THE

LOGICAL PROBLEM
OF
INDUCTION
G. H.
Professor

VON WRIGHT
of Philosophy

in the University

Second, enlarged Edition

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Helsinki

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The logical problem of induction

THE LOGICAL PROBLEM OF INDUCTION

DATE DUE
1

196?

THE LOGICAL PROBLEM

OF INDUCTION
by

GEORG HENRIK
Professor

WRIGHT

VON

oj Philosophy

University of Helsingfors

(Second Revised Edition)

New

York

BARNHS & NOBLH,

INC,

FIRST EDITION 1941

Acta Philosophica Fennica)

(Fasc. Ill

SECOND EDITION

1957

REPRINTED JUNE

1965

PRINTED IN GREAT BRITAIN

To

EINO KAILA

PREFACE TO THE REVISED EDITION

THIS book was

first

published in 1941 as a thesis for the doctor's

It
appeared as volume III

degree in the University of Helsingfors.

in the series Acta Philosophica Fennica.

The book having been out

of print for some time, I was invited by its present publishers, Basil
Blackwell & Mott, Ltd., to prepare a new edition of it.
The book is essentially a discussion of the traditional problem of
the Justification of Induction, sometimes also called the Problem of
Hume. It examines some main attempts at a solution of this problem:
the doctrine of synthetical judgments a priori, conventionalism,
inductive logic in the tradition of Bacon and Mill, the approach to
induction in terms of probability, and the pragmatist approach. All
these attempts contribute something important to our understanding
of the nature of induction. But in a certain respect they all fail to

accomplish what they, often at least, claim to achieve. In a concluding chapter I try to show why this 'failure' on the part of any
attempted justification of induction is inevitable, and why the view
that it has catastrophic implications rests on a misunderstanding.
Since this book was first published, there has been a noticeable
revival of interest in Inductive Logic. The new interest, however,
has chiefly been in theformal aspects of the relation between premisses
and conclusion in inductive arguments. The two main branches of
this formal study may be called Elimination-Theory and Confirmation-Theory. The former is a further development of the tradition in
inductive logic founded by Bacon. The latter is essentially a theory
of inductive probability. In another work, called A Treatise on

&

Induction and Probability (Routledge

Kegaji Paul, London 1951)
I have tried to contribute something to both these branches of formal
study.

were now to write afresh on the Problem of Hume, I should

probably write a very different book from this one. Not that the
present work expresses opinions which conflict with my present
views. But on many topics, here but lightly touched, I feel the need
If I

for a

more profound

discussion. This feeling attaches particularly to

vii

PREFACE
the treatment of synthetical judgments a priori in Chap.
now think is superficial.

II,

which

In order, however, not to attempt what I believe to be an unhappy

compromise between the original set-up of the discussion and a new
method of dealing with the problems, I have tried to interfere with
the original text as little as possible
beyond correcting downright
errors and adding references to more recent literature. Some sections
These
I have preferred to re-write entirely rather than to revise.
sections are the following:
3 of Chap. I, Remarks about various usages of the term 'induction \
has been re-written and considerably expanded by the inclusion,
among other things, of some comments on Aristotle's doctrine of

induction and on the mutual relation between generalization and

inference from particulars to particulars (induction versus eduction).
4 of Chap. IV, The mechanism of elimination, has been re-written
so as to

conform

elimination which

&

(Chaps. Ill
To Section

to the fuller account of the logic of induction by

Treatise on Induction and Probability
is
given in

my

IV).

1 of Chap. V, I have appended a longish Note on the role

and
Induction
Hypothesis in Science. The urge to write it came from
of
some recent attempts, which I consider unjustified, to decry the

importance of inductive inference and of the epistemological and

logical problems which it raises.
The whole of Chap. VI, Formal Analysis of Inductive Probability, is
new. The old version of it was extremely weak and guilty of many
errors. This chapter stands somewhat aside from the main theme of
the book. I hope that, in its present shape, it could serve as an
introduction to a more thorough study of an important province of
contemporary formal Inductive Logic.
From Chap. VII of the old edition the first Section has been omitted.
Its content is incorporated in Section 2 of the
present Chap. VI.
2 of Chap. VIII, Reichenbach's Method of Correction, has been
re-written. The old version was neither accurate as an account nor
fair as an appreciation of Reichenbach's opinions. To the same
chapter has been added in the revised edition Section 3, The goodness
of inductive policies reconsidered. Here the problem of the justification
of induction is viewed from an angle somewhat different from that
adopted in the rest of the book. Had I written a completely new book
viii

PREFACE
on induction,
in this section

should probably have

more pervading of the

The Notes contain mainly

made the point of view adopted

discussion throughout.

historical material.

scholarship in the history of learning or thought.

occasionally been able to give hints which may

I have no claim to
But I have perhaps
be of interest even

to the professional historian of ideas.

I have tried to bring up to date the Bibliography of the 1941
edition. I have also added to it some items of an earlier date, and

omitted from
lists

others as irrelevant.

it

works and

and probability.
students will find

It
it

The Bibliography primarily

on the

epistemological aspects of induction

cannot pretend to be complete, but I hope

articles

useful.

Dr. C. D. Broad, from whose kindness and knowledge I have much

profited throughout my work in the field of induction and probability,
has very kindly read the revised sections and contributed both
linguistic corrections,
I

and valuable comments on the subject-matter.

am most

grateful for his assistance.

I dedicate this revised edition of my

book

to

my

first

master in

philosophy, as a token of gratitude for what he has taught me and of

admiration for what he has done to encourage serious study in logic

and philosophy

in

my

country.

GEORG HENRIK VON WRIGHT

Helsingfors, Finland
May 1955

CONTENTS
PREFACE
I.

INTRODUCTORY REMARKS ON INDUCTION

1. Inductive inference and the problem of induction.
2.
3.

Induction and eduction.

INDUCTION AND SYNTHETICAL JUDGMENTS A

1

2.
3.

4.
5.

Justification a priori

of induction.

Hume's theory of causation.

Kant and Hume.
Kant and the application-problem.
The inductive problem in the school of

13

22
27
29

Fries.

6.

Some

7.

General remarks about synthetical judgments a

other theories of causation.

33
38

priori,
III.

Different forms of inductive generalizations.

Remarks about various usages of the term 'induction'.

II.

Vii

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

1.

The way

in

which conventions enter into induc-

Some examples.
Conventionalism as an 'elimination' of the

40

tive investigations.
2.

in-

46

ductive problem.
3.

4.

Conventionalism and prediction.

Conventionalism and the justification of induction

48
.

50

IV. INDUCTIVE LOGIC

1.

2.

3.

a posteriori of induction.
Induction and discovery. Induction and deduc-

54

tion as inverse operations.

idea and aim of induction

55

Justification

The

by

elimination.

60

CONTENTS
4.
5.

The mechanism of elimination.

Remarks about the comparative
methods of Agreement and

6.

7.

8.

64
value of the

73

Difference.

The general postulates of induction by elimination.

The justification of the postulates of eliminative

76

induction.

81

The

eliminative

method and

the justification of

induction.

V. INDUCTION

84

AND PROBABILITY

1.

The

85

A scheme for the treatment of inductive probability.

88

hypothetical character of induction.

2. Hypothetical induction and probable knowledge.

VI.

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

1. The abstract calculus of
probability.
2. The interpretation of formal probability.
3. The doctrine of inverse probability.

86

90
97
102

Criticism of inverse probability.

112

5.

Confirmation and probability.

117

6.

The Paradoxes of Confirmation.

122

7.

Confirmation and elimination.

127

4.

8.

Probability,

scope and

from analogy.

Reasoning
simplicity.
Mathematical and philosophi132

cal probability.

VIL PROBABILITY AND THE JUSTIFICATION OF INDUCTION

1.

2.
3.

4.
5.

and degrees of belief.

Rationality of beliefs and success in predictions.
The Cancelling-out of Chance and the Theorem of

138

Bernoulli.

144

Probability

The idea of 'probable success'.

Logical and psychological, absolute and
justification

149
relative

of induction with probability.

xi

141

153

CONTENTS
VIII.

INDUCTION AS A SELF-CORRECTING OPERATION

1.

Induction the best

mode

unknown. The ideas of

IX.

of reasoning about the

Peirce.

59

2.

Reichenbach's Method of Correction.

163

3.

The goodness of inductive

167

policies reconsidered.

SUMMARY AND CONCLUSIONS

1. The thesis of the
'impossibility' of

justifying

176

induction.
2.
3.

The
The

logical nature of Hume's 'scepticism'.

critical and the constructive task of inductive

178

183

philosophy.

NOTES

185

BIBLIOGRAPHY

227

Xll

CHAPTER

INTRODUCTORY REMARKS ON INDUCTION

1.

Inductive inference

and

the problem of induction.

BY an

inductive inference we mean roughly this: From the fact that

something is true of a certain number of members of a class, we
conclude that the same thing will be true of unknown members of that
class also.

If this conclusion
applies to

unexamined members of the

class,

we

an unlimited number of

say that the induction has led

to the establishment of a generalization. As this is the most

important type of inductive inference, induction is often defined as the
process by which we proceed from particular to general, or from less
1
general to more general propositions.
It is, however, not
necessary that an inductive inference should
lead to a generalization.
may also extend the conclusion to a
limited number of unknown members of the class,
to the next

We

e.g.

member which

turns up, thus proceeding from

particulars to a new
2
particular. Both cases of inductive inference, that from particulars
to universals and that from
particulars to particulars, are covered by
the definition of /induction as
reasoning from the known to the
unknown. 8 This" definition also includes induction as reasoning from
the past to the future. It must be observed that this time-character-

of inductive inference, which is sometimes mentioned in the

it, is of no essential importance, and that induction
may also proceed from past cases to other unexamined instances
belonging to the past.
Inductive inference, as is well known, plays an
important role both
in science and in
every-day life. When the general aim of science is
characterized by the words *savoir pour prevoir', then science is
conceived as a system of well-established inductions. Still more
fundamental is the importance of induction as the basis of almost all
our actions. When I assume that the same food that nourished me
istic

definition of

yesterday will nourish

me

today, or that
l

if I

put

my hand

into the

THE LOGICAL PROBLEM OF INDUCTION

am making

fire it will hurt, I

where there

is

Also in many
inference, we may be

inductive inference.

no question of conscious

cases,
said to 'act inductively'.

But what are the criteria that an inductive inference is legitimate,

how do we know that from what has been true of hitherto
examined members of a class we can infer something as to other

i.e.

members
are

also of the class? Is

it
possible, provided certain conditions
inference must necessarily be
that
an
inductive
prove
what are these conditions? Or, if this be not the case, is it

fulfilled, to

valid,

and

then perhaps possible to determine which inductions are likely and

which again are unlikely to be true, so that we shall be able to avoid
'bad' inductions and by keeping to the 'good' ones arrive at the
truth at least in a majority of cases?
In all these questions the validity of inductive arguments is asked
for, i.e. the logical nature of the relation that prevails between the
evidence on which the induction is based as premisses, and the

induced proposition

by

these questions

itself

we

as conclusion.

The problem constituted

shall, therefore, call the logical

problem of
With it can be contrasted the question of the factual origin
of inductions from observations, i.e. of the psychological conditions
that are essential for the discovery of invariances and laws in the flux
of phenomena, and of the practical rules of scientific methodology,
which can be abstracted from those conditions. This problem, which
falls outside the scope of the present treatise, might be called
It deserves mention that
the psychological problem of induction.
whereas philosophy of induction in England has been predominantly concerned with the former aspect of the problem, French
authors on questions of induction have mostly taken interest in the
induction.

latter.

Historically the logical problem of induction has, ever since the

appearance of David Hume's Treatise on Human Nature two
hundred years ago, been closely tied up with the question about the
justification of our use of inductive arguments. The use of induction,
we are inclined to think, would not be rational, unless we can

by substituting for the mere belief that induction will lead to

the truth some guarantees that, under given conditions and with
specified limitations, it will actually do so. If such a justification
cannot be given, the foundation of human knowledge about empirical

justify

it

INTRODUCTORY REMARKS ON INDUCTION

seems to sink into a bog of irrationality, fatal for the superb
of natural laws and well-confirmed empirical rules based upon
it.
It is the chief aim of this treatise to clarify the muddles in the
philosophy of induction, which originate from the 'sceptical' results
of Hume. The logical problem of induction, therefore, as treated
by us, will predominantly be viewed in its relation to the classical
question about the justification of induction.
It might be suggested that there is a fundamental difference
between the problem of justifying induction in cases where the
reality
edifice

inference involves a generalization, and the same problem in cases

where the inference is extended only to a limited number of cases.

This apparent difference is connected with the fact that inductions

covering an unlimited range of instances are unverifiable in the sense
that we can experience only a finite number of cases covered by them,
whereas inductions extended solely to a limited number of new cases
are at least 'in principle' verifiable,

i.e.

all the cases

may

fall

within

our experience and be verified separately. It is, however, important

to note that the problem of induction as we are going to treat it has
nothing to do with the possibility of finding a subsequent verification
of inductive inferences, i.e. of verifying them by verifying all their
single instances. The justification we are looking for is a justification
of inductive inferences before their actual experiential verification,
and this problem is essentially the same when the induction under
consideration applies to the next member only of the class in question
and when it applies to an infinite multitude of possible future
members.*
In the following pages we shall mainly speak of inductions as
generalizations, i.e. as extended to an unlimited number of instances,
except where otherwise mentioned. Our first task will be to examine
some different logical forms which inductive generalizations may
assume.
2.

Different forms of inductive generalizations.

Our

of an inductive generalization covers several

In the simplest case we infer from the fact that all
observed members of a class A have a property R that all unknown
members of the class will have the same property. The generalization
is thus of the form: 1
definition

different cases.

THE LOGICAL PROBLEM OF INDUCTION

(X)

(1)

that

is

to say,

it is

[AX+BX},

true for every x, that if x is A, it is also B. If it is

B is A, the general implication takes the form

true as well that every

of a general equivalence between Ax and Bx.

One of the predicates A and 5, or both of them, may also have more
than one argument. This is for example the case with the generalization: any two elements of the class A have the relation B to each
other, which is of the form:

W(y)[Ax&Ay+B(x,y)].

(2)

Evidently a great majority of natural laws in advanced sciences such

as physics and astronomy concern the relations in which objects

stand to each other and not the

Inductions of the form

'properties'

of single objects.

(1) are therefore to be regarded as fairly

'primitive *.

The generalization may further be one about ordered sets of

individuals. The symbolic expression of such an induction would
for example be the following:

(x)(y)[F(x y)~(Ax+By)],

(3)

where F is the relation determining which of any x's and s together

form an ordered pair in this case. The important category of inductive generalizations, which are commonly known under the name

of causal laws, are generalizations of this type.

characteristic feature of the logical structure of generalizations
of the kind (1)
(3) is that they are general implications or equivalences involving only the universal operator. Inductions of this type
we call Universal Inductions or Universal Generalizations. 1 We
shall in the next three chapters be dealing mainly with inductions of

this kind,

write

and when

them

it is

useful to state

in the simplest

form

(1),

them

in

symbols we

except where otherwise

shall
is re*

quired.

