b/h, if -^>-^j i-e. if, in the
a/h o/h
language of § 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 h-^ is not less favourable
to a/hx than x is to a/hhi, then h^^ is favourable to a/h.
For a/hh, = a/h . -^ — . ,_ ^ . „/^ ; and by hypothesis the
a/h a/hx a/hhjX
second term on the right is greater than unity and the pro-
duct of the third and fourth terms is greater than or equal
to unity.
(33.1) A fortiori, if a; is favourable to a/h and not favour-
able to a/hhi, and if h^ is not unfavourable to a/hx, then A^ is
favourable to a/h.
(34) If x is favomable to a/h, and h^ is not less favourable
to x/ha than x is to h^/ha, then A^ is favourable to a/h.
This follows by the same reasoning as (33), since by an
application of the Multiplication Theorem
148 A TREATISE ON PROBABILITY pt. n
a/hk^x a/hhj^ aj/M^a hjha
ajhx a/hhjX x/ha ' hjhax
(35) If X is favourable to ajh, but not more favourable to it
ttan h-^x is, and not less favourable to it than to a/hh^, then
hi is favourable to a/h.
-c I1.T II i «A a/hhix] (a/hx alhlu
For aMi = ah.\-j^.-l-^\.\-!—.-J—l..
[a/hx a/h ) \ a/h a/hh^x.
This result is a little more substantial than the two
preceding. By judging the influence of x and h^x on the
arguments ajh and ajhh^, we can infer the influence of h^ by
itself on the argument ajh.
5. The Multiplication Theorems :
(36) If ajh and ajh are independent, a^ajh = ajh . ajh.
For a-^ajh = ajaji . ajh = ajaji . ajh by X.,
and since ajh and ajh are independent,
ajaji = ajh and ajajh = ajh by XIII.
Therefore a^ajh = ajh . ajh.
Hence, when ajh and ajh are independent, we can arrive at the
probability of % and a^ jointly on the same hypothesis by simple
multiplication of the probabiUties ajh and ajh taken separately.
(37) If pjh =pjpji =pjpi2)zh = . . .,
PiP2P3...pJh = {pJhY.
For PiPzPs ■ ■ ■ /h =pjh . pjpji . pjpipji ... by repeated
applications of X.
6. The Inverse Principle :
(38) ^^ = -Ip^ ■ ^, provided Ih, aJi, and a^h are
ajbh o/a^h ajh
each consistent.
For ajih . hjh = h/aji . ajh,
and aJbh . hjh = l/aji . ajh by X.,
whence the result follows, since Ijh^O, unless hh is in-
consistent.
(38.1) If ajh=p-y, ajh=p2, l/ajb = qi, llajh=q^, and
ajhh + aJbh = 1, then it easily follows that
ajbh--
PiSi
P1Q1+P2S1
FUNDAMENTAL THE0EEM8 149
and ' ajbh=^^^
(38.2) If ajh = ajh the above reduces to
S1 + S2
and aJhh = ^^ ,
since ajh=i=0, unless a^h 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 6 represent the occurrence of an event B, a^
and ttg the hypotheses of the existence of two possible causes
Aj and Ag of B, and h the general data of the problem. Then p^
and P2 are the d priori probabiUties of the existence of A^ and Ag
respectively, when it is not known whether or not the event B
has occurred ; y^ and q^ the probabilities that each of the causes
Aj and Aj, if it exists, will be followed by the event B. Then
^ ^ and — — are the probabilities of the existence
of Ai and Aj 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 hh 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 Priaciple
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 com-
bination of conclusions ; given a/h^ and a/h^ we considered the
relation of aiajh to these probabilities. In this paragraph the
corresponding problem of the combination of premisses will be
150 A TEEATISE ON PEOBABILITY pt. n
treated; given a/h^ and ajh^ we shall consider -the relation of
a/hji^ to these probabilities.
(39) alh,h,h = ^-^= ..y^?./. V X. and (24.2)
u + v
where u is the a priori probability of the conclusion a and both
hypotheses h-^ and h^ jointly, and v is the d priori probability
of the contradictory of the conclusion and both hypotheses \
and ^2 jointly.
ah-^jh^ + aAj/^2 /^i/a«'2 • i + Aj/aA^ • (1 - 2')
hjahi . p
hjahj^ . p + hjah-y . (1 -^) '
where p = ajh-^ and q = ajh^.
(40.1) If 2? = ! a/hjh.
and increases with
hjdhi
These results are not very valuable and show the need of an
original method of reduction. This is suppHed 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 = {al'b}a/h Def.
We may call {a^hj the coefficient of influence of b upon a on
hypothesis h.
XVI. {a"b} ■ {ah^ = U'bh} Def.
and similarly {aJ'b} ■ {abhdh} = {a!'b''cdh}.
These coefficients thus belong by definition to a general class of
operators, which we may call separative factors.
(41) ab/h=:{a"b}.a/h.b/h,
since ab/h = a/bh . b/h.
^ The substance of propositions (41) to (49) below is derived in its entirety
from his notes, — the exposition only is mine.
CH. 3:iv FUNDAMENTAL THEOREMS 151
Thus we may also call •[a''6} the coefficient of dependence between
a and 6 on hypothesis h.
(41-1) abcjh = UHh) . a/h . b/h . c/h.
For ahc/h = \ahh'\ab/h . cjh by (41),
= {al^ . {aJ^h} . ajh . Ijh . c/h by (41).
(41.2) And in general
aicd ...lh = {aJ'bhV . . .] .ajh .Ijh . cjh .djh...
(42) {a%) = {h^,
since " a/bh . bjh = bjah . a/h,
(42.1) {aH^ = {ah^b},
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. {aHc^ defg] = [bc^aY def}
[a^bc" defg] = [ahb^fed "g} .
(43) As a multiplier the separative factor operates so as to
separate the terms that may be associated (or joined) in the
multiphcand.
Thus {ab^cdh} . {a^ = {a^bhd\},
for abcde/h = (abhd^ej -. ab/h . cd/h . e/h
= {abhd"e} . {aH} . a/h . b/h . cd/h . e/h,
and also abcde/h = [a^b^dh] . a/h . b/h . cd/h . e/h.
Similarly (for example)
{abc^'dhf} . {abh} . {a''&} = {a''&''c^cZV}-
(44) {a^b}.{ab} = {a''b}.
For ab/h = {a&} ab/h .
By a symbolic convention, therefore, we may put {ah] =1.
(44.1) If {a"6}=l, it follows that a/h and b/h are in-
dependent arguments ; and conversely.
(45) Rule of Repetition {aa^b} = {«"&}.
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/hY+^a/xh .b/xh...: {x'/hy+^a/x'h . b/x'h . . .
: {x"/hf^^a/x"h . b/x"h . . .
which may be written
152 A TREATISE ON PROBABILITY m- n
»+l n+l n.+l
Hx/ah : Ux'/ah : Ilx"/ah : . . .
= {xlhT+^Ualxh : {x'jhf+^Uajx'h : . . .
Now
x/hahc . . .: x'/hahc . . .: x"/habc . . .
= x/h . (aic . . .) /xh : x'/h . (ale . . .) jx'h : . . . by (38),
and
abc... /xh = {a^^b^h . . . }Ua/xh by (41.2),
.-. {xlkf .xjhabc ...: {x'/h)'^.x'jhabc ...: {x"lhy.x"lhabc ...:...
= {a^"&*c . . . )xlah . x/bh .x/ch...: {a^Vc . . . }x'/ah . x'/bh
. x'/ch .......
which may be written
(jc/A)" . x/habc . . . x{a'"'¥''c . . .} . x/ah . x/bh . x/ch . . .
where variations of x are involved.
The cumulative formula is to be applied when, having accumu-
lated 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/ak, x/bh, etc., we require to know two
other sets of values, viz. : (1) x/h, etc., i.e. the d priori
probabilities of x, etc., and (2) {(f'^¥h . . . }, etc., i.e. the
coeflGlcients of dependence between a, h, and c ... on hypotheses
xh, etc. It may be remarked that the values {a*6'"''c . . .},
\af%'^^c ...}... are not in any way related, even when x' =x.
What corresponds to the cumulative formula has been em-
ployed, sometimes, by mathematicians in a simplified form
which is, except under special conditions, incorrect. First, it
has been tacitly assumed that {afWh . . . }, {a'"'b''h ...}...
are all unity : so that
{x/hyx/habc . . . oc x/ah . x/bh . x/ch . . .
Secondly, the factor (x/h)'^ has been omitted, so that
x/habc . . . oc 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 = {af'b'^c . . . } a/xh . b/xh . c/xh . . .
then x/habc . . . az^a'^'^b c . . . } x/ah . x/bh . x/ch . . .
1 See p 180 below.
CH. XIV FUNDAMENTAL THEOREMS 153
This result is exceedingly interesting. Mr. Johnson is the first to
arrive at the simple relation, expressed above, between the direct
and the inverse formulae : viz. that the same coefficient is re-
quired for correcting the simple formulae of multiplication in
both cases. As he remarks, however, while the direct formula
gives the required probabihty directly by multiplication, the
inverse formula gives only the comparative probability.
(46.2) If X, x' , x" . . . are exclusive and exhaustive alterna-
tives.
xjhabc .
(x/hy.{a'^¥h
. . .}'n.x/ah
2[(£c7A)-".{a"^''6^\
. . . }Ux'/ah]
since
xl7iaic...x{xlh)-''{a'"Vc
71-1
. . . ^Ux/ah,
and
^x'/habc . . . = 1
by (24.7)
(47).
x/habc .
x/h
. . a/h . b/h .c/h...
abc .. ./h a
abc . . . /xh
jxh . b/xh . c/xh . . .
x/ah x/bh
' _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 . c/xh .
a/xh b/xh
_ a/h b/h
whence the result, smce -!-p-= -!-— , etc.
a/h x/h
(47.1) The above formula may be written in the condensed
form
{"''bo ■ • . M = pw".T7}" ^"^"'^"'''' -y t^''"*'} • ^^'"'^ • ^''''> • • • ^•
. .g. {x/hYx/habc ..._ {a^^^h"^ . . .} x/ah . x/bh . x/ch . . .
^ '' {x/hYx/habc ..." {a^>'¥\'''' ..)' x/ah . x/bh . x/ch ...'
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.)
154
A TREATISE ON PROBABILITY
This formula gives xjhabc ... in terms of xjah, xjbh, etc.,
together with the three values x/h, [af^V'^c'''' . . .}, and
(48.1)
xjhabcd ■ . ■ xjhhcd .
{(f%cd . . . } . xjah x/h
{a^lcd . . . } . xjah ' x/h
xjhabcd ..." xjhhcd .
This gives the effect on the odds (prob. x : prob. x) of the extra
knowledge a.
(49) When several data co-operate as evidence ia favour of a
proposition, they contiaually strengthen their own mutual
probabiUties, on the assumption that when the proposition
is known to be true or to be false the data jointly are not
counterdependent.
Z.e. if {a*6'"*'c . . . } and [a^lF^c . . . } are not less than
unity, and xjkh>xjh where k is any of the data a,b,o..., then
^aJ^y^c^d . . .]■ beginning with imity, continually increases, as
the number of its terms is increased.
ahc . . . jh=xdbc . . . jh+xabc . . .jh by (24.2).
= xjh .dbc... jxh+xjh . ahc . . . jxh.
'^xjh . Uajxh . hjxh . . .+ xjhUajxh . hjxh . . .
(siace {a'^'^h^'^c . . ..} and {a^'^Vh . . . } are not less than unity),
~axjh hxjh
xjh xjh
xjah xjbx
xjh
^xjh . np
ahc . . . jh
.„ ^Vaxjh hxjh
+ xjh.U\-^.^-
L xjh xjh
U[ajh.hjh...]
xjh.U
xjh
■ \+xjh
-p. Vxjah xjhh
L xjh xjh
We can show that each additional piece of evidence a,b,c...
increases the value of this expression. For let xjh . G+xjh . G' be
its value when all the evidence up to k exclusive is taken, so that
xjkh.G+xjkh.G'
is its value when k is taken. Now Gr>G' since xjah>xjh, etc.,
and xjah<xjh, etc., by the hypothesis that the evidence
favours x ; and for the same reason xjkh - xjh, which is equal
to xjh -xjkh, is positive.
.-. G (xjkh - xjh)>G'{xjh - xjkh),
i.e. xjkh . G + xjkh . G'>xjh . G + xjh . G',
whence the result.
CH. XIV FUNDAMENTAL THEOEEMS 155
(49.1) The above proposition can be generalised for the
case of exclusive alternatives x, x' , ad' . . . (in place of z, x).
For {al'h^^ ...)
= x/h . {a*6^^c . . . ) ] a'^ {b'^x} {c^'x} . . .
+ x'/h . {a^Vc ...) {aV} {&V}{cV} . . .
+ x"/h . {a'"V\ ...} {aV} \b^x"}{GV} ... + ...;
from which it follows that, if {a^Vc . . . }etc. <t:l, and if
{a'^x} -1, {b''x-l}, {c''a3-l}, etc., have the same sign, then
{aJ'bh . . . } 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 for impossibility and 1 for certainty was
first introduced by Leibnitz in a very early pamphlet, entitled
Specimen certitvMnis seu demonstrationum in jure, exhibitum in
doctrina conddtionum, published in 1665 (vide Couturat, Logique de
Leibnitz, p. 553). Leibnitz represented intermediate degrees of
probability by the sign ^, meaning, however, by this symbol a
variable between 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 ia detail in Chapters XV., XVL, and
XVII., no one has worked out anything very elaborate.
Mr. McColl published a number of brief notes on Probability of
considerable iuterest — 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 underUes his symbolism, as it does mine.^ The
probability of a, relative to the a priori premiss h, he writes - ; and
the probability, given b in addition to the d priori premiss, he writes
j-. Thus - = ajh, and 5- = ajbh. The difference t — , i-e. the change
in the probabiUty of a brought about by the addition of b to the
evidence, he calls ' the dependence of the statement a upon the state-
1 I did not come across these notes until my own method was considerably
developed. Mr. MoColl has been the first to nse the fundamental symbol of
Probability.
156 A TREATISE ON PEOBABILITY pt. n
ment b,' and denotes it by St- Thus 8t = 0, where, in my termin-
ology, 6 is irrelevant to a on evidence h. The multiplication and
,,. . , , , . , „ ah a h b a
addition formulae he gives as follows: — = -.- = -.-.
° e e a e
a + b a h ah
e e e e "
Also S- = =58-, where A= -
D a €
It is surprising how little use he succeeds in making of these good
results. He arrives, however, at the inverse formula in the shape —
Cr V
V ^ '■=" Cr v'
2j — ■ —
r=l £ ^r
where Cj^ . . . c„ 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
e
b a
, - + - .
a £
b
a
In a paper entitled " Operations in Eelative Number with Apph-
cations to the Theory of Probabihties," ^ Mr. B. I. Gilman attempted
a symbohc 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 probabUity
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 TJntersuchungen iiber die Ermittlung von Wahr-
scheinlichkeiten, made a beginning on somewhat similar Hues ; 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
^ Published in the volume of Johns HopHna Studies in Logic.
^ g, vol. i. pp. 43-48.
OH. XIV FUNDAMENTAL THEOEEMS 157
E by W(E), and the probability of tbe event E given the occurrence of
an event F by Wj.(E). He uses this symbol to give "Wj,(E) = WjCE)
as the criterion of the independence of the events E and F (F denoting
the non-occurrence of F) ; Wj,(B) = 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. Porctzki entitled
" A Solution of the General Problem of the Theory of Probability by
Means of Mathematical Logic." I have seen it stated that Schroder
iatended to publish ultimately a symbohc 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 ia order to make the convention of numerical measure-
ment 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 advcmce 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 r.a/h=: h/f, then a/h = -. h/f. Def.
XIX. If h/f=q.c/g, then ^-.hlf^^lg. Def.
r r
Thus if blh = ajh + ajh+ ... to r terms, then the probability
bjh is said to be r times the probability ajh ; hence if ahjh =0 and
a/h=b/h, the probability {a + b)/h is tivice the probabiHty a/h.
If a and b are exhaustive as weU as exclusive alternatives re-
latively to h, so that (a + b)/h = 1, since we take the relation of
certainty as our unit, then a/h=b/h—^.
We also need the following axiom postulating the existence of
relations of probabiUty corresponding to all proper fractions :
158
CH. XV FUNDAMENTAL THEOREMS 159
(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 -•
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 «//=- and h/h = \ alf+hlh^ '^^'^ .
m n ' mn
Let the probability ^ — =P, which exists by (vii.),
mn
then 71 . P = - = a// by (XIX.),
and m . P = - = b/h,
n
.: a/f + llh = n . P + ??i . P, if this probabihty exists,
= P + P . . . to w terms + P + P . . . to m terms,
= P + P .. .to m + n terms,
= {m + n)-p = '"^'^ by (XIX.).
mn J \ )
This probabihty exists in virtue of (vii.).
4. Many probabilities — ^in fact aU those which are equal to
the probabihty of some other argument which has the same
premiss 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 ia the most usual sense, unless the pro-
bability with which they are thus comparable is the relation
of certainty. The conditions under which a probabihty a/h is
numerically measurable and equal to - are easily seen. It
160 A TEEATISE ON PEOBABILITY pt. n
is necessary that there should exist probabilities a^h^^, ajh^ . . .,
aj\ . . . ajh„ such that
ajhj^ = ajh^ = . . . = ajhg = . . . = a^/h„
q r
a/h = XaJh^, and Xajhs = l.
1 1
If alh=^ and 6/A = ^, it foUows from (32) that ablh=^
only if a\h and h\h are independent arguments. Unless, there-
fore, 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 probabiUty, therefore, is concerned with arguments
which are hoQi 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 equi-
probabiUty 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 informa-
tion favouring one of them. In actual reasoning, therefore,
perfectly equal probabiUties, 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. 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 prob-
ability, 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 ques-
tion in the form — Is it more or less probable than so and so ? —
where ' so and so ' is some comparable and better known prob-
ability. 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 prob-
OH. XV FUNDAMENTAL THEOREMS 161
ability 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 probabihty
which is taken as the standard of comparison is itself non-
numerical and not, as in Darwin's instance, a numerical one,
wiU 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 com-
pletely 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 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 pur-
pose, 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.^
^ Life and Letters, vol. ii. p. 240.
^ 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 oonolu-
sion ; 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. vni.) under the title "On the Conditions by which the
Solutions of Questions in the Theory of Probabilities are limited."
3 Theory of Statistics, chap. ii.
M
162 A TREATISE ON PROBABILITY pt. u
(51) xyjh always lies between^ xjh and x/h+y/h-l and
between y/h and xjh + y/h-1.
For xyjh = xjh -xyjh by (24.2),
= xlh - y/h . xjyh by X.
Now xjyh lies between and 1 by (2) and (3),
.-. xy/h lies between x/h and xjh -y/h,
i.e. between x/h and x/h+yjh-l.
As xylh<0, the above limits may be replaced by x/h and 0, if
x/h+y/h-l<0.
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 inde-
pendent arguments. For instance, i£ y/h is nearly certain, ay/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+l
(52) XjX^ . . . Xji^^Jh is always greater than S xjh - n.
1
For by (51) x^x^ . . . x„+Jh>XjX2 . . . xjh+x^+jh - 1
>x^x^ . . . a;„ _ i/A + xjh + x^+Jh - 2,
and so on.
(53) xy/h + xy/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 -xy/h
and xy/h = y/h - xy/h,
.: xy/h + xy/h = x/h - y/h + 1 - 2xy/h,
whence the required result.
(54) xy/h - xy/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 con-
clusions (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
^ In thia and the following theorems the term ' between ' includes the
limits.
CH. XV FUNDAMENTAL THBOKEMS 163
of its not being comparable with any of our standard probabilities.
Thus oui 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 X + v = a (i.)
\ + a- = h (ii.)
\+i'+cr = c (iii.)
X + fjL+v+p = d (iv.)
X+fj, + a- + T = e (v.)
\+IJ, + v+p + cr+T+v = l (vi.)
where \, fi, v, p, a, t, v represent probabiUties which are to be
eliminated, and Hmits are to be found for c in terms of the
standard probabiUties a, b, d, e, and 1.
X, fi, etc., must all lie between and 1.
From (i.) and (iii.) a-=c-a; from (ii.) and (iii.) v=c-b.
From (i.), (ii.), and (iii.) X = a + b-c.
whence c-a^O, c-'b'>0, a + b-c^O,
substituting for a; v,\ia (iv.), (v.), and (vi.)
fi + p^d-a, fi + T=e-b, fi + p + T + v = l-c,
whence p = d - a - fj,, T = e-i-fi, v = l-c-d + a-e + b+fi,
.-. d-a-fi^O, e-l-fjb->0, l-c-d + a-e + b+iJb->0.
We have still to eliminate jx. /M^d - a, fi^e - b,
fjb^c + d + e-a-b-1,
.: d-a'>c + d + e-a-b-l ebuA e-b^c + d + e-a-b-1.
Hence we have :
Upper limits of c: — b + 1 -e,a + l-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.
CHAPTBE 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 a-Jh and aj/A are independent arguments.
In Theorem (33) of Chapter XIV. it was shown that, if a^h and
a^h are independent, a-^a^/h=ajjh .ajh. Thus where on given
evidence there is independence between a^ and a^, the probability
on this evidence of a^a^ jointly is the product of the probabilities
of Oj and a^ separately. It is difficult to apply mathematical
reasoning to the Calculus of Probabihties unless this condition
is fulfilled ; and the fulfilment of the condition has often been
assumed too hghtly. A good many of the most misleading
fallacies in the theory of Probabihty have been due to a use of
the MultipUcation 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 pro-
positions. 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 inde-
pendent 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.
164 *
OH. XVI FUNDAMENTAL THEOREMS 165
This sense, which it is not easy to define quite precisely, is
at any rate not the sense with which we are concerned when we
deal with independent probabihties. We are concerned, not with
direct causation of the kind described above, but with ' depend-
ence 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 sorne 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 know-
ledge 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 con-
nected 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 stUl obscure, and
I have preferred to define independence by reference to the con-
ception of relevance, which has been already discussed. Whether
there reaUy 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 nothing useful to
say about them. Nearly everything with which I deal can be
expressed in terms of logical relevance. And the relations be-
tween logical relevance and material cause must be left doubtful.
166 A TREATISE ON PEOBABILITY pt. n
4. It will be useful to give a few examples out of writers who,
as I conceive, have been led into mistakes through misappre-
hending 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 possibiUties,' an ' objective possibility '
being a chance event brought about by the combination or con-
vergence of phenomena belonging to ivdefendent 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 apphcations of probability would remain hypo-
thetical.
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 inde-
pendent, 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 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 amxmgst 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 1 ' the natural reply would be, ' The percentage
of vaccinated amongst the attacked should be the same as in
the general population.' . . ."
^ For some account of Coumot, see Chapter XXIV. § 3.
2 " Notes on the Theory of Association of Attributes in Statistics," Bio-
metrika, vol. ii. p. 125.
CH. XVI FUNDAMENTAL THEOREMS 167
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 propor-
tions in a given world actually are. As ia 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 difierent from
either of these objective forms of iadependence.
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 acquiescence by giving a perfectly correct defini-
tion : " Two events are said to be independent when the
probability of the happening of either of them is unafiected by
our expectation of the occurrence or failure of the other." ^ But
a moment later he interprets the term ia quite a different sense ;
for, accordiag 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 in-
dependent 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 com-
pound event depending upon them is precluded from occurring.
1 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 Jmws of Thought" {Phil. Mag. 4th series, vol. vii., 1864). 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.
MoCoU, in his " Sixth Paper on Calculus of Equivalent Statements," saw
that Boole's fallacy 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.
168 A TEEATISE ON PEOBABILITY pt. h
. . . 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 em-
ploying the term independence in a double sense are far-reaching.
For he uses a method of reduction which is only vaUd when the
arguments to which it is applied are independent in the first
sense, and assumes that it is vaUd 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 ia results so paradoxical that they might have led him
to detect his fundamental error.^ Boole was almost certainly
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 definiticJn : " Two events are independent
if the latter might have existed without the former, or the
1 Op. cit. p. 258.
^ There is an excellent instance of this, Laws of Thoughi, p. 286. Boole
discusses the problem : Given the probability p of the disjunction ' either Y
ia true, or X and Y are false,' required the probability of the conditional pro-
position, ' If X ia true, Y ia true.' The two propositions are formally equivalent ;
but Boole, through the error pointed out above, arrives at the result — — ,
1 —p-j-cp
where c ia the probability of ' If either Y is true, or X and Y false, X ia 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 neoeaaarily equivalent.
* In studying and oritioiaing Boole's work on Probability, it is very im-
portant to take into account the various articles which he contributed to the
Philosophical Magazine daring 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 aa superaeding the expoaition 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.
CH. XVI FUNDAMENTAL THEOEEMS 169
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 independenoCj 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 and approximate solutions.
Even De Morgan was careless enough in one of his writings ^
to enunciate the Multiphcation Theorem in the following form :
" The probabiUty 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 probabiUty
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 deter-
mining the probability of throwing heads twice in two consecutive
tosses of a coin. The plain man generally assumes without
hesitation that the chance is (J)^- For the d priori chance of
heads at the first toss is J, and we might naturally stfppose that
the two events are 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 Hes, then it is reasonable to put the probability
somewhat higher than (J)^. The 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,
1 " 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 defini-
tion of Independence is also adopted by Maofarlane, Algebra of Logic, p. 21.
^ " Theory of Probabilities " in the Encyclopaedia Meiropolitana.
170 A TREATISE ON PROBABILITY pt. n
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 more than refer to an investigation by Laplace and to
one real and one supposed fallacy of Independence of a type with
which we shall not be concerned in Chapter XXIX.
8. Laplace, in so far as he took account at all of the considera-
tions explained in § 7, discussed them imder the heading of Bes
inigcditis mammies qui peuvent exister entre hs chances que Von
suppose egaks.^ 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 | by an amount unknown. But
this is not the correct 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 ' =^i ; ' heads at second toss '
=h,^. Then h-i]'h = 'hj'h='^, and hrji^jh^hjhji .h^h. Hence
hjijh = [hjhj^ only if h^hji=hjh, 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
hjhih will not differ greatly from h^/h (for relative to most hypo-
theses heads at the first toss will not much influence our expectation
of heads at the second), and \ will, therefore, give a good approxi-
mation to the required probability. Laplace suggests an ingeni-
ous 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 hy, ty, h^, t^ denote heads and tails respectively with
the first and second coins respectively at the first toss, and
hy, tl, h^, t.^ the corresponding events at the second toss, then
1 Essai pMlosophique, p. 49. See also " Memoirs sur les Probabilit^s," Mim.
de I'Acad. p. 228, and op. D'Alembert, " Sur le ealoul dea probabilitSs,"
Opuscules mathemaliques (1780), vol. vii.
OH. XVI FUNDAMENTAL THEOREMS 171
the probabiKty (with the above convention) of ' heads twice run-
ning,' i.e. agreement between the two coins twice running, is
{h^^ + t^t^){\h^ + t^t^')/h = {h^h^' + ht^)l{h^h^ + t^t^, h)
Since h^h^'/ihjhjj + 1^^, h) = t^t^l{hji^ + t^t{, h) by the Principle
of Indifference, and Aj^a'^a^a'M = 0-
.-. (Aj^a' + ht/)l{\h^ + tyt^, h) = 2.hjb^ lihjij; + t^t^, h) by (24.1).
Similarly {h-Ji^^ + t^t^'yh = Ih^h^jh.
We may assume that h-jjh-^h =h-i]h, i.e. that heads with one
coin is irrelevant to the probability of heads with the other : and
hjh=hi'/h=^ by the Principle of Indifference, so that
(AA' + <iV)M = 2(i)^ = i.
= ihjihg, hjh^' + tji^, h),
since, (hj)^' +tjt-^') being irrelevant to h'Jh, h'^jQiJi^ +t-^ti, h) =
Now hjih^, hji^ + tj,^. h) is greater than J, since the fact of
the coins having agreed once may be some reason for supposing
they will agree again. But it is less than hjhji : for we may
assume that hjih^', hji-^ +tj,^, h) is less than hj{h^, Ji-Jii, h),
and also that h^Qi^, hjii, h)=h^hjh, i.e. that heads twice
running with one coin does not increase the probabiUty of heads
twice running with a different coin. Laplace's method of tossing,
therefore, jdelds with these assumptions, more or less legitimate
according to the content of h, a probability nearer to | than is
hji^lh. If ^2/(^2'; JtJi^+hPi, ^)=i) then the probability is
exactly J.
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 \, and that similarly the probability of the existence there of
any other element is also \. The probability, therefore, that
not one of the 68 terrestrial elements mil be found on Sirius
is {\)^^, and that at least one will be found there is 1 - ( J)^* or
approximately certain. This argument, or a similar one, has
been seriously advanced. It would seem to prove also, amongst
172 A TREATISE ON PROBABILITY pt. n
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 IndiSerence. The probability of iron on Sirius is not \. 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 probabihty.^
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 great velocity in every direction. Both the directions
and velocities are unknown, but the probability that a molecule
has a given velocity is a fmiction of that velocity and is inde-
pendent 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 <^{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
(f){x)(l){y)<j)(z). Thus if F{v) represents the probability of a total
velocity v, we have (j){x)<l)(y)<j){z) = ¥{v), where v^ = x^+y^ + z^.
It is not difficult to deduce from this (assuming that the
' See Von Kries, Die Principien der Wahrscheinlichkeitsrechnung, p. 10.
Stumpf {tJber den Begriffder 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 is -, where
z is the number of possible atomic weights. A. Nitsche ( VierteljscJi. f. wissensch.
PAiZo*., 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 probable, the chance
of each being ^.
^ I take the statement of this from Bertrand's Calcul des probabiliies, 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
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).
CH. XVI FUNDAMENTAL THEOEEMS 173
functions are analytical) that d){x) must be of the form
It is generally agreed at the present time that this result is
erroneous. But the nature of the error is, I think, quite difEerent
from what it is commonly supposed to be.
Bertrand,^ Poincare,^ and Von Kiies,^ all cite this argument of
Maxwell's as an illustration of the fallacy of Independence ; and
argue that <f}{x), (f){y), and <j){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 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 a; ' by
X(a;), the corresponding propositions relative to the axes of Y
and Z by Y{y) and Z{z), and the proposition ' the total
velocity is d ' by Y{v) ; and let h represent our d priori data.
Then if 'K{x)/h=^{x) it is a justifiable inference from the
Principle of Indifference tha,t Y(y)lh=(p(y) and Z{z)/h=<f){z).
Maxwell infers from this that X{x)Y(y)Z{z)/h=<ji{x)<j){y)^(z).
That is to say, he assumes that Y{y)jX(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 prob-
ability 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=Y(v)/h, where v^=x^+y^ + z^. It is here,
and in a very elementary way, that the error creeps in. The
propositions X(a!)Y(^)Z(z) and Y{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,
' Calcul des probabilitia, p. 30.
2 Calcul des probabiUtes (2nd ed.), pp. 41-44
^ WahrscheinlichkeitsrecJinung, p. 199.
174 A TREATISE ON PROBABILITY pt. n
which will yield the same value v. Thus the probability Y{v)jh
is much greater than the probability X(a;)Y(y)Z(z)/A. 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 com-
ponents parallel to the axes. Indeed I think it is a legitimate
extension of the preceding argument to put V(i;)/A=^(«) ; 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 buUets over a target — a subject round which,
on accoimt 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 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 com-
municated to the Royal Society,^ was a rule for the determination
of inverse probabilities expUcitly enimciated. It is true that
solutions to inductive problems requiring an impUcit and more
or less fallacious use of the inverse principle had already been
propounded, notably by Daniel Bernoulli ia his investigations
iato 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 distribu-
1 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.
2 " Essai d'une nouvelle analyse de la mortality causte par la petite v6role,
et des avantages de I'inoculation pour la pr^venir," Hist, de I'Acad., Paris, 1760
(pubUshed 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.