With
tions in

the Universal Inductions can be contrasted those generalizawhich we infer that something will be true not of all the

members of a

class,

but of a certain proportion of them only, Induc4

INTRODUCTORY REMARKS ON INDUCTION

tions of this kind

we

shall call Statistical Inductions or Statistical

Generalizations. 8

Such generalizations have recently come to play

an increasingly important role in scientific inquiry, both in social
sciences and mathematical physics. The
question whether Statistical
Generalizations are to be regarded merely as 'approximations' to
the truth, being in principle replaceable by a
system of Universal
Inductions, or whether they represent 'ultimate' laws of nature, is
one of the chief controversies in the philosophy of modern natural
science.*

Generalizations speak about finite proportions of

multitudes of elements. The concept of a
proportion which
has a clear-cut meaning when applied to a finite number of enumerable
elements or to a series of an infinite number of elements,
given
according to a rule determining the characteristics of each one of the
Statistical

infinite

members

in the series, becomes problematic when we

apply it, as is
the case in Statistical Generalizations, to an unlimited multitude of
what we may call 'empirically given' elements. 5 In order to show
that the conception of a proportion has a definite
meaning even in

we shall examine the following symbolic

expression which is equivalent to the statement that a proportion p
of all elements which are A, are also B:

this last-mentioned case

N
z==i

\n>m&
(Elq)(*)(xJ(ExJ\n>m&
L

(4)

In order to understand
each instance of A as

=pe
n

i
I

formula, suppose first of all that we give

occurs an ordinal number: the first, the

this
it

Then the expression (4) says that a

., the rc'th and so on.
proportion/? of all the A 's are 5, if there exists one and only one real
number p between and 1, including these two limits such that
for any element xm there exists in the series a later element xn at
second,

elements which -are both A and B to the total

n of examined elements of the class A, falls in the interval

which the

number

ratio,

of

all

e can be
any amount however small. We say, in other
a
if
words, that proportion p of the elements which are A are also
,

/>

where

THE LOGICAL PROBLEM OF INDUCTION

over and over again it happens that, when the selection of A's is
enlarged, the proportion of 5's among them sooner or later becomes
'practically equal' to p. One further condition must be added, viz.
is the only number for which this is true. We could
imagine
where more than one number had this property. In such
cases we should say that there is no definite
proportion of A s which

that

cases

are 5.

The most interesting feature of this definition of a proportion is,

for the purpose of our inquiry, that it is applicable also to series of
elements, the characteristics of which are given empirically or
and not according to mathematical rule. For the
formula shows that for this applicability it is only necessary that
the elements of the class A must be enumerable, i.e. it must be
possible to count them as they occur and to give to each one of
them an ordinal number, thus ordering them in a series. This
extensionally,

is fulfilled for all series or collections

('populations')
of empirically given elements to which inductive generalizations

condition

apply.

That

this is the case follows

from what we mean by

'empiri-

cally given'.

proportion as defined by us is nothing but the limiting value of a

certain relative frequency in an ever-increasing collection of elements. 9
It is very important to note that we can
apply the concept of a limit
(or proportion) to extensionally given series of elements without
saying anything more about the way in which the characteristics are
distributed in the series than

is

contained in the definition

itself

of

The question of

further properties of this distribution

(randomness, irregularity, 'Nachwirkungsfreiheit' and so on) occurs
first when we have to determine the connection between Statistical

the concept.

Generalizations and Probability-Laws, between propositions like

'a proportion^ of all A's are B* and *it is
probable to degree/? that

an x, taken at random, which is A will also be J5'

We have already mentioned that inductive generalizations are
.

unverifiable propositions in the sense that we can never verify more

than a finite number of their instances. This unverifiability applied

equally to Universal and to Statistical Generalizations. But in

respect of falsifiability there is not a corresponding symmetry between
the

two types of

falsified if

generalization.

a single proposition

universal implication like (1) is

turns out to be true; i.e. if

Aa &~Ba
6

INTRODUCTORY REMARKS ON INDUCTION

one individual which, although it is A, is not B, then
maintain that all A's are B. But if we had
obviously
that
10
asserted, say,
per cent of the ^4's were B, there is not a corresof
ponding possibility
falsifying the statement. For even if it hapthat
the
actual
pened
proportion of observed A's which are B
deviates to any extent from 10 per cent, it is always conceivable that,
by extending our observations to new A's, we finally arrive at a
point where the observed proportion will equal 10 per cent. And
there

is

at least

it is

false to

as this might happen over

the true proportion of A's

and over again, after every deviation,

which are B may after all be just this

percentage.
Statistical Generalizations

fore, are neither verifiable

or assertions about proportions, therefalsifiable propositions. This fact

nor

which simply follows from what we mean by saying that such and
such a proportion of the elements in a class have a certain property,
must not be considered alarming from an epistemological point of
view. 10
It must further be noted that from the logical structure of the
formulae (1)
(4) it follows that a Universal Generalization is not
a special or extreme case of a Statistical Generalization, that is to
say a Statistical Generalization where the proportion in question is
1.
For a proportion 1 or 100 per cent of the ^4's might be 5, and
there might still exist an infinite multitude of A's which are not
B. Conversely, that
per cent of the A's are B is in no way
incompatible with the existence of an infinite number of ^4's which

arej?. 11

We have still to mention those inductive generalizations, which

resemble Statistical Inductions in that their symbolic expressions
contain both universal and existential operators, for which reason
they are neither verifiable nor falsifiable propositions, but which
nevertheless do not assert anything about proportions. For example:
i.e. a velocity which is
there exists a maximum velocity in nature

or there exists a minimum quantity

all other velocities
of energy. Of this type also is the following proposition: To
every species of flower with honey there exists, for the purpose of
This
fertilization, an insect which is able to reach the honey.
in
his
prediction of
proposition may be said to have guided Darwin
the existence of the insect which fertilizes Angraeum sesquipetale.
greater than

THE LOGICAL PROBLEM OF INDUCTION

The symbolic expression of

(x)

(5)

3.

Remarks about

this

proposition would have the form:

{Ax-+(Ey) [By&R

(x, y)]}.

various usages of the term 'induction*.

Induction

and eduction.

The logician's term 'induction' is a translation of the Greek e-ncxyooyfi

which occurs in the logical works of Aristotle. It is noteworthy that
Aristotle uses the term in three different ways. 1
In the Topics, which is probably among the earliest of his logical
2
writings, Aristotle defines induction as *a passage from individuals
to univer sals'. As an example he gives 3 'the argument that supposing

the skilled pilot is the most effective,

charioteer, then in general the skilled man
task'.

known

Of

this

kind of induction he

and

likewise the

skilled

the best at his particular

it
that
says*
proceeds 'from the
is

unknown'.

to the

In the Prior Analytics treatment of induction is linked with the

6
theory of the syllogism. The account is not very clear. Aristotle
7
8
gives an example which can be rendered as follows: Man, the horse,

and the mule are

long-lived.

the bileless animals.

lived.

It

is

But man, the horse, and the mule are

Therefore

here essential that

all

the bileless animals are long'induction proceeds through an

all

enumeration of all the cases'. 8

In the Posterior Analytics, finally, induction

said 9 to impart new

knowledge by 'exhibiting the universal as implicit in the clearly
known particular'. In induction we abstract, through an act of
intuition,

10

instance of

is

a general truth from considerations of a particular

it.

It is essential

to the Aristotelian doctrine that

know-

11
ledge of particulars is possible only through sense-perception.
Text-books on inductive logic usually mention only the two

first

kinds of inductive inference distinguished above. Induction which

proceeds 'through an enumeration of all the cases' is usually called
1*
1*
complete. It is more appropriately called summary or summative
induction. Induction in the sense which Aristotle seems to contemplate in the Topics

is

traditionally called incomplete.

8

It is

also called

INTRODUCTORY REMARKS ON INDUCTION

The

problematic.
induction. 15

name

best

for

it

seems to us to be ampliattve

Induction in the sense of the Posterior Analytics has been called

It is a faculty of the intellect

abstractive or intuitive induction. 15

which

is highly significant both for

epistemology and metaphysics.
recursive
(or
mathematical) induction one understands an
By
of
the
following type: The first member of a series has a
argument

property A. It is shown that, z/ the n'th member has this property,

then the n+Tth member has it too. From these two facts we
conclude that all members of the series have the property A. This
17
It seems
type of argument is of great importance in mathematics.
to have been consciously employed by James Bernoulli 18
sometimes also called Bernoullian induction.

first

Summative and
types of argument.
it is

and

is

recursive induction are both logically conclusive

It is of the essence of
ampliative induction that

inconclusive', that the

argument

is

conclusion goes beyond ('transcends')

follow logically from them.

'ampliative'
its

means

premisses,

i.e.

that

its

does not

Ampliative induction, though in itself inconclusive, may neverbe turned into a conclusive argument, when supplemented by

theless

certain additional premisses. 19 Inductive inference, when exhibited

in 'syllogistic' form, has been termed demonstrative induction. 20 The

supplementary premisses are sometimes referred to under the name

of Presuppositions of Induction.
An argument by summative induction can be given the following
schematic form:

A and
A and

.-.

Ally's are

and A n are all of them B

and A n are all the A's

A n can be interpreted
A and
are properties (classes). A l
alternatively as classes or as individuals. The example from Aristotle,
quoted above, answers to the first interpretation.

Summative induction

is

not, as has sometimes been said, a useless

It often renders good service to 'the

or trivial kind of argument.

economy of thought' by summing up

information contained in

its

premisses.
9

in a general formula the

Another relevant use of it

THE LOGICAL PROBLEM OF INDUCTION

occurs in mathematical proofs. In order to establish a proposition,
first 'resolve' it into a finite number of 'cases' to be

we sometimes

considered separately. Thereupon we carry out the proof of the

proposition for each case or show that the proof of some cases can

be traced back to other cases which we have already settled. Finally

we conclude by summative induction that the proposition has been
established. 21

The 'problematic' element in summative induction usually is in

the second premiss, which says that the enumerated cases exhaust
the scope of the generalization. Summative induction, however,
no problem of justification similar to that of ampliative
we have described in section 1 But there is a superthe two modes of induction which may be
between
analogy

presents

induction which
ficial

very misleading with regard to the problem of how ampliative

induction is to be justified. In both cases we draw a general conclusion after the enumeration of a certain number of single instances.
This suggests the same name, viz. that of induction and inductive
in possession of this common
inference, for the two processes. Once
arrive at the further idea that, as reasoning from every one

name we

of the cases separately to all the cases leads to a certain conclusion,

so reasoning from some only of the cases to all of them leads to
something 'less' than certainty but still resembling it. This 'something' which resembles certainty without possessing its full power is
then called probability.
The idea of induction as a sort of inference and of the relation
between this kind of inference and probability thus has one of its roots
in the apparent analogy between so-called complete and incomplete,

summative and ampliative, induction.

In this book we are concerned exclusively with induction which is
22
ampliative. Some authors, among them Mill, even wish to restrict
the term 'induction' to ampliative reasoning.

As was already observed above (p, 1), ampliative reasoning

deserving the name of induction need not aim at the establishing of
general propositions. It may also conclude from some particular
cases to some other particular cases. This type of reasoning has been
called eduction,
10

INTRODUCTORY REMARKS ON INDUCTION

It is convenient to define the distinction between
(generalizing)
induction and eduction in such a
way that inference from some
instances of a class to
number of new instances of the class

any finite
counts as eduction
even if this number of new instances should
happen, with the old ones, to exhaust the class in question. It follows
from this convention that (genuine) generalization
always pertains to
a numerically unrestricted
class of cases.
('potentially infinite')

though recognizing inference from particulars to particulars, 1 *

was of the opinion that 'whenever, from a set of
particular cases, we
can legitimately draw any inference, we
make our
may
Mill,

legitimately

inference a general one'. 25 This, we believe,

expresses an important
insight. The following considerations will perhaps serve to make
this insight

more

explicit:

Let us assume that all ,4's so far observed have been B. On the
basis of this we are
the next
willing to assert that the next

(or

which turns up

few)

be B, But we are not

willing to assert the
9
general proposition that all A s are B. What could be the reason for
will also

this hesitation to generalize?

seems off-hand plausible to think that hesitation to

generalize
to fear that the circumstances under which the A's so far
have been observed might have been peculiar to the observed /4's
and that a variation in accompanying circumstances will affect the
occurrence of the property B in other ,4 's. Fn other words, we entertain
a suspicion that some feature (or features) C, other than A, is to be
It

must be due

held "responsible* for the fact that all ,4's so far observed have been
B. If C is present in an A, then this A will be 5. But if C is absent
from an A 9 B may be absent too. Thus the eduction that the next

be

B gets its

legitimacy' from our belief in a general proposition

to the effect that all A's which are C are also B and our belief that

will

whatever it may be, will accompany the next A.

truth contained in Mill's dictum thus
appears to be, that
being 'backed* by a general truth is part of what we mean by a
legitimate eduction. And this would serve to indicate that use of
eduction is 'logically secondary 9 to generalization. 26

this C,

The

In recent times Carnap 27 has emphasized the importance of

ampliathan universal inference, the importance of which
he thinks has been overrated in traditional
's
theory.
tive inference other

Carnap

attitude

is

probably influenced by the fact that in his system of

THE LOGICAL PROBLEM OF INDUCTION

inductive logic general propositions

(laws)

always have a zero

28
This fact, however,
probability relative to their confirming evidence.
would seem to us to indicate a peculiar limitation in Carnap's theory

rather than a limitation in the relevance of generalizing induction. 20

Classical theory of induction may have unduly neglected the study

of eductive inferences, but it

rates the practical

is

even more obvious that Carnap underlogical interest of generalization.

importance and

12

CHAPTER

II

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

1.

Justification

a priori of induction.

IN the next three chapters we shall be concerned with the possibility

of justifying inductions with certainty, i.e. of proving the truth of
inductive inferences either prior to the verification of any of their
instances or after the verification of some of them. In the former

we speak of justification of induction a priori, in the latter

of justification a posteriori. It must be noted that justification a
priori does not exclude the possibility that instances of the induction have been recorded previously. The observation of such
instances may even be a psychologically necessary condition for the
detection of the justification. That the justification is a priori means,
for the purpose of our investigation, only that factual instances play
no role in the proof of the truth of the induction.
There is a prima facie presumption in favour of the possibility
of justifying inductive inference by means of a priori arguments in
case

certain typical cases. An instance is afforded by so-called causal

laws. If we observe only that A is regularly followed by B, we cannot
infer with certainty that the same will always be the case. But it

A to 5,
began
causally connected with B.
with the observation of a certain regularity, and found upon closer
sometimes happens that we can 'explain' the sequence from

e.g.

by discovering that

We

is

investigation a reason for it. This reason, we say, is a justification of

the induction that the observed regularity will hold also for the future.

We

shall first discuss this

argument from causal connections. The

which we arrive we shall automatically be able to extend to

the problem as a whole of justifying induction a priori.
results at

2.

Hume's

theory of causation.

There are three fundamental types of inductive generalizations

based upon alleged causal connections. That A is the cause of B
13

THE LOGICAL PROBLEM OF INDUCTION

may

first

of

all

imply that whenever

occurs

it

will

be followed by
But the

say that A is a sufficient condition of B.

causal relation may also entail that when B has happened

B.

In this case

we

always have been preceded by A.

Here

is

it

must

called a necessary

In the third place A may be both a sufficient and

of B. In the two first cases the inductive generalicondition
necessary
zation, expressed in symbols, is a general implication; in the third
case again it is a general equivalence. 1

condition of B.

A and B
the
an
inductive
of
of
above-mentioned
any
generalization
justifies
types, we have to examine the logical structure of the relation between
cause and effect. For the purpose of this investigation we can follow
the lines of Hume. It must be noted that Hume, in speaking about
cause and effect, chiefly had in mind relations where one term is a
In order to see whether a causal connection between

of the other. 2 This, however, does not in any

way restrict the field of applicability of his theory of causation.
Hume's argument against the view that causal connection between
A and B would involve some power or force making B a necessary
consequent of A, or in other words that causal connection would
justify induction, is best illustrated with his own well-known example
of the billiard-balls. 3
sufficient condition

We commonly say that the impact of one moving ball against

another stationary one is the cause of the second ball's movement,
What is the content of this assertion? If I first consider a single case
actually experience, which is relevant to my
and effect, is the movement of the first ball, its
impact against the second one and finally the second ball's movement.
In this we find no additional experience of a causal power, except
perhaps in the purely psychological sense, which taken together with
the experience of the movement of the first ball and the impact
would assure us of the second ball's movement as an event which
must necessarily and inevitably follow.*
Again, if I consider, not a single instance of the causal law, but a

where

this

happens,

all I

assertion about cause

number of such

instances, the situation remains fundamentally the

same. In each case I experience the same succession of events, but
even if the succession repeats itself a great number of times, this
additional experience does not give us any further information as to
what will happen next. 5 If the assertion that A is the cause of B is to
14

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

imply that

always be followed by 5, the causal proposition is

in need of
justification.
In examining the validity of the
argument of Hume we have first
to note two peculiarities of it which are
open to criticism.
In the first place Hume's argument uses as a
premiss his general
empiristic thesis that any meaningful idea must be capable of
6
reduction to certain definite impressions or
sense-experiences. As
itself

will

an instance of induction and

there is no impression
corresponding to the idea of 'necessary
connection' between cause and effect, we can
safely conclude that
such a connection does not actually exist. Of course we have not
therewith denied the existence of the idea of such a connection. The
origin of this idea is a very interesting psychological problem. Hume
himself, as is well known, ascribed its origin to the force of habit. 7
Secondly, Hume in discussing causality does not distinguish
sharply the phenomenological aspect of the question from the
physical one. It is, strictly speaking, not clear whether his analysis
of causal connection applies to the way in which our
impressions of
the external objects are connected or to the connection between the

physical objects themselves. In his criticism of causality, Hume

speaks about external objects in a language of almost naive realism,

and totally ignores the problem how the language of sensations is

related to the language of things. 8 This in its turn, it
might be maintained, makes his arguments seem more plausible than if he had
sharply distinguished between the world of sensations and the world
of things.
We have next to show that Hume's criticism can be reformulated
in such a way that its two above-mentioned
peculiarities become
irrelevant to the essential points in his arguments.

we

shall conceive of

For

this

purpose

Hume's theory of

causation not as a theory

about 'matters of fact', but as an inquiry into the grammar of certain
words. The question of how far this is in accordance with the
intentions of Hume himself will be considered later.
Hume states the essence of his theory in the following words:
'There is no object, which implies the existence of any other if we

consider these objects in themselves. 9 We may re-state this formulation in the following way: From
propositions asserting the existence
of a certain object or the happening of a certain event, 10 can never
follow propositions asserting the existence of another object or the
'

THE LOGICAL PROBLEM OF INDUCTION

happening of another event, different from the first object and the
event. In this formulation two words, viz. 'follow' and 'dif11
ferent', need further elucidation.
When we say that the impact of the one billiard-ball against the
other is something 'different' from the second ball's movement after

first

the stroke, the

means

first

of

word
all

separate happenings.

here has a twofold meaning. It

experience the two events as distinct,

'different'

that

we

This difference between them

we

call their

psychological difference. But secondly, the events are different also

in the sense that the proposition asserting the taking place of the

not entailed by, or cannot be deduced from the

proposition asserting the taking place of the first event. That is their

second event

is

logical difference.

In the case of the billiard-balls, as in several other cases, logical

and psychological difference are concomitant properties. 12 But it

may also happen that two events, although they are psychologically
not different logically, i.e. the propositions asserting
after all be entailed by the propositions asserting
the other. This is not unlikely to be the case if the events are of a
rather complicated structure. (In such a case we should be inclined
different, are

one of them may

to say that the difference between the events was only Apparent'.)
It must be observed that the arguments of Hume apply only to cases

where there is at least logical difference between the causally related

events. For if the events are not logically different, then there is
actually a necessary connection between cause and effect, that is,
'necessary' in the sense of 'logically necessary'.
proposition b is said to 'follow' from a proposition a, only if

b can be deduced (derived) from a by means of principles of logic

If b follows from #, then the implication a-*b and the
alone.
If

and

we

a&b

are logically necessary propositions. 13

substitute these elucidations of what we mean by 'follow'

equivalence a*

'different' in

our formulation above of the central point in

theory, this formulation becomes tautological. For it then

says, that from propositions asserting the existence of a certain object
or the occurrence of a certain event, there never follow propositions

Hume's

asserting the existence of another object or the occurrence of another

event, if the latter propositions do not follow from the former ones,
It is at once obvious that the validity of Hume's criticism of
16

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

when reformulated

in this way, is independent of the two

which
were
mentioned
as characteristic of Hume's
peculiarities
in
its
For
form.
it
is
not affected by the truth or
theory
'original'
falsehood of his thesis about the way in which meaningful ideas
are related to impressions, nor is it dependent upon whether we
14
speak about sensations of objects or about the objects themselves.
It is also quite independent of the relation between the
physical
language and the sense-datum language. This is important, because
from the way in which Hume himself formulates his theory together
with his general thesis about the derivation of all meaningful ideas
from sense-experiences it may appear as if in some way it were
essential, for his criticism of causality, that propositions about
physical objects were translatable into the language of sense-data.
causality,

This, however, is not the case.

On the other hand it is easy to foresee that several objections will
be made against our statement of the Humean theory of causation.
It

might be objected first that, although it overcomes certain difficulties

its original formulation, it makes the theory
valueless, as the only thing that remains of it, when

present in the theory in

itself

wholly

reformulated, is a tautology, a mere truism. Second, that surely

Hume himself intended his theory to be something more than merely

an inquiry into the grammar of certain words, for which reason

our formulation omits all the philosophically interesting and controversial points about causality brought forward by Hume.

And

third, that

our conception of the term 'follow' does not cover

that kind of necessary connection which, in the opinion of the

opponents to Hume's theory, exists between cause and effect.
are not
Our answer to the first and second objections is this.

We

directly interested in the question as to whether our formulations

himself as regards the nature of causal
cover the intentions of

Hume

Nor do we wish

to decide whether our theory omits

the philosophically interesting points about causality, because it is
tautologous. The only thing we wish to do is to show that from our
reformulation of Hume's theory of causation can be demonstrated
of justifying inductions as truths a priori by reference
the
relationship.

impossibility
to causal connections.

may be truly said to have been one

of the chief aims of Hume 's own theory, we believe that our reformulation does more justice to the arguments of Hume than at first seems
As

this

17

THE LOGICAL PROBLEM OF INDUCTION

We

endeavour, moreover, to suggest that most of the

outbursts about the absurdity of the Humean theory,
fatal consequences for practical life, and about its destruc-

to be the case.

common

so

about

its

emanate from the failure to see that its

primarily grammatical and from the unfounded belief
importance
that Hume's criticism would in some way weaken the foundation

tive influence

upon

science,

is

of the actual world-order."

If we could show that causal relationship does not justify induction, then we should have automatically disposed of the third
objection against our statement of Hume's argument. For the kind
of necessary connection referred to in this objection is just that
alleged property of the causal relation which is supposed to justify
induction.

We

introduce the term 'analytical' to

and the term

'synthetical' to

mean

mean

that which

logically necessary
is

neither logically

16
necessary nor self-contradictory.

when

the

word

'follow'

is

That b follows from a means,

defined as above, that b is an analytical,

logically necessary, consequence of a. We can now re-state

our formulation of Hume's argument in the following form: If the
causal relation is synthetical, i.e. if the effect is not a logically necessary consequence of the cause, then the relation cannot be analytically valid. This formulation, obviously, is also a mere tautology. But
as actually a great many causal laws, it seems, are formulated and

i.e.

applied as synthetical propositions, it follows that the causal relations

with which we are concerned in many cases do not as a matter offact
possess analytical validity. This again is no truism, but an empirical
proposition. Now the difficulty consists in seeing that if a causal
law can actually be shown to lack logical necessity, then it cannot

guarantee a priori the truth of the inductive generalization which

it

implies.
It is at this point that Hume, in our opinion, took a great step
forward in comparison with his predecessors. The fact that reasoning

from cause to

mainly used in practical life as well as in

not reasoning involving logical necessity, was
seen and pointed out long before Hume by a great many philosophers
of various schools and periods. It can therefore be justly said that
'ware dies der Kern und Inhalt seiner Lehre, so ware er in der Tat an
effect, as

scientific inquiry, is

keinem Punkte

17
xiber die antike Skepsis hinausgelangt'.
It is

18

a most

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

interesting fact which the ancient sceptics already clearly apprehended, that causal reasoning as an inference of one fact from another
different one was never logically necessary, and their opinion in this
matter was, according to themselves, derived from the teachings of
the sophists. 18 During the Middle Ages this opinion was not unknown among scholastic philosophers. 19 Later Hobbes, 20 Male22
21
branche, and Leibniz expressed similar views.
None of these philosophers, however, seems to have realized that
if reasoning from cause to effect is void of logical necessity it cannot
justify induction. In so far as the inductive problem occurred to their
minds at all, they mostly concluded that induction must be justified
by means of some rational principle of a not-tautological character.
In other words they assumed, for the purpose of justifying induction,
the existence of a kind of necessary connection other than the
analytical one, a kind of necessity that was assumed to be a possible
property of synthetical relations. This assumption is the kernel of
the doctrine of synthetical judgments a priori. It is expressed very

by Leibniz, who

having shown that there is no

reasoning involving logical necessity which from the examination of
single instances leads to a universal synthetical truth, says: *Hinc
sine adminiculo
jam patet, inductionem per se nihil producere
ratione
universali
ab
non
sed
inductione,
pendenpropositionum
explicitly

after

tium.' 23

Hume's

greatest contribution to philosophy,

having seen that

this

assumption

is

unjustified.

we

think, consists in

And as was indicated

above we want to show that his demonstration of this can be carried

out also on the basis of our tautological reformulation of his argument.

Suppose somebody says that the event B is so connected with the

event A, to which it stands in a synthetical and not an analytical

has taken place

relation, that when
necessarily follows. What
does this assertion convey? In the first place perhaps something like
will
this:
feel perfectly sure that after the occurrence of A,

We

In so far as this statement

follow within a certain interval of time.

is meant to be merely the expression of a psychological fact, which
make of a causal
actually in most cases is inseparable from the use we
B
law, then it does not tell us anything as to whether
really will follow

or not.

It is therefore

obvious that reference to this psychological

19

THE LOGICAL PROBLEM OF INDUCTION

convey the whole meaning of the assertion that the
connection between A and B is necessary. With this we want to say
something more, something which at the same time will justify the
belief or the conviction which we have that B must follow A.
It is not immediately clear what this 'more', i.e. this rational
ground or whatever we like to call it, could be, but it is at any rate
obvious that if the justification is intended to be a guarantee for the
truth of the causal proposition a priori, one of its chief functions is
to exclude the possibility of A occurring and not being followed by
B. Suppose, however, that it were alleged that this had happened
in spite of this 'more'. How is this to be dealt with? Shall we say
that perhaps B after all was not in necessary connection with A!
If we do this, then the causal proposition was itself a kind of induction and as such in need of justification. The only alternative to this
again is that we 'save' the truth of the statement that B must follow
A by saying either that B actually followed upon the occurrence of
A, although for some reason it escaped notice, or that the event
which took place first was only 'apparently', but not 'really', A.
About this 'saving' of the causal proposition the following must
be observed. If we save the proposition by supposing that one of the
two things which fall under the latter alternative be true, then this

fact does not

supposition may later be shown to be false. So we are thrown back

on the first alternative, viz. that which made the causal law itself an
induction. In order to avoid this we must do the 'saving' in such

within a certain
that the proposition that B must follow
is made the standard for the truth of the expression
was not a "real" A\ And if this
'either B has escaped notice or

way

interval of time

done, the causal law becomes analytically valid,

of the way in which certain words are used.
is

i.e.

valid because

Now it is obvious that if the proposition that B necessarily follows

A

is to be true a
priori, then we must adopt the second of the two
above alternatives. But then, as it has been shown, the proposition
becomes analytical. This on the other hand contradicts the condition
that the relation between A and B was synthetical., Therefore the

assumption that we could

proposition by showing

it

justify the belief in

to be true

a priori

is

a synthetical causal

contradictory.
Thus the 'more than a mere psychological fact' that is contained
in the assertion that B necessarily follows upon A, cannot, whatever
20

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

'more' might be, under any circumstances
guarantee the truth
a priori. On the other hand we must not overlook
that the psychological fact has some
bearing upon the problem of

this

of

this assertion

justifying induction. If,

relation exists between

by introducing an assertion that a causal

and 5, we raise a law about an observed
the higher rank of a causal law, this fact in itself stands

regularity to
as an expression of increased confidence in the
proposition that
will follow A. (This stronger
of
confidence
or reliability, by
feeling
the way, is one of the reasons why we speak of certain
generalizations
as 'laws of nature', as opposed to others as
being mere 'general

24

or why under certain circumstances we regard the reduction

of an observed uniformity to another as an explanation of the first
one. 25 ) It is not altogether out of the way, moreover, to call this
increased confidence a justification of our previous
assumption that
the observed regularity could be generalized. We have
only to
remember that this 'justification' does not tell us anything about the
'real' truth of the induction, but is
solely an expression of our belief

facts',

in this truth.

The above conclusion as to the impossibility of justifying induction

by reference to causal relations which, although synthetical, were
true a priori, has been reached by an analysis of the
meaning of
certain words and expressions. From the way in which the
analysis
has been pursued it is seen, as will later be shown in detail, that the
result applies not only to the causal relation but to all the cases where
alleged that the truth of a proposition, which is not logically
necessary, could be guaranteed a priori.
To our analysis the following objection is conceivable.
do we

it is

How

know

that the meaning given

by us to the analysed expressions is

the 'true' one? Is it not plausible to assume that some of those
philosophers who have objected to the arguments of Hume have
used these expressions with a meaning different from ours, and that
this meaning of theirs has enabled them truly to regard causal
relatedness, for example, as a means of justifying the truth of inducassumptions a priori!
dispute about the 'true' meaning is futile. We do not pretend
that certain expressions, as used by us, mean exactly the same as when
used by other philosophers who have expressed different opinions
about the matters here under discussion. By examining some types
tive

The

21

THE LOGICAL PROBLEM OF INDUCTION

of argument which are intended to 'refute' those of

we endeavour

Hume

about

show

that this possible difference in the

causality
of
does
have any bearing upon the
terms,
not,
however,
meaning
in
difference
opinion.
possible

3.

to

Kant and Hume.

It was mentioned above that Hume in his analysis of causality did

not distinguish sharply between the world of things and the world of
between the physical and the phenomenal. Kant too,
sensations
in his earlier writings does not seem to have been aware of the
problems which arise when we begin to speak about things as being
different from our sensations of them. 1 It is, moreover, interesting to
note that as long as he had not become conscious of the importance
of these problems, his view as to the possibility of establishing the
a priori truth of any empirical proposition was in full accordance with
the opinion of Hume. 2
It is one of Kant's chief merits, however, that he saw with such
extraordinary clarity the bundle of difficult philosophical questions
which spring up when we pass from the realm of our private experito the realm of objective or rather, interences
our sensations
i.e. to the realm of that which we have called
subjective experience,

physical objects and events. Kant employs the word 'Wahrnehmung'

as opposed to 'Erfahrung' in roughly the same way as we employ the
word 'phenomenal' as opposed to 'physical'. 8 The sensations are,
so to speak, immediately given to us.

how we can

The

difficulty consists in seeing

acquire from them inter-subjective experience, or as

the problem is one of the 'Moglichkeit der Erfahrung'. 4

Kant puts it,

It was not until Kant

realized the significance of this

problem that

towards the possibility of synthetical judgments a priori

underwent a radical change. 5 In his attempt to solve the problem he
arrived at results which, in his own opinion, conflicted with the
essential point in Hume's theory of causation. 6 It will be our task to
show that Kant's results do not vitiate Hume's doctrine as reformulated above and that they consequently do not cause any modification
in our view as to the possibility of justifying induction by means of
his attitude

a priori arguments.
22

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

According to Kant the passage from our subjective sensations to
inter-subjectively valid judgments on nature is possible through a
'Synthesis', i.e. a rational process which transforms the 'material'
7
given by the sensation into an inter-subjectively valid experience.
As an illustration of the way in which this 'Synthesis' operates, we
will briefly examine the ideas laid down in Kant's famous Second
Analogy of Experience, where he tries to prove the necessity of the
Universal Law of Causation. 8 It seems plausible to re-state the
'points' in Kant's argument roughly as follows:
In speaking about cause and effect we usually presuppose the
existence of an objective time-order, i.e. of inter-subjectively valid
judgments about 'before' and 'after'. But how do we know that an
objective time exists? Hume spoke about time as a sort of container
in which events take place, one after the other. On this basis he
showed that causality is merely a regular sequence in time and not a
necessary connection between different events. Kant may be said,
with Whitehead, to accuse Hume of 'an extraordinary naive assumption of time as pure succession'. 9 He attacks Hume's theory by
showing that the assumption of an objective time rests upon the

assumption a priori of the truth of causal laws. Time, in other words,

presupposes causality.
The ideas of 'before' and 'after' have, in the first place, a purely
'subjective' meaning, as denoting certain phenomenological features
of the way in which sensations succeed one another. When we say
that physical objects or events stand to each other in the relation of

and

'before'
different.