OH. XVI FUNDAMENTAL THEOEEMS 175
tion 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 "Memoice sur
la probabilite des causes par les evfenemens " {MSmoires
present6s d I'Academie des Sciences, vol. vi., 1774). He states
the principle as follows (p. 623) :
" Si un 6venement peut Stre produit par un nombre n de
causes diS&entes, les probabiUt6s de I'existence de ces causes
prises de I'evdnement sont entre elles comme les probabilites de
I'evenement prises de ces causes ; et la probabilite de I'eadstence
de chacune d' elles est egale a la probabilite de I'evenement prise
de cette cause, divisee par la somme de toutes les probabilites
de I'dvfenement prises de chacune de ces causes."
He speaks as if he intended to prove this principle, but he only
give 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 originaUty of Bayes's Memoir are con-
siderable, 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 enuncia-
tion 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
h P
of the second :j^ and the probabihty of both together — , and, it
being first discovered that the second event has happened, from
hence I guess that the first event has also happened, the prob-
p
ability I am in the right is r-" I^ *^® occurrence of the first event
1 Quoted by Todhunter, op. cit. p. 296. Todhunter underrates the import-
ance of this passage, which he finds unoriginal, yet obscure.
176 A TEEATISE ON PEOBABILITY pt. n
is denoted by a and of the second by b, this corresponds to
ablh^albh . h/h and therefore albh =^At- ; for ahlh =— , hlh =—,
b/h N N
P
albh=--. The direct and 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 plausibiUty of their thesis. Mr. McColl^ gave
a symbolic proof, closely resembling theorem (38) when differ-
ences 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 possibiUties narrower than that with which
we started. Only part of the original field or Spieh-aum of
possibility is now admissible (zuldssig). Causes have d posteriori
probabilities which are proportional to the extent of their occur-
rence in the now restricted field of possibility.
There is much ia this which seems to be true, but it hardly
amoimts to a proof. The whole discussion is in reality an
appeal to intuition. For how do we know that the possibilities
admissible d posteriori are still, as they were assumed to be d
priori, equal possibilities ? Von Kries himself notices that there
is a difficulty ; and 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 DonMn in 1851,
in an article which aroused some interest in the Philosophical
1 "Sixth Paper on the Caloulus of Equivalent Statements," Proc. Land.
Math. Soc, 1897, vol. xxviii. p. 567. See also p. 153 above.
' Wahrsoheinlichkeitsrechnung, p. 178.
' Die Principien der Wahrscheinlichkeitsrechnung, pp. 117-121. The above
account of Von Kries's argument is much condensed.
OH. XVI FUNDAMENTAL THEOREMS 177
iine 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
fiih^hg,. . . of which the probabiUties relative to a particular state
of information are Pip^^ • • •> ^^^ if ^^^ information be gained
which changes the probabilities of some of them, suppose of
fe^_i.i 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." 2
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 interpreta-
tion, 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 hj^. . .h^to the exclusion of the rest, the probability
of what is conveyed by the new information is the same whichever
of the hypotheses hj^. . .h^ has been taken, then Donkin's
principle is valid. For let a be the old information, a' the new,
and let hja =f„ \laa' =^^' ; then
, -, , , Km' la a'/hM . )o_
p, =h,aa =—J~=-^ ,
a ja a ja
:. ^— ='^, etc., if a'/h^a = a'/h^a, which is the condition already
Pr Ps
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
^ " 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. MarkofE ( WahrscheinKchkeitsrechnwng,
p. 8) under the title ' Unabhangigkeitsaxiom.'
N
178 A TEEATISE ON PKOBABILITY pt. n
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 a priori
probabilities of the hypotheses a^ and a^ drop out if pj^ =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 un-
likely, just in the same proportion that it is likely or unlikely
that observed events should foUow 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 a priori prob-
abilities 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 qualifica-
tion, have been led into 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 enimciates
is " that which common sense leads us to adopt almost in-
stinctively."
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
^ See the passage quoted above, p. 175.
' " Essay on Probabilities," in the Cabinet Encydopcedia, p. 27.
^ 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 attri-
OH. XVI FUNDAMENTAL THBOEEMS 179
by a superficial error, and De Morgan, adopting the same mis-
taken rule, pushed it 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 : " Eequired 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 circum-
stances X, being y, and the probabiKty, when we only know of
the existence of y, being y." * 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
bated 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
f , and in the other case f ; then the opposite probabilities wiU be f and f, which
multiplied together give -J-f as the probability against the conclusion. . . ."
The Archbishop's error, in that a negative can always be turned into an
affirmative by a change of verbal expression, was first poiuted 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."
1 " Theory of Probabilities," Encydopcedia 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 : " d priori
vegetation on the planets is neither likely nor unlikely ; suppose argument
from analogy makes it ^\ ; then the total probability is J+i . i\ or ^ J." De
Morgan seems to accept 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 (toe. cit. p. 620) that " the mean strength
of any probabilities of an event which are foimded upon different judgments
or observations is to be measured by that supposed probability of the event
a 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
zjx and zjy is to be measured by zjxy.
180 A TEEATISE ON PEOBABILITY pt. n
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,
ajhi=\ and alh^=\, he concludes that ajh-Ji2=\- The plausi-
bility of this is superficial. Consider, for example, the following
instance : A^ = A is black and B is black or white, i^j =B is black
and A is black or white, a = both A and B are black. Boole also
concluded without valid reason that 0/^1^2 increases, the greater
the & priori improbability of the combination h-Jiz-
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 win be worth whUe 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 wUl 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 win speak the truth is y. X and Y agree in a particular
statement. The chance that this statement is true is
xy
xy + {\-x){l-y)
For the chance that they both speak the truth is xy, and the
chance that they both speak falsely is (1 -a;)(l -y). As, in this
^ 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 Probabilites, p. 31.
^ E.g., Boole, Laws of Thought, p. 279.
De Morgan, Formal Logic, p. 195.
Condorcet, Essai, p. 4.
Lacroix, Traite, p. 248.
Cournot, Exposition, p. 354.
Poisson, Becherchea, p. 323.
This list could be greatly extended.
OH. XVI FUHTOAMENTAL THEOREMS 181
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 a^ stand for " X^ asserts
a," and let a/aJi=Xi, where h, our general data, is by itself
irrelevant to a, i.e., x^ is the probability that a statement is true
of which we only know that X^ has asserted it. Similarly let us
write hlbji=x^, where b^ stands for " Xg asserts 6." The above
argument then assumes that, if X^ and Xj are witnesses who are
causally independent in the sense there is no collusion between
them direct or indirect, ahlaJ}J>' = ala-Ji . hjhji^x^x^.
But ab/ajbji=afajbbji . bja^bji, and this is not equal to XyX^
unless ajajbbji=ajaji and b/ujb^h =blbji. It is not a sufficient
condition for this, as seems usually to be supposed, that X^ and Xg
shoidd 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 6j, it is evident that the formula breaks down. In the one
extreme case, where the assertions of the two contradict one
another, ahlaJ)Ji=Q. In the other extreme, where the two agree
in the same assertion, i.e. where a = b, alaj)bji = 1 and not = aja-Ji.
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 the problem include the hypothesis that X^ and
Xg are each asked and reply to a question to which there is only
one correct answer. Let a^ = " X^ asserts a in reply to the ques-
tion," and mi = "Xi gives the correct answer to the question."
mjJa]h=Xi and mJaJi=X2,
Xi and ajg being, in the conventional language of thi-: problem,
the " credibilities " of the witnesses. We have, since the wit-
nesses agree and since a follows from wi^aj and m^ follows from aa^,
a/aih^mi/aih;
ajaifn^ = 1 ; mjaafi = 1 .
AlsOjSiace the witnesses are, in the ordinary sense, "independent "
182 A TKEATISE ON PROBABILITY pt. h
witnesses, aja^ah=a^ah and ajajah=a2/dh ; that is to say, the
probability of X2's asserting a is independent of the fact of X^'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
t^'ieis / ,. , I. m^ajaji
a^a/aji + a^d/aji aja-^ah . Xy + aja^ah . (1 - JCg)'
If this is to be equal to ^, ^ ^^ -, we must have
XjX2 + {l-X{){l-X2)
a^ja-^ah x^
Now -J^ -J^ by the hypothesis of " independence »
aa^fh d/h ajaji d/h
da^jh a/h djaji a/h
X.2 d/h
1 - X2 a/h
This then is the assumption which has tacitly slipped into the
conventional formula, — that a/h=dlh = ^. It is assumed, that
is 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, d 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, d 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 d 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 d priori improb-
ability of some assertions outweighs the force of testimony
otherwise reliable, depends on the avoidance of it. The correct
view is also taken by Laplace in his Essai philosophigue (pp.
CH. XVI FUJIDAMENTAL THEOEEMS 183
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
EncyclopSdie) 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 re-
liability " or " coefficient of credibility." If a witness's credi-
bility is represented by x, do we mean that, if a is the true answer,
the probability of his giving it is x, or do we mean that if he
answers a the probability of a's being true is a; ? These two things
are not equivalent.
Let a^ stand for " a is asserted by the witness " ; ^ for our
evidence bearing on the witness's veracity ; and Ag ^or other
evidence bearing on the truth of a. Let a/hji^, i.e. the d priori
probability of a apart from our knowledge of the fact that the
witness has asserted it, be represented by p.
n Ih
Let ajaji^=x^ and a^la\=x^; so that !»i=— t^-^^z- ^^
Oil hi
general ajJi^d^ a-^l\. Do we mean by the witness's credibility
XiOX x^'i
We require a/ajiji^.
Let ai/dhi^ = 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.
x^ x^p
X2P + aJd]ijh2 . (1 -p) X2P+r{l -f)
for ajahjh2=ajahj^ and ajdhjh2=ai/ahi, since, given certain
knowledge concerning a, h^ is irrelevant to the probability of a^.
19. Generally speaking, all problems, in regard to the com-
bination 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.
184 A TEEATISE ON PEOBABILITY pt. n
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 for x
as a, where a is given by the equation
xjah^xlh + a-yi/xjh . x/h ;
so that a-y^xjh . xjh measures the amount by which A's assertion
of X increases its probability.
20. One of the most ancient problems in probability is con-
cerned 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 ia his Theologiae Christianae Principia
Mathematica, published in 1699.^ He proves that suspicions of
any history vary ia 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 iu 1789." ^ About the same
time Locke raised the matter in chap. xvi. bk. iv. of the
Essay Concerning Human Understanding : " Traditional testi-
monies the farther removed, the less there 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
* See Todhunter's History, p. 64. It has been suggested that the anonymous
essay iu the Phil. Trans, for 1699 entitled " A Calculation of the OrecUbility
of Human Testimony " is due to Craig. In this it is argued that, if the
credibilities of a set of witnesses are p^ . . . p^, then if they are
successive the resulting probability is the product p^p^ . . . y„ ; if they are
concurrent, it is : ^ _ (^ _^^)(i -p,) . . . (1 -p,).
This last result follows from the supposition that the first witness leaves an
amount of doubt represented by 1 - j), ; of this the second removes the fraction
Pj, and so on. See also Lacroix, Traite elimentaire, p. 262. The above theory
was actually adopted by BicquiUey.
^ 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 THEOEEMS 185
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 vaUd 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 weU attested, and by never so credible witnesses, will
not be admitted as a proof in Judicature." If this is stiU a good
rule of law, it seems to radicate an excessive subservience to the
principle of the decay of evidence.
But, although Locke affirms sound maxims, he gives no theory
that can afEord 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
diSerent 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,
Pi and ^2 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. Maofarlane 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 ana-
lytical 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 himseK who has ever attempted to use it. The
term ' cause ' is frequently used ia these examples where it might
have been better to use the term ' hjrpothesis.' 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 a priori probabilities of two causes Aj and A^
are c^ and Cg respectively. The probability that if the cause A^
^ Such examples as these might sometimes be set to teat 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.
186
OH. xvn FUNDAMENTAL THEOREMS 187
occur, an event E will accompany it (whether as a consequence
of Ai or not), is p^, and the probability that E will accompany Ag,
if Ag present itself, is p^. Moreover, the event E cannot appear
in the absence of both the causes A^ and Ag. Required the prob-
ability 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 equation
[1 - ei(l -pj) - ^] [1 - 62(1 -p^) - u] ^ (M-CiPiKw-Ca^a)
l-u Ci^i + CaPa-'"
which is not less than c^^ and c^2 ^^^ iiot greater than
1 -Ci(l -pj), 1 -C2(l -p^), or C1P1 + C2P2.
This solution can easily be seen to be wrong. For in the
case where A^ and Ag cannot both occur, the solution is
u==CjPi+c^2'' whereas Boole's equations do not reduce to
^ Phil. Mag. 4th series, voL t1
* Cayley's solution was defended against Boole by Dedekind (Orelle'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 ProbabiUties," 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.
WUbraham gave as the solution u^Cjp^+c^^-z, where z is necessarily less
than either c,yj or c^p.^- 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 piu:e 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."
188 A TREATISE ON PROBABILITY pt. n
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 «!, Oj, e assert the occurrences of the two causes and the
event respectively, and let h be the data of the problem.
Then we have aJh=Ci, a^h=C2, e/aji^pi, ejaji=p2' "^^
require ejh. Let ejh^u, and let ajajeh=z. Since the event
cannot occur in the absence of both the causes,
eldja2h = 0.
It follows from this that a-^a^eh^O, unless ejh=0,
i.e. {a^ + a^leh = \,
whence a-^eh + a J eh = 1 + a^ajeh by (24).
Now ajeh = ^^^ and aJeh = ^,
. . w ,
where z is the probability after the event that both the causes were
present.
If we write ea-^a^jh—y,
y = a-^ajeh . e/h = uz,
so that u = (cj^i + C22'2) ~ 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 solu-
tion which are independent of y and z: then, since y^O,
Again
e/h = eajjh + eajh^djh + eajh^l - c^ + CjPj^ by (24.2) and (4).
Similarly e/^ ^1 - Cg + C2P2. From the same equations it appears
that e/h'>Cjjp-i and '^c^p^-
'' Boole's error is pointed out and a correct solution given in Mr. MoColl's
" Sixth Article on the Calculus of Equivalent Statements " {Proc. Land. Math.
Soc. vol. xxviii. p. 562).
OH. XVII FUNDAMENTAL THEOEBMS 189
.". u lies between
the greatest of \ ^^ and the least of ] 1 -Cj^{l -pi)
^'^' [l-c,{l-p,).
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 d priori probabilities of the causes Cj
and Cg are to be eUminated. The only limit we then have is
U<Pi+P2-
(56.3) Suppose that one of the a priori probabilities Cg is to be
eliminated. We then have limits Cjp{^u<i 1 - q + c^p^^. If, there-
fore, Ci is large, u does not differ widely from Cjp^.
(56.4) Suppose Pi is to be eliminated. We then have
If therefore c^ is large or Cg small, u does not difier widely
from CiPj.
(56.5) If a^aji=a^h, i.e. if our knowledge of each of the
causes is independent, we have a closer approximation. For
y = eafy/h = eja^a^h . a-yja^h . Ui/h = e/aja^h . CiC^,
.: u = CiPi + CiPz - C1C2 ■ e/ojCCih,
.: U>CiPi + C2P2-CiC2.
(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 certaia causes A^, Aj, . . . A^, and if q is the a priori
probability of the cause A^ and p^ the probabihty that, if the
cause Ai be known to exist, the event E will occur : required the
probability of E.
This is Boole's problem VI. {Laws of ThougM, 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 «.'"
degree which he cannot solve. I know no other discussion of the
problem. The solution is as follows :
ejh = eajh + ea-Jk = edjh + eja-^h . ajh = edi/h + c^^p^ (i. )
edjh = edja^/h + edi/a^h . ajh = edja^/h + c^ . edja^h,
edjja^h = e/a^h - eaja^h =P2 — • «%*2/^j
190 A TREATISE ON PROBABILITY
aud edid^ajh = ea^d^laji . Cg = c.^{ela^ - edjd^afflj
=C3Pa-edja^Jh,
.: ejh = ed^d^ajh + c^pi + c^^ + HPz ~ eSiaJh - eoid^^jh.
In general
ed^a^ 1 . . d^_i/h = ed^d^ . ■ ■ d^_idjh + ed^dz ■ . . d^_iajh
= eai . . . dr/h + ea^ . . . d^_-jaji . c^
= edi. . . djh + c^{elajh -ed^.. .d^_ i/«^r^}
= eSi . . . (i^jh + c^p^ -edy. . . d^.^a^/h,
.: finally we have e/h = ea^ . . . djh + %c^Pr ~ ^^^ • • • dj._ia^/h.
1 2
But since the n causes are supposed to be exhaustive
edi...djh = 0,
.-. ejh = Ic^Pr ~ Seffli . . .d^_ ^ajh (ii.) ■
1 2
Let edi. . .d^, -fljh = n^ ;
n n
then e/h = %c^p^-Xn^ (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 a^jaji = a^l'h = c^, then
ea^ . . .dr_iajh = ed^. . . d^_ilafi . c^
--c^ . e/fflj . . . dr_ia^h[l -% . . . d^_-jajij
= c.[l - n(l -Ci) . . . (1 -c^_i)]e/di . . . d^_ia^h.
1
Let e/di . . .d^_ ^aji = m,.,
then e/h = 2 c^p^ - S c^[l - II (1 - c^)]m^.
r=l r=2 s=l
These results do not look very promising as they stand, but
they lead to some useful approsmations on the ehmination of
m^ and n^ and to some interesting special cases.
OH. xvn
FUNDAMENTAL THEOREMS
191
(57.2) From equation (i.) it follows that e/h^c-iPi and
n
e/h^l -Cy{l -pj) ; and from equation (ii.) that e/A^Sc^j?,. ;
.•. e/h lies between ,^
I'^iPi 1
the greatest of i ] and the least of ^ 1 - '^iCl -^i)
^CnP^o :
(57.3) Further, if the causes are independent it follows from
(57.1) that
e/h^tc^p^. - 2c^[l - n(l - cj],
1 2 1
so that e/h lies between
the greatest of
(57.4) Now consider the case in which Pi=Pz = - ■ •=Pn = ^>
i.e. in which any of the causes would be sufficient, and in which
the causes are independent. Then m,. = 1 ; so that
r=n r=n s=r — 1
e/h=tc,-tcll- n (1-0]
r=l r=2 s = l
, =1-(1-Ci)(l-C,)...(l-Cj.
(57.5) Let Ci, c^. . .c^ be small quantities so that their
squares and products may be neglected.
Tlieii e/h=tc,p,,
I.e. the smaller the probabihties of the causes the more do they
approach the condition of being mutually exclusive.^
(57.6) The d post^iori probability of a particular cause a^
after the event has been observed is
e/aji . a^/h
te,p,-icll-n\l-c,)]
\ ' ^ and the
'f ^ least of
tc^Pr
l-cS.-p^)
OnPn
i-c^Ci-pJ
a^jeh = -
ejh
jPj^
ejh
(This is Boole's problem IX., p. 357).
^ Boole arriTes at this lesult, Laws of Thmghi, p. 345, but I doubt his proof.
192 A TEEATISE ON PEOBABILITY pt. n
(58) The probability of the occurrence of a certain natural
phenomenon under given circumstances is f. There is also a
probability a of a permanent cause of the phenomenon, %.e. of a
cause which would always produce the event under the circum-
stances supposed. What is the probability that the phenomenon,
being observed n times, will occur the w + 1"' ?
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 x^ asserts the occurrence of the event at the
r** trial and t the existence of the ' permanent cause ' we have
x^/h=p, t/h = a, x^lth = \,
and we require ««+i/a:i • • • a;„A =y™+i-
It is also asstmied that if there is no permanent cause the prob-
ability of Xg is not affected by the observations x^, etc., i.e.
xjx^ . . . xfh = xjih,^
- xJi/h_xJh-Xst/h p-a
Xjtn = -jpr — ■
i/h i/h 1-a
x^jx^ . . . x^_ih = xj/xi . . . x^ _-Ji + xjtjxi . . . x^_Ji
= tjxi . . . x^_-Ji+x^/ixj^ . . . x^_jh . i/aji . . . x^_ih
_ x^. . .x^_-^t,jh f-a Xy. . .x^.-ilth.ijh
Xy. . .x^_i/h 1-a Xy. . .x^_.Jh
p-a\l -a
p-a\'^ 1
(1-a)
yiyi-'-Vr-i i-« y-iyi- •■Vr-i
a-l-(^-a)'
,1-a
I.e. y, = ^
yiSi%---yr-\
a + (jp-a)|__
Also 2/1 =p and y^ = ?,
yi
This assumption, which is tacitly introduced by Boole, is not generaUy
justifiable. I use it here, as my main purpose is to iUustrate a method. The
same problem, wOhout this assumption, will be discussed in dealing with Pure
Induction.
CH. xvn
FUNDAMENTAL THEOEBMS
193
so that
Vn+l'-
VI -ay
'. + {p-a)
p-a
(58.1) If p =a, ?/„ = 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.
(58.2)
a{p - a)
¥n+l-yn = r
-ffA^-a/
p -a
1-
l-ffl
a + {p-a.)
P-
1-a
71-1
a + (p-a)
p -a
1-a
(by easy algebra) ;
and p is always >a and <1.
p -a
So that {p - ffi)( j is positive and decreases as r increases,
As n increases y„ = 1 - e, where
-ip-a)
1-
p -a
p -a
1 -a
\n~2
a + (p-a)
p-a
71-2
so that for any value of rj however small a value of n can be
found such that e<»y so long as a is not zero.
(58.3) t^ the d posteriori probability of a permanent cause
after n successful observations is
. , J Xt . . . xjth . t/h
a
^1^2 ■ ■ . «/„
a + {p-a)
p-a
tn = l- e', where e' =
„-.,H
a + (p-a)
p-a
1-a
194 A TEEATISE ON PEOBABILITY ft. h
So that t^ approaclies 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 wUl yield us
the most probable value of the quantity, on the basis of this evid-
ence. 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 fimctions of the original measurements which we com-
monly 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
&!,... 6^ .. . 6„, and let a^ represent the conclusion that the
value is, in fact, b^. Further let x^ represent the evidence that
a measurement has been made of magnitude y^.
If a measurement y^ has been made, what is the probabUity
that the real value is b^ 1 The appUcation of the theorem of
inverse probability yields the following result :
%xja^. ajh
r=l
(the number of possible values of the quantity being n), where
Ti stands for any other relevant evidence which we may have,
in addition to the fact that a measurement cCj, has been made.
Next, let us suppose that a number of measurements 2/i • • • 2/m
^ The substance of §§ 3-7 has been printed in the Journal of the Royal
Statistical Society, vol. Ixxiv. p. 323 (February 1911).
CH. XVII FUNDAMENTAL THEOREMS 195
have been made ; what is now the probability that the real value
is bg ? We require the value of aJxyX^ . . . xji. As before,
, -, X-, . . . . xjaji. ajh
Xxj^ . . . xjaji. a^jJi
r = l
At this point we must introduce the simplifying assumption
that, if we Imew 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 x^jx^ . . . Xga^h=x^/aji. This assumption is ex-
ceedingly important. It is tantamount to the assumption that
ova law of error is unchanged throughout the series of observations
in question. The general evidence h, that is to say, which justifies
oui 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 difierent
magnitudes have been made under analogous circumstances in
the past. The above assumption will not be justified if the
additional experience, which a knowledge of the errors in the new
measurements would supply, is sufficiently comprehensive, rela-
tively to our former experience, to be capable of modifjdng 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 x^, etc., are independent of
one another relatively to evidence aji, etc., it follows from the
ordinary rule for the multiplication of independent probabilities
that s=m
x^ . . . . xjaji = nxjaji.
3=1
ajk. Uxjafi
Hence aJxjXz . . . x^h = ^^^ "^^
S UxJa^h. aJh
r=lLs=l
196 A TREATISE ON PROBABILITY pt. n
The most probahh value of the quantity under measurement,
given the m measurements y^, etc. — ^which is our quaesitum — is
therefore that value which makes the above expression a maxi-
mum. Since the denominator is the same for all values of b^,
we must find the value which makes the numerator a maximum.
Let us assume that aj/A=a!2/A= . . . -ajh. We assume, that
is to say, that we have no reason d priori (i.e. before any measure-
ments have been made) for thinking any one of the possible
values of the quantity more likely than any other. We require,
q=m
therefore, the value of 6,, which makes the expression Ilxjaji
3=1
a maximum. Let us denote this value by y.
We can make no further progress without a further assump-
tion. Let us assume that xjaji — ^namely, the probability of a
measurement y^ assuming the real value to be 6, — ^is an algebraic
function / of y^ and b^, the same function for all values of y^ and
6j within the limits of the problem.^ We assume, that is to say,
xjaji =f{yq,bg), and we have to find the value of &„ namely y,
q=m
which makes Ilf(yg^,y) a maximum. Equating to zero the
9=1 _
differential coef&cient of this expression with respect toy, we
have 2 •i-i^9!£i=0,2 where /'=-;^. This equation may be
3=1 f{yq,y) dy
f
written for brevity in the form S^'=0.
Jq
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
^ Gauss, iu obtaining the normal law of error, made, in effect, the more
special assumption that xjaji is a function of e, only, where e^ is the error and
ej=6j-yg. 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 /(j/,,?/) can vanish.
CH. xvn FUNDAMENTAL THEOREMS 197
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)-
f
And the form oif{y^,y) determines the algebraic relation ^-^^ =
between the measurements and the most probable value. It
may be well to repeat that f{y^,y) denotes the probability to
us that an observer will make a measurement y^ in observing a
quantity whose true value we know to be y. A law of error tells
us what this probability is for aU possible values of y^ 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 ?
f
^•L«=0 must be equivalent to l,{y-yq)=0, since the
Jq
J 5=™
most probable value y must equal — Xy^.
f
.: •'-2 = <l>"(y){y - yq) where (j)"{y) is some function which
Jq
is not zero and is independent of y^.
Integrating,
iog fg=/'^"(y){y-yq)dy+ir{yq) where yjriy^) is some func-
tion independent of y.
=<P'{y){y -yq) -<l>(y) +i^iyq)-
So that /^=e"^'(J'K!'-s'»)-«s')+'f'(i'«)-
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 (/)(«/)= -AV and ■fiy^^ -khj^+logk, we obtain
/g=Ae~*'^''~*''', the form normally assumed.
=Ae "*'"''', where z^ is the absolute magnitude of the error in
the measurement y^.
This is, clearly, only one amongst a number of possible solu-
tions. But with one additional assumption we can prove that
this is the only law of error which leads to the arithmetic mean.
198 A TREATISE ON PROBABILITY pt. u
Let us assume that negative and positive errors of the same
absolute amount are equally likely.
In this case/j must be of the form Be*^*~^')',
••• <t>'{y){y - y,) - 4>(y) + fiy^) = ^(y - yaf-
Differentiating with respect to y,
^{y-Vif
But ^"(y) is, by hjrpothesis, independent of y^.
.•. ^(w-«.)^= -^2 ^]iej.e A; is constant ; integrating,
0{y-yq?= -^%-2//+logC and wehave/g=Ae-*^'(s'-«' (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 ?
In this case S — = must be equivalent to Tly^ =y^, i-s- to
/a 5=1
Slog^ = 0.
y
Proceeding as before, we find that the law of error is
/ =Ae"'''(*^'°s-+/-^''*+'''^*'\
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
'i>'{y) log h+\& dy + yjr{y;)=<l>{y-y,f
y ^ y
oT<^"{y)\ogy^=~<^{y-y,f,
which is impossible.
The simplest law of error, which leads to the geometric mean,
seems to be obtained by putting ^'{y)= -ky, ■\{r{y^)=0. This
^vea/.=A(|)\-'..
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 McAIister {Proceedings of the
Royal Society, vol. xxix. (1879) p. 365). His investigation de-
pends upon the obvious fact that, if the geometric mean of the
CH. xvn FUNDAMENTAL THEOEEMS 199
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 y^^, etc., and the
quantity by y, let their logarithms be denoted by \, etc., and by
I. Then, if Z^, etc., obey the normal law of eiTOi, f (1^,1) = Ae~*^'<'«~'>'.
Hence the law of error for y^, etc., is determined by
/(y„2/)=Ae-'='('°8^'-"'e^>'
=Ae-«='("'ef)',
and the most probable value of y must, clearly, be the geometric
mean of y^, 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 1/^(^3) = - k%log y^)^,
and (p'iy) = 2k^ log y, we have
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, fimctions of the quantities y, which are given by
z='E{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
1 Sktw Frequency Curves, p. 22, published by the Astronomical Laboratory
at Groningen (1903).
200 A TREATISE ON PROBABILITY pt. n
those wMch have been akeady 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 withj
the geometric law which can be derived from the normal arith-)'
metic law. Similarly if we start from the simplest geometric
law of error, namely, / =A ( — 1 e ^, we can easily find, by
writing log y = l and log y^ = l^, the corresponding arithmetic'
law, namely, j^=^^'^V-k)-i''A^ which is obtaiued from the
generalised arithmetic law by putting ^(i)=AV and -^{1^=0.
And, in general, corresponding to the arithmetic law
f =Ae"'''^*^^^~'''^""'''^''^'*''''^'''^
we have the geometric law
where
y=\ogz, 2/g=log2g, /^L^& = 0(logz)and-fi(Zj)=-^(logz).
(iii.) What law of error does the harmonic mean imply ?
In this case, %^-^ =0 must be equivalent to S( ) =0'
h \% y/
Proceeding as before, we find that/^ = Ae''"'(''>L7,-^]"-/"^^'*+''''''«^■
A simple form of this is obtained by putting 4>'{y) = - ^^^ and
^{yd=-1<hJr Then /g=Ae7/*-^=)'=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
^ It may be added that Professor Kapteyn's monograpli 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.
CH. xvn FUNDAMENTAL THEOEEMS * 201
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 ''^ ^th, and, if the
number is even, all values between the — th and the ( — + 1 Ith are
• 2 V2 /
equally 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 median y oi y^ y^ . . . y^ is
m
found, that is to say, by making 2 1 y„ - «/ 1 a miniTmim where
\yq-y\ is the difference always reckoned positive between y^
and y.
We can now return to the investigation of the law of error
corresponding to the median.
m
Write \y-yq\=Zg- Then since Xz^ is to be a minimum we
1
™« — w
must have %^ — -^=0. Whence, proceeding as before, we have
1 ^3
/,=Ae/V*"<'^*+*<'"'-
The simplest case of this is obtained by putting
4>"{y)==-k^
ir{y,)=y^^h%,
whence /^=Ae-*''2'-^'i=Ae-'^'"«-
This satisfies the additional condition that positive and nega-
tive errors of equal magnitude are equally hkely. 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.
The median law of error, /^ = Ae'^''", where z^ is the absolute
amount of the error always reckoned positive, is of some historical
202 • A TEEATISE ON PROBABILITY pt. n
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 probabilite des
causes par les evenemens." ^ This memoir was not subsequently
incorporated in his Theorie analytique, and does not represent his
more mature view. In the Thiorie 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 ad-
vantages, in comparison with all other laws of error, for the pur-
poses 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 remem-
brance. 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 iu a somewhat simplified
form : " Determiner le milieu que Ton doit prendre entre trois
observations denudes d'un meme phenomene." He begins by
assuming a law of error z = ^{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 ^(z)=— e~™^. If this formula
is to follow from his arguments, y must denote the ahsol/ute 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
^ Memoires presentes a I'Acadimie des Sciences, vol. vi.
CH. XVII FUNDAMENTAL THEOEEMS 203
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 followiag general
formulae are easily obtained :
(v.) If ^6{yi^,y) =0 is the law of relation between the measure-
ments and the most probable value of the quantity, then the law
of error f^{y^,y) is given hj f^=A.e^^^y^^^"^v'>'^y+'^^y^\ Since /^ lies
between and '^, fO{y^y)<l)"{y)dy +^^[y^ +log A must be negative
for all values of y^ and y that are physically possible ; and, since
the values of y^ are between them exhaustive,
where the summation is for all terms that can be formed by giving
y^ every value d 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"*^'-^^""^'^', where the most probable
value of the quantity is given by the equation
^{y - yg)f'{y - y,Y = O. where f'iy - yf = -J{y - y^f^
"'(y ~ yq)
The arithmetic mean is a special case of this obtained by putting
fiy-yq)^ = {y-yqY I ^^^ ^^^ median is a special case obtained
by putting f(y - y/ =+-y/{y- y^f.