'after', temporal relatedness means something entirely

In order to see how the mind acquires the idea of an

objective time,

we have

to examine

more

closely the nature of

experiences of successions among sensations.

I cast my eye on the wall of a house and view

it

our

from top to bottom.

My sensations of the various parts of the wall are, in the phenomenological sense, temporally related.

floating

down

a stream.

I also

Here again

follow with

my

my

eye a vessel

sensations of the various

positions of the vessel are temporally related. Nevertheless I am

inclined, in the former case, to say that the various parts of the
(physical) house exist simultaneously, whereas the various (physical)
positions of the vessel exist successively. In the one case 'before' and
'after' in the succession of sensations correspond to 'before' and
23

THE LOGICAL PROBLEM OF INDUCTION

realm of things, in the other case not.
arrive at the notion of this difference?

'after' in the

mind

How

does the

We

get the answer by pointing, with Kant, to a remarkable

difference in the succession of sensations in the two cases. In the case

of the house this succession can be reversed: instead of experiencing

the parts of the wall from top to bottom, I might have experienced
them in the opposite direction. But in the case of the vessel, the
succession of sensations cannot be reversed. As Kant himself
remarks: 'Ich sehe z.B. ein Schiff den Strom hinab treiben. Meine
Wahrnehmung seiner Stelle unterhalb folgt auf die Wahrnehmung
der Stelle desselben oberhalb dem Laufe des Flusses, und es ist
unmftglich, dass in der Apprehension dieser Erscheinung das Schiff
zuerst unterhalb, nachher aber oberhalb des Stromes wahrgenommen
werden sollte. Die Ordnung in der Folge der Wahrnehmungen in der
Apprehension ist hier also bestimmt, und an dieselbe ist die letztere

dem

vorigen Beispiele von einem Hause konnten

Wahrnehmungen in der Apprehension von der Spitze desselben
anfangen und beim Boden endigen, aber auch von unten anfangen
und oben endigen, imgleichen rechts oder links das Mannigfaltige der
empirischen Anschauung apprehendieren. In der Reihe dieser

gebunden.

In

ineine

Wahrnehmungen war also keine bestimmte Ordnung.'

It is on the existence of such irreversible series of successive
10

sensa-

tions that the possibility of an objective time is founded. If there

existed no irreversibility in the flux of sensations, it would not be
possible to talk about 'before' and 'after' in any other than the
11
phenomenological sense of the words.
The statement that a series of sensations is irreversible is a law or
rule determining the order in which sensations succeed one another.

As

ordering rules irreversibility-statements are, in the terminology

of Kant, causal laws. 12 Thus he was entitled to say that the 'Synthesis
which, from the material given by the sensations, takes us to the idea
of an objective time, is possible for the reason, that the succession
of sensations is, in characteristic cases, governed by causal laws.
This is the meaning of the above assertion that 'time presupposes
'

causality*.

The

'Synthesis', giving to the sensations their 'Gegenstandlichkeit',

always carried out in a manner analogous to that which leads to

objective time from the time-sensations. In order that inter-subjective

is

24

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

experience may be possible,
about a world of things as

in order that we
may be able to speak
opposed to a world of sensations, it is a
i.e.

necessary condition that there shall exist certain invariant relations

or laws governing the stream of sensations. These invariances
these laws prevailing in the world of sensations
constitute the
objective, physical or 'real' world. The physical is the invariance in
the phenomenal. 13 Nature is the sum or the
of all the laws

system

which regulate our subjective experiences, 14 If there were no invariance, no law and order in the realm of sensations, there would
15
be no physical world and no
Kant's
inter-subjective experience.
would
thus
not
be
'Erfahrung'
possible.
Kant is not the first philosopher to have the idea that the
physical
is the invariance in the
phenomenal. Leibniz had already formulated
the idea clearly. 16 What is new in Kant is, above
all, the idea of 'defrom
the
of
ducing'
conception
inter-subjective experience itself a
set of rules to which the invariances
defining the physical world have
to conform. With one of these rules we are
already familiar, namely
the Universal Law of Causation, the deduction of which has been
outlined above. 17 It states that 'alles, was
geschieht (anhebt zu sein),
18
setzt etwas voraus, worauf es nach einer
Regel folgt'.
To the rules which are deduced from the general idea of 'Erfahrung' there correspond certain general concepts, called the cate1*
The category corresponding to the Universal Law of
Causation is that of Causality. The rules are said to subsume the
experiential content of the sensations under the general concepts of
20

gories.

the categories.
According to

Kant

these rules, in conformity with which the

subsumption takes

place, are synthetical judgments a priori** They

are synthetical since they,
apparently, assert something about the
course of nature. And they are a priori since their truth is a

necessary
condition for the possibility of
inter-subjective experience. They
cannot be contradicted by experience, because experience itself
presupposes them.

The Universal Law of Causation

therefore

according to Kant,
and does the
doctrine of Kant have any bearing upon the
problem of Hume?
The answer to these questions can be got from studying again
the arguments in the Second Analogy. The
synthetical aspect of
synthetical

and a priori.

is,

How is this to be understood,

25

THE LOGICAL PROBLEM OF INDUCTION

causality lies in the fact that there exist sequences of phenomena
conforming to causal laws, or else, that causality prevails in this
world as far as our experience goes. The aprioristic aspect consists

again in that the prevalence of causality (for past and for future
experience alike) can be used to define an objective time-order. Now

which makes us think of the Universal Law of

Causation as being at the same time synthetical and a priori, seems
to be this:
The actual possibility of inter-subjective knowledge indicates that
time in the objective sense of the word exists as a matter of fact.
the seductive element

From

this follows, prior to

any further experience, the unrestricted

prevalence of causality, since causality was, so to speak, a defining

characteristic of time.
In this reasoning, however, there is a serious fallacy. All we
actually know is that hitherto., on account of certain uniformities of
the way in which phenomena have occurred to us, it has been possible
to arrange experiences in an inter-subjectively valid order of time,
defined by these uniformities themselves. From this we may conclude
that if uniformities of the same kind, i.e. causal uniformities or
causality, are continuously going to exist, then it will always be
possible to arrange experiences in this objective order. It does not,
however, follow from the transcendental deduction of the Universal
Law of Causation that uniformities of the kind mentioned are going
to pervade also the realm of our coming sensations. That such will
be the case, i.e. that the law of causation will be true, is an inductive
generalization on the basis of what is hitherto known to us. This
generalization is a synthetical proposition the truth of which has
not been proved a priori.
The Universal Law of Causation, therefore, can a priori be made
a necessary condition for the existence of an objective time, but the
truth of it qua synthetical, i.e. as a proposition about the continuous
existence of time, cannot be established in advance. The apparent
possibility of doing so, by reference to time as a matter of fact,
disappears if we consider that we are thus referring, not to the
existence of time in general, but to the existence of time up to a certain
follow the unconditioned validity
point. And from this does not
of causality, or in other words, the continued existence of time.
j

Analogous arguments apply

to the other categories

26

and to the

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

judgments about the form of experience corresponding to them. In
laying down defining criteria of inter-subjective experience these
judgments are a priori and analytical. As generalizations about
certain characteristic features of the

phenomenal world they are

again inductive, synthetical judgments, the truth of which cannot be

proved a priori.

The foregoing reasoning also makes it clear

why the transcendental
deduction does not lead to a vitiation of Hume's results as to
causality and the justification of induction. We know that certain

among thein causal laws, have held true hitherto, and that
henceforth no inductions or no inductions of a certain
type are
going to hold true also, then
will not be
inductions,
if

inter-subjective experience
possible any more. But that which would be necessary in order to
justify induction, does not follow from this, viz, that causality and
induction are actually going to hold also for the
future, or, generally
speaking, that inter-subjective experience is going to continue.
4.

Kant and

the

application-problem.

Let us suppose that we had established a

priori the truth of the
Universal Law of Causation, formulated as for
example by Kant
above. 1 The following question
now
be
asked:
In what way, if
may
at all, can the
of
this
universal
truth
be
knowledge
applied to the
establishment of actual causal connections between concrete events?
Or, in other words, in what way can it provide us with a justification
of specific inductions? This problem we shall call the
applicationproblem of the Universal Law of Causation.
It is easy to see that the
question asked can only be answered in
one way, viz, that it is not possible by means of the Universal Law
of Causation alone to establish general
propositions about the actual
course of events. 2 This seems to be admitted by Kant, when he
says:
Auf mehrere Gesetze aber als die, auf denen erne Natur
uberhaupt
'

Gesetzmassigkeit der Erscheinungen in Raum und Zeit beruht,

auch das reine Verstandesvermogen nicht zu, durch blosse
Kategorien den Erscheinungen a priori Gesetze vorzuschreiben,
als

reicht

Besondere Gesetze, weil sie empirisch bestimmte

Erscheinungen
Es
betreffen, konnen davon nicht vollstandig abgekitet werden
muss Erfahrung dazu kommen, urn die letztere uberhaupt kennen
zu lernen. '*
.

27

THE LOGICAL PROBLEM OF INDUCTION

But what Kant
that, if the

in his Kritik der reinen

Universal

Vermmft has overlooked

Law of Causation does not help us

is

to a justifica-

single induction, then it does not provide us with a

of Hume's problem. If Kant's transcendental
solution
satisfactory
deduction does not enable us to guarantee the truth of any inductive

tion of

any

generalization, then it leaves the logical problem of induction at

4
precisely the same point where we had left it with Hume.

On the other hand it seems obvious that Kant believed himself to

have solved the problem of Hume, and never realized that the obstacle
put to his theory by the application-problem deprived it in fact of
5
any bearing upon the inductive problem, In his later writings,
however, he became aware of some difficulties on this point.
There are some very obscure passages in the Prolegomena where
Kant tries to answer the question as to how the Universal Law of
Causation is applied to concrete cases. 6 From these contexts one
gets the impression that he believed the category of causation to be
applicable to special cases in such a way that it could raise observed
regularities to the higher level of universal and necessary laws. It is,
however, uncertain what Kant here really means by 'necessity' and
'universality'. Does he mean simply the property of inter-subjective
validity, belonging to physical propositions as opposed to phenomenological ones, which arises from the application of the category
to the material given by the sensations, or is he thinking of necessity
and universality in that sense which justifies induction? Some
7
arguments could be put forward in favour of the first interpretation,
but it seems to us most likely that Kant himself did not clearly
separate the two aspects from each other, and therefore can be said
to have tried in some obscure way to include, even if he had the first
one predominantly in mind, the second one also in his argumentation.
This view is strengthened by the fact that Maimon, when he criticized
Kant's treatment of the problem of Hume and showed that the
Universal Law of Causation alone is not sufficient for the establishment of any single causal law, seems to have been inspired by those
very examples by means of which Kant in the Prolegomena apparently

show just

the opposite. 8
Later, however, in the Kritik der Urtheilskraft Kant is quite clear
as to the impossibility of deducing special laws of nature from the
tried to

Universal

Law

of Causation. 9 For the establishment, therefore, of

28

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

true inductions other
principles than those deduced from the general
idea of 'Erfahrung' are needed. 10 In his
attempt to formulate these
principles Kant approaches the rules and canons, like the principles
of Uniformity of Nature or Limited
Independent Variety, laid down
11
in various attempts to establish a so-called 'inductive
With
logic'.

these attempts we shall be concerned later on. It is here sufficient to

note that Kant explicitly
regarded these principles as not

provable a priori, but merely as 'subjective' 12 assumptions.

5.

being

The

Our

inductive problem in the school

of Fries.
result so far is then that Kant was not able to

Universal

Law

Furthermore

it

show

the

of Causation to be synthetical and true a

priori.
has been proved that even if Kant had shown this,

the difficulties raised by the

application-problem would have deprived
his result of all
importance for the problem of Hume.
Jacob Friedrich Fries and his followers, both his immediate

and the adherents of the recent Neo-Friesian school, are

of importance for the discussion of the
present problem both on
account of their ideas on the way in which synthetical
judgments
a priori are to be established, and because of the
special attention
disciples

given by them to the application-problem of those judgments and its

bearing upon the justification of induction.
According to the Friesians the idea of deducing synthetical judgments a priori from the concept of inter-subjective experience, or
from any other general principle, is fundamentally unsound. For
either

it

makes those judgments

analytical,

or

it

raises the further

problem as to the truth of the principle from which the deduction is

made. 2 If this principle is to be established by reference to a new idea

we

are involved in an infinite retrogression. 3 Therefore, as Fries

4
rightly points out, it is not possible by the transcendental method of
Kant to establish any judgment as being both synthetical and a
priori,

This impossibility follows from the idea that those judgments must
be proved, i.e. deduced from superior principles. Fries speaks of this
idea as 'das rationalistische Vorurtheil'. 5
Fries himself tried to establish synthetical judgments a
priori by
reference to a source of knowledge, called by him 'unmittelbare
Erkenntnis'. 6 This immediate knowledge is a sort of mental fact,

which

is

7
'urtheilsmassig wiederholt' in the

29

form of synthetical

THE LOGICAL PROBLEM OF INDUCTION

judgments a priori. It is, in other words, a purely empirical, singular,
and a posteriori detectable fact that we are in possession of general
knowledge which is synthetical, in the sense that it contains information about what is going to happen, and a priori in the sense that we
know it to be true generally prior to actual testing. To ask for a
proof of these judgments would be wholly to misunderstand the
question, as our knowledge of them is simply a fact. As such they
are to be regarded as starting-points for philosophical investigations
and not as something which these investigations themselves ought to
8

establish or justify. 9
Even if the criticism of Fries

and his followers rightly points to

defects in Kant's theory of synthetical judgments a priori, it
ought not to be difficult to see that the new theory offered by the
Friesians is no more successful as regards the establishment of these

some

judgments than the rejected one. That such is the case can be shown
by an example.
Suppose the judgment that all A's are B to be synthetical and a
priori according to the theory of Fries. Suppose further that somebody claims to have found an A which is not B. This, a Friesian
would say, is impossible as we know that all A's are B. But what does
it mean that we 'know' this? It may mean, for
example, that we use
the predicates A and B in such a way that always when A is present
we say that B is present also. If somebody claims to have found an
A without B, we should tell him that either A was no 'real' A, or A
was only 'apparently' lacking B. In this case, however, the proposition that all A's are B would be analytical as it provides us with a
standard for when the proposition 'either A is not a real A' or 'A is
only apparently lacking 5' is true. Therefore this possibility must be
ruled out.

But if, by 'knowing' that all A's are B, we mean something else,
whatever it may be, we cannot exclude the possibility of there being
an A which 'genuinely', and not merely 'apparently' lacks B. This
is seen by
considering how the phrase 'apparently lacking -B' may be
defined. It can be defined as above, i.e. the predicate B is used in
such a way that it is attributed to all A's, for which reason every
(real) A in which B is not as yet detected is said 'apparently* to lack
B. This, however, is not the only possible, nor the most natural
definition of the phrase. 'A apparently lacks 5* can also mean that it
30

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

B has first been detected in A upon closer

and
that
we
investigation
suppose the case in question to be of this
kind. Here again, however, the statement that A
apparently lacks
B is a hypothesis about what is going to happen after closer investigation of the circumstances, and it is not possible to forecast whether
has often happened that

these investigations will actually lead to the detection of

or not. If
we cannot decide this, however, we cannot decide whether A lacks

'genuinely' or only 'apparently so that if somebody claims to have

found an A without 5, we cannot decide in advance whether the
case is one where B really is lacking, or whether it has

merely escaped
This again means that we cannot exclude the
possibility that the judgment 'A is not J5' is true, or in other words

our attention so
that

far.

we cannot be

sure a priori that all ^4's are B.

therefore, 'knowing' that all A'$ are B is to

mean something
which makes the proposition not analytical
which can perfectly
well be the case
then the proposition can never at the same time
be true a priori. Thus the Friesian way of establishing synthetical
If,

judgments a priori is as insufficient as that of Kant.

It has been mentioned that the Friesians have
paid some attention
to that problem which presented a serious obstacle to the doctrine
of Kant, viz. the question of how to arrive at special inductions

from the general principles established by a priori reasoning. Apelt's

work Die Theorie der Induction illustrates the difficulties in which
the theory of synthetical judgments a priori is necessarily involved
when it has to show its applicability to concrete cases.
shall

We

here give a short account of Apelt's ideas.

Pure induction, i.e. the examination of successive instances of, say,
&$ which are J?, is not able alone to assure us of the truth of the
10
But if we can deduce this
general proposition that all A's are 5.
inductive proposition from a system of general knowledge already
known to us, it becomes a law, i.e. a generally and necessarily true
11
From this it would appear that in the
proposition aboxit nature.
opinion of Apeit not only certain principles of a very general kind,

such as the Universal Law of Causation, but also every special law
How is this to be
of nature were a synthetical proposition a priori
understood?
The explanation seems to be something like the following: From
a system of general principles alone it can be deduced that if there is
31

THE LOGICAL PROBLEM OF INDUCTION

A then it must be B, but not that A's actually exist. Thus for the
establishment of the general proposition that all A's are B induction
is necessary in order to show that A's being B actually exist, and
deduction again is necessary in order to prove the universality of this
fact. 13 Thus induction may be called the bridge which leads from the

any

facts to the laws,

truths.

14

from the contingent experiences

to the necessary

15

This interpretation of the theory is confirmed by the instances

given by Apelt. He says for example that the planetary movement
of Mars can be deduced from the general law of gravitation, but that
the actual orbit of the planet has to be determined empirically. 16
As another instance of how induction works Apelt mentions Bradley 's
detection of the aberration of light which was made empirically and
afterwards proved to be a consequence of general physical principles,
17
this proof giving it the character of a general and necessary law.
It is clear that this theory of induction is entirely based upon the
assumption that induction from facts and deduction from general
principles always give concordant results. There is one extremely
interesting passage in Apelt's work where he tries by means of an
example to show the impossibility of a contradiction between the
principles and the facts. Daniel Bernoulli and Laplace stressed that
the law that force and acceleration are proportional is no a priori
truth, since it is conceivable that experience might show the force to be
proportional, say, to the second power of the acceleration. Against
18
'Dies ist jedoch eine Irrung. Wenn ein Naturthis Apelt says:
forscher einen Fall fande, bei welchem die beobachtete Grosse der
Veranderung einer Bewegung mit der anderweit bereits bekannten
Intensitat der Kraft nicht iibereinstirnmte, so wiirde er das Gesetz

/= $

nicht in Zweifel zeihen, sondern er wiirde vermuthen, dass

ausser der zur Erklarung der Erscheinung angenommenen Kraft
noch andere Krafte mit im Spiele seien. Es lasst sich gar keine
die diesem Gesetze zuwider ware, eben weil
aus der Erfahrung folgt, sondern vor jeder bestimmten
'
Erfahrung schon a priori feststeht
This is a most beautiful instance of how the old doctrine of
synthetical judgments a priori approaches the doctrine, called
19
Here
conventionalism, which will be examined in the next chapter,
it is sufficient to note that if the impossibility of a contradiction

Beobachtung machen,
es nicht

32

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

between principles and facts is a consequence of the
way in which we
have to interpret the facts, then these
principles become analytical.
If for example an
apparent contradiction between the observations

and the law/= jt is to be interpreted as being due to the

presence of
hitherto unobserved forces, then the law is
but
the
definition
nothing
which enables us to decide when we have to
of
unobserved
speak
forces and when not. For if the law is not made the definition of the
presence of such forces, but only a 'symptom' or 'indicator' of them,
then it is not a priori certain that these forces
really are there and that
it is not the law after all that is false. But if the truth of a
principle
is a truth
guaranteed per definitionem then the principle is analytical.
The following we can say in conclusion about this
theory of induction: The school of Fries is in advance of Kant in that it has seen that
if synthetical
judgments a priori are to help us to a solution of Hume's
then
it must be
problem
possible to deduce all laws of nature from
these judgments, and thus make them too
synthetical and a priori
The attempt to do this, however, meets with insurmountable difficulties when we come to the
question of how to exclude the possibility
that experience will contradict the a
priori principles and laws. Out
of this difficulty there are only two ways. Either we retain the
aprioristic nature of the principles and laws, in which case they
become analytical, or we retain their synthetical nature, in which case
have also seen
they can no longer be known to be true a priori.

We

that in the only case where one of the adherents of the Friesian
theory faces this question, he apparently decides in favour of the
former alternative, without, however, realizing that therewith he has
also given up the doctrine about judgments being both
synthetical

and a priori. This

decision of his also indicates the course which our

further investigations as to the possibility of justifying induction by
means of a priori arguments have to take. Before this, however, we
have to show that certain other more recent theories, which are

put
forward in opposition to Hume's view about causation, also lead to
difficulties which point in the same direction for their solution.
6.

of

Some

other theories of causation.

Although the failure of the above attempts to solve the problem

Hume by reference to synthetical judgments a priori is quite
33

THE LOGICAL PROBLEM OF INDUCTION

it does not follow that modern
had
unanimously accepted Hume's results. There are,
philosophy
on the contrary, a multitude of theories put forward against his.
The most important ones are perhaps those of Whitehead on causal
perception, of Meyerson on scientific explanation, and of Bradley
and Bosanquet on concrete universals.

commonly admitted nowadays,

It

might be said that

all

these theories have the idea in

common

that a right understanding of the nature of causal relationship

of the concrete cases where such
requires a much more careful analysis
1
than
Hume
manifested
And it is alleged
are
gives.
relationships
that this closer examination of single instances of causal relationship
which Hume denied the relation between

will reveal to us the necessity

cause and effect to possess. 2

Thus

in the analysis of particular facts

We shall next discuss the different

universal truths are discoverable.' 3

meanings that this thesis might have, and its bearing on Hume 's results.
The proposition that it is possible to arrive at general knowledge
from singular facts or to anticipate an event B after the observation
of the event A can mean all sorts of different things. It may first of all
mean that as a matter of fact singular facts suggest to us general
conclusions. This is no mere truism but a point of considerable
importance,

if

we

consider the following:

the effect

Hume says, on one occasion, that

is

totally different

from

there anything in the one to suggest the smallest

hint of the other'.* This phrase interpreted in its most natural sense,

the cause

is

nor

is

certainly false. Not only is there in general something in the cau$e

'points' in direction to the effect and lets us anticipate it, but

which

a most important and interesting feature of

and the way in which we react to things.
We are, in our daily life, constantly confronted with situations which
are not very similar to situations with which we are familiar and in
which we are compelled, without the aid of previous experience, to
anticipate the right course of action. As a matter of fact we do this
in a manner which clearly shows the untenability of Hume's view
that the belief in causal connection arises as a mental habit produced
5
by the repeated impressions of the same succession of events. On this
point Whitehead 's theory about immediate causal perception gives
a much better account of the way in which we arrive at the knowledge
of causal laws than Hume's theory about the force of habit. 4
that such

is

the case

the world in which

is

we

live

34

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

If, therefore, the thesis that general knowledge is detectable in
single
facts is interpreted as above, it
expresses an important truth. The
manner, moreover, in which Whitehead treats it offers an

of how

explanation

we

as a matter of fact arrive at inductive

knowledge, and this
is
much
more
than
that
explanation
satisfactory
given by Hume. But
it is also clear that with this
of
the
statement we do not
interpretation

obtain anything of relevance to the problem whether it is

possible
prior to verification to guarantee the truth of general propositions.
From this it follows that the above interpretation is not sufficient
for the purpose of contradicting Hume's result as to the
question of
7
induction.
But
the
same
can
also
be
justifying
phrase
interpreted
That the single instance of an induction may contain
differently.
general information can also mean that an analysis of the single case will
show this to have such a constitution that the truth of the general proposition follows from it. Or, if it isa
question of a causal law, theanalysis
show
the
effect
a
to
be
may
necessary consequence of the cause.
This interpretation seems to fit in peculiarly well with Meyerson 's
theory of 'scientific explanation'. Expressed in our previous terminology, the essence of this could be formulated as follows: Hume

maintained that the effect is something 'different' from the cause

and can therefore not be a necessary consequence of it. We have seen
that this is true if 'different' means 'logically different'. But difference
may also mean psychological difference and of such cases Hume's
statement need not necessarily be true. In other words, cause and
effect may have a different appearance, but nevertheless upon closer
investigation may be shown to be the same in the sense that the effect
logically follows from the cause, or is contained in it. Now the
theory of Meyerson seems to suggest that always where there is a
question of real causal relationship, the effect upon closer investigation can be explained as a necessary consequence of the cause. 8 If
this is possible the effect is said to be 'identical' with the cause. 9
Now it is not our intention to show Meyerson 's statement to be
false. We shall, on the contrary, soon have occasion to show that it
is true, if not for all, at least for a great many such cases where we

We

only want to show that if the

effect is identical with the cause, in the sense of being a logical consequence of it, then the causal law, i.e. the inductive proposition based

speak about causal relationship.

upon

this

connection must be analytical.

35

THE LOGICAL PROBLEM OF INDUCTION

a single A and a single

stand in necessary causal
to
each
can
other
be
relationship
generalized to the statement that
when A occurs it will always be followed by B, if the A which occurs

The

fact that

10
Sameness here of
repeatedly really is the same A in all the cases.
course does not mean spatio-temporal identity but sameness in the

sense of having in common all the characteristics relevant to the

causal property. Under such circumstances the inductive proposi-

were an A apparently not followed by 5,

that
this A although it may have a certain
conclude
safely
resemblance to the previous A's is not of the same kind as those A's
which should produce B. If it 'really' were the same then it would
11
Sameness in other words has been defined so as to
produce B.

tion

is

true, because if there

we can

include the property of having the occurrence of B as a logical consequence. On the other hand, if we do not define sameness in this way
?

which

is

the truth of the inductive

that
It is

we cannot be
proposition. And if we

perfectly well conceivable,

sure in advance of

define sameness so
sure of this then the proposition becomes analytical.
this because it follows from the definition of two A's
being 'the

we can be

same'.

Thus

the theory of Meyerson is to lead to the establishment of

causal
general
propositions a priori known to be true then these
if

propositions must be analytical. This is a consequence of his theory

of which Meyerson himself does not seem to have been aware. 12

The circumstance which forced

the theory of

Meyerson

to

make

inductive propositions analytical if their truth is to be guaranteed

a priori was the difficulties presented by the definition of sameness,

seems to us plausible to say that the theory of causation put forward

by Bradley and Bosanquet is an attempt to meet this difficulty with
the explicit purpose in mind of making the inductive propositions,
based upon causal relationship and known to be true a priori,
13
synthetical and not analytical.
This is done in the theory of 'concrete universals'. To suppose A
to be a concrete universal or, in the terminology of Mill, a natural
14
kind, is to suppose that although any instance of A may have an
unlimited variety and multitude of properties, these properties are
bound up with each other in such a way that the repetition of a few

It

of them brings with it the repetition of the whole, probably infinite,

number of properties belonging to the universal. 15 Thus in our
36

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

example above it seems unnecessary for the purpose of establishing
the truth of the
proposition that A will always be succeeded by B to
define sameness in a
way which makes the proposition analytical, but
it is sufficient to assume that the A s
are instances of the same
natural kind. If it is once shown, as is
supposed to have been the case,
that one instance of the universal has the
power of producing B then
any instance of it will have the same
And in order to know
9

power.

that

two A's belong

to the

same natural kind

it is

necessary only to

know that they have a limited number of

properties, say XYZ, in
common. As soon as XYZ is repeated in A, A will be followed
by B.

Let us assume this theory to be true in so far as that concrete

universals really exist. There remains,
nevertheless, a difficulty which
none of the supporters of the
to have noted. How is it
seems
theory
are sufficient for the
possible to know that the properties

XYZ

determination of the objects which

belong to the same natural kind?
Suppose that A once produced B and that another A, having at least
XYZ in common with the previous one does not
B. Then

we must assume
the members of

produce

XYZ was not sufficient for the determination of

kind. We should say for
example that a fourth

that

the

needed for this determination, and that the absence

of this in the second case
prevented B from following. If this characteristic is
supposed to be T and it again happens that an A which has
XYZT in common with the first one does not produce B, we have to
look for a fifth characteristic and so on. In other words, if we want
a priori to be sure that A will
always be followed by B, we have to
define the universal which
guarantees this truth in such a way that if
an A is not followed by B then there must exist some ultimate
property of the universal, not yet observed or taken into account,
which is not possessed by the A in question. But in this case the
truth of the inductive
be followed
proposition that A will
characteristic

is

by

always

becomes

analytical. If the theory of concrete universals was intended to establish the proposition as
synthetical and true a priori,
it has failed to do so.

We

may thus conclude that in whatever way we interpret the thesis

that singular facts provide us with
general information, we cannot
overthrow Hume's results as to the impossibility of
guaranteeing a
Either an
priori the truth of inductive propositions
qua synthetical.

inductive proposition remains synthetical, and in this case

t>

37

it is

not

THE LOGICAL PROBLEM OF INDUCTION

possible to decide anything as to its truth a priori, or we can guarantee
its truth, in which case the
proposition becomes analytical. If we lay

emphasis primarily upon the guaranteeing of the truth, then we are

driven to accept the alternative that justifiable inductions are analytical sentences. This
consequence, as was shown above, also applies
to the older doctrines of synthetical judgments a priori.

7.

General remarks about synthetical judgments a priori.

The reader should have observed above that when we have had to
show the inefficiency of a certain philosophical theory to guarantee
a priori the truth of a general synthetical proposition, we have always
made use of one and the same 'technique'. Finally, we must say a
few words about the justification of this 'technique', and about the
general significance of the results at which we arrive by its aid.
The method used by us has been, generally speaking, as follows:

Suppose that the proposition 'all A's are 5' is said to be synthetical
and a priori. Suppose also that somebody maintains that there is
an A which is not B. How is such a situation to be judged?
The possibility that the latter of the two assertions about A and B
is true can be excluded, it seems, in two
ways. Either we use the
terms A and B in such a way that the phrase 'A is not B' would
contradict that use; or, if this is not done, the proposition is shown
some other reason., (e.g. careless observation, a mistake

to be false for

in the records, a deliberate lie or anything). But in order to guarantee

a priori that some such reason will be present we must determine

exactly

which

what

is

not

is

to be called a reason against the existence of an

in such a way that the presence of such a reason

follows in any situation where it is maintained that an A is not B.