We can obtain other special cases by putting
f{y-yqf={y-yqy>
when the law of error is Ae~*'^*""*«^' and the most probable values
are the roots of my^ - By^Xy^ + ^y'Zy'^q - Xyq=^ 5 and by putting
fiy-yqT=iog{y-y^f, when the law of error is ^_ ^p ^^^
the most probable values the roots of X = 0. In all these
y-y,
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
204 A TREATISE ON PEOBABILITY pt. u
supposing that the quantity we measure is more likely to have
any one of its possible values than any other.
(6) 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
ia the next.
(c) That the probability of a measurement of given magnitude,
when in addition to the d 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 con-
tinuous, without sensible error.
(e) That the d 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 d priori.
Subject to these assumptions, we have reached the following
conclusions :
(1) The most general form of the law of error is
leading to the equation S Oiy^y) =0, connecting the most probable
value and the actual measurements, where y is the most probable
value and y^, etc., the measurements.
(2) Assuming that positive and negative errors of the same
absolute magnitude are equally likely, the most general form is
/,=Ae-'^"^*'-^'>', leading to the equation 'Z{y-yq)f'{y-y,f=0,
where /'2: = —/z. Of the special cases to which this form gives
rise, the most interesting were
(3)/g=Ae-*='<''-8"J"=Ae-*^"^'', where z^=\y-y^\, leading to
the arithmetic mean of the measurements as the most probable
value of the quantity ; and
(4) /j=Ae~*'^«, leading to the median.
(5) The most general form leading to the arithmetic mean is
/j=Ae*'('-'>f^-^'>-'''(^'+'''(*«>, with the special cases (3), and
(6) / =^^'^^y-y^)-i''^ _
(7) The most general form leading to the geometric mean is
f^^^^'{y)\osyi+fm.ay+^^y,)^ with the special cases :
CH. xvn FUNDAMENTAL THEOREMS 205
(8)/,=A(^iy'%-"^and
(9)/,=Ae-K'°«^'^
(10) The most general form leading to the harmonic mean is
f,=AeM~-i]-Af+^<y'\ with the special
case
,, (v-yi)' -kW
(ll)/,=Ae y^ =Ae
(12) The most general form leading to the median is
/«=Ae
with the special case (4).
In each of these expressions, /^ is the probability of a measmre-
ment y^, 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 probability, I am unwUUng 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.
206 A TEEATISE ON PEOBABILITY pt. n
between the accuracy of wMcli we have no reason to discriminate,
we often think it reasonable to act as iE 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 siim of these three estimates is taken, viz.
3900, and because they were three bodies, the third, i.e. 1300, is
taken as the mean value asked for. This is the axiom : aequali-
bus 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 embarassee des difficultds d'analyse, elle n'a evite que la
compUcation 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 tjrpe of law of
error for the d, priori probabilities of given errors. It was also
found that the assumption of this law led on to a more com-
plicated rule, known as the Method of Least Squares, for com-
bining 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.^
^ De vardye van de hf-renten na proportie van de hsrenten. The Hague, 1671.
* 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 aJl in the scientific sense of the term. It is simply
a rule or direction. . . ."
CH. xvn FUNDAMENTAL THEOREMS 207
The analytical theorems of Laplace and Gauss are compUcated,
but the special assumptions upon which they are based are easily
stated. 1 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, (6) that the errors are so small that their cubes
and higher powers may be neglected. Assumption (a) is arbi-
trary,^ and Gauss did not state it explicitly. These two assump-
tions, together with certain others, lead us to the result. For
let <p(z) be the law of error where z is the error, and let us assume,
as it always is assumed in these proofs, that (f>(z) can be expanded
by Maclaurin's Theorem. Then (^{a;)=(^(0) +z^'(0) +^0"(O) +
z-
3
0"'(o) -I- . . . It is also supposed that positive and negative
O I
errors are equally probable, i.e. (f>{z)=(f>{-z), so that ^'(0) and
^'"(0) vanish. Since we may neglect z* in comparison with z^,
^{z)=(j){0)+^z^^"{0). But (neglecting z* and higher powers)
tea i_^
a + te^ = ae « , SO that <f){z) =ae » .
Gauss's proof looks much more compUcated than this, but he
obtains the form ae " by neglecting higher powers of z, so that
this expression is really equivalent to a + bz^. By this approxi-
mation he has reduced all the possible laws to an equivalent
form.3 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 estabhsh the
normal law of error and the arithmetic mean as a law of logic,
1 For an account of the three principal methods of arriving at the Method
of Least Squares and the Arithmetic Mean, see Ellis, Lea^t Squares. Gauss's
first method is in the Theoria Molua, and his second in the Theoria Combina-
ticmis Observationum. Laplace's investigations are in chap. iv. of the second
Book of the Theorie analytique. Laplace's method was improved by Poisson
in the Oonnaissance des temps for 1827 and 1832.
2 It does not follow, as G. Hagen argues {Orundzuge der Wahrscheinlichkeits-
rechnung, 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.
3 This is pointed out by Bertrand, Calcul des probabilite.", p 267.
208 A TREATISE ON PROBABILITY pt. n
others have claimed for it the testimony of experience and have
deemed it a law of natmre.^
That this cannot be so, is evident. For suppose that x^Xc^ ■ ■ -^n
are a set of observations of an unknown quantity x. Then, by
this principle, x = -"Zx^ gives the most probable value of x. But
n
suppose we had wished to determine x^, our observations, assum-
ing that we can multiply correctly, would be x^^, x^ . . . x^,
and the most probable value of ^=-%x^. But (_Sa;^)-4= -Sa;^^
n n n
And in general, -Sflcc,) ^f{-%x^. Nor is this a consideration
n n
which 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 d'vre(A 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 rtile 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 dis-
tribution is very far from being a general law of frequency
distribution either for errors of observation or for the distribution
of deviationsfrom type such as occur in organic populations. . . .
It is not even approximately correct, for example, in the distribu-
tion of barometric variations, of grades of fertility and incidence
of disease."
^ This is, of course, a very common point of view indeed. Cf. Berirand,
op. cit. p. 183: "Malgre les objections pr^c^dentes, lafoimule de Gauss doit
etre adoptte. L'observation la oonflrme : cela doit suffire dans les applications."
' This was noticed by Galton.
' E.g. Letters on the Theory of ProbabUitiea, p. 114.
* On " Errors of Judgment, etc.," Phil. Trans. A, voL cxeviii. pp. 23S-299.
The following quotation is from his memoir On the Oeneral Theory of Skew
Correlation and Nonlinear Regression, where further references are given.
OH. xvn FUNDAMENTAL THEOREMS 209
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, i£ 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 pre-
sented 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 in-
convenient ; and our natural course wiU be to measure certain
other quantities which we know to bear fixed and invariable
relations to the imknowns we wish to determine. In surveying,
for instance, or in astronomy, we constantly prefer to take
measurements of angles or distances m. which we are not interested
for their own sakes, but which bear known geometrical relation-
ships to the set of ultimate unknowns.
p
210 A TEEATISE ON PEOBABILITY rr. n
If we wish to determine the most probable values of a set of
imlmowns x^, x^, x^ . . . x^, 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 :
a-jX-i^+a^^+ . . . +a^y='Vi,
b-^x-i^+bzx^+ . . . +b^^=Y2,
' mi
KjX-^ + 102X2 + . . . + tC^j. — V ^
where Vi, etc., are the quantities directly observed, and the a's,
b's, etc., are supposed known (»i>-r).
We have ia such a case n equations to determine r unknowns,
and siace 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 iu 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 dijfferent 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 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~^^, 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
^ See Meniman's Method of Least Squares, p. 181, for an historical sketch,
from which the above is taken. In 1877 Merriman published in the Trans-
actions of the Connecticut Academy a list of writings relating to the Method of
Least Squares and the theory of accidental errors of observation, which com-
prised 408 titles — classified as 313 memoirs, 72 books, 23 parts of books.
OH. xvn FUNDAMENTAL THEOREMS 211
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 simimarise a set of facts and give us the
actual value of a composite quantity, as, for example, the index
number of the cost of Uving. 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 com-
posite 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 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 ia respects
relevant to the probable value of the quaesitum. Thus there
may be reasons, quite apart from the actual resxdts of the indi-
vidual 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 independ-
ence between the instances, which the method of averages
212 A TREATISE ON PROBABILITY pt. n
presumes, is imperfectly satisfied, and consequently that our
mode of combining tlie instances in an average must be modified
accordingly.
Some modern statisticians, who, really influenced perhaps by
practical 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 ia 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.
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 con-
clusion. 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 wheat 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
OH. XVII FUNDAMENTAL THEOREMS 213
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 mmiber of numerical quantities, which, while they
do not measure our quaesitwm 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 very confusiag questions can arise both as to
what sort of a thiQg 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 ia 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 fluctua-
tions in our quaesitum. I must not burden this book with a
discussion of the problems of Index Numbers. But I venture to
tMnk that they would be sooner cleared up if the natures and
purposes of differing index numbers were more sharply distin-
guished — 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 iastrument
of precision, and those which combiae results, not of the quaesitum
itself, b^t 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. Tfie 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
214 A TKEATISE ON PEOBABILITY pt. n
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 pf 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 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 reliabiUties 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 particvilar law of error as appropriate to the observa-
tions. 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. 6. Hagen's Grundz&ge der Wah/rscheinlichkeitsrechnung, p. 63 : " Die
Tauschung, die man duroh Versohweigen von Messungen begeht, lasst sioh
eben so wenig entsohuldigen, als wenn man Messungen falsohen oder flngiren
woUte,"
PART III
INDUCTION AND ANALOGY
215
CHAPTEK 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. — HtTME."^
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 justifica-
tion 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 satis-
factory 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 when.
2. What are its distinguishing characteristics ? What are
the quaHties which in ordinary discourse seem to afford strength
to an inductive argument ?
' Philosophical Essays concerning Human Understandijig.
217
218 A TREATISE ON PROBABILITY pt. m
I shall try to answer these questions before I proceed to
the more fundamental problem — What ground have we for re-
garding 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 assump-
tions, if we could adopt them, he 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 fixm reliance and security the same taste and rehsh iu 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 ex-
periments.
It will be useful to call arguments inductive which depend
iu 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 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
OH. xvm INDUCTION" AND ANALOGY 219
as mucli 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
iastances, 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 char-
acteristics 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 con-
clusion different from that which it infers from a hundred in-
stances, 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.
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
220 A TREATISE ON PROBABILITY n. in
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 general-
isation 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 in-
ductions are themselves stisceptible 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 Correla-
tion. If, for instance, we base upon the data, that this and that
and those 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 — pre-
sents 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
' What Mill calls ' approximate generalisations.'
CH. xvm INDUCTION AND ANALOGY 221
directed to the solution of essentially inductive problems. But
it has been seldom apprehended clearly, either by these writers
or by others, that the vahdity of every induction, strictly inter-
preted, 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 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 vahdity 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.
CHAPTEE 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. —
1. Hume rightly maintains ttat 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 /(aj) is true for all those values of x for
which <j){x) is true, then we have a generalisation about ^ and /
which we may write g{^,f)- If, for example, we are dealing with
the generalisation, " All swans are white," this is equivalent to
the statement, " ' a; is white ' is true for all those values of x for
which ' a; is a swan' is true." The proposition <f){a).f{a) is an
instance of the generalisation ^(^, /).
By thus defining a generalisation in terms of prepositional
functions, it becomes possible to deal with all kinds of generalisa-
. ' A Treatise of Human Nature.
222
OH. XIX INDUCTION AND ANALOGY 223
tions in a uniform way ; and also to bring generalisation into
convenient 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 generalisa-
tion g{(f), /), therefore, asserts that one analogy is always accom-
panied by another, namely, that between all objects having the
analogy <f) there is also the analogy /. The set of propositional
functions, which are satisfied by both of the two objects, con-
stitute the positive analogy. The analogies, which would be
disclosed by complete knowledge, may be termed the total positive
arutlogy ; 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 iastances 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 conmion to some of the iastances, but
not to all.
A simple notation, in accordance with these definitions, will
be useful. If there is a positive analogy ^ between a set of in-
stances «! . . . a,t, whether or not this is the total analogy
between them, let us write this —
aj , . . On
1 Hence A (^) = 0(Oi) . ^(aj) . . . 0(a„)= 11 <ti(x).
224 A TEEATISE ON PEOBABILITY pt. m
And if there is a negative analogy (j)', let us write this —
A (<^')-^
ai . . . On
Thus A {<!>) expresses the fact that there is a set of
ttl . . . ttn
characteristics (^ which are common to all the instances, and
A (<^') that there is a set of characteristics <p' which is
ai. . .On
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 has been found to exist, another
characteristic/ has been found to be associated with it. We argue
from this that any instance, which is known to share the first
analogy <p, is likely to share also the second analogy/. We have
found in certain cases, that is to say, that both <f> and/ are true
of them ; and we wish to assert/ as true of other cases in which
we have only observed <p. We seek to establish the generalisation
?(^> /)> °^ *^® ground that and / constitute between them an
olaserved 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 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{^,f), — ^upon the comprehensiveness,
that is to say, of the condition </> and the conclusion/ respectively.
The more comprehensive the condition <^ and the less com-
prehensive the conclusion /, the greater d priori probability do
we attribute to the generalisation g. With every increase in <j)
this probability increases, and with every increase in / it will
diminish.
1 Hence A (0') = S 4>'{x) . 2 (f>'{x).
ai. ..On x=ar x=a/
OH. XIX INDUCTION AND ANALOGY 225
The condition ^(=^1(^2) ^ more compreliensive than the
condition ^1, relative to the general evidence h, if ^2 is a condition
independent of ^^ relative to h, 4>z being independent of ^j, if
g{<f>i, </)2)/^=t=l, i.e. if, relative to h, the satisfaction of (p^ is not
inferrible from that of (f>i.
Similarly the conclusion /( =fifi) is more comprehensive than
the conclusion /i, relative to the general evidence h, if /2 is a con-
clusion independent oifi, relative to h, i.e. if g{fi,f2)lh=^l.
If (j} =<^ii^a 3.nd/=/j/2, where <j)i and <p2 are independent and
fi and /a are independent relative to h, we have —
^'(^i, /)/^ =ff{M2, f) • 9M2, f)l^
^9{<l>,f)IK
and g{^, f)jh =g{<f}, fJz)lh
=9Wi,f2)lh.g{<l>,-A)lh
so that g{^, A)lh^g{<t>, f)lh^9{<^x, f)/h.
This proves the statement made above. It will be noticed
that we cannot necessarily compare the a priori probabilities
of two generaUsations in respect of more and less, unless the con-
dition of the first is included in the condition of the second, and
the conclusion of the second is included in that of the first.
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
d priori to the evidence by which we support it. Since, when-
ever the conclusion / is complex, i.e. resolvable into the form
fifz where g(fi, f^jh =4= 1, we can express the probability of the
generalisation g{(f>,f) as the product of the probabilities of the
two generalisations g{^fi, /a) and g{(j>, fi), we may assume in what
follows, that the conclusion /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
^ If <jy{a) is a proposition and \j/{a) = 1i . B{a), where ^ is a proposition not
involving a, then we must regard 6(a), not ^(a) as the statement about a.
Q
226 A TREATISE ON PROBABILITY pt. m
all the instances which are known to satisfy the condition <j),
are also known to satisfy the conclusion / 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 ^ or
in /, i.e. that the positive analogy between the instances is
exactly co-extensive with the analogy ^/ 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
practical utiUty ; for i£ the analogy covered by the generalisa-
tion, 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 aU 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 re-
garded as the same if they only difier 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 generalisa-
tions 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
OH. XIX INDUCTION AND ANALOGY 227
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 aU alike, and yet which are not covered
by the generalisation. If ^^ is the part of the positive analogy
between the instances which is not covered by the generalisation,
then the probability of this type of argument from analogy can
be written—
g{<l>,f)l A (</.<^,/).
The value of this probability turns on the comprehensiveness
of ^y. There are some characteristics ^j common to all the
instances, which the generaUsation treats as unessential, but
the less comprehensive these are the better. <f>y stands for the
characteristics ia which all the instances resemble one another
outside those covered by the generaUsation. 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 compre-
hensiveness of ^1- When, however, our knowledge of the
instances is complete, it is not necessary to make separate
mention of the negative analogy A (^') in the above formula.
Oi., .On
For ^' simply includes all those functions about the instances,
which are not included in ^^i/, and of which the contradictories
are not included in them ; so that ia stating A {cjxpif), we
state by implication A {<})') also.
aj . . . On
The whole process of strengthening the argument in favour
of the generalisation g{(f), f) by the accumulation of further ex-
perience 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 (p^ between the instances which our generahsation
disregards. Thus the' advantage of additional instances, derived
228 A TEEATISE ON PEOBABILITY pt. m
from experience, arises not out of their number as such, but out
of their tendency to limit and reduce the comprehensiveness of
<f>i, or, in other words, out of their tendency to increase the negative
analogy (/>', since ^j^' comprise between them whatever is not
covered by (pf. The more numerous the instances, the less com-
prehensive are their superfluous resemblances likely to be. But
a single additional instance which greatly reduced 0^ would in-
crease the probability of the argument more than a large number
of instances which afEected ^^ 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 accoimt 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 —
/ aj. . .On aj . . . ttn
where ^^i/ stands for the characteristics in which all n instances
a^^ . . . a^ are krumn to be alike, and (f)' stands for the char-
acteristics 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 ^^ or increases the
negative analogy ^'. 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 om: 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
OH. XIX INDUCTION AND ANALOGY 229
the more complete our knowledge of the instances, the less
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 argu-
ment will be discussed in detail in the following chapter on
Pure Induction.
8. When om; 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 (pi,
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 multipHcation
of instances. A sub-analogy of this kind between the instances
ttr . . . a^ may be written A (aI^j.) ; and the formula, if it
Or . . -Cba
is to take account of all the relevant information, ought, there-
fore, to be written —
g{<j>,f)/ A {cpcpj) A (,^')n( A {f,)],
/«!... On tti . . . O/t V Or . . . Oa J
where the terms of 11/ A (i^s:)]. stand for the various sub-
analogies between sub-classes of the instances, which are not
included in (/)^i/ or in (j)'.
9. There is now another complexity to be introduced. We
must dispense with the assumption that the whole of the analogy
covered by the generaUsation 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 con-
tradicts it ; and such instances afford some support to the
generalisation. Suppose that ,,<j> and ,,/ are part of <j> and / re-
spectively, then we may have a set of instances h-y. . .b^ which
show the following analogies :
A (6«^5<^15/) A (,.^')n/ A (,,|r,)|,
6l...6n. hi...hin ybr.-.'ba j
where ^^i is the analogy not covered by the generalisation, and
so on, as before.
230 A TREATISE ON PROBABILITY m. m
The formula, therefore, is now as follows :
g{<l>,f)l n I A („</.„</.i^) A umm A (t,)1
/ a,6. . . (^aj . . . On ai...a,i j ya^'bi... j
In this expression „^, ^/are the whole or part of ^,/; the product
n is composed of the positive and negative analogies for each
of the sets of instances Oj . . . a„, 6^ . . . 6„, etc. ; and the
product n contains the various sub-analogies of different sub-
classes of all the instances a^. . .a^, b^. . . b^, 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 <^, the conditioji of the general-
isation, or/, its conclusion, or part of or of /is knoivn to be false.
Suppose that there are instances of which </> is true and /false, it
is clear that the generalisation is ruined. But cases in which we
know fart of ^ to be true and/ to be false, and are ignorant as
to the truth or falsity of the rest of ^, weaken it to some extent.
We must take accoimt, therefore, of analogies
ttj' . . . a'nf
where ^<^, part of <^, is true of all the set, and „,/, part of / is
false of all the set, while the truth or falsity of some part of ^ and
/ is unknown. The negative evidence, however, can strengthen
as well as weaken the evidence. We deem instances favourably
relevant in which <^ and/ 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 aj>af, where „^ and ^ are parts
of (^ and /, but also for sets which show analogies a^a/>
^ Even if we want to distinguish between the sub-analogies of the a set and
the sub-analogies of the 6 set, this information can be gathered from the pro-
duct n.
^ I am disposed to thiak that we need not pay attention to instances for
which part of is known to be false, and part of / to be true. But the
question is a Uttle perplexing.
CH. XIX INDUCTION AND AINALOGY 231
or analogies „^^, where ((<^ and „/ are the whole or part of <^
and/, and ^/ are the contradictories of ^ andf.^
It should be added, perhaps, that the theoretical classifica-
tion of most empirical arguments in daily use is complicated by
the account which we reasonably take of generalisations previ-
ously 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 generahty, namely —
g{<p,f)/ A {<}>^J) 1 {<f>')U( A {^,)l
J ai. . . On aj , , . On \ar-.-(ta J
in which no partial instances occur, i.e. no iastances in which part
only of the analogy, required by the generalisation, is known to
exist. In this third degree of generahty, it wiU be remembered,
our knowledge of the characteristics of the instances is in-
complete, 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 formula the incompleteness
of our knowledge is implicitly recognised in that 4)^if^' 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 con-
junctions „^ J, „0 J, or „0 J, where „0 and J are part of ^ and/.
An argument, therefore, from experience, in which, on the
basis of examined instances, we establish a generalisation apphc-
able 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 0i known to be common to
all the instances, but ignored as unessential by the generalisation.
(2) By increasing the differences ^' known to exist between
the iastances.
1 Where the conclusion /is simple and not complex (see § 5), some of these
complications cannot, of course, arise.
232 A TREATISE ON PROBABILITY pt. m
(3) By diminishing the sub-analogies or unessential resem-
blances i^j. 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 know-
ledge 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 ^^ as well as (^ is a necessary condition
of/. The object of (2) is to avoid the possibility that there may
be some resemblances additional to <f), common to aU the instances,
which have escaped our notice. The object of (3) is to get rid
of indications that the total value of ^^ may be greater than the
known value. When (f>^if is the total positive analogy between
the instances, so that the known value of (j}^ 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
(^1 falls short of its total value and we cannot infer tfi' from it,
it is better to regard (2) as fundamental ; in any case every
reduction of (^^ must increase (j)'.
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 difEering 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 assxmiptions 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 thouglit that the essence of inductive argu-
ment lies in the multiphcation 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 in-
stances 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 iu-
creasing the number of iustances 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
233
234 A TREATISE ON PROBABILITY pt. m
force of the argument largely depends, not on the mere number
of these predictions, but on the knowledge that the circumstances
in which they are fulfilled differ widely from one another in a
vast number of important respects. The variety of the circum-
stances, 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 thiag, to diminish the
characteristics conmion to all the examined instances and yet not
taken account of by our generalisation. Our method, however,
may be one which certainly achieves this object, or it may be one
which possibly achieves it. The former of these, which is obvi-
ously the more satisfactory, may consist either in increasing our
definite knowledge respecting instances examined already, or ia
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 de-
finite 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 \mregulated induction of ordinary
experience to depend on the latter. It is when our definite
knowledge about the instances is hmited, 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 judg-
ment, is negative, and consists in showing that a Une of advance,
which might have seemed promising, turns out to be a blind
alley, and that we are thrown back on known Analogy. Pure
c!H. XX INDUCTION AND ANALOGY 235
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 d, jpriori data of the investigation ;
let g represent the generalisation which we seek to establish ;
let XyX^ . . . a5„ represent instances of g.
Then x^gh = 1, x^gh = 1 . . . xjgh = 1 ; given g, that is to
say, the truth of each of its instances follows. The problem is
to determine the probability gjhx^^ • ■ -x^, 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 nothiag
in our ci priori data which leads us to distinguish between the
d priori likelihood of the different instances ; we assume, that is
to say, that there is no reason d priori for expecting the occurrence
of any one instance with greater reliance than any other, i.e.
XjJh=X2/h= . . . =xjh.
Write g/fix^os^ . . .x„ =p^
and. x^_^ijnXjOS2 . . • ^n~¥n+i-i
then
Pn glhxy...x^ xjhx^i . . . x^_i
Pn-i g/hxi . . . x^_-^ g/hXj;. . .Xn_i.x,Jhxj^...x^_-i^
X^jhXi . . ■ X^_i
« 1 1
^^, and hence 2?„ = .p^, where Po=g/h, i.e. Pq
is the d priori probability of the generalisation.
1 In the most general sense we can regard any proposition as the generalisa-
tion of all the propositions which foUow from it. For if h is any proposition,
and we put 0(a;)= ' x can be inferred from h ' and/(a;)=a;, then ff(0, /)=fc. 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 con-
sequences which the conclusion and the premisses have in common.
236 A TREATISE ON PROBABILITY m. m
It follows, therefore, that^„>^„_i so long as y„4=l.
Further,
XjX^ . . . xjh = xjhx-ips^ ...«„_!. x^x^ ...x^_-i]h .
=y„ . x-iX^ . . . x^_\]h
=ynyn-i ■■■yv
. ^ - Po _ Po _
yiyz ■■■yn x^^z • ■ • xji''
^ Po
x^x^ . . . x^jh + XjX^ . . . x^jh
^ Po
glh + XjX2...xJgh.gjh
= Po
Po+XjX2...xJgh{l-po)
This approaches unity as a limit, if XjX^ . . ■ xjgh . —
Po
approaches zero as a liinit, 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 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 in-
stances, a knowledge of the latter would enable us to predict it.
If it difiers 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 liinit, 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 probabihty of the generalisation
increases, may be such that it will approach certainty as a
^ Sinoo Pn>-Pn-i so long as ^n + 1.
OH. XX INDUCTION AND ANALOGY 237
limit when the number of independent instances of it are in-
definitely increased ? We have already shown, as a basis for
this investigation, that p^ approaches the limit of certainty for
a generalisation g, if, as n increases, x^x^ . . ■ xjgh becomes
small compared with p^, i.e. if the d priori probability of so many
instances, assuming the falsehood of the generalisation, is small
compared with the generalisation's d 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
for all values of r, and P(,>r;, where e and 17 are finite proba-
' bilities, separated, that is to say, from impossibiUty 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.^
I argued in Chapter III. that not aU 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 applica-
tion 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
^ The proof of these conditions, which is obvious, is as follows :
x^x^... Xn/gh = x„lxix^ . . . x„.^gh . Xj^x^. . . x„.Jgh<:{l - e)",
where e is finite and PD>ri where j; is finite. There is always, under these
(1 - e)" 1
conditions, some finite value of n such that both (1 - f)" and — s^re less
than any given finite quantity, however small.
^ Hence a series of probabflities p^p^ ■• -Pr approaches a limit L, if, given
any positive finite number e however small, a positive integer n can always be
found such that for all values of r greater than n the difEerenoe between L and p^
is less than e.y, where 7 is the measure of certainty.
238 A TREATISE ON PROBABILITY pt. m
its conclusion can be shown to be one of a finite number of alter-
natives, wHch are between them exhaustive or, at any rate, have
a finite probability, and to which the Principle of Indifference
is appKcable ; 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 probabiKty of a pure induction may tend towards
certainty as the number of instances is increased, are (1) that
x^/x^x^. . .Xr_jffh falls short of certainty by a finite amount
for all values of r, and (2) that p^, the d priori probabihty of our
generalisation, exceeds impossibihty by a fimite amount. It is
easy to see that we can show by an exactly similar argument that
the foUowing more general conditions are equally satisfactory :
(1) That x^jxyx^ ■ ■ ■ x^-iffh falls short of certainty by a finite
amount for all values of r beyond a specified value s.
(2) That pg, the probability of the generaUsation 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 fimite
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 ex-
perience 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 otherwisCj however small, then the
probability can be strengthened and can tend to increase towards
certainty by the mere multipUcation 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 impHcitly on an argument in inverse prob-
ability, are all vitiated by unjustifiable assumptions relating
to the magnitude of the a priori probability p^. Jevons, for
OH. XX IKDUCTION AND ANALOGY 239
instance, avowedly assumes tliat 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 judg-
ment as his. The arguments against it and the contradictions
to which it leads have been dealt with in Chapter IV. The
demonstration of Laplace, which depends upon the Eule 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 con-
siderations 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 nwnher of conclu-
sions, 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 ioitial
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 probabiHty 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 per-
ceiving that lesser claims in the beginning might yield them
240 A TEEATISE ON PEOBABILITY n. m
what they wanted in the end. And transcendental 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 meta-
physical 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 beKef 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
grotmds 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 sugges-
tions 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 in-
stances 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 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
^ A paper by Mr. G. E. Moore entitled, " The Nature and Reality of Objects
of Perception," -wMoh 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 descvibe^ a]bpT@,
OH. XX INDUCTION AND ANALOGY 241
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 argu-
ment 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 emmciation, given in the two preceding chapters, of the
Principles of Analogy and Pure Induction there has been no
reference to experience or causaUty or law. So far, the argument
has been perfectly formal and might relate to a set of proposi-
tions 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 con-
sider, 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
nattirally apply them to aU kinds of argument aHke, including
formal arguments as, for example, about numbers. When we
are told that Fermat's formula for a prime, namely, 2^" + 1 for
all values of a, has been verified in every case ia which veri-
fication is not excessively laborious — ^namely, for a = l, 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 refer-
ence 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 2^" + 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
^ This formula has, in fact, been disproved in recent times, e.g. 2^° + 1 =
4, 294, 967, 297 = 641 x 6, 700, 417. Thus it is no longer so good an illustration
as it would have been a hundred years ago.
242
OH. XXI INDUCTION AND ANALOGY 243
at random is a true source of primes. To maintain that there is
no appreciable gromid in such a case is paradoxical. If, on the
other hand, a partial verification 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 cL 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 re-
mainder. Would this fact, by itself, raise any kind of presump-
tion 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 aU 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 probabUity 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 ornot theuseof empirical argument appears plausible
to us in these particular examples, it is certainly true that 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
^ Pp. 229-231 (one volume edition). Jevons uses these illustrationa, not
for the purpose to which I am here putting them, but to demonstrate the falli-
bility of empirical laws.
244 A TREATISE ON PROBABILITY pt. m
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 imreasonable for us to pursue them. If no finite prob-
ability exists that a formula, for which we have empirical verifica-
tion, 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 beheve, therefore, that, if we trust the
promptings of conamon 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 d priori probabiHties, 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 modem mathematician
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 particu-
lar 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,
' See Jevons, loc. cit. p. 231.
OH. XXI INDUCTION AND ANALOGY 245
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 argu-
ment 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 utihty. But the
processes of induction and addition would remain reasonable.
It may confuse, secondly, the vahdity of attributing probabiUty
to the conclusion of an argument with the question of the actual
truth of the conclusion. Induction tells us that, on the basis 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 wiU 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 irrationaUty 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
246 A TKEATISE ON PEOBABILITY n. m
or the ravings of insanity, but simply hypotheses, justifiable as
such at the time when they were propounded, but which a fuUer
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 attri-
buted the badness of the crops to the iron ploughshares, and dis-
carded them for the old wooden ones. The method of reasoning
of the farmers is not difEerent from that of science, and may,
surely, have had for them some appreciable probability ia its
favour. " It is a curious superstition," says a recent pioneer ia
Borneo, " this of the Dusuns, to attribute anything — ^whether
good or bad, lucky or unlucky — that happens to them to some-
thing 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 DiSerence ?
The following passage from Jevons's Principles of Science well
illustrates the tendency, to which he himself yielded, to depreci-
ate 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 Umit
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 cHmacteric, and legal majority at twenty-one
years, the third climacteric." Eeligious systems from Pythagoras
to Comte have sought to derive strength from the virtue of seven.
1 Oolden Bough, p. 174.
OH. XXI INDUCTION AND ANALOGY 247
" And even in scientific matters the loftiest intellects have occa-
sionally 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 ioferririg 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 super-
stitious 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 attri-
buted 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 all hold to be weU estab-
lished, may require for its justification the combination of
experience with a jast appreciable a priori possibUity 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
difierent 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
1 " This is animism, or that senae 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 aU 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 a priori probability, if some of those
universal conclusions of common senae, which the most sceptical do not kick
away, are to be supported with rational foundations.