In either case the truth of the proposition that all A's are B follows

from the use or the

proposition,

if its

definition of certain terms. 1

truth

is

to

Consequently the
be guaranteed a priori, must be analytical

The whole reasoning hangs upon

the assumption that

it is

possible

deny a synthetical judgment a priori. It is uncertain how far this

assumption would be in explicit accordance with known theories
about such judgments. In general it is not possible from the contexts
in which those theories are expounded to find a clear answer to this

to

38

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

question. Kastil, in discussing different theories of synthetical judgments a priori, assumes it to be concordant at least with some of them

an

affirmative answer to the question whether such a denial

2
possible or not. But it must be observed that even if the answer,
as given by any philosopher, were in the negative, it would have only

to give
is

an apparent bearing upon the method of reasoning employed by us.

This is seen in the following way:

The word
things.

to

'possible', in this context,

It is plausible to

when we

several different
like

word

in such a sense that the negation of a synthetical

in his terminology, 'impossible'. But
becomes,
priori
that such a negation is 'possible', we simply intend to

employ the

judgment a

may mean

assume that some philosopher would

say
say that as a matter offact it might happen that somebody asserts a
proposition of a form contradictory to the form of the general
proposition, and this possibility cannot be denied by anybody.

We can thus end this chapter with the conclusion that the only way

guarantee a priori the truth of general propositions is to make them

we shall see what bearing this result
analytical. In the next chapter
has upon the question of justifying induction.
to

39

CHAPTER

III

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

1.

The way

Some

.in

which conventions enter into inductive investigations.

examples.

WE know from chemistry that the

44 C. We have obviously arrived

melting-point of phosphorus
at this result in

is

what may be

termed an inductive way, i.e. we have melted different pieces of a

substance known to us under the name of phosphorus and found that
they all, suppose the experiment to have been carefully performed,
melted at the same temperature of 44 C. From this fact we generalize
that all pieces of phosphorus will melt at 44 C or, as we also express
This inductive
it, that the melting-point of phosphorus is 44 C.
is of the form (x) [Ax-+Bx], if A is the
generalization
property of
44
at
and
of
B
the
C.
melting
property
being phosphorus
In considering what justified our making this generalization, it
immediately occurs to us that the justification is not solely in the
experimental facts as such, but that the clue to it essentially lies in
the multitude of circumstances and qualifications which determine
the correctness and the significance of the experiments. If somebody,
for example, had made a number of haphazard experiments under
strange conditions and based a generalization upon his results we
should not have attached much weight to it, even if the number of
1
experiments were great and all had led to concordant results. Let
us therefore consider the essentials of what we call a correct experiment.
The fundamental condition is that we have reliable criteria which
enable us to decide when it is with a piece of phosphorus that we are
dealing and when not. It is quite conceivable that we are not able to
enumerate exactly the criteria used by us in our experiments, and a
scientist in the first place would hardly bother about such an enumeration. But certainly we have relied upon some criteria, as we have
chosen a definite substance and not at random for the experiments.
Let us assume these criteria to have been K, L, A/, e.g. macroscopic
40

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

properties such as colour, smell, taste, etc. We shall for the present
be concerned only with these criteria, and shall assume all other
things about the conditions of the experiment to be settled.
Suppose then that we find a substance having the characteristics

K, L,

M, which does

not melt at 44 C. Does

previous generalization as

this fact
imply that our
to the melting-point of
phosphorus is

falsified?

Obviously what has happened could be regarded as a falsification

of the law. But there is also another way left open which, in the
practice of science under similar circumstances, is very often resorted

We simply declare that the last examined substance cannot have

been a piece of phosphorus at all. The property of melting at 44 C,
which originally was an observed 'empirical' property of substances,
already known to us under the name of phosphorus, is thus made a
standard for what may be called phosphorus and what may not. If
to.

this is

44

done the generalization that all pieces of phosphorus melt at

can never be falsified, i.e. is absolutely true under any circum-

stances whatsoever.

Here we have a case of an 'inductively' established generalization

being absolutely true. If we consider wherein the reconcilability of
absolute truth and induction consists, we find that it has its root in
the fact that the word 'phosphorus' has been used during the course
of the investigation in what may be termed a quasi-ambiguous way.
At the outset we used it to denote a substance characterized by a
number of properties with which we were already familiar a certain
colour, smell, taste, macroscopical structure and so on. In enunciating the law about the melting-point we were in the first place enun-

an empirical discovery, 2 viz. that the substances with the

the
properties mentioned were found to have a further property
ciating

in common.
In so far as the generalization 'all
pieces of phosphorus melt at 44 C' is to mean that, whenever in the
future we find a substance with the first-mentioned properties, it will
exhibit the further discovered property too, then this generalization

melting-point

a hypothesis which later experience may confirm or refute.

Although the word 'phosphorus' was used at the outset for substances exhibiting the properties mentioned, it is by no means

is

certain that we, even at the beginning, wished to define phosphorus

as a substance exhibiting these characteristics. 3
simply asserted

We

41

THE LOGICAL PROBLEM OF INDUCTION

that these substances were phosphorus, as though phosphorus were
something fixed and given, about whose definition we never need to

certain co-existence of a number of easily observable

has
made us familiar with something called phosphorus,
properties
and which of these properties are defining or fundamental ones and
which again empirical and accidental, is a question which never before
occurred to us. The first time that we were confronted with it was
L, M, which
perhaps in the above situation, where the properties
we used to regard as criteria of phosphorus, are present, but a further

bother.

property which hitherto always accompanied them is absent.

Now the question occurs: what then is phosphorus? Is the substance now under examination phosphorus or not? In such a situation it is quite conceivable that we should renounce every pretension

as the 'true' criteria for what may

of regarding the properties K, L,
be called phosphorus and should find the experimentally discovered
property, especially if it is exactly measurable and sharply distinguishable from other characteristics, more convenient for this
purpose. But at the same time we make the inductive generalization

true per definitionem,

proposition.*
On the other

hand

i.e.

absolutely true as being an analytical

the fact that

we

in this

way

'save' the truth

of

the inductively established proposition by making it analytical, does

not necessarily imply that we announce the melting-point itself as a
defining property of phosphorus. This certainly is a possibility close
at hand, but there is also another way open.
not
In making our decision as to whether an instance of K, L,

not a falsification of the law of the meltingto

a multitude of circumstances. It is very
we
have
regard
point,
that
to
assume
among those 'circumstances' are to be
plausible
found assumptions which are themselves inductive. For example,
does not melt at
we may suppose that if a substance with K, L,
44 C then it differs from phosphorus also in other properties, 5 in,
say, its microphysical structure, and this difference is the 'cause of
some substances with K, L,
melting at the temperature in question
and others not melting. The 'probability which we attach to assump6
tions of this kind will influence our decision.
Thus we can decide to regard the law about the melting-point as
true, not because the melting-point itself defines phosphorus, but

melting at 44

is

or

is

'

'

42

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

because

why

indicates the presence of another

property that explains
phosphorus melts at exactly this temperature. Here it is imit

portant to observe that even if we do not know of any such property,

or if every property that is assumed to cause the characteristic
melting-point of phosphorus is shown later not to be the cause
looked for, it may nevertheless be plausible, on what we know, for
example, of other substances and their melting-points, to
postulate

its

existence.

This possibility is important for the following reason. If we, as a

matter of actual fact, had to save the truth of a general proposition,
such as that about the melting-point of phosphorus, by saying that a
substance which does not melt at the temperature in question cannot
be phosphorus, then we should almost certainly not want to say
anything decisive about the definition of phosphorus. We would
rather say something like this: Perhaps the melting-point can be
used for the purpose of defining phosphorus, but perhaps also there
will be found some 'deeper' quality of the substance which will
explain why phosphorus melts at just 44 C. But irrespective of which
alternative finally is to be chosen, we wish under all circumstances
to adhere to the norm that phosphorus melts at 44 C. Whether it is
because the temperature defines phosphorus, or whether it is because
it only indicates some true criterion of that substance, is a
question
not as yet considered, and one that need not be settled in this connection. 7

We shall next consider another case, slightly different from the

former one, which gives a new illustration of the way in which
conventions enter into inductive investigations. As an example we
famous instance of the billiard-balls.
have observed that the impact of one billiard-ball against
another is, so far as our experience goes, followed by the movement
of the second ball. From the observed fact we might conclude that
the impact of the first ball is the cause of the second ball's movement,
implying that whenever the first ball strikes the second the latter will
move, which again is an inductive law or generalization. What of its
justification? Is it really possible that the law, which seems to us so
obvious, could be false; that one day it might happen that, although
one ball strikes another, the second one is left unmoved? Or is there
something to exclude this possibility a priortf
shall take the

We

43

THE LOGICAL PROBLEM OF INDUCTION

Let us suppose, in order to find the answers to these questions, that
has actually happened that a ball is struck by another but is left
unmoved. We should then under no circumstances immediately say
that the previously enunciated law has been falsified. Instead of this
we should investigate the more closely the circumstances under which
the impact has taken place in order to find an 'explanation' of what
it

happened, i.e. to show generally speaking, that what happened was

in accordance with some general law operating against the law which
we were in the first place considering. Suppose, for example, that we
find that the second ball was fixed to the table and could not move
at all. This would justify us in saying that the law was not false, but
that the cause could not operate because of the presence of a counteracting cause, the ball being fixed to the table.
All this may seem extremely trivial. In fact

it is of fundamental
us
that
the
as
has
shown
law,
originally enunciated,
importance.
was still incomplete in its formulation, that instead of saying that
whenever one ball strikes another the second one will move, we
intended to say that whenever one ball strikes another the second
ball will move, provided certain circumstances are present, and certain
conditions fulfilled. Of these further conditions and qualifications
there are obviously a great number. They specify, first of all, the
quality of the material used in the experiment: one ball must not be
of iron, the other of paper; the impact must have a certain minimum

It

force; the surface over

which the

balls

move must be

a kind.

And

upon by

forces of a certain kind, exceeding a

of such and such

further they exclude the possibility of counteracting

causes: the ball must not be fixed to the table; it must not be acted

maximum amount, and

so on.

Thus the inductive generalization, which we intended to formulate

on the basis of what experience has taught us about the effect which
the impact of one billiard-ball against another has produced in the
past, is in fact far more complicated than its usual enunciation
indicates.
For practical purposes, however, it will generally be
sufficient to state the law in the simplest formulation, perhaps with

a very few qualifications, because the further additions to it are either

concerned solely with exceptional circumstances, which very seldom

need be taken into account, or they are such that the conditions laid

down

in

them are so

trivial

and

'self-evident' that their fulfilment is

44

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

taken for granted without special mention. Besides this, the
leaving
out of the additional qualifications seems to be
merely a matter of
convenience and quite harmless from an
epistemological point of
view. At the back of our minds we have the idea that
although the
law 'in practice' is left incomplete in its formulation, it is
always
'theoretically' possible to formulate it in full, if needed. And, it is
added, if really all relevant circumstances are taken into account,
8
then the law certainly will hold for the case in
question.
This idea, however, ought to be more
examined.
Let us
closely
ask the following question: How would it be
to
know
that
possible
all the conditions
for
the
formulation
in
full
of
the
law
necessary
have been taken into account? This is
in more than one

way.

We

possible

may, for

certain definite

decide after the enumeration of a

of conditions that all 'relevant' circumstances

instance,

number

have been taken into account.

If then the impact is not followed by

the supposed effect, we must
speak of a falsification of the inductive
law. But it is not certain that this way would recommend itself as

being plausible or in conformity with the actual practice of science.

decision as to when all relevant circumstances have been taken
into account would always retain an air of arbitrariness, which in
actual scientific procedure we wish to avoid.
Our attitude to the question as to the presence or not of all relevant
circumstances in a given enumeration of conditions is therefore

The

usually as follows: Perhaps

all

relevant circumstances have been

enumerated, perhaps not; this is a question on which future experience

will give us elucidation. It is conceivable moreover that,
depending
upon a number of circumstances of which several are themselves
experiences and assumptions, we regard it as highly
desirable to make the truth of the generalization about the billiardballs itself the standard for deciding when all conditions necessary

inductive

for the validity of the experiment are fulfilled. In other words, as

soon as the ball is struck, but left unmoved, we say that there must
still exist some condition, 'relevant' to the truth of the law, which

has not yet been taken into account, and which is absent in this case.
Thus the truth of the induction serves as the norm which guides us in
our search for new qualifications to be added for the purpose of
getting a complete and exhaustive formulation of the law aimed at
in

making the

generalization.
45

THE LOGICAL PROBLEM OF INDUCTION

The above examples
in
It

are meant to illustrate the two

typical ways
which conventions may be introduced in inductive lines of
thought.
must finally be observed that the two ways do not in
occur

general

separately, but are usually both present in connection with the same
induction. As will be remembered, we
supposed in the example of
the melting-point of
that all conditions as to the correct-

phosphorus
from those as to the true

ness of the experiment (apart

criteria of
phosphorus) were settled. Actually in settling these conditions we
should be confronted with problems such as this: under what
conditions should the measurement of the
temperature take place in
order to be correct? This
question again resembles that of the
or
not
of
presence
counteracting causes in the instance of the billiardballs.

Here

instance

also conventions similar to those in the last-mentioned

may enter, and so the two ways are combined.

thus found that conventions play a fundamental role in

investigations which rightly and truly are called inductive. GeneralIt is

from experience get part of their strength and

convincing
power from the analytically binding force of conventions about the
use of certain words and expressions. But what
bearing has this

izations

upon
2.

the inductive problem as such?

Conventionalism as an 'elimination* of the inductive

problem.

The importance of conventionalism or analytical steps of

thought in

inductive investigations is already

implicitly contained in the doctrines
of Mill and Whewell on induction. For example the role
played by
possible 'counteracting causes' in causal inductions has been noted
1
by both the authors. When Whewell constantly emphasizes that
the process of induction has
something to do with the formation of
concepts and that with every scientific induction there is introduced

new idea, he moves along lines of thought which are not always
very far from the ideas which were illustrated above in the example
of the melting-point of phosphorus. 2
Bacon had mentioned
3
induction as an operation by which
concepts are defined. Jevons*
and Mach 5 stress the connection between induction and the classifi6
7
cation of natural phenomena, and
Sigwart and Broad give good
of
how
induction
is used for the formation of scientific
examples
a

concepts.

Mach

also pointed out a certain resemblance

46

between the

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

inductive procedure which results in the
definition of a concept and
the mathematical method called
recursive induction. 8 Induction as
a step in the formation of
concepts is also related to the Aristotelian
intuitive' induction. 9

The first philosopher who

clearly stated the general importance
of conventions for the foundations of
science and for inductive
investigations was Poincarf." He did not, however, use conventionalism for the
purpose of offering a general theory of induction ll
This has been done
by certain other philosophers, who think that
conventionalism lines of
thought when developed to a certain
extreme would lead to an elimination of the
inductive problem

The argument is roughly the

12
following.
The problem of induction, as
put forward by Hume and
with by most
philosophers after his time, has its
in a
origin

as dealt

miscon-

ception of the nature of scientific truth. 15 It is an

over-simplification,
not in accordance with the real use of scientific
propositions, to
regard every generalization at which we arrive by induction as
being
purely synthetical. On the contrary, in so far as induction can claim
to reach absolute truth it is because there
has taken place a transition

from synthetical

to

analytical,

from

confirmed empirical generalizations to

well-established

and

linguistic conventions,

well-

which

obtain their unrestricted

validity from being analytical and tautologous. As this transition from synthetical to analytical is sometimes
difficult to perceive, is
very seldom explicitly formulated, and is often
hidden in the ambiguous forms of
language, we easily arrive at the
mistaken idea of something which is at the same time

synthetical
necessarily true." From the attempt to reconcile these contradictory attributes of one and the same sentence originates the inductive problem in its 'classical' form. When we have
seen that this
attempt was undertaken on the basis of a
the

and

misunderstanding
induction is not
to show how propositions which are
synthetical can also be known
to be true for unexamined
instances, but to show how universal and

whole problem disappears,

is

eliminated.