248 A TREATISE ON PROBABILITY pt. m
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 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 efiect,
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 the previous chapter 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 spaoe.^ It has also been pointed out that ordinary in-
ductive 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.
"^ 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 bUlion years.
* Is this inteipretation of the Principle of the Uniformity of Nature affected
by the Doctrine of Relativity?
CH. XXI INDUCTION AND ANALOGY 249
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 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
efEect, 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 efEect, 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, accord-
ing 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 impUed 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
250 A TEEATISE ON PROBABILITY pt. m
of a phenomenon which has appeared as part of a more complex
phenomenon, may be some reason for expecting it to be associated
on another occasion with part of the same complex. Yet if
different wholes were subject to different laws qud 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 nimiber of legally
atomic units and the laws connectiag them, it would be possible
to deduce their effects pro tanto without an exhaustive knowledge
of all the coexistiag 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 assump-
tions which we do actually make, and may serve as an introduction
to what follows.
CHAPTER XXII
THE JUSTIFICATION OF THESE METHODS
1. The general Une 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 aU) 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 proposi-
tions, 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 indepen-
dent variety should be actually finite. "We must distinguish,
therefore, between systems which may be termed finite and
infinite respectively, the terms finite and imfmite referring not to
251
252 A TREATISE ON PROBABILITY w. in
the number of members in the system but to the amount of in-
dependent variety in it.
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 vahdly obtained by means of
an inductive argument. I now proceed to approach the question
from a shghtly 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.w in-
stances which have r known qualities, a^a^ ... a^ in common,
these r qualities constituting the known positive analogy. From
these qualities three (say) are picked out, namely, %, a^, a^, and
we inquire with what probability all objects having these three
qualities have also certain other qualities which we have picked
out, namely, a^_i, a^. We wish to determine, that is to say,
whether the quaUties «,._!, a^ are bound up with the qualities
%, fflgj <^3- 111 tliiis approaching this question we seem to
suppose that the qualities of an object are bound together in
a Hmited number of groups, a sub-class of each group being an
infaUible symptom of the coexistence of certain other members
of it also.
Three possibilities are open, any of which would prove
destructive to our generaHaation. It may be the case (1) that
a^_iOV a^is independent of all the other quaUties of the instances
— ^they may not overlap, that is to say, with any other groups ;
or (2) that ajO,^^ do not belong to the same groups as a^.^o,^ \
or (3) that a^a^a^, while they belong to the same group as a^_ia„
are not sufficient to specify this group uniquely — they belong,
that is to say, to other groups also which do not include a^_y and
a^. 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 a^-A are
numerous and comprehensive, because this increases the likeli-
hood that some at least of them fall under heading (1), and also
because it increases the likelihood of (3). We trust it if the
terms typified by a^a^a^ are numerous and comprehensive,
because this decreases the likelihood both of (2) and of (3). If
CH. xxn INDUCTION AND ANALOGY 253
we find a new instance which agrees with the former instances in
a^a^^a^_-^^ but not in a^, we welcome it, because this disposes of
the possibility that it is a^, alone or in combination, that is bound
up with a^_-ja^. 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 qtialities 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 fiist 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 thiak, that the almost innumerable
apparent properties of any given object all arise out of a finite
number of generator properties, which we may call ^t^-^^. ■ . .
Some arise out of ip^ alone, some out of ^^ in conjunction with (p^,
and so on. The properties which arise out of ^^ alone form one
group ; those which arise out of ^x^a ^ 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
254 A TREATISE ON PROBABILITY pt. m
properties arise (say) out of three generator properties ^i<^2^3,
then this set of properties may be said to specify the group
4'i4'z4'3' 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 <p and/, either/
belongs to the group ^ or it arises out of generators partly distinct
from those out of which ^ arises. For the reason already ex-
plained there is a finite probabihty that / and (f) belong to the
same group. If this is the case, i.e. if the generalisation g{^f)
is valid, then / will certainly be true of all other cases in which
(p is true ; if this is not the case, then / wiU not always be true
when tj) is true. We have, therefore, the preliminary conditions
necessary for the apphcation of pure iaduction. If x^, etc., are
the instances,
g/h =p^, where p^ is finite,
xjgh=l, etc.,
and xJx^X2, . . . x^_jgh = l -e, where e 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 (^ 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 / belongs to this group is relatively
large. If, on the other hand, / is complex, there will be, for the
same reasons mutatis mutandis, a relatively smaller initial prob-
ability than otherwise that /belongs to any other given group.
CH. xxn INDUCTION AND ANALOGY 255
When the argument is mainly by analogy, we endeavour to
obtain evidence which makes the initial probability jJq relatively
high ; when the analogy is weak and the argument depends for
its strength upon pure induction, p^ is small and j3,„, which is
based upon numerous iastances, 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 UkeHhood that two characters belong
to the same group. The numerous characters of an object which
are known to us may be represented by a^a^ . . . a„. We select
two sets of these, a^ and a^, and seek to determine whether a^
always belongs to the group specified by a^. Our previous know-
ledge will enable us, in general, to rule out many of the object's
characters as being irrelevant to the groups specified by a^ and a^,
although this will not be possible in the most fundamental in-
quiries. We may also know that certain characters are always
associated with a^ or with a^. But there wiU be left a residuum
of whose connection with a^ or a^ we are ignorant. These
characters, whose relevance is in doubt, may be represented by
a.
■r+l ■
.a^^i- If the analogy is perfect, these characters are
eliminated altogether. Otherwise, the argument is weakened
in proportion to the comprehensiveness of these doubtful char-
acters. For it may be the case that some of a,._,.i . . .a^^i are
necessary as well as a„ in order to specify all the generators
which are required to produce a^.
5. We may possibly be justified in neglecting certain of the
characters a^+i . . . a^-i 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
256 A TEEATISE ON PEOBABILITY ft. m
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 aUke 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 ia their being 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 posi-
tion is irrelevant, I do not mean to deny that a generahsation, 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 con-
dition. A generahsation 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 generahsations extend, do
not have an infinite number of independent quahties ; that, in
other words, their characteristics, however numerous, cohere
together in groups of invariable connection, which are finite
in number. This does not hmit the number of entities which
are only numerically distinct. In the language used at the
beginniag of this chapter, the use of inductive methods can be
CH. xxn INDUCTION AND ANALOGY 257
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 char-
acter can arise in more than one way, can belong to more than
one distinct group, and can arise out of more than one generator.
(f) might, for instance, be sometimes due to a generator a^, and
ai might invariably produce/. But we could not generalise
from (j> to/, if ^ naight be due in other cases to a different
generator ag which would not be competent to produce /.
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 ^ 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 prob-
ability 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 argu-
ments, and this superiority may possibly explain the great con-
fidence 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 con-
nected with some of our acts. We do not, as in other cases,
1 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.
S
258 A TEEATISE ON PROBABILITY pt. m
merely observe invariable sequence or coexistence between con-
sciousness 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 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 con-
nection 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
imiverse is limited in such a way that there is no one object so
complex that its quaUties fall into an infinite number of inde-
pendent 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 some-
times 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 probabfiity in favour
of any instance of the generalisation in question, without in-
quiring 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 to it, we must obtain, by some
means or other, a finite probability that the set of characters,
OH. xxa INDUCTION AND ANALOGY 259
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 hypo-
thesis strongly suggests that our conclusions should be in the
form of inductive correlations, rather than of universal general-
isations. Perhaps our generalisations should always run : ' It is
probable that any given ^ is/,' rather than, ' It is probable that
all ^ are/.' Certainly, what we commonly ^eem to hold with con-
viction 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 estabUsh a given numerical degree of probability, which
corresponds to the vagueness in the degree of probabHity 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 assump-
tions 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, the finiteness of the system, to which the argument is appUed,
for certain. But we need not in fact go so far as this.
If our conclusion is C and our empirical evidence is B, then,
in order to justify inductive methods, our premisses must include,
in addition to E, a general hypothesis H such that C/H, the
260 A TEEATISE ON PROBABILITY pt. m
d priori probability of our conclusion, has a finite value. The
effect of E is to increase the probability of C above its initial
d 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 CAddence 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 considera-
tion, the inductive hypothesis, and the process of strengthening
C/H by the addition E the indiictive 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 assump-
tion. If, therefore, we have any reason (H') for attributing
d priori a finite probability to the Inductive Hypothesis (H), then
the actual conformity of experience d posteriori with expectations
based on the assumption of H can be utilised by the inductive
method to attribute an enhanced value to the probabihty 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 Hjrpothesis, not at its d priori value, but at
the value to which experience in general has raised it. What
we require d priori, therefore, is not the certainty of the Inductive
Hypothesis, but a finite probability in its favour.^
Our assumption, in its most limited form, then, amounts to
this, that we have a finite d 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
d 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.
^ 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. 68 and 147. In these passages the
necessary conditions for the above are elucidated. I am, therefore, assuming
that in the case now in question these conditions actually are fulfilled.
CH. XXII INDUCTION AND ANALOGY 261
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 beUeve, 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
land 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 imderlying 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 inde-
pendent variety, amounts to very much the same thing. If the
fimdamental laws of connection changed altogether with varia-
tions, for instance, ia 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 inde-
pendent 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 difficiilt 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
262 A TEEATISE ON PROBABILITY pt. hi
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 d 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 ill question, and is to this extent dependent on experience.
But if it is experience 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,
iu 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 imderstand the
meaning." Knowledge, so obtained, 1 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 ' yeUow is a
colour ; ' we might also know that colour cannot exist without
extension, or that two colours cannot be perceived at the same
OH. xxn INDUCTION AND ANALOGY 263
time in the same place. Other philosophers might use terms
difEerently 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 unex-
plored field where no certaia 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 pro-
positions about them which go far beyond a mere expression of
this meaning; — propositions of the kind which some philo-
sophers 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 appre-
hended by us. If this view is correct the Inductive Hypothesis
is not the kind of thing about which we can have direct know-
ledge 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 fixst 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 beUeving, 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 ' neces-
sary ' connection ^ with some set of objects at a previous time.
This behef 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
^ I do not propose to define the meaning of this.
264 A TREATISE ON PEOBABILITY pt. m
about numbers arises out of our perceiving directly, when we
understand the meaning of a number, that they are of the re-
quired character.^ 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 Uke 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 behef that this conviction
gets its invincible certainty from some valid principle darkly
present to our minds, even though it stiU eludes the peering
eyes of philosophy.
'^ Since numbers are logical entities, it may be thought less unorthodox to
make such an assumption in their case.
CHAPTER XXIII
SOME HISTOEICAL 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
Mni, 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 detaUs, it becomes clear that they
were reaUy 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
^ See note at the end of this chapter on " The Use of the Term Induction."
' He speaks of himself as being " in hao re plane protopirus, et vestigia
nullius sequutus " ; and in the Praefatio Oeneralis he compares his method to
the mariner's compass, until the discovery of which no widg sea could be
crossed (see Spedding and Ellis, vol. i. p. 24).
265
266 A TKEATISE ON PROBABILITY pt. m
not do this, and against such pretensions Macaulay's well-known
essay was not unjustly directed. MiU, however, expressly dis-
claimed in his preface any other 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 hj 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 fife. 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 In-
duction 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 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
1 This view is taken in the edition of James Spedding and Leslie EEis.
Their introductions to Bacon's philosophical works seem to me to be very greatly
superior to th« accounts to be found elsewhere. They make intelligible, what
seems, according to other commentaries, fanciful and without sense or reason.
OH. xxm INDUCTION AND ANALOGY 267
simple treatment, and, in seeking for ease and certainty, to
treat far too lightly the problem of justifyrag what he had
claimed. MUl shirks, almost openly, the difficulties ; and scarcely
attempts to disguise from himself or his readers that he grotmds
induction upon a circular argument.
3. Some of the most characteristic errors both of Bacon and
of MiU 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 estabhshing a
conclusion which is absolutely certain, and that an argument
is invaUd if the generalisation, which it supports, admits of
exceptions in fact. " Absolute certainty," says Leslie Elhs,^ " is
one of the distinguishing characters of the Baconian induction."
It was, in this respect, mainly that it improved upon the older
induction fer enumerationem simpUcem. " The induction which
the logicians speak of," Bacon argues in the Advancement of
Lea/rning, " is utterly vicious and incompetent. . . . For to con-
clude 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 Uable 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 MUl, although there
are passages in which he abates them,^ for his own rules of pro-
cedure. An induction has no validity, according to him as
according to Bacon, unless it is absolutely certain. The follow-
ing passage ^ is significant of the spirit in which the subject
was approached by him : " Let us 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 con-
clusion 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 uncertaiuty which is present, more
1 Op. dt. vol. i. p. 23.
2 When he deals with Plurality of Causes, for instance-
' Bk. iii. chap. iii. 3 (the italics aie mine).
268 A TREATISE ON PROBABILITY pt. m
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 MiU 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 beween
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 funda-
mental 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 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 con-
cerned 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
1 This misappieliension may be connected with Mill's complete failuie 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. xsni INDUCTION AND ANALOGY 269
Bacon attached supreme importance, and wHcli he held to con-
stitute the essential superiority of his method over those which
preceded it, entirely consists in the determination of what char-
acters (or natures as he would call them) belong to the positive
and negative analogies respectively. The first two tables with
which the iuvestigation begins are, first, the table essentiae et
praesentiae, which contains all known instances in. which the
given nature is present, and, second, the table decUnationis 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 oi 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 pre-
cision and their certainty ; and he underestimated the import-
ance of pure induction. But there was, at bottom, nothing about
his rules impracticable or fantastic, or 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 investi-
gators. But he saw further than Bacon in allowing for the
PluraUty of Causes, and in admittiag that an element of pure
induction was therefore made necessary. " The Plurality of
Causes," he says,^ " is the only reason why mere number of in-
stances is of any importance in inductive inquiry. The tendency
of imscientific 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 recog-
nised as immaterial, nothing whatever is added to the evidence of
1 Ellis, vol. i. p. 33.
2 Ellis, vol. i. p. 89.
' Book iv. chap. x. 2,
270 A TEEATISE ON PROBABILITY pt. m
the conclusion. A single instance eliminating some antecedent
wliicli 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 multi-
plication 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 deter-
mination 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
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 inter-
mingled 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 exemphfy
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 prob-
ability which Analogy seems to supply, primarily depends upon
a certain limitation of independent variety and upon the deriva-
tion of aU 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
tmless we have in each case some such limitation in the number
of the ultimate alternatives.
It was ui 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 differ-
CH. xxm INDUCTION AND ANALOGY 271
ence of things whicli are few in munber, and the degrees and
co-ordinations whereof make all this variety." And in Valeriiis
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 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 bound-
less 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 Hmited number
of simple elements — ^which is, according to EUis,^ the central
point of Bacon's whole system — ^was a real contribution to philo-
sophy.
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 MUl. 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 fiarst
1 Quoted by Ellis, vol. i. p. 41.
2 Vol. i. p. 28.
272 A TEEATISE ON PROBABILITY pt. m
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 MiU came Hume. Hume's sceptical
criticisms are usually associated with causality ; but argument
by induction — ^inference from past particulars to future generalisa-
tions — ^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 aU possible lines of proof seemed
equally unpromisiag. 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 solu-
tion 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 frag-
mentary 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 generaHsations 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 pre-
^ Couturat, Opuscules et fragments inedits de Leibniz, p. 232.
' Couturat, La Logique de Leibniz d'apris des documents inedits, pp.
262, 267.
OH. xxm INDUCTION AND ANALOGY 273
viously untried is a just ground of secure confidence, of which
he cites as a nearly perfect example the key to a crj^togram.^
10. Whewell and Jevons furnished logicians with a store-
house 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 recogni-
tion of Mill's inadequacy renders their exposition, which, ia 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.
1 Letter to Conring, 19th March 1678.
NOTES ON PAET III
(i.) On the Use of the Teem Induction
1. Induction is in origin a translation of tte Aristotelian eTrayioyij.
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
reahse 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 generalisation 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 simpKcem 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 _^sJ 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 empiri-
cal, and so as to exclude generalisations, such as those of mathematics,
which have been proved formally. Jevons was led, partly by the
linguistic 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
* See Ellis's edition of Bacon's Worlcs, vol. i. p. 37. On the first occasion
on which Induction is mentioned in the Novum Organum, it is used in this
secondary sense.
274
NOTES INDUCTION AND ANALOGY 275
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 tjrpes of arguments which combine, in one form or
another, pure induction with analogy.
(ii.) On the Use of the Teem 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 con-
nection 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 efEect, and even to speak of a 'probable cause, where there is no
implication of necessity and where the antecedents wiQ 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
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 Umited
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.
^ Cf. Czuber, Wahrscheinlichkeitsrechnung, p. 139. In dealing with Inverse
Probability Czuber explains that he means by possible cause the various Be-
dingungskomplexe from which the cause can result.
2 See Chapter XXIV. §3.
276 A TREATISE ON PEOBABILITY pt. ni
We must first distinguisli 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 pa/rticula/r 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 aU, in this sense, timeless. We may, therefore, divide our data
into two portions k and I, such that k denotes our formal and
nomologic evidence, consisting of propositions whose predication
does not involve a particular time reference, and I 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
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/ah = 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 h can invaUdate this.
(ii.) If b/ah = 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 efEect to sufficient cause, is equivalent to the
proposition that, if & 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 bjak=\. No use has been made so far of our
existential knowledge I, which is irrelevant to the definitions pre-
ceding.
(iv.) If bjakl = 1 and bjkl 4= 1, A is a sufficient cause of B undej
conditions I.
(v.) If bjakl = and bjkl =!= 0, A is a necessary cause of B under
conditions I.
(vi.) If there is any existential proposition h such that bjahk = 1
and bjhk =t= 1, A is, relative to k, a possible sufficient cause of B.
(vii.) If there is an existential proposition h such that b/shk =
and b/hk =1= 0, A is, relative to k, a possible necessary cause of B.
(viii.) If b/ahkl = 1, b/hk 4= 1, and h/akl 4= 0, A is, relative to k,
a possible sufficient cause of B under conditions I.
(ix.) If b/ahkl = 0, b/hU=^0, h/akl ^0, and h/akl ^0, A is,
relative to k, a possible necessary cause of B under conditions I.
^ Die Prindpien der WahrscheirdichkeitsrecJinung, p. 86.
NOTES INDUCTION AND ANALOGY 277
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 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 caxisally dependent or inde-
pendent.
4. These definitions preserve the distinction between ' causally
independent ' and ' independent for probability,' — ^the distinction
between causa essendi and causa cognoscendi. If hJahM^hldhhl,
where a and h 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 5, relative to data kl.
If a and 6 are causally dependent, according to definition (x.), 6 is a
possible causa essendi, relative to data hi.
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 pheno-
menon upon the expectation of another.
PART IV
SOME PHILOSOPHICAL APPLICATIONS OF
PEOBABILITY
279
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 Eandomness 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 foimdations 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 com-
monly held to arise out of the antithesis between ' cause ' and
' chance,' between events, that is to say, which are causally con-
nected 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 '
281
282 A TREATISE ON PROBABILITY pt. iv
event to one, part of whose cause is a volition following on a
conscious desire for the event.^
2. The method of this treatise has been to regard subjective
probability as fundamental and to treat all other relevant con-
ceptions as derivative from this. That there is such a thing as
probability in this sense has been admitted by all sensible philo-
sophers 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 ia favour of the view that it is this (what-
ever 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 distiaction 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 deter-
minist 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 explica-
tions que j'ai donnees . . . ," he writes in the preface to his
Exposition, " sur le double sens du mot de probabilit6, qui
tantot se rapporte a vme certaine mesure de nos connaissances, et
tant6t a une mesure de la possibility des choses, independamment
de la connaissance que nous en avons : ces explications, dis-je,
me semblent propres a resoudre les difficultes qui ont rendu
jusqu'ici suspecte a de bons esprits toute la theorie de la proba-
bility mathematique." It will be worth while to pause for a
moment to consider the ideas of Cournot.
1 This is discussed in Chapter XXV. § 4.
2 D'Alembert, oolleoting (largely from Hume, many passages being trans-
lated almost verbatim) in the Encyclopedie methodigue the most up-to-date
commonplaces of the subject, found it natural to write : " II n'y a point de
hasard a proprement parler ; mais il y a son Equivalent : I'ignorance, oil nous
sommes des vraies causes des 6v6nemeus, a sur notre esprit I'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.
CH. XXIV PHILOSOPHICAL APPLICATIONS 283
3. Cournot, while admitting that there is such a thing as sub-
jective chance, was concerned to dispute the opinion that chance
is merely the offspring of ignorance, saying that in this case
" le calcid 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 com-
bination 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 diSerent series.
A ' chance ' 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 contri-
buted something, I think, to the stock of our ideas. He has
1 " Le mot hasard," Cournot writes in his Essai sur les fondements de nos
connaissances, " n'indique pas une cause substantielle, mais une id6e : cette idee
est oelle de la oombinaison entre plusieurs series de causes ou de faits qui se
developpent chacun dans sa s6rie propre, ind6pendamment les uns des autres."
This is very like the definition given by Jean de la Placette in his Traite desjeux
de hasard, to which Cournot refers : " Pour moi, je suis persuade que le hasard
renferme quelque chose de r6al et de positif, savoir un concours de deux ou
plusieurs evenements contingents, chacun desquels a ses causes, mais en sorte
que leur concours n'en a aucune que Ton connaisse."
2 Ussai sur les fondements de nos connaissances, i. 134 : " La nature ne se
gouverne pas par une loi unique ... ses lois ne sont pas toutes ddriv^es les
unes des autres, ou d6riv6es toutes d'une loi supMeure par une n6oessit6 pure-
ment logique . . . nous devons les ooucevoir au-contraire oomme ayant pu
6tre d^cr^ttes s6par6ment d'une infinite de mani^res."
284 A TREATISE ON PROBABILITY pt. iv
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 con-
fused. But in Bertrand's criticism of him a relevant distraction,
though not elucidated, is brought before the mind. " The
motives for believing," wrote Condorcet, " that, from ten million
white balls mixed with one black, it will not be the black ball
which I shall draw at the first attempt is q/" 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 lo ooo ooo ' i^either 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 extraguished ; he
would be within his rights, just as Condorcet is within his ; both
woTild exceed theic rights in accusiug of error those who think
differently." Before commenting on this distinction, let us have
before us also some interesting passages by Poincare.
5. We certainly do not use the term ' chance,' Poincare 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
^ Coumot's work on Probability has been highly praised by authorities as
diTerse 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 Bevue de metaphysiqiie et de morale, devoted to Cournot and pub-
lished 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 hasa/rd. Cournot's philosophy of
the subject is developed, not so much in his Exposition de la theorie des chances,
as in later works, especially in his Essai sur lee fondem^nls de nos connaissances.
Ooumot 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 illuminatiag 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.
' Calcul des probabilites, p. xix.
CH. XXIV PHILOSOPHICAL APPLICATIONS 285
ignorant." But Poincare immediately adds : " Is this definition
very satisfactory ? When the first Chaldaean shepherds f oUowed
mth 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 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,
Poincare 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 stUl, 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 Poincare
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 wUl fall, but not on
which side — chance wiU 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
^ Galcul des probabilitea (2nd edition), p. 2. This passage also appears in an
article in the Bevue du moia for 1907 and in the author's Science et methode, of
the English translation of which I have made use above,— at the cost of doing
incomplete justice to Foincar^'s most admirable style.
286 A TREATISE ON PROBABILITY pt. iv
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 fortuit-
ous phenomenon:" " The greatest chance is the bicth 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 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 compre-
hension of the true character of chance."
Poincare 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 Couinot, 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
Poincare, 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 d 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' con-
firms 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
CH. XXIV PHILOSOPHICAL APPLICATIONS 287
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 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 im-
probable. 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 agaiast it ; when, that is to
say, the probabihties 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 occur-
rence 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 stiU be subject to chance in the sense just defined, and
we may be able to infer this without actually haAong the addi-
tional information in question. If, however complete otir
knowledge of certain kinds of things might be, there would still
exist independence between the propositions, the conjunction
288 A TREATISE ON PROBABILITY pt. iv
of which we are investigating, then we may say there is an
objective sense in which the actual conjunction of these pro-
positions 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 alterna-
tive 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 distinc-
tion,* between nomologic knowledge and ontologic knowledge,
between knowledge of laws and knowledge of facts or existence.
Given certain facts /(a) about a and certain laws of connection, L,
we can infer certainly or probably other facts 0(a) about a. If
a comply knowledge of laws of connection together with /(a)
yields no appreciable probability for preferring <j){a) to other
alternatives, then I suggest that an actual connection between ^
and/ 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, though appreciable, are very smaU ; and
we are not insistent to assert the rule of chance if a comparatively
sUght addition to our ontologic knowledge would render the
probability or the grounds for preference appreciable.
1 Logic of Chance, p. 245.
» See Part III. Note (u.) § 2, p. 275.
OH. XXIV PHILOSOPHICAL APPLICATIONS 289
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
equi-probable, 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 stand-
point of chance, in nearly the same position as laws. A connec-
tion is not due to chance, therefore, if a knowledge of the per-
manent facts of existence could lead to their prediction.
To sum up again therefore, — ^if within a given field of observa-
tion or experience a knowledge of those facts of existence which are
permanent or invariable within that field, together with a know-
ledge of all the relevant fundamental causal laws or general
principles, and of a few other facts of existence, would not
permit us, given/(a), to attribute an appreciable probability to
(^(a) (or an appreciable probability to the alternative ^i(a)
rather than <f>2{<^)) ! then the conjunction of ^(a) (or of ^i(a)
rather than <f>2{a) with /(a)) is due to objective chance.
9. If we return to the examples of Poincare, the above defini-
tion appears to conform satisfactorily with the usages of common
sense. It is when an excu^ 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 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 cele-
brations on such a day such as would be not unlikely to accelerate
u
290 A TEEATISE ON PROBABILITY n. iv
death, we should say that a man's djong on his birthday was not
altogether a matter of chance. If we Imew 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 connec-
tion 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 be 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 Peicce,^ 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 in-
stances as often as any other set of the same niimber." 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 know-
ledge 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.*
1 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.
' " Law of Error," Gamb. Phil. Trans., 1904, p. 128.
* Bat 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, there-
fore, is sometimes nearer than his definition to the popular usage.
CH. XXIV PHILOSOPHICAL APPLICATIONS 291
I do not believe, therefore, that this kind of definition is a
useful one. The term' must be defined with reference to prob-
ability, 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 d 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 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 diflSicult 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 8{x) . ^{x),
relative to evidence ^, if ' a; is a ' is irrelevant to the probability
d}(x)lS{x) . h. Thus ' the number of the inhabitants of France is
odd ' is, relative to my knowledge, a random instance of the
propositional function ' aj is an odd number,' since ' a is the
number of the inhabitants of France ' is irrelevant to the prob-
ability 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
1 In the above S(a!) stands for ' a; is a number,' 0{a!) stands for ' x is odd,'
a stands for ' the number of inhabitants of France,'
292 A TEEATISE ON PROBABILITY m.iv
probability ^ that any instance taken at random of the
generalisation ' all numbers are odd ' (or of the corresponding
generalisation * aU 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 scientific-
ally and statistically important. 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 qualifica-
tion I shall use it as the opposite of ' random selection ' in the
wider unqualified sense.
CHAPTER XXV
SOME PEOBLEMS ARISING OUT OF THE DISCUSSION OP 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 ia the preceding chapter.
The first of these is concerned with the iaclinations to the
ecliptic of the orbits of the planets of the solar system. This
problem has a long history, but it wiQ be sufficient to take De
Morgan's statement of it.^ If we suppose that each of the orbits
might have amy iaclination, we obtain a vast number of combina-
tions of which only a smaU 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 iacUnations being what they are."
The answer to this was pointed out by D'Alembert ^ in criticis-
^ Article on Frobahilitiea in Encydopae^a MetropoUtana, p. 412, § 46. De
Morgan takes this without acknowledgment from Laplace, Theorie analytique
des probabilites {1st edition), pp. 257, 258. Laplace also allows for the fact
that all the planets move ia the same sense as the earth. He concludes : " On
verra que I'existence d'une caiiae commune qui a dirig6 tous ces mouvemens
dans le sens de la rotation du soleil, et sur des plans peu inclines k celui de son
^quateur, est indiqu^e avec une probability bien sup^rieure k celle du plus
grand nombre des faits historiques sur lesquels on ne se permet auoun doute."
Laplace had in his turn borrowed the example, also without acknowledgment,
from Daniel Bernoulli. See also D'Alembert, Opuscules maihematiques, voL iv.,
1768, pp. 89 and 292.
2 Op. cit. p. 292. " II y a certainement d'infini oontre un a parier que les
Plandtes ne devraient pas se trouver dans le mSme plan ; ce n'est pas une raison
pour en conclure que cette disposition, si eUe avoit Ueu, auroit n^cessairement
d'autre cause que le hasard ; car il y auroit de mSme I'inflni contre un k parier
293
294 A TREATISE ON PROBABILITY w. iv
ing 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 A 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 d 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 Philo-
sophical 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 con-
sequence 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 pre-
sumption against the scattering of the several contiguous stars
that may be observed " by mere chance as it might happen."
And 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 iaduction was confirmed
by the later investigations of HerscheU adds interest to the
speculation. But apart from this the argument is evidently
que les Plan^tes pourroient n'avoir pas une certaine disposition d^terminte k
volenti. . . ."
D'Alembert is employing tte instance for his own purposes, in order to build
up an ad hominem argument in favour of his theory conoeming ' runs ' against
D. Bernoulli (see also p. 317).
'^ See also Todhimter'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 par-
ticular on Michell's Problem of the Distribution of the Fixed Stars," Phil.
Mag., 1851.
OH. XXV PHILOSOPHICAL APPLICATIONS 295
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,
i£ 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 " un-
caused." 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 ^ con-
ditions 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 steUar laws and posi-
tions 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 inter-
action 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 d priori that binary stars,
when they arise, do arise in this way, then, since the frequent
coincidence of a given set of independent causes relatively few
in number is more likely than that of a set relatively numerous,
the observation of binary stars will raise this probability d pos-
teriori 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 d priori
probability which may exist in favour of the second hypothesis.
1 See § 3 of Note (ii.) to Part III.
296 A TEEATISE ON PEOBABILITY m. iv
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 biaary 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 iateraction of a small number of
iadependent 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, arisiag out of
this, that there are not likely to be forces tendiag 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 argu-
ment is more complex and much less conclusive than he seems to
have 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 asser-
tions 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 argu-
ment. 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
OH. XXV PHILOSOPHICAL APPLICATIONS 297
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 volimie about the sun as centre contain
the same number of stars.^ He argues that the observed facts
render probable the following disjunction : Either the distribu-
tion 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 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 Ji/nal 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 a 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.
^ " On the Improbability of a Random Distribution of the Stars in Space,"
Proceedings of Royal Society, series A, vol. 84, pp. 47-70, 1910.
2 It is, therefore, independent of direction, and the distribution is random
even if the stars are massed in particular quarters of the heavens. The defini-
tion is, therefore, exceedingly arbitrary.
' This should run more correctly, I think, "not a sphere vnth the sun as
centre."
298 A TREATISE ON PROBABILITY pt. iv
Symbolically : Let h be our origiijial data, a tbe occurrence
of the event, 6 the existence of the supposed conscious agent.
Then ajh is assumed very small in comparison with ajhh ; and
we require hjah, the probability, that is to say, of b after a is
known. The inverse principle of probability already demon-
hlh
strated shows that hjah = ajhh.— ^, and bjah is therefore not
determinate in terms of ajbh and ajh alone. Thus we cannot
measure the probability of the conscious agent's existence after
the event,' unless we can measure its probability b^ore 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 ; 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 argu-
ment 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 d 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-
' The probability that a remarkable success in naming playing cards is due
to psychic agency, was discussed by Professor Edgeworth in MetreUke. This
was, I think, the first application of probabilities to these questions. See also
Proceedings of the Society for PsycMcal Research, Parts VIII. and X. ; Professor
Edgeworth's article on Psychical Research and Statistical Method, Stat. Joum.
vol. Lzxxii. (1919) p. 222 ; and Experiments in Psychical Research at Leland
Stanford Junior University, by J. Coover.
OH. XXV PHILOSOPHICAL APPLICATIONS 299
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 ia which our iaitial 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 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 aU 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 differ-
ently 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
d 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 com-
bine 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 quaKties have never been observed but whose existence we
postulate for purposes of explanation, stands in it also. If the
300 A TEEATISE ON PEOBABILITY vr.iv
analysis of Part III. is correct, we can never attribute a finite
probability ^ to the 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, neverthe-
less, in a position altogether different from that of such generalis-
ations 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 differ-
ence 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 be-
tween 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 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
' 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 con-
sequences, 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 re-
minded 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.
OH. XXV PHILOSOPHICAL APPLICATIONS 301
which no known agency could afford even an improbable ex-
planation, 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 iaappropriately 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 in-
frequently claimed. If we can directly perceive these ' spirits,'
as many of those who are described in James's Varieties of
ReUgious 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 Eesearch.
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 ia every relevant particular
some one object which we know to exist, or is it sufficient that we
shoidd know instances of aU its supposed qualities, though never
in combination ? It is clear that some qualities may be irrelevant
— ^position in time and space, for example — ^and that ' every
relevant particular ' need not include these. But can the initial
probability exist if our hypothesis assume^ 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 1 Could any evidence, for example,
persuade Ixs that a tree felt the emotion of amusement, even if
it laughed repeatedly when we made jokes ? Yet the analogy
302 A TEEATISE ON PEOBABILITY m. iv
which we demand seems to be a matter of degree ; for it does not
seem imreasonable 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 wanderiag 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 prob-
ability that some phenomenon is due to ' abnormal ' causes,
our mathematics Tvill be apt to lead us into unjustifiable
conclusions.
6. Uninstructed common sense seems to be specially unre-
liable 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 im-
probable on the available evidence. But it will often occur —
whenever, in fdct, 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 a 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 bmld with success. Yet Michell's argument wd the argu-
OH. XXV PHILOSOPHICAL APPLICATIONS 303
ment from design derive a good deal of their plausibility, I thiak,
from the • remarkable ' character of the actual constitution
whether of the heavens or of the universe, la forgetfulaess 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 ia need of an explana-
tion, and that any sufficient explanation has a high probability
ia its favour. That an explanation is particularly required,
possesses a measure of truth ; for it is likely that our original
data were much lacldng in completeness, and the occurrence of
the extraordinary event briags to light this deficiency. But
that we are not justified iu adoptiag mth 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 hy us ; and, therefore, any explanation which
deals with such cases falls ia prepared soil. Eesults which,
judgiag from ourselves, conscious agents would be Kkely to pro-
duce fall into this category. Eesults 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 are or are not likely to be due to
' chance.' This may explaia, for example, why the particular
concurrence of atoms that go to compose the human eye, why a
series of correct guesses ia naming playing cards, why special
symmetry or special asymmetry amongst the stars, seem to
require explanation in no ordinary degree. Prior to an explana-
tion 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 miad than are the causes of others, and the attempt to
explain them deserves, therefore, to be more carefully considered.
THs supposition, derived by analogy or induction from those
cases in which we believe the causes to be known to us, has, per-
haps, 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
304 A TEEATISE ON PROBABILITY w. iv
channels of communication, is a fact worthy of investigation,
because it is more likely to be susceptible of a simple causal ex-
planation, 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 6e/ore the examination of some or all of
the available instances than if it is proposed after there examina-
tion. 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 ia favour of such a conclusion
are well put by Peirce : ^ "All 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
^ C. S. Peirce, A Theory of Probable Inference, pp. 162-167 ; published in
Johns Hopkins Studies in Logic, 1883.
OH. XXV PHILOSOPHICAL APPLICATIONS 305
on this evidence to Dr. Lyon Playfair's argument about the
specific gravities of the three allotropic forms of carbon :
Diamond . . . 348=^12
Graphite . . . 2-29 = Vi2
Charcoal . . . 1-88= t/l2
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 argument^, 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, 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 in-
ductions had to be thought of before we examined the cases to
which we apply them, we should, doubtless, make fewer induc-
tions ; 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 d 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
X
306 A TREATISE ON PEOBABILITY n. iv
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 Kes 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.
The same point arises not uncommonly in statistical inquiries.
If a theory is first proposed and is then confirmed by the examina-
tion 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 considera-
tions, 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 preconcep-
tions 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.
CHAPTEE XXVI
THE APPLICATION OF PEOBABILrrY 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 ex-
pressing 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." ^
1 Herod, vii. 10.
307
308 A TREATISE ON PROBABILITY pt. iv
3. The first contact of theories of probability with modern
ethics appears in the Jesuit doctrine of probabUism. According
to this doctriae 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 denoimce
it.^ It may be suspected, however, that the object of this
doctrine was not so much duty as safety. The priest who per-
mitted 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 prob-
abilism. " 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 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 ia 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
^ 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 nnreasonable 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 ; it he be not, he is not
boimd under the tie of necessity to know anything of it. That that is
necessary shall be certainly conveyed to him : God, that beat can, will certainly
take care for that ; for if he does not, it becomes to be not necessary ; or if it
should still remain necessaiy, 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."
2 The Part Royal Logic (1662), Eng. Trans, p. 367.
3 Essay concerning Human Understanding; book ii. chap. xxi. § 66.
CH. XXVI PHILOSOPHICAL APPLICATIONS 309
still need the art of thinkiiig 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 con-
cerned with the summing up of consequences, the place of prob-
ability 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 infimite 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 justi-
fication for saying that it is even probably right to choose the
greater good within the region over which a probable forecast
may extend."
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 pro-
duces more good than the other in the near future, and if we have
no means of discriminating between their results in the distant
^ Nouveaux Essais, book ii. chap. xxi.
310 A TREATISE ON PROBABILITY rr. iv
future, then by what seems a legitimate application of the
Principle of Indifference we may suppose that there is a prob-
ability 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
wiU 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 wUl, in particular instances, produce an overbalance
of happiness upon the whole ; since so many and distant things
must come iato the account. And that which makes it our duty
is that there is some appearance that it wiQ, 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 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 simiiltaneous 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 BtiU 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
1 This passage is from the Analogy. The Bishop adds : " ... and also
that such benevolent endeavour is a cultivation of that most excellent of aU
virtuous principles, the active principle of benevolence."
OH. xxYi PHILOSOPHICAL APPLICATIONS 311
and indiAddual personality. Or we may suppose that, when
conscious units are ia 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 numeric-
ally measurable. This theory goes on to maintain that what
we ought to add together, when, ia. order to decide between two
courses of action, we sum up the results of each, are the ' mathe-
matical 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 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 prob-
ability.
The first assumption, that quantities of goodness are duly
subject to the laws of arithmetic, appears to me to be open to a
certain amoimt 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
' Priority in the conception of mathematical expectation can, I think, be
claimed by Leibniz, De incerti aestimatione, 1678 (Couturat, Logique de Leibniz,
p. 248). In a letter to Plaooius, 1687 (Dutens, vi. i. 36 and Couturat, op. cit.
p. 246) Leibniz proposed an application of the same principle to juris-
prudence, 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 proporiiion. The doctrine, seems
sensible, but I am not aware that it has ever been acted on.
312 A TEEATISE ON PKOBABILITY pt. iv
been advocated in Part I. of this treatise. Lastly, if both these
points be waived, the doctrine that the ' mathematical expecta-
tions ' 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 attain-
ment of a state of mediocrity. Putting on one side the first of
these groimds 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 numeric-
ally measurable. It follows from this that the ' mathematical
expectations ' 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 ia 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 ia
virtue of an iatuitive judgment directed to the situation as a
whole, and not ia 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 attaimng 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 prob-
ability and goodness jointly. It may be remarked, further,
that the difficulty exists, whether the numerical iadeterminate-
ness 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,
OH. XXVI PHILOSOPHICAL APPLICATIONS 313
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 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 iaformation 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 iaformation 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 measuxes what our
preferences ought to be — whether, that is to say, the undesir-
ability of a given comrse of action increases in direct proportion
to any increase in the imcertainty of its attaining its object, or
whether some allowance ought to be made for ' risk,' its undesir-
abUity 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 prob-
ability a maximum ? Those who rely on the conception of
' mathematical expectation ' must hold that this is an indisput-
able proposition. The justifications for this view most commonly
advanced resemble that given by Coudorcet in his " Eeflexions
1 Ars Conjectandi, p. 215 : " Non suffloit expendere unum alterumve argu-
mentum, sed oonquirenda sunt omnia, quae in cognitionem Mstram venire
possunt, atque uUo modo ad probationem rei facere videntur."
a Essay concerning Human Understanding, book ii. chap. xxi. § 67 : " He
that judges without informing himself to the utmost that he is capable, cannot
acquit himBeii of judging amiss."
314 A TEBATISB ON PEOBABILITY m. rv
sur la rSgle g6n6rale, qui prescrit de prendxe pour valeur d'lrn
6veneinent incertain, la probability de cet ^venement nniltipli6e
par la valeur de rSvenement en M-rnSme," ^ where he argues
from Bernoulli's theorem that such a rule wiU lead to satisfactory
results if a very large number of trials be made. As, however,
it will be shown lq 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 doiag
so), it wiU be worth while to quote the main passage in which he
declares his scepticism : " II me sembloit " (in reading Bernoulli's
Ars Conjectandi) " que cette matiere avoit besoin d'etre trait^e
d'une maniere plus claire ; je voyois bien que I'esp^rance 6toit
plus grande, 1° que la somme esperee etoit plus grande, 2° que
la probability de gagner I'etoit aussi. Mais je ne voyois pas avec
la mSme evidence, et je ne le vols pas encore, 1° que la probabUite
soit estimee exactement par les m^thodes usitees ; 2° que quand
elle le seroit, I'esperance doive etre proportionnelle k cette proba^
bUite simple, plut6t qu'a une puissance ou mSme a une fonction
de cette probabilite ; 3° que quand il y a plusieurs combinaisons
qui donnent diEEerens avantages ou diSerens risques (qu'on
regarde comme des avantages n^gatifs) il faiUe se contenter
d'ajouter simplement ensemble toutes les esperances pour avoir
I'esperance 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 1 We
may doubt whether the moral value of speculative and cautious
action respectively can be 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
1 Hist, de VAcad., Paris, 1781.
" Opuscules matMmaiiques, vol. iv., 1768 (extraits de lettres), pp. 284, 285.
See also p. 88 of the same volume.
OH. XXVI PHILOSOPHICAL APPLICATIONS 315
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 con-
clusion that, other things berag 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 weU 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 beiag 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 foUows. If
A is the amount of good which may' result, f its probability
{p + q=\), and E the value of the 'mathematical expectation,'
so that B=33A, then the 'risk' is E, where Il=^(A-E) =
'p{l-p)k. = 'pqk. = qSi. This may be put in another way: E
measures the net immediate sacrifice which should be made ia the
hope of obtainiag A ; g' is the probability that this sacrifice will
be made in vain ; so that gE is the ' risk.' ^ The ordiaary theory
supposes that the ethical value of an expectation is a function
of E only and is entirely independent of R.
We could, if we liked, define a conventional coefficient c of
weight and risk, such as c=— tjz r, where w measures the
(l+?)(l+w)
' weight,' which is equal to unity when jj = 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
^ The theory of Eisiko is briefly dealt with by Czuber, Wahrscheinlichheits-
rechnung, vol. ii. pp. 219 et seq. K R measures the first insurance, this leads to a
Risiko of the second order, Rj = gR = g^R. This agaia may be insured against,
and by a sufficient number of such reinsurances the risk can be completely
shifted : E+Ri-t-R2+ . .. =£(1+2+22+ . . .) = j— = -=A.
° If pA = p'A', w>w', and ?=?', then cA>c'A'; if pA=p'A', w=w', and
g<g', then cA>c'A'; if pA=p'A', w>w', and ?<?', then 6A>c'A'; but if
pA =p'A!, w = to', and q > q', we cannot in general compare cA and c'A'.
316 A TREATISE ON PROBABILITY pt. iv
complicated fiinction 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 mtdti-
plyiag together the magnitudes obtaiaed in the two direct judg-
ments ; 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 — 1£
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, " eclairer les Sciences morales
et politiques par le flambeau de I'Algebre." 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 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, 2'"^ shillings if no
head appears until the r*^ 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 ?
1 Por 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.
OH. XXVI PHILOSOPHICAL APPLICATIONS 317
n
The mathematical answer is 2(|f2''"\ if the number of tosses
1
00
IS not in any case to exceed n in aU, and %{yfT'''^ if this restriction
1
IS removed. That is to say, Paul should pay - shillings in the
first 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 ofEered will occur at
once to the reader. The conditions of the game iw/ply contra-
diction, 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 imagina-
tion ; 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.
D'Alembert's principal explanations are, first, that true ex-
pectation 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
mathematicien : tons deux estiment I'argent par sa quantite
numerique." 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 a; is the ' physical ' fortune and y the ' moral ' fortune,
dy=k — '
X
or y=klog-, where k and a are constants.
On the basis of this formula of Bernoulli's a considerable
318 A TEEATISE ON PROBABILITY m. iv
theory has been built up both by Bernoulli^ himseK and by
Laplace.^ It leads easily to the further formula —
x = {a+Xj)p^{a+X2)p.2. . .,
where a is the initial ' physical ' fortune, p^, etc., the probabilities
of obtaining increments a^, 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 a^, 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 aU sums greater
than 2^ (16,777,116) are regarded as ' 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 infimte 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 unwiUing 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
1 " Specimen Theoriae Novae de Mensura Sortis," Comm. Acad. Petrop.
vol. V. for 1730 and 1731, pp. 175-192 (pubUshed 1738). See Todhunter, pp.
213 et seq.
' Theorie analytique, chap. x. " De I'esp&ance morale," pp. 432-445.
OH. XXVI PHILOSOPHICAL APPLICATIONS 319
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 ehmin-
ated 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 nimiber of tosses is ia no case to exceed a finite number,
the risk E, as already defined, may be very great, and the relative
risk = will be almost unity. Where there is no limit to the
number of tosses, the risk is infinite. A relative risk, which
approaches unity, may, it has been already suggested, be a factor
which must be taken mto account in ethical calculation.
10. In establishing the doctriae, that all private gambling
must be with certaiaty a losing game, precisely contrary argu-
ments 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. J{a) is
the chance that A will be ruined. Thus f{a) = — j-,^ so that
a+o
the poorer a gambler is, relatively to his opponent, the more
likely he is to be ruined. But further, if & = oo , 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 argu-
ment supposes, for ever ? ^ At the end of aiVLj 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
1 Oalcul des probabilites, p. 129.
2 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.
320 A TREATISE ON PROBABILITY pt. iv
individually must lose and that everybody collectively must wia.
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 milKonaires gain nothing by gam-
bling with one another, and until the poor man departs from the
path of prudence the millionaire does not find his oppprtimity.
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 million-
aires are often fortunate fools who have thriven on unfortunate
ones.^
11. In conclusion we may discuss a little further the concep-
tion 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 certaia 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 aU 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
beiag 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.
1 From the social point of view, however, this moral against gambUng 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.
OH. XXVI PHILOSOPHICAL APPLICATIONS 321
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 I'application du calcul des probaljilites a I'inoculation de
la petite v6role," ^ 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 difEer 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 prob-
ability, 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 foUow a course of conduct which may with the
slightest probability lead to an infinite good, imtil it is logically
disproved that such a result of our action is impossible. The
Emperor who embraced the Eoman Catholic religion, not because
^ Opuscules mathematiques, vol. ii.
2 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 advan-
tage ; 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 attain-
ment of salvation is of higher value than all the blessings of the world put
together. . . ."
Y
322 A TREATISE ON PROBABILITY pt. iv
lie believed it, but because it offered iusurance 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 re-
sembling them ia some particular. I am thinking of such argu-
ments as BufEon's when he names ^^^^^ as the limit, beyond
which probability is negligible, 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 fiity-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 iascribed 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 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 wiU generally lead to truth,
rather than falsehood, is at the best only probable. The pro-
position 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.
1 See p. 76.
OH. XXVI PHILOSOPHICAL APPLICATIONS 323
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 afEorded only the TwiUght, as I may so say, of Prob-
ability, 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
325
CHAPTER XXVII
. THE NATUEE OP STATISTICAL INFERENCE
1. The Theory of Statistics, as it is now understood,^ can be
divided into two parts wHcli are for many purposes better kept
distinct. The first function of the theory is purely desoriptive.
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 fimction 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 pre-
liminary 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
1 See Yule, Introduction to Statistics, pp. 1-5, for a very interesting account
of the eTolution of the meaning of the term statistics.
327
328 A TEEATISE ON PEOBABILITY pt. v
may take less pains over the transitional argument, which is
to permit him to use this description for the purposes of
generalisation.
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 imder
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 graphic-
ally. We then determine by some method of curve fitting a
mathematical frequency curve which passes with close approxima-
tion 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 observa-
tions 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
accoimt, 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 difficidt to
measure such degree of correspondence as may appear to exist
between the variations in two series, the terms of which are in
some manner associated in couples,— by coincidence, in this case.
0H.3:xvn STATISTICAL INFERENCE 329
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 in-
ductive scientist must take cognisance. He has recorded the
results of the observations in circimistances 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 iaductive generahsation.^ This tendency,
which has existed in some degree, as, I thiak, 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 supplement-
iag and improving a statistical description, and for the
purpose of indicating the precision of some generalisation.
The term ' correlation ' itseK is used both to describe an
observed characteristic of particular phenomena and in the
enunciation of an inductive law which relates to phenomena
in general.
3, I have been at pain^ to enforce this contrast between
statistical description and statistical induction, because the
chapters which foUow are to be entirely about the latter, whereas
nearly all statistical treatises are mainly concerned with the
former. My object wiU be to analyse, so far as I can, the logical
1 Cf. Whiteliead, Introdw^ion to Mathematics, p. 27 : " There is no more
common error than to assume that, because prolonged and accurate mathe-
matical calculations have been made, the application of the result to some fact
of nature is absolutely certain."
330 A TEEATISE ON PKOBABILITY pt. v
basis of statistical modes of argument. This involves a double
task. To mark down those which are iavalid amongst argu-
ments 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 ty^e of argument we seek to infer an imknown
statistical frequency from an d 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 fre-
quency. 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.
Each of these types of argument can be further compUcated
by being appHed not simply to the occurrence of a simple event
but to the concurrence imder given conditions of two or more
events. When this two or more dimensional classification re-
places the one dimensional, the theory becomes what is some-
times termed Correlation, as distinguished from simple Statis-
tical 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
CH. xxvn STATISTICAL INFERENCE 331
investigation goiag. In Chapter XXIX. the first type of argu-
ment, as classified above, is analysed, and the conditions which
are required for its vaHdity are stated. The crucial problem
of attacking the second and third types of argument is the
subject of my concluding chapters.
CHAPTEE XXVIII
THE LAW OF GREAT NUMBERS
Natiira quidem suas habet consuetudines, natas ex reditu causarum, sed non
nisi lis iirl ri iroXii. Novi morbi inundant subinde humanum genus, quodsi
ergo de mortibua quotounque experimeuta feceris, non ideo naturae rerum llmites
posuisti, ut pro future variare non possit. — Leibhiz in u letter to Bernoulli,
December 3, 1703.
1. It has always been known that, while some sets of events
invariably happen together, other sets generaVy 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 suf&cient for him to judge
it reasonable to be there. But this Irind 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 Ught a new fact
which had not been suspected previously — ^namely, that ia certain
cases of partial association the degree of association, i.e. the pro-
portion 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 imder 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,
332
cH.xxvm STATISTICAL INFERENCE 333
which, is not one of complete equality, tends everywhere, when
the number of recorded instances becomes large, to 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 ci priori probability is known throughout,
then (subject to certain conditions which he himself did not make
clear) in the long run a certain determiaate frequency of occurrence
is to be expected. Siissmilch [Die gottliche Ordnung in den
Veranderungen 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 ia particular."
Kant foimd in them (as many later writers have done) some
bearing on the problem of Eree 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,^ " sont soumises a une loi imiver-
seUe qu'on pent appeler la loi des grands nombres. . . . De ces
exemples de toutes natures, il resulte que la loi universelle des
grands nombres est A6]k pour nous un fait g6n6ral et incontestable,
r&ultant d'exp&iences qui ne se d4mentent jamais." This is
the language of exaggeration ; it is also extremely vague. But
1 In Idee zu einer allgemeinen Qeachichie in weltbiirgerlicher Absicht, 1784. For
a discussion of this passage and for the coimection between Kant and Siissmilch,
see liottin's Queielet, pp. 367, 368.
2 See p. 345.
" Becherches, 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 con-
tributions to Comptes Rendua 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.
334 A TREATISE ON PEOBABILITY m. v
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 d 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 d 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 ecole fran9aise." " Ma jeunesse
et mon zMe," he wrote in later years, " ne tard^rent pas a, me
mettre en rapport avec les hommes les plus distingues de cette
epoque ; qu'on me permette de citer Fourier, Poisson, Lacroix,
specialement connus, comme Laplace, par leurs excellents Merits
sur la thdorie mathematique des probability. . . . C'est done
an milieu des savants, statisticiens, et ^oonomistes de ce temps
que j'ai commenc6 mes travaux." ^ Shortly afterwards began
his long series of papers, extending down to 1873, on the apphca-
tion of Probability to social statistics. He wrote a text-book
on ProbabiUty 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 livelihood) a professorship of mathematics, Quetelet had studied
as an aft 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, pro-
jects, 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
^ For the details of the life of Quetelet and for a very fuU discussion of his
writings with special reference to Probability, see Lottin's Quetelet, statieticien et
cH.xxvin STATISTICAL INFERENCE 335
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 consider-
able 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 weU calculated to strike the imagination — ^the regularity of
the number of suicides, " I'efirayante 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 chimerique
et, par consequent, tout a 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 appHcations 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 Mendehsm
have opened out for them a great field of application throughout
biology.
1 Compare, for instance, the following passage from Becherches sur le penchant
au crime : " H me semble que ce qui se rattaohe k I'esp^ce humaine, consid&6e
en masse, est de I'ordre des faits physiques ; plus le nombre des individus est
grand, plus la volenti individuelle s' efface et laisse pr^dominer la seiie des faits
g6n6raux qui dependent des causes g6n6rales. . . . Ce sont ces causes qu'il
s'agit de saisir, et dte qu'on les oonnaltra, on en d6terminera les effets pour la
sooi6t6 comme on determine les effets par les causes dans les sciences physiques."
336 A TREATISE ON PROBABILITY pt. v
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 nwme/rous, the observed degree of
frequency is therefore stahh. Observation shows that some
statistical frequencies are, within narrower or wider limits, stable.
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 ia succes-
sive sets, gives the clue to the abstract inquiry upon which we
are about to embark.
CHAPTER XXIX
THE USE OF i PRIORI PROBABrLITIES FOR THE PEEDIOTION OF
STATISTICAL FREQUENCY — THE THEOREMS OF BERNOULLI,
POISSON, AND TCHEBYCHEFF
Hoc igitur est illud Problema, quod eTulgandum hoc loco proposui, post-
quam jam per vioennium pressi, et cujus turn novitas, turn summa utilitas cum
paii conjuncta diffloultate omnibus reliqnis hujus dootrinae capitibus pondus
et pretium superaddere potest. — Bebnottlli.*
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 parti-
cular case. But, as we shall see, the theorem is only vahd 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
a 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, and endeavour to
1 Ars Oonjeciandi, p. 227.
' In the simplest cases, dealt with by Bernoulli, these probabilities are all
supposed equal.
337 Z
338 A TREATISE ON PEOBABILITY m. v
determine the proper limits witMn which each method has
validity.
2. Bernoulh'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 probabihty
that the proportion of the event's occurrences will diverge from
the most probable proportion p by less than a given amount 6,
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 -m times out of the m occasions is (subject to certain conditions
to be elucidated later) j?"?™"" multipUed by the coefficient of
this expression in the expansion of (p + g')™, where p + q = \. If
n '
we write n = mp-h, this term is 7-—^ tt-.P^Q^''^- It
{mp-h)'.{mq+h)l^
is easily 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 approxi-
mation is required. Provided that m is large, we can simplify
n '
the expression _ " rr-fP"?™"" by means of Stirling's
Theorem, and obtain as its approximate value
1 ^'
-e 2nvpq^
y/2irmpq
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 objec-
tion, which 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
* See, e.g., Bowley, Elements of Statistics, p. 298. The objeotion about to
be raised does not apply to these closer approximations.
OH. XXIX STATISTICAL mFERBNCE 339-
the value for h=0; while the expression under approximation
is unsyminetrical. It is easily seen that this want of sjonmetry
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 iu § 15.
" Par ime fiction qui rendra les calculs plus faciles " (to quote
Bertrand), we now replace the iuteger A by a continuous variable
z and argue that the probabiHty that the amount of the diverg-
ence from the most probable value m^ will lie between z and z + dz,
is
1 ^'
2m^q dz.
A^27rmpq
This ' fiction ' will do no harm so long as it is remembered that we
are now dealing with a particular kind of approximation. The
probabiHty that the divergence h from the most probable value
mp will be less than some given quantity a is, therefore.
1 /"+» _51_
If we put =t, this is equal to
s/'^mpq
\/2mpg
Thus, if we write a = ijlm/pq 7, the probability ^ that the
number of occurrences will lie between
mp + ij'im.pq 7 and m/p - ^'im.pq 7
2 p
is measured by ^ — -=■ e dt. This same expression measures
1 The replacement of the integer 7i 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 proba-
hilites, chap. iv.). Some writers, using rather mere precision, give the result as
Mo
JttJ a ^fHirmpq
(e.g. Laplace, by the use of Euler's • Theorem, and more recently Czuber,
340 A TREATISE ON PROBABILITY ft v
the probability that the 'proportion of occurrences will lie
between
p+ \,/ — 7 and p- ^1 — 7.
^ m ^ m
2 r*
The different values of the integral —7== e"*"(^« = 6(«) are given
in tables.^
The probability that the proportion of occurrences will lie
between given Hmits varies with the magnitude of / , and
this expression is sometimes used, therefore, to measure the
' precision ' of the series. Given the d priori probabiUties, the
precision varies inversely with the square root of the number of
instances. Thus, while the probability that the absolute diverg-
ence will be less than a given amount a decreases, the probabiUty
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. BemouUi himseM was not acquainted with Stirling's
theorem, and his proof differs a good deal from the proof outhned
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
r
t, so that - is the probabihty of the event at each experiment,
V
then, given any degree of probability c, it is possible to make such
a number of experiments that the probability, that the propor-
tionate number of the event's occurrences wiU he between
r+l , r-1 . , „
and , is greater than c.^
Wahrscheinlichkeitsrechnung, vol. i. p. 121). As the -whole formula is approxi-
mate, the simpler expression given in the text is probably not less satisfactory in
practice. See also Czuber, Entwicklung, pp. 76, 77, and Eggenberger, Beitrdge
zur Darstellung des BernouUischen Theorems.
^ A list of the principal tables is given by Czuber, loc. cit. vol. i. p. 122.
^ Ars Gonjectandi, 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, Theorie analytigue, livre ii. chap. iii. For an account of
Laplace's proof see Todhunter's History, pp. 548-553.
OH. XXIX STATISTICAL INFEEENCE 341
4. We seem, therefore, to have proved that, if the d priori
probabiUty of an event under certain conditions is p, the pro-
portion of times most probable d 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 un-
likely to differ widely from p. This amounts to the principle
which ElUs 1 and Venn have employed as the defining axiom of
probabiHty, save that if the series is ' long enough ' the proportion,
according to them, will certainly be p. Laplace ^ believed that the
theorem afforded a demonstration of a general law of nature, and
in his second edition pubUshed in 1814 he replaces ^ the eloquent
dedication, A NapoUon-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 r^sultat du calcul des probabilites, confirme
par de nombreuses et funestes experiences."
5. Such is the famous Theorem of BernoulH which some have
believed * to have a universal validity and to be applicable to all
' properly calculated ' probabiUties. 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 ex-
ception 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 probabihty of heads is ^ and the probability of
tails is I ; yet the probabihty of m heads and m tails in 2m tosses
"■ 091 the Foundation of the Theory of ProbabiUties : "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 d 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."
' Bssai philosophique, p. 53 : " On pent tirer du th^ordme pr^c^dent cette
consequence qui doit Stre regardee comme une loi g^nSrale, savoir, que les
rapports des effets de la nature, sout k, fort peu pres constans, quand ces effets
sont consid^r^s en grand nombre."
• ' Introduotion, 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 prob-
ability is increased, the greater the length we give to our series."
342 A TREATISE ON PROBABILITY m. v
is zero, and it is certain d priori that there wUl be either 2m
heads or none. Clearly Bernoulh'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 probabihty of one occurrence, p^ is the probability of r
occurrences running. Our discussion of the theorems of multi-
plication wiU 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 vaUd 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 alto-
gether 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 miUion, 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 milUon trials, the probabihty in the hght of this additional
knowledge that the event will occur on the next occasion is stiU
no more than one in a milhon.
Such a condition is very seldom fulfilled. If our initial prob-
abihty 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 ^ve a concrete instance of a case in which the
conditions for the application of BemouUi's Theorem are com-
pletely 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 initial probabihty. If we
2 fy
wish to employ the expression — = I e dt we are in a worse
position. For this is an approximate formida which requires for
its vahdity 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 :
OH. XXIX STATISTICAL INFERENCE 343
Let ^^ represent the statement that the event has occurred
on m out of n occasions and has not occurred on the others ; and
let ■^■^'h =p, where h represents our d 'priori data, so that f is the
d, priori probabiUty of the event in question. Bernoulli's Theorem
then requires a series of conditions, of which the following is
typical : m+i^n+ilm^n • h=jXjJh, 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 a priori probabihty before the first occasion.
Let us select one of these conditions for closer consideration.
If y^ represents the statement that the event has occurred on each
of r successive occasions, 2/r/^=«/r/2/r-i^-2/r-iA ^^^ so on, so
s=r
that y^/A= Ylyjy^_-Ji. Hence if we are to have y^jh^p^, we
s=l
must have yjys-i^=p for all values of s from I to r. But in
many particular examples ys/ys-i^ increases with s, so that
y^jh>p''. Bernoulh's Theorem, that is to say, tends, if it is
carelessly appHed, 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 probabihty of heads at
the first toss is \ ; but if heads 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 \. For the d 'priori
probabihty of its being a conjurer's penny, or otherwise biassed
so as to fall heads almost invariably, is not usually so infinitesim-
ally small as (1)"°°. We can only apply Bernoulh's Theorem
with rigour for a prediction as to the penny's behaviour over a
series of a thousand tosses, if we have d priori such exhaustive
knowledge of the penny's constitution and of the other con-
ditions of the problem that 999 heads miming would not cause
us to modify in any respect our prediction d priori.
7. It seldom happens, therefore, that we can apply Bernoulh'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 of the Theorem is not hkely to be legiti-
mate, and some degree of approximation will be involved in
utilising its results.
Not so infrequently, however, artificial series can be devised
344 A TEEATISE ON PEOBABILITY n. v
in wMch the assumptions of Bernoulli's Theorem are relatively
legitimate.^ Given, that is to say, a proposition %, some series
ttia^ . . . can be found, which satisfies the conditions :
(i.) cbjh = fflj/A . . . = a^/h.
(ii.) a,/as . . . a^ . . . h=ajh.
Adherents of the Frequency Theory of ProbabiUty, who use the
principal conclusion of Bernoulli's Theorem as the defining pro-
perty 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 appHcable. 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
probabihty of a in certain conditions c is | ' is not in general
equivalent, as has sometimes been supposed, to ' It is 500 to I
that in 90,000 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. BernoulH's Theorem supphes the simplest formula by
which we can attempt to pass from the d 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
BernoulH's Theorem involves two assumptions, one (in the form
in which it is usually enunciated) tacit and the other exphcit.
It is assumed, first, that a knowledge of what has occurred at
some of the trials would not affect the probability of what may
occur at any of the others ; and it is assumed, secondly, that these
probabilities are aU equal a priori. It is assumed, that is to say,
that the probabihty of the event's occurrence at the rth trial is
equal a priori to its probabihty at the nth. trial, and, further, that
it is unaffected by a knowledge of what may actually have
occurred at the nth trial.