To justify

necessary truth originates from synthetical propositions changing

their nature into
analytical.

There is much to be said for the view held

by certain philosophers
that the whole of science, even that
part of it which is based upon
induction, is not a system of general synthetical
propositions, but of

THE LOGICAL PROBLEM OF INDUCTION

statements which are analytically true. When in this system a
general proposition is discarded, it is not because it has been falsified,
i.e. contradicted by an
experiential proposition, but because enlarged
the employment of some new mode of
recommended
has
experience

Thus this view of scientific truth accounts also

of
'falsification' and need not presuppose a
phenomenon
that
has
been rendered unchangeable once for all.
of
science
'system
And as it seems at the same time to eliminate the problem of induction, which occurs with the 'usual' view of the nature of scientific
truth, it gets a further air of plausibility from this.
We shall call the view that general truth in science is always
typical representative of this
analytical, radical conventionalism.
view is Le Roy. 16 Ideas similar to those of Le Roy have in recent
17
Radical conventionalism,
years been expounded by Ajdukiewicz.
with explicit reference to induction has been developed by Schuppe, 18
19
and Dingier. 20 The conventionalism of Dingier is
Cornelius,
peculiar in that he prescribes further conditions for the conventions
which make up the bulk of exact science, as, for example, that
21
geometry must be Euclidean and mechanics Newtonian.
It is not our intention to discuss here the merits and demerits of
the respective systems of conventionalism. We shall only show that
it is not
possible to eliminate the inductive problem as a whole (even
in science) by taking a radically conventionalistic view of the system
scientific expression.

15

for the

'

of natural laws.

3.

Conventionalism and prediction.

Suppose that we decide to regard every general proposition,

established by induction, as true/w conventionem, i.e. either as part
of a definition or as an incomplete proposition, the completion of
which is guided by its truth, or as a combination of both these cases.

Then

it

would be

possible to co-ordinate with these analytical

propositions (one or more) synthetical general propositions after the

following pattern:
Take the sentence 'the melting-point of phosphorus is 44 C' as

enunciating a defining property of phosphorus. To this analytical

proposition we thereupon co-ordinate the following general syn48

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

thetical proposition:

'if a substance has all the

defining properties
of phosphorus,
except perhaps the melting-point which is still
unexamined, then the substance is phosphorus.' Or if, as
certainly
would be more in accordance with actual scientific
practice, the
defining properties of phosphorus are not explicitly enunciated: 'if
a substance has such and such
properties, which, irrespective of the
further question whether
or
they are defining

properties
empirical ones,
are regarded as reliable criteria of
phosphorus, then the substance will
melt at 44 C'. Again, in the case of the
proposition concerning the
billiard-balls, we co-ordinate the following synthetical sentence: 'if
we know only that such and such conditions are fulfilled when we make
the experiment, then the
impact of one ball against another will
be followed by the movement of the second one'.
The mere fact that we are able to do the co-ordination is, of course,
trivial. But the further fact
that, in a great number of cases, we use
the co-ordinations to stress the
reliability ascribed by us to certain
inductive generalizations has a
the
of

deep bearing upon

induction. 1

problem

In order to understand

this, consider the following alternative:

never justified in
co-ordinating to the analytical
propositions such synthetical propositions? This would imply that it
would be impossible to make any reliable predictions in science since

What

if

we were

an analytic proposition in itself never

justifies predictions. From the
mere knowledge that A,B, and C define a substance it does not follow
that we can regard, say, the
presence of A and as a reasonable basis
for predicting the presence of C.
Only if A and B are reliable 'signs'
of

C are we justified in co-ordinating to

the definition the synthetical

proposition that if a substance has the properties A and B then

it has
also the property C.
Therefore, the fact that this coordination actually takes place in a great number of cases is

nothing but an expression of the other fact that we regard reliable

predictions as possible in science. This, by the way, is what makes
science important and useful.
But here we are immediately confronted with a new question. How

do we know that the

synthetical propositions co-ordinated to the

analytical ones really are reliable? (What has been said, above, implies
that they are regarded as such, but this is of course no
proof
that they actually are so.) Or, in other words, how do we know that
49

THE LOGICAL PROBLEM OF INDUCTION

science can be used in future for predictions, not simply that
as a matter of fact, been possible to use it for this purpose?

it

has,

These questions are nothing but the re-occurrence of the inductive

problem within the conventionalistic conception of science itself.
We cannot settle the questions by making the co-ordinated synthetical
2
propositions themselves analytical. For then we could again create
new co-ordinations of synthetical propositions, and the inductive

problem would only have been pushed to a new

dismiss the question by saying that, according

level.

Nor can we

to radical conven-

tionalism, only analytical propositions are formulated as laws of

science, and that consequently the above-mentioned co-ordinated

do not belong to the system of science.

not whether we pretend that the co-ordinated
propositions are laws of nature or not, or even whether they are
formulated at all, but solely the fact that we regard certain predictions
as reliable and act accordingly. This fact remains even within the
system of radical conventionalism and its 'justification' constitutes
synthetical propositions

What

matters

is

the inductive problem, 3

4.

Conventionalism and the justification of induction.

We have thus seen that conventionalism, even in its most radical

form, does not eliminate the inductive problem. But, nevertheless,
conventionalistic points of view contribute to a clarification of
important aspects of the problem.
(1) We are now aware of the important fact that the general question about a 'justification of induction' covers not one, but (at least)

two

different things.

One

is this:

how can we- prove an

is: how can we
prove

inductive

The other

that such
a generalization is a reliable basis for making predictions? There
is a
strong temptation to regard an answer to the former question as
also an answer to the latter, and this tendency explains why we are
liable to overlook the difference between the two aspects of the
problem of how induction is to be justified. For if I have proved that
it is true that all A's are 5, have I then not also
proved that any
prediction of B on the basis of A will be true?
As we know, there really is a means of proving that all A $ are

generalization to be true?

50

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

by making the proposition true per conventionem. Thus it is
possible to justify induction, if by 'justification' we mean only a
proof of the absolute truth of the inductive proposition. But this

B,

viz.,

does not tell us anything as to the

reliability of preA's are Bper conventionem, then to
say that an A
will be B is not to predict
anything about A, but simply to state a
tautologous fact about it. Therefore, if the first question contained
in the problem of the
justification of induction is answered by resortto
conventionalism, then the second question is still left open.
ing
We have said that if it is true per conventionem that all A*$ are -B
then there is no further question of
'predictions' of B on the basis
of A. This, however, does not exclude the same verbal mode of
expression 'all A's are 5' from being used in future for predicting B
on the basis of properties of the A 9 s other than B. But the
reliability
of those predictions is, of course, not to the
slightest degree increased
'justification'

dictions.

For

if all

'

is made
by the fact that the proposition 'all ^t's are
absolutely
true per conventionem. 1
(2) Those who emphasize that conventionalism eliminates the
inductive problem seem to have been so
impressed by the fact that
it is
possible to account for the absolute validity which we some-

times attribute to inductive generalizations, that

they overlook the
fact that it is not this alone which we had in mind when we demanded

a justification of induction. 2 The general

propositions, according to
which predictions are made, still remain to be justified. But in the
justification demanded for them, we seem to be content with something 'less' than absolute truth. Scarcely anybody would pretend
that predictions, even when based
upon the safest inductions, might
not fail sometimes. We are satisfied in knowing that they are
highly
'probable* at any rate. Thus conventionalism may be said to be able
to dispose of the element of absolute truth contained in induction,

and what then remains to be accounted for is the element ofprobability

which is attached to the inductive predictions.
(3)

The

idea that conventionalism could eliminate the inductive

problem, however, originates from more than a mere failure to see that
to justify induction means not only to establish the truth of general
propositions, but also to give rational grounds for the reliability of
predictions. There is a deeper reason, why conventionalism seems to
dispose of the inductive problem as a -whole.
51

THE LOGICAL PROBLEM OF INDUCTION

as it is actually used, both in everyday life and in
so constituted that in most cases it is not settled whether

Language
science,

is

a given proposition is 'really' synthetical or analytical, nor which

criteria of a certain object are defining criteria, and which again
are empirical. For this reason it is not usually immediately clear
whether a general proposition, as used by us, is 'really' used qua
as a linguistic standard for interpreting facts, or qua
as a means of predicting experiences.
synthetical,
Therefore, if I have arrived by induction at a general law, I need
not immediately decide as to its analytical or synthetical nature. 1

analytical,

i.e.

i.e.

for instance, to begin with, use it for the

purpose of making
successful
and
as a conevery
prediction
may regard
predictions,

may

firmation of the law.

Not

until

am

confronted with a situation in

and this may actually never

which a prediction fails to hold
do
I have to consider whether the law has been falsified or
happen
whether it is more plausible to 'save' its truth by 'explaining away the
failure of the prediction. As both alternatives are always possible I need
never fear that experience will compel* me to withdraw the inductive
generalization once established. It is always in our power to decide
whether a law has been falsified or not, and this fact, which conventionalism reveals to us, explains the feeling of unshakable validity
which we sometimes attach to those inductions which, on the other
'

4
hand, are themselves 'confirmed' by successful predictions.
(4) It must also be observed that if in the case of an apparent
'falsification' of a law, we decide to make the law analytical, this
usually happens because there exists some empirical invariance or
uniformity which, in spite of the exception to it in this case, is
regarded as 'strong' enough to justify the introduction of the
convention. 5 The convention, so to speak, serves the purpose of

strengthening an already assumed empirical law;

it

adds absolute

validity to something which is already in itself 'almost' absolutely

true. Psychologically, therefore, the transition from the synthetical
to the analytical which takes place when a convention is used to

may mean only a very slight increase in the

feeling of confidence which we associate with the proposition, and
this fact may obscure the fundamental change in the logical nature
'justify'

induction

of the generalization which is introduced by the convention. So the

inductive proposition, which has been transformed into an analytical
52

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

one, retains an empirical 'flavour' which gives to the conventionalistic
decision an air of
being concerned not only with the future use of
words, but with future facts as well.
(5) There is a tendency, very much furthered by the introduction
of exact symbols and notations, to
regard the bulk of human knowledge as expressible in a definite set of propositions. 6 As was seen
from the above discussion of the
example of the billiard-balls, the

idea underlying this

tendency is to some extent delusive. It leads
us to regard as pure
knowledge about facts, knowledge which actually
of
its
firmness
from verbal circumstances. The question
gets part
of the justification of induction must be understood
the
against

background of language, as an expression of knowledge, being not

only ambiguous and 'unsettled', but also in a certain sense inexhaustiveS

The importance of

this inexhaustiveness of
language for
revealed to us by the conventionalistic
which the question was discussed above.

the problem of induction

points of view from

is

53

CHAPTER

IV

INDUCTIVE LOGIC

1.

Justification a posteriori

of induction.

IN the previous chapters it was shown that the only way to guarantee a priori the truth of inductive generalizations is to make them
analytical. And from this it followed that we cannot justify a priori
predictions from inductive laws.
The idea of a justification a priori of induction was to make the
proof of inductive propositions independent of the empirical testing

of instances of those propositions. With this idea can be contrasted

that of proving inductive truths with the aid of verified instances of
the generalizations. The justification of induction is thus a posteriori. 1
Of attempts to justify induction a posteriori there are two fundamentally different types.
Of the one type are the theories of induction which contrast the
process leading from singular facts to inductive laws with the
process of deducing those facts from the laws, the latter of these two
'inverse' processes being regarded as the justification of the 'inductive leap'

made

in the former.

The

logical element in induction

is

inductive philosophy of Whewell is the most

noteworthy representative of this type. Of the other type are the
attempts to formalize the very process of generalizing from given data,
i.e. to make inductive
propositions follow from singular instances

thus

deduction.

according to given

Bacon and

The

rules.

Of

this type is the theory

of induction of

Mill.

Theories of the second type thus aim at the creation of a logic of

main branch of formal study, viz.
deductive logic. This, however, must not mislead us to the idea that
the logic involved in inductive reasoning is, in any circumstances,
of a different kind from the logic used in what is commonly
known under the name of syllogistic or deductive reasoning. It is
important to observe from the very beginning that the logic of all
induction, 'parallel' to the other

54

INDUCTIVE LOGIC

known

attempts at a so-called logic of induction

as the logic of that
process of thought which
reasoning, even if this fact

is

is
is

exactly the same

called deductive

sometimes obscured by a misleading

terminology.
In the following sections

we shall examine the above two types

an inductive logic. Here, as in the preceding
two chapters, we are concerned only with a
justification of induction
to
Considerations
about
the
leading
certainty.
probability of propositions are still outside the
of
our
sphere
investigation.
of

2.

'classical*

attempts at

Induction and discovery.

Induction

and deduction as

inverse

operations.

According to a well-known definition induction is 'the operation

1
of discovering and proving general
propositions'. In this definition
the two fundamentally different
aspects, viz., that of discovering a
general proposition and that of proving it, are parallelized in a way
which has been fatal for the philosophy of induction. A careful
separation of them on the other hand contributes much to a clarification of the ideas about induction and its
justification.
The problem of how to discover a generalization from a set of
particular data is related to the question which, in the logic of
Jevons, is called the inverse (or inductive) problem. 2 This problem,
again, could be re-stated in the terminology of modern symbolic
logic roughly as follows: Given a certain number of propositions,
a, b c
., construct a truth-function in the form of an
equivalence
y

which

true for certain given combinations, and for them only, of

truth-values of those propositions. Assume for
example that the
propositions are a and b and that the equivalence is to be true for
is

all
possible combinations of truth-values of the propositions, except
the case when a is true and b false. Then a truth-function fulfilling

these conditions

is

<2==a&.

Given the tmth^function, any of the propositions asserting one of

the prescribed combinations of truth- values (in our example the
propositions a&b, ^a&<^b, <~^a&b) can be deduced from it. Given
the latter propositions, the construction of the truth-function again
inverse of this deduction. In the opinion of Jevons the

becomes the

55

THE LOGICAL PROBLEM OF INDUCTION

sole

to

way

perform

truth-function. 3

this inverse

operation was to 'guess' at the

We

suppose the truth-function fulfilling the condiwe

tions in question to be T; by deducing the particular data from

verify (or falsify) this supposition.

The analogy between this and induction

of an induction

we have

is

obvious. 4

In the case

also a set of particular data the 'law' for

which we are

in search of, i.e. a proposition from which these

are deducible. The invention of this law cannot
data
particular
be
usually
performed 'mechanically' but is the outcome of skilful
5
As soon as any law, L,
guessing, guided by scientific 'intuition'.
has been guessed at, we can test the result of the guessing by trying

to deduce the given data from the law. If the deduction can
carried through, then the supposition that I, is a law of the kind

look for has been

verified.

be

we

by which we establish inductive proposimust be observed that this

case
induction.
of
When from the
does
not
every
accompany
process
fact that such and such X's are B, I infer that all A s are B, there does
not occur any discovery'., in the sense that I guess at a law from which
the observed particulars are subsequently deduced. The law, so to
speak, follows 'directly' from the given data.
On the other hand it is clear that such cases as the last-mentioned

The process just

outlined,

tions, is the typical process of discovery. It

are of a rather 'primitive' kind, and that the many beautiful instances
of induction which science affords us are generally peculiar on
account of the element of discovery which they contain. This is the
case in every instance of quantitative induction in which empirical
measurement has provided us with a set of corresponding values of the
variables,

dence.

and we wish

to detect a law or function for this corresponis that in which the values

A sub-class of these instances again

are pictured in a diagram and we look for a curve connecting them.