A formtda, which dispenses with the expHcit assimiption of
equal d priori probabihties at every trial, was proposed by
Poisson,^ and is usually known by his name. It does not dispense,
1 In the discussion in Chapter XVI., p. 170, of the probability of a diverg-
ence 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 apphoation of
Bernoulli's Theorem is more legitimate than in the natural series.
^ Becherches, pp. 246 et seq.
CH. XXIX . STATISTICAL INFEEENCE 345
however, with the other inexphcit assumption. The difierence
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 vahd apphcation 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 draw-
ing, or of balls drawn from a series of urns, each containing black
and white balls ia the same known proportion. The typical
example for Poisson's Theorem is that of balls drawn from a seriefs
of urns, each containing black and white balls in diff&reni known
proportions.
Poisson's Theorem may be enunciated as follows : ^ Let s
trials be made, and at the Tith trial (\ = 1, 2 . . . s) let the prob-
abilities for the occurrence and non-occurrence of the event be
f^, % respectively. Then, if ='p, the probabiHty that the
s
number of occurrences m of the event in the s trials will lie
between the limits sf±}, is given by
P
P = :^\ e ^'sdx +
where k
ky/irs] Q ki,/TTS
X I
By substituting — -^=1 and ^=7, this maybe written
k;Js ks/s
in a form corresponding to that of Bernoulli's Theorem,^ namely :
The probability that the number of occurrences of the event
will lie between sp±yk^is given by
^TT J g hy/Trs
9. This is a highly ingenious theorem and extends the applica-
tion 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 char-
acterised by differing statistical frequencies ; no further com-
\ For the proof see Poisson, Recherches, loc. cit., or Czuber, WahrscheMich-
keitsrechnung, vol. i. pp. 153-159.
2 For the analogous form of Bernoulli's Theorem see p. 339 (footnote).
346 A TREATISE ON PROBABILITY «. v
plication 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 know-
ledge whether or not the first ball drawn was black afEects 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 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 probabihties 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 character-
istics 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 chamge 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 types.^
' Numerous instances could be quoted. To take a recent English ex-
ample, 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
resuU of any one throw or toss does not afiect, 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 con-
ditions.
^ The types which I distinguish under four heads (the BemouUian, the
CH.XXIX STATISTICAL INFERENCE 347
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 um, 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 o- is the number of balls and s the number
of drawings, he reaches the interesting conclusion (assuming that
0-, s and a--s are all large) that the probabiUty of the number of
black balls lying within given Hmits is the same as it would be
if the balls were replaced after each drawing and the mmiber
of drawings were s instead of s.
cr
In addition to the assumptions already stated. Professor
Czuber's solution appKes only to those cases where the limits, for
which we wish to determine the probabihty, are narrow compared
with the total number of black balls pa: Professor Pearson ^ has
worked out the same problem in a much more general manner,
so as to deal with the wTiole range, i.e. the frequency or prob-
ability of all possible ratios of black balls, even where s>p<r. The
various forms of curve, which result, according to the different
relations existing between p, s, and a, supply examples of each
of the different types of frequency curve which arise out of a
PoisBonian, and the two described above) Bachelier {Caloul des probabilites,
p. 155) classifies as follows :
(i.) Wben the conditions are identical throughout, the problem has uni-
formite ;
(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 independance ;
(iii.) When they vary in a manner which depends upon what has happened
at the earlier stages, it has connexiU.
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 objeo-
tions,^-or rather the value of the results is limited in the same way.
1 It is of no consequence whether the balls are drawn successively and not
replaced, or are drawn simultaneously.
a Loc. cit. vol. i. pp. 163, 164.
3 " Skew Variation in Homogeneous Material," Phil. Trans. (1895), p. 360.
348 A TKEATISE ON PROBABILITY pt. v
classification according to (i.) skewness or symmetry, (ii.) Kmita-
tion of range in one, both or neither direction ; and he designates,
therefore, the curves which are thus obtained as generalised prob-
ability curves. His discussion of the properties of these curves is
interesting, however, to the student of descriptive statistics
rather than to the student of probabihty. The most generaUsed
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 I 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 /i be then taken out, the probabihty that a given
proportion of these i^ balls will be black is independent of
the nurnber and the colour of the I balls originally d/rawn out. For,
he argues, iil+fj, balls are drawn out, the probabihty of a com-
bination, which is made up of I black and white balls in given
proportions followed by /jl balls, of which m are white and n black,
must be the same as that of a similar combination in which the
fi balls precede the I balls. Hence the probabihty of m white
balls in fi drawings, given that the I balls have already been
drawn out, must be equal to the probabihty of the same result,
when 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 I balls originally drawn is hnown and where it is not.
The /aci of their having been drawn in certain proportions, pro-
vided that only the total number drawn is known and the pro-
portions are unknown, does not influence the probabihty. Poisson
states in his conclusion that the probabihty is independent of the
number and colour of the I balls originally drawn. If he had
added — as he ought — ' provided the number of each colour is
^ "Zur Theorie der Stabilitat statistisoher Eeihen," p. 216, published in
the Skandinavisk Aktuarietidskrip for 1919.
" Loc. cit. pp. 231, 232.
CH.XXIX STATISTICAL INFEEENCE 349
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 in-
volved.
In the cases where Bernoulli's Theorem is appHed to practical
questions, the A 'priori probabiHty is generally obtained empiric-
ally by reference to the statistical frequency of each alternative
in past experience under apparently similar conditions. Thus
the d priori 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 dis-
cussed later. But for the purposes of this example let us assume
that the d priori probability has been calculated on this basis.
Thus the d priori probability p ( =-) oi. an event is based on
the observation of its occurrence r times out of s occasions on
which the given conditions were present. Now, according to
BemoulU's Theorem directly apphed, the probabUity of the
event's occurring n times running is ^j" or ( - I . But, if the
event occurs at the first trial, the probability at the second
■■ For an attempt to solve other problems of this type see Bachelier, Oalcul
des probabilites, chap. ix. (Probabilites 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
shaE have a given composition ?
* Cf. Yule, Theory of Statistics, p. 258 : " We are not able to assign an
d 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."
350 A TEEATISB ON PROBABILITY pt. v
■ , 1
becomes — -, and so on. Hence the probability P, properly
s + 1
calculated, of n successive occurrences is
r r+l r+2 r+n-1
s s+1 s+2 s +n- 1
{r + n-l)\ (s-1)!
(s+m-l)! (r-l)!
Theorem, provided that r and s are large ;
1 +
Hence
P =
r\"\ r
si I n-W+''-^
1 + —
1 + -
n,
=j)''Q", where Q=^^ ^T^+l
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 com-
pared with s. It is very important to notice that two conditions
are involved. Not only must the experience, upon which the
d, '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 ex-
perience, upon which we foimd the initial probability P, is very
extensive, we must not, if P is very small, say that the probabihty
of n successive occurrences is approximately ^™, 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 351
compared with s, n small compared with r, and m small compared
with 8-r.^
The case solved above is the simplest possible. The general
problem is as follows : If an event has occurred x times in the
T ■¥ X
first y trials, its probabiUty at the y + 1th is —■ — ; determine the
s+y
d priori probability of the event's occurring p times in q trials.
If the d priori probability in question is represented by (pip, q), we
, ,, , r+p-1 s + q-1-r-p
have <j>{p,q)= ^_f_^ (l>{p-l,q-l)+ — s + g-i 9iP>9-^)-
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 prob-
r .
ability of a proportion differing widely from - is greater than
s
in Bernoullian conditions ; 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
r
tend back towards - (as when we draw balls without replacing
s
them from a bag of known composition), we should have sub-
normal 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 argument is as foUows :
In the period 1866-1877 there were registered in Austria
m= 4,311,076 male births
ji= 4,052,193 female births
s = 8,363,269;
1 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.
2 Bachelier (CaUul des probabilitea, p. 201) classifies these two kinds of con-
ditions as conditions acceUratrices 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.
352 A TREATISE ON PROBABILITY w. v
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
Wo' = — = 6,141,587,
m
and that there is a probability P = -9999779 that vl will lie
between the Umits 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
P = -9999779 = 1 --7—rr7r) to estimate the number of female
45250/
births within such narrow Umits. And we see that the con-
ditions laid down in § 11 have been flagrantly neglected. The
number of cases, over which the prediction based on Bemoulh's
Theorem is to extend, actually exceeds the number of cases upon
which the a priori probabihty has been based. It may be added
that for the period, 1877-1894, the actual value of n' did lie
between the estimated hmits, 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
beheving that 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 <i priori probabihty 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 d priori probability.
But this question belongs to the subject-matter of later chapters.
OH. XXIX STATISTICAL INFEEENCE 353
and, quite apart from the manner of calculation of the d priori
probability, the argument is invahdated by the fact than an
d 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 BemouUi and Poisson,
it is necessary to call attention to a very remarkable theorem by
TchebychefE, 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 : ^
Let x,y,z. . . represent certaia magnitudes, of which x
can take the values XjX^ . . -x^. with probabilities p^^ • • -Pk
respectively, y the values y^y^ ■ • -Vi "^^^ probabilities q^q^ . . .qi,
z the values z^z^ . . .z^ with probabilities r-^r^ . ■ -r^ and so on,
so that
k I m
'Zp = l, 'tq = l, 2r = l, etc.
Ill
k I m
Write tp^x^ = a, tq>y^ = 'b, 'Zr^z^=c, etc.,
1 1 1
k I ™
and tp^xj' = aj^, tq^^ = K tr^z^ = c, etc.,
1 1 1
so that we can describe a as the mathematical expectation or
average value of x, and % as the mathematical expectation or
average value of a;^ etc.
The probability that the sum x-vy-k-z-v ... will have for
its value x^-^y^+z^^- . . . is p^q,?^--- (provided that the
values oix,y,z.. . are independent). Hence
1 lYom Jaarn. Liouville (2), xii., 1867, " Des valeurs moyennes," an article
translated from the Russian of TchebychefE. This proof is also quoted by
Czuber, loc. cit. p. 212, through whom I first became acquainted with it. Most
of TchebyohefE's work was published previous to 1870 and appeared originally
in Russian. It was not easily accessible, therefore, until the pubUoation at
Petrograd in 1907 of the collected edition of his works in Prench. 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.
2a
354 A TREATISE ON PROBABILITY m. v
%.+yA+«^+ ■•• -a-b-c- ...fpAK'^^---
summed for all values of «, \, /i is the average expectation for
K+yA + 2^ + --- -a-h-c- ...f.
k
Now X(a;^^ - 2aa;„ + a^)p^ = Sp^xJ^ - iatp^^ + oFtp^
= «! - 2a^ + a^ = «! - a^.
Also Sg';tr^ . . . = 1, summed for all values oi\, fi. . ., and
h h
t2.{x, - a){y^ - h)p, = t2{x^^ - bx, - mj^ + ab)p^
1 1
= 2(^x22? A - S^JP A - ay^^'/c + «&5:^ J
= 2{ai/)^ -ah- ay^ + a&) = 0.
Therefore t^x^ +y^-¥zji + . . .-a-b-c- .. . fp^q^r^ . . .
= «! + &i + Ci + . . . - a^ - 6^ - c^ - . . .,
whence ^('^'' +yx + g^ + ----Q'-^-c-- O^j^.g^^'V ■ • ■ 1 ^
where the summation extends over all values of k,\, fi. . . and
a 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
{x,+y>, + z^ + ... -a-b-c-... )^
a2(ai + &i + Ci-l-...-a2_&2_c2_ . . .)
and write unity for this expression in the remaining terms, both
these processes diminish the magnitude of the left-hand side.
Hence ^p^q>,r^. . ■<~^ where the summation covers those
of values only for which
(x, + y^+Zu. + ... -a-b-c... )^
51.
a%ai + bi+c.t^ + ...-a^-¥-c^...)
If P is the probability that
{x^+y^ + «^ + ... -a-b-c... f
a\ai + b^+Ci + ...-a^-b^-c^-...)
is equal to or less than unity, it follows that
OH. -X-IfTX
STATISTICAL INFEKENCE
i-p<i.
a?
i.e.
p>i-i.
Hence the
probability that the snm
a +
6 + C + .
1+C + .
^.+yK^\^---
lies between the
, . . - a ^flSj + 61 +Ci + . .
.-a2-62.
-c^-...
and
. . . + a. J a-. + &i + Ci + . .
.-a2_&2-
-c^-...
355
is greater than 1 — ^, where a is some number greater than
imity.
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 a =-?!_; then the probabiUty that the arithmetic
mean ^ lies between the hnuts
a + b+c + . . . 1 laj^ + bi + Ci + . . . a^ + b^ + c^ + . . .
n t\/ n n
is greater than 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\^ = 1 - ?] occurs and an amount B lost when it fails, then in
s trials the probabihty that the total winnings (or losses) will lie
between the Umits
s{'pA. - qS)±a{A. + B) yfspq
is greater than 1 — 5-
a
14. From this very general result for the probable limits of
a sum composed of a number of independently varying magni-
tudes, Bernoulli's Theorem is easily derived. For let there be
^ For a proof see Czuber, he. cit. vol. i. p. 210,
356
A TREATISE ON PROBABILITY
s observations or trials, and s magnitudes XjX2 . . .Xg corre-
sponding, such that x = \ when the event under consideration
occurs, and a;=0 when it fails. If the probability of the event's
occurrence is p, we have a=jp, 6=|), etc., and 01=3?, ii=f, etc.
Hence the probability P that the number of the event's occur-
rences wiU lie between the limits sp±a^sp-sp'^, i.e. between
the limits sp±ay/spq where q = l-p, is >!-— . If we
a
compare this formula with the formula for Bernoiilli's
Theorem already given, we find that, where this formula
gives P>1 — ^, Bernoulli's Theorem with greater precision
gives P=©(— T^j. The degree of superiority in the matter
of precision supplied by the latter can be illustrated by the
following table :
a2.
a
-i-
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 a is small, BemoulU's
formula gives a value of P very much in excess of 1 — -. But
Bernoulli's formula involves a process of approximation which is
only valid when s is large. TchebychefE's formula involves no
such process and is equally vaUd for all values of s. We have
seen in § 11 that there are numerous cases in which for a
different reason BernoulH's formula exaggerates the results,
and, therefore, TchebychefE's more cautious limits may some-
times prove useful.
The deduction of a corresponding form of Poisson's Theorem
from TchebychefE's general formula obviously follows on similar
lines. For we put^ «=i'i> ^=P2> etc., and ai=Pi, bi=P2, etc.,
1 I am using the same notation as that used for Poisson's Theorem in § 8.
OH. XXIX STATISTICAL INFERENCE 357
and find that the probability that the number of the event's
occurrences will lie between the limits
A. /A \
1 11
-.a V 2
i.e. between the limits sp±,a s/ "Zp^q^,
i.e. between the limits sp ± ij2ak ^s,
is greater than t — r-
In CreUe's Journal'^ TchebychefE proves Poisson's Theorem
directly by a method similar to his general method, and also
obtatQS several supplementary results such as the following :
I. If the chances of an event E in /* consecutive trials are
PjP^ ■ ■ -P^ respectively, and their sum is s, the probability that
E will occur at least m times is less than
— /
n-s)S/
'm{fi-m)fsYf fi-sV-'^+''
2{in-s)'\ fj, \fj,/ \/ji,-mJ
provided that m>s + l;
II. and the probability that E will not occur more than n times
is less than
5 - n)\/ fjL
provided that n< s-1.
III. Henc6 the probability that E will occur less than m times
and more than n is greater than
1 M(/"' -
n - s)\/ fi
-m)/sWA'-s V"™+^
2(m-s)\/ A* V"''/ V/*
2(s-m)V fi \n) \fjt'-
m)/sY+Y//,-sY'''
provided m>s + l, n<s-l.
15. Tchebycheff's methods have been set out and his results
admirably extended by A. A. MarkofE.^ And some develop-
* Vol. 33 (1846), Demonstration elemenlaire d'une proposition genercUe de la
theorie des probabilites.
' The reader is referred to Markoff's WaJirscheinlichkeitsrecJmung, and par-
ticularly to p. 67, for a striking development, along mathematical lines, of
358 A TEEATISE ON PEOBABILITY pt. v
ments along the same lines by Tschuprow (" Zur Theorie der
Stabilitat statistischer Reihen," Skandinavisk AMuarietidskrift,
1919) have convinced me that TchebychefE's discovery is far
more than a technical device for solving a special problem, and
points the way to the fmidamental method for attacking these
questions on the mathematical side. The Laplacian mathe-
matics, although it stiU 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 unsym-
metrical. The fact, that the usual approximation for the prob-
ability of a divergence h from the most probable number of
occurrences (the notation is that of § 2 above) takes the form
1 ^^
7 e '^m>s, which is the same for +A 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 diverg-
ence 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 much more likely (about 5 times as likely) not to appear at
all than to appear exactly 20 times. That this must be so wiU
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 smaUness 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
TchebychefE's leading idea. Further references to later memoirs, which, being
in the Russian language, are inaccessible to me, wiE be found in the Biblio-
graphy.
OH. XXIX STATISTICAL INFEEENCE 359
frequency of any given defection as compared with the corre-
sponding excess. Thus the actual frequency in a series of trials
of an event, of which the probability at each trial is less than ^,
is likely to fall short of its most probable value more often than
it exceeds it. What is ia fact true is that the mathematical
expectation of deficiency is equal to the mathematical expecta-
tion 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 pubhshed 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 Mr.
Simmons's article. His principal theorem ^ is as follows : If
r is the probability of the event at each trial and n{a + 1) the
number of trials, n and a beiag integers,^ the probabihty 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 = l, constantly
diminishing as n increases, lying always between times the
. / a 1 Y^''+^'> , la-1 .
greatest term m H and ^ — - times the
^ \a-l-l a + 1/ Ba + l
^ " A New Theorem in Probability." Mr. Simmons claimed uoTelty 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 fre-
quency, 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 (Abhand-
lungen, p. 120).
* 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.
360
greatest term in
A TREATISE ON PROBABILITY
I \(7i+l)(a+l)
■+-
, and being approxi-
1 a-1
\a + l a + 1
mately equal, when n is very large, to
The following table gives the value of the excess A of the
probability of a frequency less than pm ovei; the probabiHty of
a frequency greater than pm for various values of p the prob-
ability and m the number of trials
calculated by Mr. Simmons :
p = — r, m=n{a + l)
as
P'
m.
A.
1
3
3
•037037
1
3
15
•02243662
1
3
24
•0182706
1
4
4
•034687
1
i
20
•03201413
1
10
10
•084777
1
10
20
•068673713
1
100
100
•101813
1
ioo
200
•081324387
1
1000
1000
•103454
Thus unless not only m but mp also is large the want of symmetry
is Ukely to be appreciable. Thus it is easily found that in 100
sets of 4 trials each, where p = j, the actual frequency is likely to
exceed the most probable 26 times and to fall short of it 31 times ;
and in 100 sets of 10 trials each, where ?> = tt;; to exceed 26 times
and to faU short 34 times.
Mr^. Simmons was first directed to this investigation through
OH. XXIX STATISTICAL INFEEENCE 361
noticing in the examination of sets of random digits that " each
digit presented itself, with unexpected frequency, less than — of
the number of times. For instance, in 100 sets of 150 digits each,
I f oimd that a digit presented itseK 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 statis-
tician's while to bear in mind this appreciable want of symmetry
in the distribution about the mode or most probable value in
many even of those cases in which BemouUian conditions are
strictly fulfilled.
17. I will conclude this chapter by an account of some of the
attempts which have been made to verify d posteriori the con-
clusions of Bernoulli's Theorem. These attempts are nearly
useless, first, because we can seldom be certain d priori that the
conditions assumed in BemouUi'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 BemouUi'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
BemouUi's Theorem seem to be fulfilled most nearly. The earliest
recorded experiment was carried out by BufEon,^ who, assisted
^ Mr. Yule (Introchiction to Statistics, p. 254) recommends its indulgence :
" The student is strongly recommended to cany out a few series of such ex-
periments personally, in order to acquire confidence in the use of the theory."
Mr. Yule himself has indulged moderately.
> Essai d^arithmetigue morale (see Bibliography), published 1777, said to
have been composed about 1760.
362 A TREATISE ON PROBABILITY m. v
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.' ^ In Bitffon'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 nmnber 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 pubhshed in 1850
and the latest in 1893.^ In his first set of experiments Wolf
completed 1000 sets of tosses with two dice, each set continviing
mitil 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
foUowing, namely a proportion of -83533 of unlike pairs, as against
5
the theoretical value -83333, i.e. -. In his second set of experi-
^ Formal Logic, p. 185, published 1847. De Morgan gives Buffon's results,
as well as his pupil's, in fuU. Buffon's results are also investigated by Poisson,
Becherches, pp. 132-135.
2 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, Inirochiction to Statistics, p. 254.
' See Bibliography. Of the earliest of these investigations I have no first-
hand iaiowledge 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, Wahrscheinlichlceitsrechnung, vol. i.
pp. 139-152, and Czuber, Entwicklung der Wahrscheinlichkeitstheorie, pp. 88-91.
OH. XXIX STATISTICAL INFERENCE 363
ments WoK 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 ia the VierteTjahrsschrift der
Natwforsohenden Gesellschafi in Zurich. 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 ex-
planation 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 experi-
ments, — ^that Wolf's dice were very ill made. Indeed the ex-
periments 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 fife
280,000 results of tossing individual dice. It is not clear that
WoK 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
ready-made data provided by the published results of lotteries
and roulette.^
^ Czuber quotes the principal results (he. 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 (" Wiirfelversuche," Zeitscknflfur Math, und Phys. vol. 62 (1.913), pp.
149-156), who recorded 24 series of 180 throws each with four distinguishable
dice.
^ Tor the publication of such returns there has always been a sufficient
demand on the part of gamblers. An Almanack romain sur la loierie royale de
France was published at Paris in 1830, which contained aU 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
364 A TREATISE ON PROBABILITY pt. v
Czuber ^ has made calculatioDS based on the lotteries
of Prague (2854 drawings) and Briinn (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 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 equi-
probability of aU 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 ci ^iori 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
Theorem shows that the proportionate deviation decreases while the absolute
deviation increases. Cf. Houdin's Les Trickeries des Qrecs devoiUes : " 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 east or turn-up.
This is the most elementary of the theories on probabilities ; it is termed the
maturity of the chances." Laplace (Essai philosophdqiie, p. 142) quotes an
amusing instance of the same beUef not drawn from the annals of gambling :
" J'ai vu des hommes d&irant ardemment d'avoir mi fils, n'apprendre qu'avec
peine les naissances des gar9ons dans le mois oil ils aUaient deyenir pdres.
S'imaginant que le rapport de oes naissances k oeUes des filles devait Stre le
mSme i la fin de chaque mois, ils jugaient que les gardens d6ji. n6s rendaient
plus probables les naissances prochaines des fiUes."
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 MorUe Carlo Facts and Fallacies.
'^ Zum Oesetz der grossen Zahlen. The results are summarised in his Wahr-
scheinlichkeitsrechnung, vol. i. p. 139.
2 Kollektivmasslehre, p. 229. These results also are summarised by Czuber,
loc. cit.
^ The Chances of Death, voL i.
OH. XXIX STATISTICAL INFEEENCE 365
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 intro-
duces this experimental result in support of his thesis that the
world is so constituted that long ruas do not as a matter of fact
occur in it.^ Not merely are long runs very 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 d priori and, as a
general principle, there is no adequate evidence to support it from
experience. Its origLa is to be found, perhaps, ia the fact that
* Naturphilosophische Untersuchungen zur Wahrscheinlichkeitstheorie.
' Dr. Marbe's monograph has given rise in Germany to a good deal of dis-
cussion, not directed towards showing what a preposterous method this is for
demonstrating a natural law, but because the experimental result itsdf 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 A priori of the various sequences. The problem is discussed by
Von Bortkiewicz, Bromse, Bruns, Grimsehl, and Griinbanm (for exact references
to these see the BibUography), and by Lexis {Abhandhmgen, pp. 222-226) and
Ozuber (WaJM-ecJieinlichkeitsrechnung, 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 mathematigues (1761). Works on Probability
usually contain some reference to D'Alemberfc, but his sceptical opinions, re-
jected 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. 316). The second is connected with the first, and
was also used to support his incorrect evaluation of the probabihty 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 sceptioaJ doubts the hardly less iucorreot con-
clusions of the orthodox theory of that time.
366 A TEEATISE ON PROBABILITY pt. v
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-caUed random events some slight degree of causal and material
dependence between successive occurrences may, nevertheless,
exist. In these cases ' rims ' 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 experi-
ments as these this is the only hind 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 prob-
able degree of regularity of habit on the part of the spinner could be sufSoient
to produce regularity in the result.
CHAPTER XXX
THE MATHEMATICAL USB OP STATISTICAL FREQUENCIES FOE
THE DETERMINATION OP PROBABILITY A POSTMRIORl — THE
METHODS OP LAPLACE
Utilissima est aestimatio probabilitatum, quanquam in exemplis juridiois
politioisque plerumque non tam subtili oalculo opus est, quam acourata
omnium ciroumstantiarum 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 probabihties of the various possible
frequencies of the event over the whole series, without discussing
in detail by what method the initial probability had been deter-
mined. In statistical inquiries it is generally the case that this
initial probability is based, not upon the Principle of In-
difEerence, 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 found-
ing judgments of probabihty 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 mathe-
matical charlatanry by which, for a hundred years past, the basis
of theoretical statistics has been greatly undermined,
367
368 A TREATISE ON PROBABILITY pt. v
2. Two direct methods have been commonly employed,
theoretically inconsistent vdth 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 deter-
mine from d 'priori considerations, he points out, by how much it
is more probable that we shaU throw 7 rather than 8 with two dice,
but we cannot determine by such means the probability that a
young man of twenty wiU outHve an old man of sixty. Yet is it
not possible that we might obtain this knowledge d 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 asymp-
totically to some determinate degree of probability with the
increase of observations ? Nesdo, Vir AmpUssime, an specula-
tionibus istis soliditatis dUquid inesse Tibi videatur.
Leibniz's reply goes to the root of the difficulty. The calcula-
tion of probabilities is of the utmost value, he says, but in statisti-
cal inquiries there is need not so much of mathematical subtlety
as of a precise statement of all the circumstances. The possible
contingencies are too nimierous to be covered by a finite number
of experiments, and exact calculation is, therefore, out of the
quesition. Although nature has her habits, due to the recurrence
of causes, they are general, not invariable. Yet empirical calcula-
tion, although it is inexact, may be adequate ia affairs of practice.^
■■ For the exact references see Bibliography.
* Leibniz's actual expressions (in a letter to Bernoulli, December 3, 1703) are
as f oUows : Utilissima est aeatimatio probabilitatum, quanquam in exemplis
juridicis politicisque plerumgue non tarn subtili calculo opus est, quam accurata
omnium circumstantiarum enumeratione. Cum empirice aestimamus proba-
OH. XXX STATISTICAL INFEEENCE 369
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
a posteriori as lying between 201 : 100 and 199 : 100. " Certus
sum," he concluded the controversy, " Tibi placituram demonstra-
tionem, 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 probabiUties a 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 Ber-
noulli, I have not been mainly influenced by the historical interest
of it. The view of Leibniz, dweUing mainly on considerations
of analogy, and 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 exphcit use of the Inversion of Bemoulh's
Theorem. The investigations carried out by D'Alembert, Daniel
BernoulH, and others rehed 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
bilitates per experimenta suooessuum, quaeris an ea via tandem aestimatio
perfeote obtineri possit. Idque a Te repertum soribis. Difficultas in eo mihi
inease videtur, quod oontingentia seu quae infinitis pendent ciroumstantiis, per
finita experimenta determinari non possunt ; natura quidem suas habet oonsue-
tudines, natas ex reditu causarum, sed non nisi us ^Tri to ttoM. Novi morbi
inundant subinde humanum genus, quodsi ergo de mortibus quotounque ex-
perimenta feoeris, non ideo naturae rerum limites posuisti, ut pro futuro variare
non possit. Etsi autem empirice non posset haberi perfeota aestimatio, non
ideo minus empirioa aestimatio in praxi utilis et sufficiens foret.
^ The relevant passages are on pp. 224-227 of the Ars Conjectandi.
2b
370 A TEEATISE ON PROBABILITY pt. v
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 probabihty in its favour. But they
did not endeavour to pass from the observed frequency of occur-
rence 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 BernouUi's Theorem. Bernoulh's Theorem, in the form in
which Laplace proved it, states that, if j? is the probability d
m
priori, there is a probability P that the proportion of times — —
of the event's occurrence in fjt,{=m + n) trials wiU lie between
p±j 1^, where P==-^fV'^&+ , ^ - e-y\ The in-
version of the theorem, which he assumes without proof,
states that, i£ the event is observed to happen m times
in /x trials, there is a probability P that the probability
. , .,,,., m l2mn
01 the event p will he between —±7 / — ;-, where
,1, '\/ /J?
P = — -= e'*^dt + — e~'>'^ The same result is aleo given
J^
1^'
by Poisson.^ Thus, given the frequency of occurrence in fi
trials, these writers infer the probability of occurrence at
subsequent trials within certain Hmits, just as, given the
d priori probabihty, Bernoulli's Theorem would enable them
to predict the frequency of occurrence in /a trials within corre-
sponding limits.
1 For an account of the treatments of this topic both by Laplace and by
Poiason, 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 Bule 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 mathe-
matical methods by which this ctuestion has been approached, and this minor
point, which is of historical interest mainly, I forbear to enter into.
OH. XXX STATISTICAL INFERENCE 371
If the number of trials is at alii numerous, these limits are
narrow and the purport of the iaversion 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 + n
AM
measures the frequency, also measures the most probable
value of the probability. The simplicity of the process has re-
commended it, since the time of Laplace, to a great niunber of
writers. Czuber's argument, criticised on p. 351, with reference
to the proportions of male and female births in Austria, is based
upon an imqualified 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
incapable of it, unless some illegitimate assumption has been
introduced. But, apart from this, there are some obvious objec-
tions. We have seen in the preceding chapter that Bernoulli's
Theorem itself cannot be applied to all Mnds of data indiscrimin-
ately, but only when certain rather stringent conditions are ful-
filled. 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 occur-
rence 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 imexamined
instances to which we intend our conclusion to be applicable.
An. imanalysed statement of frequency cannot tell us this.
This method of passing from statistical frequencies to prob-
abilities is not, however, hke 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 appHcations of
it are no longer ventured upon, but the theory which underlies
it is still widely adopted, especially by French writers upon
probabihty, and seldom repudiated.
372 A TEEATISE ON PROBABILITY pt. v
5. The f omiula in question, which Venn ^ has called the Rule
of Siuscession, declares that, if we know no more than that an
event has occurred m times and failed n times under given con-
ditions, then the probability of its occurrence when those con-
ditions are next fulfilled is -. It is necessary, however,
m+n + 2 ■'
before we examine the proof of this formula, to discuss in detail
the reasoning which leads up to it.
This preliminary reasoning involves the Lapladan theory of
' unknown probabilities.' The postulate, upon which it depends,
is introduced to supplement the Principle of Indifference, and
is 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 probabilite d'un 6venement
simple est inconnue, on peut lui supposer egalement toutes les
valeurs depuis zero jusqu'a I'unite. La probabilite de chacune
de ces hypotheses tiree de I'evenement observe est . . . tme
fraction dont le numerateur est la probability de I'eveUement dans
cette hypothese, et dont le denominateur est la somme des pro-
babilites semblables relatives k toutes les hypotheses. . . ." ^
Thus when the probability of an event is unknown, we may
suppose all possible values of the probability between and 1 to
be equally likely d 'priori. The probabiUty, after the event has
occurred, that the probability a priori was - (say), is measured
1 . ^
by a fraction of which - is the numerator and the sums of all the
■' r
possible a 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 and 1 as the d priori
probabihty of drawing a white ball. If we could assume that
these constitutions are equally probable a priori, we should
obtain probabilities for each of them d posteriori according to
Laplace's rule.
^ Logic of Chance, p. 190. " Esaai phihso'phigue, p. 16.
STATISTICAL INFEEENCE 373
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 hkely, and each of which
leads to the event in question with a different degree of probability,
so that for every value between and 1 there is one and only one
hypothetical constitution of things, the assumption of which
invests the event with a probabihty 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, ia the first place, does Laplace mean by an miknown
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 nimierical value and that, in those
cases where there seems to be no numerical value, this value is
not non-existent but imknown ; 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 and 1 is equally
probable. With the possible interpretations of the term ' un-
known probability,' and with the theory that every probabihty
can be measured by one of the real numbers between 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 HI., but in a manner favourable to
Laplace's theory, it remains doubtful whether we could legitim-
ately attribute a value to the probabihty of an unknown prob-
abihty's having such and such a value. If a probabihty is
unknown, surely the probabihty, 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 emplojnnent of both of two inconsistent
methods. Let us consider a number of alternatives %, a2, etc.,
having probabihties pi, ^2> ^tc. ; if we do not know anything
about ai, we do not know the value of its probability p^^, and we
374 A TEEATISE ON PEOBABILITY pt. v
must consider the various possible values of f^, namely \, \, etc.,
the probabilities of these possible values being q^., q^, etc. respect-
ively. There is no reason why this process should ever stop.
For as we do not know anything about b^, we do not know the
value of its probability q^, and we must consider the various
possible values of g'j, namely Cj, Cg, etc., the probabihties of these
possible values being r^, r^, 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 probabUity 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 probabihties
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 Oj, since a^ and its contradictory form a pair
of exhaustive alternatives two in number, the probability of these
alternatives is equal and each is -. But in the reasoning 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 a^, its prob-
ability Pj^ may have any of the values b^, b^, etc. where b^ is any
fraction between and 1 ; and, as we know nothing about the
probabilities q^, q^, etc. of these alternatives b^, b^, 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
Eule of Succession, it will be seen that the hypothetical a priori
probabilities 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 and 1 ; and that these fall into equal
groups, each group corresponding to one of the real numbers
OH. XXX STATISTICAL INFERENCE 375
between and 1, this number measuring the degree of probability
with which we could predict the event, if we knew that an ante-
cedent condition belonging to that group was fulfilled. It is
then assumed that all of these possible antecedent conditions are
d 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 nimiber of black and white balls. But for the assump-
tion that we have in general the kind of knowledge which is
necessary about the possible antecedents, no reasonable founda-
tion 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 happen-
ing of an event a number of times which Ues 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 imme-
diately made the expression of the result of a set of circumstances."
It cannot be allowed that this is generally tfue ; ^ and even in
those cases in which it is true we are thrown back on the d 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 a 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 a!"(l-a;)''. If,
however, x is unknown, all Values of it between and
1 Cabinet Encyclopaedia, p. 87.
'^ For instance, it is not true even in the standard instance of balls drawn from
an urn containing black and white in unknown proportions, unless the number
of balls is infinite.
376 A TREATISE ON PROBABILITY pt. v
1 are d 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 d posteriori that
X lies between x and x + dx is proportional to a;™(l - x)'^dx,
and is equal, therefore, to Aa;™(l -xfdx 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
A.™(1-.)V. = 1 .•.A=-i>±^.
5 ^ r(m + i)r{7!, + 1)
Hence the probability that the event will occur at the (m+n + l)th
trial, when we know that it has occurred m times in m + n
trials, is
aI x'^+\l-x)''dx.
J
If we substitute the value of A fotind above, this is equal to
m + 1 1
m + n + 2
The class of problem to which the theorem is supposed to
apply is the following : There are certain conditions such that we
are ignorant d priori as to whether they do or do not lead to the
occurrence of a particular event ; on m out oi 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 -. In the cases where
m+n+2
n=0, i.e. when the event has invariably occurred, the formula
1 The theorem is sometimes enunciated by contemporary writers in a much
more guarded form, e.g. by Czuber, WahrscJieinUchkeitsrechnuTig, vol. i. p. 197,
and by Baohelier, Ccdcul des probabilites, p. 487. Bachelier, instead of assuming
that the d priori probabilities of all possible values of the probability of the
event are equal, writes ui{y)dy as the d priori probability that the probability is
y, so that after m occurrences is m+?i trials the probability that the probability
lies between y and y + dy is "^ ' - — w-f,^X. ' ^^ °^^ ^^^ ^° ^^^ of ffi d
priori, he suggests that the simplest hypothesis is to put fi = l, which leads, as
above, to Laplace's Law of Suocesmon. He also proposes the hypothesis
Ci>{y) = a + a^ + Ogj/" + .... in which case the denominator is a series of Eulerian
integrals. There is a discussion of the Law of Succession, and of the contra-
dictions and paradoxes to which it leads, by E. T. Whittaker and others in
Part VL vol. viii. (1920^ of the Transactions of the FtKvMy of Actuaries in
Scotland.
STATISTICAL INFEEENCE 377
Wl "4" 1
yields the result -. In the case where the conditions have
m + 2
been observed once only and the event has occurred on that
2
occasion, the result is -. If the conditions have never been met
1
with at all, the probabiUty of the event is -. And even in the
case where on the only occasion on which the conditions were
observed, the event did not occur, the probabihty is -•
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 d priori probability of the event's happen-
ing once, then aj" is the a 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 prob-
ability of its happening a second time ; its successive occurrences
are not, therefore, independent. If the d, priori probabihty of the
event is -, and if, after it has been observed once, the probability
. . . 2 .
that it win occur a second time is -, then it follows that the a
1112
priori probabihty of its occurring twice is not „ x „, but ^ x ^,
i.e. - ; and in general the d, priori probabihty of its happening
■ • • ■ /iV, 1
n times m succession is not -■ but
,2/ n + l
10. But refinements of disproof are hardly needed. The
principle's conclusion is inconsistent with its premisses. We
begin with the assumption that the d priori probabiUty of an event,
about which we have no information and no experience, is un-
known, and that all values between and 1 are equally probable.
We end with the conclusion that the d priori probability of
such an event is -. It has been pointed out in § 7 that this
A
contradiction was latent, as soon as the Principle of Indifference
was superimposed on the principle of unknown probabihties.
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
378 A TEEATISE ON PROBABILITY pt. v
whatever complexity, in favour of which no positive argument
exists, the hke 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 - ? Or if we do suppose it,
we are involved ia contradictions, — ^for 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 ia 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 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 corre-
sponding, 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 Probabihty,^ 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
1 This second condition is often omitted {e.g. Bertrand, Galcul des proba-
bilites, p. 172).
2 The correct solution for the case of a finite number of bails, on the hjrpo-
thesis that each possible ratio is equally likely, is as foUows : The probability
of a black ball at a further trial, after black balls have been successively with-
1 3
drawn and replaced p times, is - -^^ where there are ii, balls and s, represents
the sum of the rth powers of the first n natural numbers. This reduces to
^— g,— the solution usually given,— when n is infinite. More generally, if
p black balls and q white balls have been drawn and replaced, the chance
r=n
that the next baU will be black is -
1 r=o
3 See Chapter VIII.
OH. XXX STATISTICAL INFERENCE 379
explicitly rejected on the ground that it does not accord with
experience.^ But Professor Karl Pearson, who accepts it, has
made the necessary restateinent,^ 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 hjrpothesis of the equal dis-
tribution of ignorance is, within the hmits of practical Hfe, justi-
fied by our experience of statistical ratios, which d priori are
unknown, i.e. such ratios do not tend to cluster markedly round
any particular value. ' Chances ' lie between 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 judg-
ment of probable future statistical ratios. . . .
" Let the chance of a given event occurring be supposed to lie
between x and x + dx, then ii on n=p + q trials an event has been
observed to occur p times and fail q times, the probability that
the true chance hes between x and x + dx is, on the equal
distribution of our ignorance,
^ xP{l-xydx
f x^l ■
J
- xfdx
" This is Bayes' Theorem. . . .*
^ Logic of Chance, p. 197.
* " On the Influence of Past Experience on Futuxe 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 ho 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.
* Professor Pearson's use of this title for the above formula is not, I think,
liistorioally correct. 5ayes' Theorem is the Inverse Principle of Probability
itself, and not this extension of it.
380 A TREATISE ON PROBABILITY pt. v
"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 d priori chance being between x and x+dx
and accordingly the total chance C^, whatever x may be of the
event occurring r times in the second series, is
a ^'^
a>^+'-{l-x)'>+'dx
TrVs
'' x^il-xfdx
This is, with a shght correction, Laplace's extension of Bayes'
Theorem." i
13. This argument can be restated as foUows. Of all the
objects which satisfy (ji{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 0, is in fact also/.
Now if we do not know the value of p and have no relevant in-
formation which bears upon it, we can assume d priori that all
values of p between and 1 are equally likely. This assumption,
which is termed the ' 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 pro-
duce 10,000 positive theories which are always false corresponding
to every one which is always true, and 10,000 correlations of posi-
^ The rest of the article is concerned with the determination of the probable
error when Laplace's Bule of Succession is used not simply to yield the prob-
ability of a single additional occurrence, but to predict the probable limits within
which the frequency will lie in a considerable series of additional trials. Pro-
fessor 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 funda-
mental 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.
CH. XXX STATISTICAL INFEEENCE 381
tive 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 quahties ; 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 imity, — of those for positive
theories and for correlations between positive quahties in the
neighbourhood of zero, and of those for negative theories and for
correlations between negative quahties in the neighbourhood of
imity. Moreover, we are seldom in so complete a state of ignor-
ance regarding the natm:e of the theory or correlation under
investigation as not to know whether or not it is a positive theory
or a correlation between positive quahties. In general, therefore,
whenever our investigation is a practical one, experience, if it
tells 1:1s 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
d 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 a priori to exceed as to fall short of (say) fifty per cent.
As Professor Pearson apphes this method to investigations where
it is plain that the quahties 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 worth while to poiat 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 and 1 are equally probable, all values of z
between and 1 are not equally probable. For the value -
is more probable than any other, and the possible values of
382 A TEBATISE ON PROBABILITY pt. v
z become increasingly improbable as they differ more widely
from -•
It may be added that the conclusions, which Professor
Pearson himself derives from this method, provide a reAudio
ad ahswrdwm of the arguments upon which they rest. He con-
siders, for example, the following problem : A sample of 100 of a
population shows 10 per cent afiected 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 groimds 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 know-
ledge 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 ia the above statement.
The method is, in fact, much too powerful. It invests any posi-
tive 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 Ride 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
^ Laws of Thought, p. 369. * Logic of Chance, p. 197.
' Galcul des probabilites, p. 174.
* Article in Cabinet Encyclopaedia, p. 64. ^ Principles of Science, p. 297.
° Logic, pp. 373, 374 ; Lotze propounds a " simple deduction " " as convin-
cing" to him "as the more obscure analysis, bywhioh it is usuaEy 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 withtoore qualifications than in the form discussed above.
* Loc. cit.
OH. XXX STATISTICAL INFERENCE 383
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 agaiast 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 conditions is sufficiently complete, and, although,
when the number of cases dealt with is small, its results are in-
credible, 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 beiag taken of the experience of the human
race, the probabihty 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 probabihty 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.
1 Lehrbuch der Wahrscheinlichlceitsrechming, p. 208.
CHAPTEE 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 know-
ledge 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 arith-
metic 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 com-
petent 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 con-
sonant 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
egarSs. It is reasonable to presume that, subject to suitable con-
ditions and qualifications, an inversion of Bernoulli's Theorem
must have validity. If we knew that our material coidd be
likened to a game of chance, we might expect to infer chances
from frequencies, with the same sort of confidence as that with
CH. XXXI STATISTICAL INFEKENCE 385
which we inier 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 Inversion of
Bernoulli's Theorem.
3. The problem is usually discussed in terms of the happemng
of an event under certain conditions, that is to say, of the co-
existence 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 prepositional .functions which may be said to concur or co-
exist 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 aoix that, if A(a) holds, B(a) wiU
hold also ?
Let us suppose that the occurrence of an instance of A(a!) is a
sign of one of the events ej{x), e^ix) ... or ej(x), and that these
are exhaustive, exclusive, and ultimate alternatives. By ex-
haustive it is meant that, whenever there is an instance of A(a;),
one of the e's is present ; by excliisive, 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 uUimate, that no one of the e's is a disjimction of two or more
alternatives which might themselves be members of the e's.
Let us assume that these alternatives are initially and throiighmet
the argument equally probable, which, subject to the above con-
ditions, 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 e^^x), e^Kx) . . . e^{x), the set
e^(x), e^{x) . . . ei(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 I and m, so that the ratio Ijm 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(a;) is adequate to establish a perfect analogy
between them ; the instances a, etc., of B(a;), that is to say, must
2 c
386 A TEEATISE ON PROBABILITY pt. v
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 prob-
ability 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 oiu" problem is to determine whether evidence
can be forthcoming of such a kind, that, as this evidence is in-
creased 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 satisfjring given conditions.
Let/(g') 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 d 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 /'(gf')/)^ . f{q), — ^which expression repre-
sents the likelihood d priori of the frequency q', given q.
An application of the inverse formula allows us to infer from
the above the d posteriori probability of q, given q', namely :
f{q)lh-Aq')lh.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
OH. XXXI STATISTICAL INFEEENCE 387
ratio q are d priori equally likely. If this supposition were
legitimate, the formula could be reduced to the algebraical ex-
pression
/(g')A-/(g)
all the terms of which can be determined numerically by Ber-
noulli's Theorem. It is easy to show that it is a maximimi when
q=q', i.e. that q' is the most probable value of l/m, and that,
when the iastances are very numerous, it is very improbable that
IJm differs from q' widely. If, therefore, the number of instances
is increased ia 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 prob-
ability, that A is ia 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 d priori equally likely. It is
not even equivalent to the assumptions that aU integral values
of I 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 state-
ment that it can arise in exactly a thousand ways. We must,
therefore, be content with some 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 I 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 d priori a finite probability in favour
of each of these possible values. Let /j, be the finite number
which m cannot exceed. Then there is a finite probability for
each of the intervals ^
■1*2 2,3 ^L-l . ,
- to -, - to -, ... - — to 1
fl fl fji fl fl
1 The intervals are supposed to include their lower but not their upper
limit.
388 A TEEATISE ON PEOBABILITY pt. v
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
which represents the a 'posteriori probability of q, given q'. Since
by sufficiently increasuig the number of instances, the sum of
terms f{q')lhf{q) for possible values of q within a certain finite
interval in the neighbourhooji of q' can be made to exceed the
other terms by any required amount, and since the sum of the
values of f{q)lh 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
Ij/i 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 g 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 to be j' 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 fjL 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 argimient, 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 preposter-
OH. XXXI STATISTICAL INFERENCE 389
ous claim, which would have been universaUy rejected long ago,
if those who made it had not so successftdly concealed them-
selves 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 oux modified conclusion, material assumptions have
been introduced. In the fiist 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 mitial prob-
ability is to be obtained ; and I have no better suggestion to ofEer
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 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(a!) may be the result of random sampling, and yet it may
stiU be the case that there are material circumstances, common
to all the examined instances of B(a;), yet not covered by the
statement A(a;)B(a;). 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 generalisa-
tion. But we are most likely to forget it. For having overcome
the difficulties peculiar to correlation,^ it is, possibly, not im-
1 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 generaliaaUon,
and the statement that A{x) is accompanied by B(a;) in a certain proportion
of cases a correlation. This is not qmte identical with its use by modern
statisticians.
390 A TREATISE ON PROBABILITY pt. v
natural for a statifltician to feel as if he had overcpme all the
difficulties.
In practice, however, our knowledge, in cases of correlation
just as in cases of generalisation, wiU seldom justify the assump-
tion 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 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 analysed statement that B
has accompanied A as often as not ha. 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 A 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 Cor-
relation, 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 mathe-
matical 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
391
392 A TREATISE ON PROBABILITY pt. v
account nothing whatever except the numbers of the instances,
is a rational proceeding. The passage already quoted from
Leibniz {In exempUs juridicis politicisque plerumque non tamen
subtiU cahulo opus est, quam accurata omnium drcumstantimum
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 investi-
gation 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 ihejlmtua-
tion 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 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
^ Giaunt in his NaMrcd and Political Observations upon the Bills of Mortality
has been quoted as one of the earliest statisticians to pay attention to these
considerations.
OH. xxxn STATISTICAL INFERENCE 393
ratios over different parts of their series of instances as weU 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 econo-
mist, 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 its practical application to the problems of sex ratio and
mortality. The fact that his general theory is so closely inter-
mingled -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 general-
ised significance. After 1879 Lexis added nothing substantial to
his earlier work, and later developments are mainly due to Von
* A list of Lexis's principal -writings on these topics wiH be found in the
Bibliography. There is little of first-rate importance which is not contained
either in the volume, Zwr Theorie der Massenerscheinungen in der menschlichen
OesdUchaft, or in the AbJmndlwngen zur Theorie der Bevolkerunga- und Moral-
Statistik. In this latter volume the two important articles on " Die Theorie der
Stabilitat statistischer Eeihen" and on "Das Geschlechtsverhaltnis der
Geborenen und die 'Wahrscheinlichkeitsrechnung," originally published in Con-
rad's Jahrbuche, are reprinted.
394 A TREATISE ON PROBABILITY pt. v
BortMewicz. 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 ProbabiUty writers of the
school of Lexis are ui 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 iu logical considerations. I am not
acquainted with much work, which has been appreciably influ-
enced by Lexis, written in other languages than German (including
with Germans, that is 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
* The reader may be specially referred to the Kritiscke Betrachtungen zur
theoretischen Statistik {first instalment — ^the later instalments being of less interest
to the student of Probability), the Anwendungen der WalirscheiTilichkeUsrechnung
auf StaUstik, and Somogeneitdt und StabUitdt in der Statistik. Of other German
and Russian writers it wiU be sufficient to mention here TsohuproW, who in
" Die Aufgaben der Theorie der Statistik " (SchmoUer's Jahrbuch, 1905) and "Zur
Theorie der Stabilitat statistischerReihen " {Skanditiavisk AMuarietidshrift) 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 Wdhrschein-
lichkeitsrechnung (vol. ii. part iv. section 1)' supplies a useful mathematical
commentary.
* Journal des aetuaires fran^is, 1874, and Theorie mathematique 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, Cakul, pp. 312-314 ; Borel,
SUments, 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.
ca. xxxu STATISTICAL INFERENCE 395
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 stiH 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 giviug 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 Bortkiewicz, though not for that reason the less influenced
or illuminated by their eminent contributions to this problem.
4. It win be clearer to begin with some analysis due to Von
Bortkiewicz,- and then to proceed to the method of Lexis him-
self, although the latter came first in point of time.
A group of observations may be made up of a number of sub-
groups, to which different frequencies for the character under
investigation are properly applicable. That is to say, a proper-.
tion — of the observations may belong to a group, for which, given
the frequency, the a 'priori probability of the character under
observation in a particular instance would be ^i, a proportion —
may belong to a second group for which f^ is the probability, and
so on. In this case, given the frequencies for the sub-groups,
the probability f for the group as a whole would be made up as
foUows :
We may call f a general jprohaJbility, and p, etc., special prob-
abilities. 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 p^ =P2 = . • . ■ =p, then p, for that particular way of resolv-
ing the total group into partial groups, is, in Bortkiewicz's termin-
ology, indifferent. lip 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
1 What follows is a free rendering of some passages in his Kritiache
^ This is clearly a very loose statement of what Bortkiewicz really means.
396 A TEEATISE ON PROBABILITY w. v
dealing with d priori probabilities, we can resolve a total prob-
ability 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 d 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.
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
Even if this particular way of resolving the series of observations
is indifferent, the actually observed frequencies p-^', p^', etc., may
nevertheless be unequal, since they may fiuctuate round the
norm p' through the operation of ' chance ' influences. If,
however, n-^, Wg, etc., are large, we can apply the usual Bemoullian
formula to discover whether, if there was a norm p', the diverg-
ences of Px,p^, etc., from it are within the limits reasonably attri-
butable on BernouUian 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.
BortMewicz goes on to say that probabilities having definitive
interpretation {definitive Bedeutung) may be designated ele-
mentary probabilities {EhmentarwahrscheinUchkeiten). But the
probabilities which usually arise in statistical inquiries are not
of this type, and may be termed average probabilities {Durch-
schnittswahrscheinlichkeiten). 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
OH. xxxn STATISTICAL INFEEENCE 397
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 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 ia England
next year is very unlikely to diverge widely from m. For this
would involve the unwarranted assumption, in BortMewicz'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
difEerent 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 iuto
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^', mg", .... etc., which are distributed round the average
frequency m. For simplicity let us consider the series of fre-
quencies TOi', m^', OTj', .... obtained by breaking up our
material according to the date of the birth. If the observed
divergences of these frequencies from their mean are not signifi-
cant, we have the beginnings of aii 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 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-
398 A TEEATISE ON PEOBABILITY pt. v
frequencies show sucli wide and discordant variations from the
mean as to suggest that some significant Analogy has been over-
looked. In this event the lack of symmetry, which characterises
the oscillations, may be taken to indicate that some of the sub-
groups are subject to a relevant influence, of which we must take
account in our generalisation, to which some of the other sub-
groups 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
m a
/•
particular frequency d, priori is ii'=p.
'2v{l-v)
p = -4769, V is the underlying probability, and g is the number of
instances to which the frequency refers. This follows from the
usual Bernoullian assumptions. Now let E be the corresponding
expression derived a posteriori by reference to the actual devia-
tions of a series of observed frequencies from their mean, so that
' I am here following fairly closely his paper, " tJber die Theorie der Stabilitat
statistioher Beihen," reprinted in his Abhandiungen zur Theorie der Bevolkerungs-
und 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. xxxn STATISTICAL INFEEENCE 399
R =P» / ~^> where [S^] is tlie sum of the squares of the devia-
tions 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 deviationB be-
tween 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 fre-
quencies 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 y/r^ +p^. In this event R carmot
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
fundamentally from one another according as R=r, >r, or <r,
R
Lexis puts — = Q, and takes Q as his measure of dispersion.^ If
Q = 1, we have normal dispersion; if Q>1, we have supernormal
dispersion; and if 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 comp^Tlson with p (the
physical component), and, therefore, R = ^r^ +p^ becomes
approximately R =p. On the other hand, if p is not very large
and the base number of instances is small, p be6ctaies ,npgligible
^ In Tsohuprow's notation {Die Aufgaben Ser ^heorie det' Statistifc, p. 45),
Q = P/C, where P (the Physical modulus) =a/ *-^ ^ ;ind C (the Com-
binatorial modulus) =»/ jj . M being the nuihber of iiistances in each
set, n the number of sets, J)^ the frequency for set k, and p the mean of the
n frequencies.
400 A TREATISE ON PROBABILITY pi. v
in comparison with r, and we have a delusive appearance of
normal dispersion.-^ Lexis weU 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 f. 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 Bemoullian 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 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 16i to 24 and shaken out at random, the modulus
in the num]l)er of dactyls would be 1-38, whereas we have con-
stantly ^obta^ined a smaller number, on an average (the square
root of the average fluctuation) 1-2." On Lexian principles
these statistibal results would support the hypothesis that the
^ This is part of the explanation of Bortkiewicz's Law of Small Numbers.
See also p. 401.
* " On Methods of Statistics," Jubilee Volume of the Boyal Statistical Society,
p. 211.
OH. xxxn STATISTICAL INFERENCE 401
series under investigation is ' organic ' and not subject to
Bemoullian 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 ia
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 connexite, leading to sub-
normal 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 super-
normal dispersion cannot also arise out of connexitS, 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 Vr^ +p^.
In short. Lexis has not pushed his analysis far enough, and he
has not fuUy 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 con-
ceiving 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 BortMewicz, 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
1 " Uber die Wahrsoheinliohkeitsreoluiuiig," p. 444 (see Bibliography).
' There are numerous references to this phenomenon in periodical literature ;
but it is sufficient to refer the reader to Von Bortkiewicz's Das Oeaelz der kkinen
Zahlen.
2d
402
A TREATISE ON PROBABILITY
to the remarkable regularity of comparatively rare events. Von
Bortkiewicz has enlarged Quetelet's catalogue with modem
instances out of the statistical records of bureaucratic Germany.
The classic instance, perhaps, is the number of Prussian cavalry-
men 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.)
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
G.
2
2
1
1
1
3
2
1
1
1
1
I.
. ,
2
3
. ,
2
1
1
1
, ,
2
3
1
II.
2
, ,
2
. ,
, ,
1
1
, ,
, ,
2
1
1
2
m.
, ,
1
1
1
2
2
, ,
, ,
, ,
1
, ,
1
2
1
IV.
1
, ,
1
1
1
1
1
, ,
1
..
V.
. ,
, ,
2
1
1
1
1
1
1
1
1
VI.
1
■ ■
2
1
2
1
1
3
1
1
1
VII.
1
1
, ,
1
1
1
, ,
2
, ,
2
1
, ,
2
vm.
1
, ,
1
1
, ,
, ,
. *.
1
. ,
1
, ,
1
IX.
, ,
, ,
2
1
1
1
. ,
2
1
1
, ,
1
2
, ,
.
X.
1
1
1
, ,'
2
2
. ,
2
1
3
, ,
1
1
XI.
2
4
, .
1
3
1
1
1
1
2
1
3
3
1
XIV.
1
1
2
1
1
3
, ,
4
, ,
1
, ,
3
2
1
, ,
2
1
XV.
1
••
••
••
•■
■■
1
1
1
•■
••
2
2
••
••
••
The agreement of this table with the theoretical results of a
random distribution of the total number of casualties is remark-
ably close : ^
Casualties in a
Year.
Number of Occasions on which the Annual
Casualties in a Corps reach the Figure
in Column 1.
1
2
3
4
5 and more
Actual.
144
91
32
11
2
Theoretical.
1431
921
33-3
8-9
2-0
0-6
Other instances are furnished by the numbers of child suicides
in Prussia, and the like.
It is Von BortMewicz's thesis that these observed regularities
^ Bortkiemoz, <rp. cit. p. 24.
OH. xxxn STATISTICAL INFERENCE 403
have a good theoretical explanation behind them, which he
dignifies with the name of the Law of Small Numbers.
The reader wiU 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 ^r^ +p^,
of which one is due to fluctuations from the average of the con-
ditions 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
vl +{n-l)c^, 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 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 synd/romy), 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 Homo-
geneity 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
^ 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.
2 " Homogeneitat und Stabilitat in der Statistik," published in the Skandi-
navisk 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.
404 A TREATISE ON PROBABILITY pt. v
we also assume that the larger mass is less homogeneous. At this
poiat, it would have helped, if Von BortMewicz, excluding from
his vocabulary homogeneity, paradromy, 7'ji, and the like, had
stopped to teU in plain language where his mathematics had led
him, and also whence they had started. But hke many other
students of Probability he is eccentric, preferring algebra to earth.
9, Where, then, though an admirer, do I criticise aU 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 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 j)j etc.,
where
If we are basing our calculations on p and do not know p-^, f^,
etc., then these calculations are more likely to be borne out by
the result if the iastances 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 /or which he has not allowed. If the d priori calculations
are based on the average over a field which is not homogeneous
CH. xxxn STATISTICAL INFERENCE 405
in all its parts, greater stability of result will be obtained if the
instances are drawn from all parts of tlie non-bomogeneous
total field, than if tbey are drawn now from one homogeneous
sub-field and now from another. This is not at all paradoxical.
Yet I believe, though with hesitation, that this is aU 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 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 iastabUity would be
displayed by this particular material ; for the frequency varies
from 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 measure-
ment of the conception of stability, which, quantitatively at
least, is not a perfectly precise one. The so-caUed 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 BernoulUan methods in an approximative
form, it does not lead to reliable results ia 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 regidarity 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 generalisa-
tion 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 wiU 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 Indudion, 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 neighbour-
hood of a given value. In either case it is asserting a char-
acteristic 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
406
CH. xxxm STATISTICAL INFEEENCE 407
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 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 general-
isation, and a proportion / 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) / discovered in the aggregate of the instances. Eor
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 appreci-
able 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 circum-
stances 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. Neverthe-
less 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
minion instances, that its probability is 1/10 for the next instance,
or that it is likely to have a frequency near to 1/10 in a further
408 A TEEATISE ON PEOBABILITY pt. v
set of observations, is a far feebler argument ; indeed it is hardly
an argument at aU. 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
CLTCimistances 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 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 ^ is true, the frequency of which is to be
CH. xxxm STATISTICAL INFERENCE 409
the subject of otir generalisation ; and if a proportion / of the
series s have the property <j}, we may say that the series s has a
frequency /for the property (p.
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, / 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 deter-
minate principle of division D, into subfields F^, Fg, etc, ; and
let the series s^ be taken from F^, s^ from Fg, and so on. Where
Fj, Fg, etc., have a finite number of constituents, s^, Sg, etc., may
possibly coincide with them ; if s^, Sg, etc., do not coincide with
F;^, Fg, etc., but are chosen from them, let us suppose that they are
chosen according to some principle of random or unbiassed
selection — s-^, that is to say, will be a random sample from F^.
Now it may happen that the frequencies /i,/g, etc., of the series
Si, Sg, etc., thus selected cluster round some mean frequency /. If
the frequencies show this characteristic (the measurement and pre-
cise determination of which I am not now considering), then the
series of series s^, Sg, etc., has a stable frequency for the classifica-
tion D. ' Great numbers ' only come in because it is difficult to
ascertain the existence of stable frequency imless the series s^, Sg,
etc., are themselves numerous and tmless each of these comprises
numerous individual instances.
Let us then apply a different priaciple of division D', leading
to series s^', Sg', etc., and to frequencies/^^/g', etc. ; and then again
a third principle of division D" leading to frequencies /^'j/g", etc. ;
and so on, to the full extent that our knowledge of the differences
between the individual instances permits us. If the frequencies
/ij/g, etc., fi'tfz, etc., fijfz", etc., and so on are all stable about/,
we have an inductive ground of some weight for asserting a
statistical generahsation.
Let. the field F, for example, comprise all Englishmen in their
sixtieth year, and let the property (f), 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 Fj, Fg, etc., on in-
numerable different principles. F^ might represent Englishmen
410 A TREATISE ON PROBABILITY pt. v
in their sixtieth year in 1901, Fg 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 Fi, Fj, etc., be constituted by the districts in which they
live. If we take large random selections s^, s^, etc., from F^, Fg,
etc., respectively, and find that the frequencies /^j/g, etc., fluctuate
closely round a mean value /, this can be expressed by the
statement that there is a stable frequency / 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 aU, 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
XXVTI. 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 1 First, it might be the case that he was fed on the movable
feasts of the 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
CH. xxxm STATISTICAL INFEEENCE 411
there from day to day, the proportion of days for scraps might
agaia 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 ia 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 ia like circumstances instead of one
dog only.
It has been assumed so far that we have an opportunity of
observing what happens on emery day of the /ear. If this is
not the 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 tjrpes.
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, inappUcable 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
412 A TREATISE ON PEOBABILITY pt. v
tte 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 iaquiry. 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 ^, is j? ; ' or, more precisely, if a is
a random member of S(a;), the probability of ^(a) is p.
It will be convenient to recapitulate from Chapter XXIV. § 11
the definition of ' an instance taken at random ' : Let ^{x)
stand for ' x has the characteristic ^' and S(a;) for ' a; is a member
of the class S ' ; then, on evidence A, a is a random member
of the class S for characteristic ^, if ' a; is a ' is irrelevant to
<f){x)/S{x) . h,^ i.e. if we have no information about a relevant
to </)(«) 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 <p{a) relative to evidence h. Let us discard
that part of our knowledge h{a) which is irrelevant to cj){a),
leaving us with relevant knowledge h'{a). Let the class of
instances a^, ajj etc., which satisfy h'{x) be designated by S. Then,
relative to evidence A, a is a random member of the class or
series S for the characteristic (j}.
Let us denote the proposition ' x is, on evidence h, a random
member of S for characteristic <^ ' by B,{x, S, <ji, h) ; then our
statistical generalisation is of the form <p{x)/R{x, S, ^,h).h =p.
If R {a, S, (^, h) holds, then, on evidence h, S is the appropriate
statistical series to which to refer a for the purposes of the charac-
teristic (j>.
It is not always the case that the evidence indicates any
series at all as ' appropriate ' in the above sense. In particular,
* The use of variables in probability, as has been pointed out on p. 58, is
very dangerous. It might therefore be better to enunciate the above : a is a
random member of S for characteristic tj>, if ^(a)/S(o) . ft = 0(6)/S(6) . fe where
S(6) . h contains no information about b, except that fi is a member of S
OH. xxxin STATISTICAL INFEEENCE 413
if evidence h indicates S as the appropriate series, and evidence
h' indicates S' as tlie 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 capable of justifying the conclusion that jp is the probability
that a random member of the series S will have the character-
istic ^ ?
In the simplest case, S is a finite series of which we know the
truth frequency for the characteristic <^, namely f} Then by a
straightforward application of the Principle of Indifference we
have f =/, so that j>{x)fR{x, S, ^,h).h =/.
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 (j}{x) is true, there cor-
responds a determinate number of members of which ^(a;)' 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 (j){x) is true and false respectively in fixed and determinate,
proportions. If this determinate proportion is known to be/, we
have, as before, p =/. The tjrpical 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 alterna-
tive 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 pto ^ where <j} is the fall
of the six-face. It is not necessary, in order to obtain this
result, to assert that S is a finite series with an actual determinate
frequency /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 —
"• I.e. if /is the proportion of the members of the series for which 0(a;) is true.
414 A TREATISE ON PROBABILITY pt. v
the determination by an analysis of tlie 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 ^ ; and it 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 Sj, S2, etc., respectively. The first part of our inquiry is
the problem of arguing from these observed frequencies to the
probable constitutions of Sj^, 83, etc., i.e. of determining the values
of <f>{x)fR{x, Sj, <j), 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 S^, Sg, etc., to the
probable constitution of S, where S, Sj, Sg 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 S^, Sg, 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 hues ex-
plained in Chapter XXXI. Moreover, if the series S^, Sj, etc.,
are finite series and the observed selections cover a great part
of their members, we can reach an at least approximate con-
clxision without raising all the theoretical dLEGiculties 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 iavolve,
our actual evidence not warranting in general more than an
CH. xxxni STATISTICAL INFERENCE 415
approximate result, are not, I think, fundamentally erroneous.
The most usual error in modem method consists in treating too
Hghtly what I have termed above the inductive problem, i.e.
the problem of passing from the series S^, Sg, 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 p^, f^, etc., with
more or less exactness, by examining either all the instances of
the series S^, Sg, etc., or random selections from them, i.e. <^{x)[R
{x, Sj, ^,h).h =pi, etc. This can be expressed for short by saying
that the series S^, Sg, etc., are subject to probable-frequencies
Pi> P2> ®*^-5 ^or the characteristic ^. Our problem is to infer from
this the probable-frequency p of the unexamined series S. The
class characteristics of the series S^, Sg, 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 S^, Sg, are to form the basis of a statistical induction,
they must show a stable value ; that is to say, either we must
have pi =p^ = etc., or at least p^, p^, etc., must be stably grouped
about their mean value. Our next task, therefore, must be
to discover whether the probable-frequencies pj^, p^, 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, xmtil a primdfade 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 imder iaquiry is
a member of identically the same series as that from which our
samples were drawn, i.e. where S = S^, which in social and scientific
inquiries is seldom the case.
What is the meaning of the assertion that p^, p^, etc., are
stably grouped about their mean value ? The answer is not
simple and not perfectly precise. We could propound various
formulae for the measurement of stability and dispersion, respect-
ively, and the problem of translating the conception of stability,
which is not quantitatively precise, iato a numerical formula
involves an arbitrary or approximative element. For practical
416 A TREATISE ON PROBABILITY pt. v
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. 399. 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 d priori, if
we had assumed that the members of Sj, Sg, etc., were obtained
by random selection out of a single universe U, that it is sub-
normal if the dispersion is less than one would have expected on
the same hjrpothesis, 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 p^, p^, 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 S^, S^, etc., are within limits which
we should expect d 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 imder 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 Sj, Sg, 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, iu relation to the characteristic
under inquiry, to the properties of S^, Sg, etc., in relation to this
characteristic depends on the differences between S, S^, Sj, 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, difiiculties.
In general this inductive part of our inquiry will be best
advanced by classifying the aggregate series of iostances 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 S^, Sg, etc., which show the most
CH. xKxui STATISTICAL IKFEEENCE 417
marked and definite class cliaracteristics. Our knowledge of the
differences between tte particular observed instances wHch
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 char-
acteristics, not all of which are shared in common by aU 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 S^, Sj, etc.,
which have frequencies f^, f^, 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^', Sg', etc.,
with frequencies /i', /2',etc. ; and soon, so far as our knowledge of
the possible relevant 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 f^, f^, etc., jf/, f^', etc., are stable about
a value p, and if, on the basis of the above positive and negative
analogies, we have a normal inductive argument for assimilating
the unexamined series S to the examined series S^, Sg, etc., S^', Sg',
etc., in respect of the characteristic under 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 argu-
ment. We first find that the observed frequencies in a set of
series are such as would have been not improbable d 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 d priori if, relative to our know-
ledge, S was also made up of random members of the hypo-
thetical 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 fore-
going analysis is along the right liaes and that it carries the
2e
418 A TREATISE ON PEOBABILITY pt. v
inquiry a good deal fuitlier 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 Laterpretation 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 equi-probable,
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 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
stabUity 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 flxed 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
OH. xxxm STATISTICAL INFEEENCE 419
hand sets of enviromnents ; the proportions of the different
kinds of persons, the proportions of the different kinds of environ-
ments, 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-caUed ' 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 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 r£ 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 MendeUan 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 ia 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
^ The fluctuations in the proportion of the sexes which, as is weE 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.
420 A TEEATISE ON PROBABILITY ft. v
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 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 tjrpes ; 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 propor-
tions. 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 ia
defect. ' Organic series ' have been incidentally discussed else-
where 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 general theoretical
scheme is not easy.-^
1 The following more precise definitions bring these ideas into line with what
has gone before : consider the terms di, Oj . . . a„ of a series s(x) ; let ' a^ia g'
OH. xxxm STATISTICAL INFERENCE 421
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 expres-
sion 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.
In the notation employed in the earlier part of this chapter I
have classified each examiaed instance a according as it did or
did not possess the characteristic <ji, i.e. satisfy the propositional
function <j>{x), or, in other words, according as ^(a) was true or
false. Thus only two possible alternatives were contemplated,
and <ji 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, aU the instances
satisfied a function ■\lr{x), and the question was, what proportion
=3, and let 9',/A=p,, where h is our data. Then, if g^lg, . . . gt ■ ■ ■ ^—Pr for s-U
values oir,s, . . .,t. . ., the terms of the series are independent relative to h. If
Pi=Pi=. . .=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 l/nSp,. If the terms are not independent, the series is an organic
series.
The same terminology can then be applied to the series Si, S^, . . . S„, regarded
as members of the series of series S(a;). Let the frequencies of the series for the
characteristic under inquiry be aii, x^, . . . x„, and let xjh = Oj{Xj), i.e. Si{Xj) is the
probability of a frequency x^ in the first series. Then if a;,/a!, . . . A = 9,(a;,) for all
values of r, s, etc., the frequencies are independent ; and if 9i(a;) = 64,x^ = . . . e{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.
422
A TREATISE ON PROBABILITY
of them also satisfied the function ^(x). A typical example was
that of sex-ratio, — yjr{x) being the birth of a child and <p{x) its
sex, where there is no question of degree in either ■\fr{x) or ^(a;).
It might.be the case, however, that the characteristics under
examiuation were capable of degree or quantitative variation ;
for example ^fr{x) might be the age of the mother and ^(a;) the
weight of the child at birth. In this case we should have a series
1^1(33), ^Ir^ix), etc., corresponding to the various age-periods of the
mothers, and a series 4>i(x),^2{^)> etc., corresponding to the various
weights of the children. Now if we concentrated our attention
on 1^1(0;) and <pi(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 '\jr-i(x) a certain
proportion satisfy ^^(a;) also. But clearly we can push our
observations further and we can take note what proportion of the
instances which satisfy i/ri(a;) satisfy (^^{x), ^3(x),and so on, respect-
ively ; and then we can do the same as regards the instances
which satisfy yltgix), y}ra{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 /„ is the proportion
of instances satisfying ■\lrg{x) which also satisfy <f>r{x) we have a
table as foUows :
■i'M
■+.(«)
■^,W
0i(a;)
/u
/.a
/18
02(a')
Ai
/.a
/.3
M'^)
/s.
/a.
/sa
We could, further, increase the complexity and completeness
of our observations to any required degree. For example we
might take account also of 6{x), the age of the father, and con-
struct a threefold table where /,^ is the proportion of instances
satisfying (f>r(x), T|r,(a!), d^ix) ; and so on up to an n-fold table.
Clearly it is not necessary for the construction of tables of
CH. xxxm STATISTICAL INFEEENCE 423
this kind that ^(x) and ^^(a;) should stand for degrees of the same
quantitative characteristic ; they might be any set of exclusive
alternatives ; for example, -\|r(a;) might be the colour of the baby's
eyes, and <^(a;) its Christian name.
But ia order that the correlation table may be of any
practical interest for the purposes of iof erence, it is necessary —
and this, I think, is one of the critical assumptions of correla-
tion — ^that <f)j{x), (j)2{x) . . . and also ^i{x), ^2(0;) . . , should
be arranged in an order that is significant, i.e. such that we have
some d priori reason for expecting some connection to exist
between the order of the ^'s and the order of the ^'s. The point
of this will be illustrated by concentrating our attention on the
simplest type of case where ^(x) and <f){x) are quantitative
characteristics arranged ia order of magnitude. Now suppose
it were the case that the younger mothers tended to bear heavier
babies, then, if ^^(a;) (piix) are the ages increasing upwards and
^i(^) ^a(^) tJie weights diminishing downwards, /^ would probably
be the greatest of the f^^'s and, generally speaking, /^^ would be
greater than/,+i ^ ; also /ga might be the greatest of the f^'s, and
so on ; so that the frequencies lying on the diagonal of the table
would be the grea.test and the frequencies would tend to be less
the farther they lay from the diagonal. If we had some reason
d 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 1
Dr. Bowley faces this problem a little more definitely than do
424 A TEEATISE ON PEOBABILITY pt. v
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 particiilar 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 it as a measure of the probability of a
statistical generalisation as to the association between quanti-
tative variations of ^(x) and i|r(a:) respectively. If, for ex-
ample, 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 de-
scription (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
X,=A + aU,+ . . . +^U,+iV, + ,V, + . . . -F^V,
Y,=iU,+2U,+ . . . +^U,-i-iW, + 2W,+ . . . ,W,
where jUf, jUj . . . ^Vj, gYj . . . iWj, 2^^ .... 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 a 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
d priori as to how X^ and Yj are constituted, then we have
calculable expectations as to the limits within which the value
^ Introduction to the Theory of Statistics, p. 191 : " The coefficient of correla-
tion, lite 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. xxxm STATISTICAL INFEEENCE 425
of r, namely the correlation coefficient between X and Y, wUl
probably turn out to lie, when we have observed it.
Dr. Bowley's next move is more dubious. If the constitu-
tions of the independent groups are similar in a certain statistical
respect {i.e. if they have the same standard deviations), then,
Dr. Bowley concludes, r=— — , which "expressed in words
shows that the correlation 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, wiU 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 d 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 observa-
tions made ia prescribed conditions, will have, d 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 vari-
ables. 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 appre-
hends 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 a priori improbable (and the greater the
number of the individual observations, the greater the improba-
bility since, if the quantities are independent, there is, then, all
the more opportunity for ' averaging out ') ; therefore, inversely.
426 A TREATISE ON PROBABILITY m. v
if the frequencies do group themselves 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 probabiUty, he wiU remember that this conchision cannot
be reached unless d 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 argu-
ment 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 asstmiption 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 groimds. 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 investi-
gators are sensible.
If we abandon the method of inverse probability in favour of
the less precise but better fovmded 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 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
CH. xxxm STATISTICAL INFERENCE 427
of pre-existing knowledge relevant to the particular inquiry, tlie
calculation of a small number of correlation coefficients may be
crucial. But otherwise we must proceed as in the case of fre-
quency coefficients ; that is to say we must have before us, in
order to found a satisfactory argument, many sets of observa-
tions, 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 general-
isation 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 some-
times 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 wiU perhaps excuse
me if I have sometimes pressed on a little faster than the diffi-
culties were overcome, and with decidedly more confidence them
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 validity of which the boasted knowledge of modem science
depends, can only exist — and I do not now pause to inquire
again whether such an argument must be circular — i£ 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 neoessest
materiem quoque finitis difEerre figuris.
The physicists of the nineteenth century have reduced matter to
428 A TEEATISE ON PEOBABILITY pt. v
tlie 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 reaUy 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 contem-
porary 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 — " I'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
429
BIBLIOGEAPHY
INTKODUCTION
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 ProbabiUty than can be found elsewhere. I have
hesitated a httle 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 Hterature of the subject ; and a list which I drew up
for my own convenience, without much attention to biblio-
graphical 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. ProbabiUty 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 ProbabiUty. It has been a matter of some difficulty,
therefore, to know precisely where to draw the Une. Where
the main subject of a book or paper is ProbabiUty 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 ProbabiUty is not the main subject or where an appUca-
tion of Probability is concerned, the chief interest of which is
431
432 A TREATISE ON PROBABILITY
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 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 hterature
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 niemoirs on this topic published before
1877.
Of bibhographical sources for ProbabiHty proper, Tod-
hunter's History of the Mathematical Theory of Probability
and Laurent's Calcul des probabilites are alone important. Of
mathematical works published before the time of Laplace,
Todhunter's list, and also his conmientary and analysis, are
complete and exact, — a work of true learning, beyond criticism.
The bibhographical catalogue at the conclusion of Laurent's
Calcul (published in 1873) is the longest Ust pubhshed hitherto
of general works on ProbabiUty. But it is unduly swollen by
the inclusion of numerous items on Insurance and Errors of
Observation, the bearing of which on ProbabiHty is very
sUght ; ^ 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 Ust is too long, and I have not always success-
fully 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 bibho-
grapher 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
accomphshed by the slow mills of the collective judgment of
1 Laurent's list eontaius 310 titles, of which I have excluded 174 from my
list as being insufiSciently relevant.
BIBLIOGRAPHY 433
tlie learned ; and I liave already 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 miuds on which
so Uttle has been written. It is now fifty-five years since
Dr. Venn, still an accustomed figure in the streets and courts
of Cambridge, first pubHshed his Logic of Chance ; yet amongst
systematic works in the English language on the logical founda-
tions 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, Coumot, Quetelet, Gauss, Mill, Boole,
Tchebychef, Lexis, and Poincare, to name those only who are
dead, are catalogued below.
Abbott, T. K. " On the Probability of Testimony and Arguments." PhU.
Mag. (4). vol. 27, 1864.
Adbain, 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 Abbe with historical notes in Amer. Joum.
Soi. vol. i. pp. 411-415, 1871.]
Ammon, O. " Some Social AppUoations of the Doctrine of Probability."
Joum. Pol. Econ. vol. 7, 1899.
AmpJire. Considerations sur la th^orie math6matique du jeu. Pp. 63. 4to.
Lyon, 1802.
ANciiiLON. " Doutes sur les bases du caloul des probabilit^s." Mem. Ac.
Berlin, pp. 3-32, 1794-5.
Abbtjthnot, 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 ratiooiniis in ludo aleae.]
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 fuU account of this book and discussion of the authorship, see
Todhunter's Histoly, pp. 48-53.]
" An Argument for Divine Providence, taken from the constant Regular-
ity observ'd in the Births of both Sexes." Phil. Trans, vol. 27, pp. 186-
190 (1710-12).
[Ajgues that the excess of male births is so invariable, that we may con-
clude that it is not an even chance whether a male or female be bom.]
2f
434 A TREATISE ON PROBABILITY
Abistotlb. Anal. Prior, ii. 27, 70* 3.
Bhet. i. 2, 1357 a 34. [See Zeller's Aristotle for further references.]
Arnauld. (The Port Royal Logic.) La Logique ou I'Art de penser. 12mo.
Paris, 1662. Another ed. C. Jourdain, Hachette, 1846. Transl. into
Eng. with introduction by T. S. Baynea. London, 1851. xlvii + 430.
See especially pp. 351-370.
Babbaqe, C. An Examination of some Questions connected with Games of
Chance. 4to. 25 pp. Trans. B. Soc. Edin., 1820.
Baohelibe, Louis. Caloid des probabilitis. Tome i. 4to. Pp. vii + 517.
Paris, 1912.
Le Jeu, la chance, et le hasard. Pp. 320. Paris, 1914.
[Bailby, Samubl.] Essays on the pursuit of truth, on the progress of know-
ledge and on the fundamental principle of all evidence and expectation.
Pp. xii-i-302. London, 1829.
Baldwin. Dictionary of Philosophy. Bibliographical volumes ; s.v. " Prob-
abiUty."
Baniol, a. "Le Hasard." Revue Internationale de Sociologie. Pp. 16.
1912.
Baebbteao. Traite du jeu. 1st ed. 1709. 2nd ed. 1744.
[Todhunter states (p. 196) that Barbeyrac is said to have pubUshed a
discourse " Sur la nature du sort."]
Bates, Thomas. An Essay towards solving a Problem in the Doctrine of
Chances. Phil. Trans, vol. M. 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 Losung eines Problems der Wahrschein-
lichkeitsrechnung. Herausgegeben von H. E. Timerding. Sm. 8vo.
Leipzig, 1908. Pp. 57.
BfiotJBLiN. " Sur les suites ou sequences dans le loterie de Gfenes." Hist, de
I'Acad. Pp. 231-280. Berhn, 1765.
" Sur I'nsage du principe de la raison suffisante dans le calcul des pro-
babilites." Hist, de I'Aoad. Pp. 382-412. Berlm, 1767. (Publ. 1769.)
Bellavitis. " Osservazioni suUa theoiia deUe probabiUtJi." Atti del Instituto
Veneto di Soienze, Lettere, ed Arti, Venice, 1857.
Benaed. "Note sur une question de probability." Journal de I'ificole
royale poUteohnique. Vol. 15, Paris, 1855.
Bentham, J. Rationale of Judicial Evidence.
See Introductory View, chap, xii., and Bk. i. chaps, v., vi., vii.
Bbenoxtlli, Daniel. " Specimen theoriae novae de mensura sortis." Comm.
Acad. Sci. Imp. Pet. vol. v. pp. 175-192, 1738.
Germ, transl. 1896, by A. Pringsheim : Die Grundlage der modemen
Wertlehre. Versuch einer neuen Theorie der Wertbestimmimg von Gluoks-
fallen (Einleitung von Ludvig Kck). Pp. 60. Leipzig, 1896.
" Recueil des pieces qui ont remport^ le prix de l'Aoad6mie Royale des
Sciences." 1734. iii. pp. 95-144.
[On " La cause physique de I'inolinaison des plans des orbites des planetes
par rapport au plan de I'^quateur de la revolution du soleil autour de son
axe."]
" Essai d'une nouvelle analyse de la mortality causae par la petite
v6role." Hist, de I'Aoad. pp. 1-45. Paris, 1760.
De UBU algorithm! infinitesimalis in arte conjectandi specimen. Novi
Comm. Petrop., 1766. xii. pp. 87-98. A 2nd memoir. Petrop., 1766.
xii. pp. 99-126. See a criticism by Trembley, Mem. de I'Aoad., Berlin,
1799.
BIBLIOGRAPHY 435
Beenottlli, Daniel. — continued.
Disquisitiones analytiquae de novo problemate oonjeeturali. Novi
Comm. Petrop. xiv. pp. 1-25, 1769. A 2nd memoir, Petrop. xiy. pp.
26-45, 1769.
" Bijudicatio maxime probabilis plurium observationum disorepantium
atque verisimiUima induotio inde formanda." Acta Aoad., pp. 3-23.
Petrop., 1777. Crit. by Euler, pp. 24-33.
Bbrnottlu, Jac. Ars conjectan(U, opus posthumum. Pp. ii -t- 306 -I- 35.
Sm. 4to, Basileae, 1713.
[Published by N. Bernoulli eight years after Jac. Bernoulli's death.]
Part I. Reprint with notes and additions of Huygens, De ratiooiniis in
ludo aleae.
Part II. Doctrina de permutationibus et combinationibus.
Part III. Explicans usum praecedentis doctrinae in variis sortitionibus
et ludis aleae. [Twenty-four problems.]
Part IV. Tradens usum et appUoationem praecedentis doctrinae in
oiviUbus, moraUbus et oeoonomicis.
Traotatus de seriebus infinitis. [Not connected with the subject of
ProbabiUty.]
Lettre a un amy, sur les partis du jeu de paume.
[The most important sections, including BemouUi's Theorem, are in
Part IV. For a very full accoimt of the whole volume see Todhunter's
History, chap, vii.]
Engl. Transl. of Part II. only, vide Maseres.
Pr. transl. of Part I. only, vide Vastel.
Germ, transl. : Wahrscheinlichteitsreclmung. 4 Telle mit dem Anhange :
Brief an einem lEYeund iiber das BaUspiel, libers, u. hrsg. v. R. Haussner.
2 vols. Sm. 8vo. 1899.
[See also Leibniz.}
Bbbnotjlli, John. De alea, sive arte oonjectandi, problemata quaedam.
CoUected ed. vol. iv. pp. 28-33. 1742.
Bbrnottlli, John (grandson). " Sur les suites ou sequences dans la loterie de
GSnes." Hist, de I'Aoad, pp. 234-253. Berlin, 1769.
" Memoire sur uu probldme de la doctrine du haaard." Hist, de I'Acad.,
pp. 384-408. Berlin, 1768.
Bebnotjlli, Nicholas. Specimina artis coujectandi, ad quaestiones juris
appUcatae. Basel, 1709. Repr. Act. Erud. Suppl., pp. 159-170, 1711.
Bbetrand, J. Calcul des probabiUtes. Pp. lvii + 332. Paris, 1889.
" Sur I'applicatiou du calcul des probabiht^s k la theorie des jugemeuts."
Comptes rendus, 1887.
"Les Lois du hasard." Rev. des Deux Mondes, p. 758. Avril 1884.
Bbssbl. " Untersuohung iiber die Wahrsoheinlichkeit der Beobaohtungsfehler."
Astr. Nachriohten, vol. xv. pp. 369-404, 1838.
Also Abhandl. von Bessel, vol. ii. pp. 372-391. Leipzig, 1875.
BiCQuiLLBY, C. F. DB. Du caloul des probabilites. 164 pp., 1783. 2nd ed.
1805.
Germ, transl. by C. F. Biidiger. Leipzig, 1788.
BiENAYMfi, J. " Sur un principe que Poisson avait cru deoouvrir et qu'U avait
appeM loi des grands nombres." , Comptes rendus de I'Aoad. des Sciences
morales, 1855.
[Reprinted in Journal de la Soo. de Statistiques de Paris, pp. 199-204,
1876.]
" Probabilite de la Constance des causes couolue des effets observes."
Procds-verbaux de la Soc. Philomathique, 1840.
" Sur la probabilite des resultats moyens des observations, etc." Sav.
:fitraugers. v., 1838.
436 A TREATISE ON PROBABILITY
BiBNAYMii, J. — continued.
" Theor^me sur la probabilite des resultats moyens des observations.''
Prooes-verbaux de la Soo. Philomathique, 1839.
" Considerations a I'appui de la d6couverte de Laplace sur la loi de pro-
babiUte dans la m^thode des moindres carr^s." Comptes rendus des
seances de I'Academie des Sciences, toI. xxxvii., 1853.
[Reprinted in Journal deLiouville, 2nd series, vol. xii., 1867, pp. 158-176.]
" Eemarques sur les differences qui distinguent I'interpolation de Cauohy
de la m^thode des moindres carrfe." Comptes rendus, 1853.
" Probabilite des erreurs dans la methode des moindres oarres." Joum.
IdouviUe, vol. xvii., 1852.
BiNBT. " Eeoherches sur une question de probabilite " (Poisson's Theorem).
Comptes rendus, 1844.
Blasckke, E. Vorlesungen iiber mathematische Statistik. Pp. viii + 268.
Leipzig, 1906.
BoBBK, K. J. Lebrbuch der WahrscheinUohkeitsreohnung. Nach System
Kleyer. Pp. 296. Stuttgart, 1891.
BoHLMANN, G. " Die GrundbegrifEe der WahrsoheinUohkeitsrechnung ia ihrer
Anwendung auf die Lebensversioherung." Atti del IV Congr. intern.
dei matematici, Borne, 1909.
Boole, G. Livestigatious of Laws of Thought on which are founded the
Mathematical Theories of Logic and Probabilities. Pp. ix+424. London,
1854.
" Proposed Questions in the Theory of Probabilities." Cambridge and
Dublin Math. Journal, 1852.
" On the Theory of ProbabUities, and in particular on Michell's Problem
of the Distribution of the Kxed Stars." Phil. Mag., 1851.
" On a General Method in the Theory of ProbabiUties." Phil. Mag.,
1852.
" On the Solution of a Question in the Theory of ProbabiUties." Phil.
Mag., 1854.
" Reply to some Observations published by Mr. Wilbraham in the Phil.
Mag. vii. p. 465, on Boole's ' Laws of Thought.' " PhU. Mag., 1854.
" ^Further Observations in reply to Mr. Wilbraham." Phil. Mag., 1854.
" On the Conditions by which the Solutions of Questions in the Theory
of ProbabiMties are limited." Phil. Mag., 1854.
" On certain Propositions in Algebra connected with the Theory of
ProbabiUties." Phil. Mag., 1855.
" On the AppUcation of the Theory of ProbabiUties to the Question of
the Combination of Testimonies or Judgments." Edin. Phil. Trans, vol.
xxi. pp. 597-652, 1857.
" On the Theory of ProbabiUties." Roy. Soo. Proc. vol. xii. pp. 179-
184, 1862-1863.
BoECHAEDT, B. Einfuhrung in die WahrsoheinUohkeitslehre. vi-H86.
BerUn, 1889.
BoBDONi, A. SuUe probabiUtil. 4to. Giom. deU' L R. Instit. Lombardo di
Soienze. T. iv. Nuova Serie. Milano, 1852.
BoREL, E. !6l6ments de la theorie des probabiUtes. 8vo, pp. vU-l-191.
Paris, 1909. 2nd ed. 1910.
LeHasard. Pp. iv -I- 312. Paris, 1914.
"Le Calcul des probabiUtes et la methode des majorit^s." L'Ann^e
psyohologique, vol. 14, pp. 125-151. Paris, 1908.
"Les ProbabiUtes d6nombrables et leurs appUcations arithmetiques."
Rendiconti del Ciroolo matematico di Palermo, 1909.
" Le Calcul des probabiUtes et la mentaUt6 individuaUste." Revue du
Mois, vol. 6, pp. 641-650, 1908.
BIBLIOGEAPHY 437
BoEBL, E. — continued.
" La Valeur practique du calcul des probabilites.'' Revue du Mois, vol.
1, pp. 424-437, 1906.
" Les Probabilites et M. le Danteo." Kevue du Mois, vol. 12, pp. 77-91
1911.
BoRTKiEwicz, L. VON. Das Gesetz der kleinen Zahlen. 8vo, pp. viiiH-52;
Leipzig, 1898.
" Anwendungen der WahrscheinUchkeitsreohnung auf Statistik." En
cyklopadie der mathematisclieii Wissensohaften, Band 1, Heft 6.
" WahraoheinUchkeitstheorie und Erfahrung." Zeitsohrift fiir Philo
Sophie und philosophisohe Kritik, vol. 121, pp. 71-81. Leipzig, 1903.
[With reference to Marbe, Bromse, and Grimaehl, q.v.]
" Kritisohe Betraohtungen zur theoretiaohen Statistik." Jahrb. f
Nationalok. u. Stat. (3), vol. 8, pp. 641-680, 1894; vol. 10, pp. 321-360,
1895 ; vol. 11, pp. 671-705, 1896. '
" Die erkenntnistheoretisohen Grundlagen der WahrscheinUohkeits
reohuung." Jahrb. f. Nationalok. u. Stat. (3), vol. 17, pp. 230-244,
1899.
[Criticised by Stumpf., q.v., who is answered by Bortkiewioz, loc. cit.
vol. 18, pp. 239-242, 1899.]
" Zur Verteidigung des Gesetzes der kleinen Zahlen." Jahrb. f . National
ok. u. Stat. (3), vol. 39, pp. 218-236, 1910.
[The literature of this topic is not fuEy dealt with in this BibUography,
but very fuU references to it will be found in the above article.]
" Uber den Prazisionsgrad des Divergenzkoeffizientes." Mitteil. des Ver-
bandes der osterr. und ungar. Versichernngsteohniker, vol. 5.
" Eealismus und FormaUsmus in der mathematischen Statistik." AUg.
Stat. Archiv, vol. ix. pp. 225-256. Munich, 1915.
Die Iterationen : ein Beitrag zur Wahracheinlichkeitstheorie. Pp
xuH-205. Berlin, 1917.
Die radioaktive Strahlung als Gegenstand wahrscheinlichkeits
theoretisoher Untersuchungen. Pp. 84. Berlin, 1913.
" WahrscheinUchkeitstheoretische Untersuchungen fiber die Knaben
quote bei Zwillings Gebieten." Sitzungsber. der Berliner Math. Ges., vol,
xvii. pp. 8-14, 1918.
Homogeneitat und Stabilitat in der Statistik. Pp. 81. (Extracted from
the Skandinavlsk Aktuarietidskiitt.) Uppsala, 1918.
BoSTWiCK, A. E. " The Theory of ProbabiUties." Science, iii., 1896,
p. 66.
BotiTBOirx, PiBBRB. " Les Origines du calcul des probabilites." Revue du
Mois, vol. 5, pp. 641-654, 1908.
BowLEY, A. L. Elements of Statistics. Pp. xiH-459. 4th ed. London,
1920.
Bradley, F. H. The Principles of Logic. Bk. i. chap. 8, §§ 32-63, pp.
201-20. London, 1883.
Beavais. " Analyse math6matique sur les probabiUtes des erreurs de situa-
tion d'un point." M6m. Sav. vol. 9, pp. 255-332, Paris, 1846.
Brendel. Wahischeinlichkeitsrechnung mit Einschluss der Anwendungen.
Gottingen, 1907.
Broad, C. D. "The Relation between Induction and Probability." Mind,
vol. xxvii. (1918). Pp. 389-404, and vol. xxix. (1920) pp. 11-45.
Bromse, H. Untersuchungen zur WahracheinUchkeitslehre. (Mit beaonderer
Beziehung auf Marbes Sohrift (q.v.).)
Zeitschrift fiir PhUosophio und philosophisohe Kritik. Band 118.
Leipzig, 1901. Pp. 145-153.
(See also Marbe, Grimsehl, and v. Bortkiewioz.)
438 A TREATISE ON PROBABILITY
Beunn, Dr. Hermann. "tJber ein Paradoxon der Wahrsoheinliohkeitsrech-
nung." Sitzungsberichte der pliilos.-philol. Klasse der K. bayrische
Akademie, pp. 692-712, 1892.
Bbtjns, H. Wahrschemliohkeitsreolumng und Kollektivmasslehre. 8vo. Pp.
viii + 310 + 18. Leipzig, 1906.
"Das Gruppensehema fur zufallige Ereignisse." Abhandl. d. Leipz.
Ges. d. Wissensoh. vol. xxix. pp. 579-628, 1906.
Bbtaut, Sophib. " On the Failure of the Attempt to deduce inductive Prin-
ciples from the Mathematical Theory of Probabilities." PMl. Mag. S. 5,
No. 109, Suppl. vol. 17.
BtJiTON. " Essai d'arithmetique morale." Supplement k I'Histoire NatureUe,
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452 A TEEATISE ON PEOBABILITY
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"The Chances of Death" and other