Kepler's discovery of the planetary path of Mars is a good example

of quantitative induction of the last kind. Observation had informed

the position of the planet in various points of its path, and
from this information the path itself was to be induced. We know
that Kepler, after having first rejected no less than nineteen assumptions as to the true path, discovered the law which agreed with the
observations, i.e. from which the observed positions could be

him of

deduced.

56

INDUCTIVE LOGIC

The process of discovery accompanying induction is of interest

from a psychological point of view. It introduces order and
perspicuity in a multitude of previously disconnected facts, and it
also

our handling of the given information. It concentrates,

handy formulation of a law, a mass of knowledge which before
had to be summed up in elaborate records of observations. Inducfacilitates

in the

tive discovery, in other

words, is an important step towards the ideal

8
of
'economy
thought'.
Induction as an operation inverse to deduction bears a certain
resemblance to the well-known way of making geometrical construc*
tions on given data, which is sometimes called the
analytical method.
This method consists in that we suppose the
problem to have been
solved, and then deduce the given data from the solution. By tracing
the thread of deductive steps in the
opposite direction we thereupon
10
carry out the construction.

The idea that induction as a logical method is analogous to this

way of reasoning in mathematics is very old. It is contained already
in Zabarella's account of a logic of induction. 11 Galileo
expounds
the same idea in his account of the resolutive method which, accord-

ing to him, described the way in which mathematical laws of nature

are discovered. This method consists in a certain
happening being
resolved
into components, each one of which is
analysed
supposed to obey some comparatively simple mathematical law. It is
then shown that when the respective values of the components of the
event in question are calculated separately under these
suppositions
and the components are then put together, their 'resultant' approximates to the actual happening. Galileo regarded the coincidence of
the calculated and the actual course as the 'verification' of the
discovered law. 12 The method is illustrated, for example, in his

discovery of the mathematical paths of falling bodies. Ideas very

similar to these on the nature of the scientific method are
expressed

by Leibniz.

1*

The inductive logic, or logic of discovery, expounded by Whewell

must be understood wholly against the background of the above idea
of induction and deduction as inverse operations and the analogy
between induction and the analytical method in mathematics. 14
According to Whewell the logic of induction is *the analysis of
doctrines inductively obtained into their constituent facts, and the
57

THE LOGICAL PROBLEM OF INDUCTION

arrangement of them in such a form that the conclusiveness of the
16
The 'Inductive Tables' 16 in which
may be distinctly seen
this function of the inductive logic is performed are tables giving a
hierarchy of propositions, beginning with particular data and
ascending from them to laws of greater and greater generality. Each
17
ascending step is 'a leap which is out of reach of method', i.e. the
more general proposition is not deducible from the less general ones,
whereas the descending steps form a chain of successive deductions.
These deductions, says Whewell, 'are the criterion of inductive
truth, in the same sense in which syllogistic demonstration is the
criterion of necessary truth'. 18 The general propositions are thus
3

induction

10
by induction and proved by deduction.
the following question must be asked: Does

discovered

Now

this scheme of
foreshadowed in the description of the scientific
method given by two of the greatest geniuses of European thought
and developed to systematic strictness by Whewell, give a justification of induction?
The question ought not to be answered off-hand. The answer
depends upon what we expect a justification of induction to be. It is
not unplausible to assume that when, in advanced sciences, we ask
for a 'justification' of the inductions made, we primarily have in
mind a proof that the known data follow from the assumed laws.
This may be the case because, as has already been said, inductions
in advanced sciences, like astronomy and mathematical physics, do
not usually follow from the data as a matter of course but are the

an inductive

logic,

90
products of 'discovery', i.e. of a sort of 'scientific guessing', and
must therefore afterwards be shown to fit the facts.
Snell detected the well-known law for the refraction of light,

^^==A:.
sm
p

Before one had this law one had to look for each r
pair of

corresponding angles from tables. When Snell hit upon the law he
certainly could not see at once that it really fitted each of the recorded
pair of angles. The proof that each pair of the tables really followed
from the law, was the justification of his discovery. Whether for this
justification already recorded facts were used or whether the law
was tested on new facts was in the first place a point of minor
importance.
58

INDUCTIVE LOGIC
If, therefore, by a justification of induction we mean a
proof that
from the inductive proposition follow the data
upon which the
induction was made, then the scheme of an inductive
logic outlined
by Whewell provides us with the justification sought for. 21 The fact
that in advanced sciences the
justification of induction, in which we

are primarily interested,

may be a justification in this sense, has as

a rule been overlooked by inductive
logicians simply because they
have chiefly confined their attention to those
'primitive' types of
inductive inference where the general
proposition follows as a matter
of course from the given data." Whewell was an
to this
exception

and that

a much
account of scientific induction than do other 'classical'
treatments of the inductive problem.
But Whewell, on the other hand, overlooked the
significance of
another aspect of induction. This aspect, the
importance of which
again becomes more readily apparent if we confine our attention to
those 'primitive inductions where the element of
discovery disappears*
is that of
generalization. It is stressed with great acuteness by Mill
in his polemics against Whewell. 23
If the discovery of the law from which
given data follow is an
induction, then it must be possible to deduce from it also data other
than those already given. 24 Kepler's discovery that the available
observation-points of the path of Mars were situated on an ellipse
rule,

is

why

his philosophy of
discovery partly gives

better

was, as such, no induction.

It was made an induction

by the further
assumption that this ellipse would give to us the path of the planet
also between the observed points, and make it
possible for us to

calculate

tion

Mars 's position for any future time. 25 This

which gives to the discovery

its

further assump-

inductive character

is

a generali-

zation.

But the truth of the generalization, involved in the discovery,

cannot be proved in Whewell 's scheme of an inductive logic.* 8 And,
consequently, if by the justification of induction we mean a proof
that the general proposition to which induction has led us is true,
then this kind of inductive logic does not justify induction.
We have therefore seen that the question whether Whewell had
been able to justify induction or not is answered affirmatively or
9
negatively depending upon what kind of 'justification we look for.

From the same considerations it also follows that the kindred question
59

THE LOGICAL PROBLEM OF INDUCTION

whether a law of nature can be verified or not has a twofold meaning, and therefore can be answered as well in the negative as in
If by 'verification' we mean a proof that the suplaw
really is a law for the given data, i.e. that the data can be
posed
deduced from it, then it is possible to verify it. But if 'verification'
is to mean a proof that the law as a generalization is true, then we do
not as yet know of a corresponding way of verifying it. Now we must
not forget that actually the term 'verification' has been used by

the affirmative.

several philosophers
case.

We

and

cover primarily the first

at the opinion that laws
arrived
they
must be added that those philosophers

scientists so as to

thus understand

how

of nature were verifiable. But

it

as a rule overlooked that this kind of verification

was not a

verifica-

tion of the law as a general proposition, but of the supposition that

the law fitted the given facts. They therefore forgot that the problem

of justifying induction has a further aspect

dealt with.
If

we

consider

why

the inductive logic of

which has not yet been

Whewell does not justify

induction in the sense of proving the truth of general propositions,

we immediately detect that this is because, although the given data
follow from the law, the converse of this, apparently, does not hold
true,

i.e.

the law does not follow from the data.

It is, therefore, the

task of an inductive logic, which has to prove the truth of the inductive generalizations themselves, to show that under certain circumstances we are entitled to infer the truth of the law from the data.

Can

3.

this task really be accomplished?

The idea and aim of induction by

The idea

elimination.

that the process of generalizing from given data could

i.e. that the generalization could be inferred from the

be formalized,

data according to fixed rules, seems to have its origin in the following
observation:
We cannot establish the truth of an inductive generalization merely

by collecting a huge number of instances confirming

it.

This kind of

induction, inductio per enumerationem simplicem, 'puerile quiddam

l
est\ to use the words of Bacon, as it is a task constantly 'pericuh
&b instantia contradictoria exponitur\* i.e. because the number of
60

INDUCTIVE LOGIC
verifying instances as such cannot eliminate the possibility of a
falsifying case. On the other hand it looks as though the examination
of only a very few cases were sometimes sufficient for the establish-

ment of -an unshakable generalization. This

evidently must be because

the cases, besides being instances of the
generalization, show some
other characteristic features which are relevant to the
validity and
of
the
inductive
inference
made
from
them.
It is the
legitimacy
business of an inductive logic to give a
scheme
of these
general
features which the given data must
in
order
to
serve
as a
possess
valid basis for a generalization. 3

The inductive method,

described in the classical attempts to formathe process of generalizing and in contrast with enumerative
induction, we shall call induction by elimination.* It is our intention
in this discussion to examine the eliminative method from the
viewpoint

lize

of modern

symbolic logic. This examination will not only enable us

exactly to estimate the value of this method for the problem of
justifying induction, but will also lead to interesting discoveries
about the logical nature of eliminative induction in
These
general.
discoveries, we think, are a testimony of the value of logistics when
used as a means of analysing and re-interpreting 'classical' doctrines
and ideas on philosophical questions.

The task of induction, if we confine our attention only to Universal

Inductions about one-place predicates, 5 could be described as that of
connecting two characteristics (properties, predicates), A and B, by
universal implication (or universal equivalence).
This task, first of all, presents two different aspects, according to
whether the characteristics are properties of the same individual
6
(object, event) or whether they belong to different individuals. In
the former case the general implication which we want to establish
is of the form:
(x)

(i)

[AX-+BX],

in the second case again of the form:

(x)(y

(2)

where

is

a function, determining the pairs of corresponding

and^'s.
61

;t's

THE LOGICAL PROBLEM OF INDUCTION

correlates the ,x's and /s as individuals

If in (2) the function
in
we shall call the generalization a
other
each
time,,
succeeding

Universal implications and equivalences of the form

(2), which are not Causal Laws, might in
(1)
accordance with a classical terminology be called Uniformities of

Causal Law.

and those of the form

Co-existence.

The most conspicuous

difference between Bacon's

and Mill's

respective systems of inductive logic

treats of Uniformities of Co-existence, the

is

that the former

latter

causal

uni-

formities.

But the task of connecting two characteristics by universal implication also presents two different aspects in another sense. Either
9
and look for another
I take one of the characteristics as 'given
which it universally
which
the 'given' one
by

characteristic,

implies.

Or,

look for a charac-

universally implied. The

former case is that of finding a necessary condition of a given characteristic, the second that of finding a sufficient condition. The connectteristic

is itself

ing of two characteristics by universal equivalence is again the

establishing of a necessary and sufficient condition of a given
8
shall, therefore, call the characteristic, which is
property.

We

*given*, the conditioned one,

the conditioning one.

Now

and the

characteristic

we are looking for,

the idea of eliminative induction could be described shortly

Suppose I look for a conditioning property of a given

as follows:

conditioned property, say A. As a rule there will be a number of

concurrent hypotheses as to this conditioning property. Is it the
characteristic B or C or D or any other which is connected with A
by universal implication? This question, obviously, we cannot
answer by collecting a great number of instances confirming one or
other of these possible general implications, i.e. by resorting to
enumerative induction. Because if we do so, then we can never, in
spite of the confirmations, eliminate the possibility that not the

confirmed hypothesis but some other is the true one. On the other
that we need only find one single instance where one
of these possible general implications does not hold in order to
invalidate and consequently also eliminate one of the concurrent
hypotheses as to the necessary or sufficient condition of which we are
in search. It is this important asymmetry in the possibilities of verifying and falsifying Universal Generalizations which is at the root of

hand we know

62

INDUCTIVE LOGIC
the idea of an Inductive method,
proceeding from elimination of
concurrent hypotheses.
Francis Bacon, who gave the first
substantially correct description
of eliminative induction, saw the
advantage of the eliminative
method over enumerative induction in that the former was a method
This association between
enabling us to reach absolute certainty.
absolute certainty and induction
elimination
is
also of
by

most

typical

later treatments of that

method. For a critical examination of

the Idea that eliminative induction could attain absolute
certainty
it is, however,
important to observe that the word 'certainty' in this
connection may mean no less than three different
things. The failure
to separate these different
from
one
another
has been the
meanings
cause of much confusion and
misunderstanding.
In the first place absolute certainty
may mean that under certain
circumstances we can prove a general
proposition to be the only
generalization which Is in accordance with certain data.
In the second place absolute certainty
may mean that the Inductive

method provides us with the premisses from which the generalizations

themselves can be deduced according to
logical rules. From this it
does not follow that the generalizations, reached
the eliminative
by

method, must be

The eliminative method would take us to this

conclusion only if we had the additional knowledge that the premisses
which this method provides are themselves true. That the conditions
true.

for this conclusion are fulfilled

Is the third
meaning of the phrase
that eliminative induction leads to absolute certainty.
It would not be inappropriate to
say of the eliminative method
that it justifies induction If it could reach
certainty In any of these

three senses. But

it is

als

clear that only if

it

does so In the third

sense, do we get that kind of justification in which we are primarily

interested here, viz. that which excludes the occurrence of a contra-

dictory instance to the law established by induction. Bacon explicitly

attributed this power to the eliminative method when he said that it
'ex aliquibus generaliter concludat ita ut instantiam contradictoriam
inveniri non posse demonstretur'. 11
It follows from the logical nature of the eliminative method, as
described above, that elimination as such only informs us of the
falsehood of certain hypotheses. From this information in itself
nothing can ever be concluded as to the (conditional or uncondi-

63

THE LOGICAL PROBLEM OF INDUCTION

tional) truth of some not-eliminated hypothesis. Pure elimination,
therefore, at least cannot attain certainty in the second and third of

the

above

senses.

however, not certain a priori that elimination in

attain even certainty in the first sense. For this it is a
It is,

itself

can

minimum

requirement that it is logically possible, by elimination alone, to

invalidate all concurrent hypotheses except one as to a necessary or
sufficient condition of a given characteristic. It will be our next task
to inquire whether the logical mechanism of elimination really can
achieve this last aim without the aid of some general postulates
about the nature of the universe.

4.

The mechanism of elimination.

Necessary Conditions. We begin with a description of how the

logical mechanism of elimination works when we are looking for a
necessary condition of a given characteristic A.
By & positive instance of the conditioned property A we mean any
individual x, of which this property can be truly predicated.
of positive instances of the
We consider a (finite) set x, y, z,
conditioned property. To the individual x there answers a set of
z
(other than A), which can be truly predicated of
properties Xi,
of properties of
this individual. 1 Similarly, there is a set 7i, Fa
of properties of z, etc. The properties in each set
y, a setZi, Z a
are assumed to be logically independent of each other. This means
that no one of the properties is such that its presence in the individual
logically follows from the presence of some (or all) of the other
.

2
properties in the individual.

These

sets

of properties

we

call

basic sets of initially possible

necessary conditions of A.
By the positive analogy between a

number of sets of properties we

the (set of) properties, which are common members of all the
sets. The (set of) properties again which are members of some but
not all the sets of properties are said to constitute the negative
8
analogy between the sets.

mean

a property P belongs to the negative analogy

between the basic sets of initially possible necessary conditions of A,
then it cannot (actually) be a necessary condition of A. For, from
It is clear that, if

64

INDUCTIVE LOGIC
the definition of the
negative analogy it follows that there exists at
one positive instance of the conditioned
property A, in which

least

is
lacking. And anything which is absent in the presence of 4
cannot be a necessary condition of A. Or, in other
words, if a
property belongs to the negative analogy between the basic sets,
then the supposition that it would be
universally implied by the

conditioned property A is invalidated and

consequently also eliminated from the class of alternative
hypotheses as to the necessary
condition of A.

It follows from this that each increase in the

negative analogy
mentioned, and conversely each decrease in the corresponding
positive analogy, effects the elimination of some possible hypothesis
as to the necessary condition of the conditioned
property A. Therefore, the object of applying the eliminative method of induction to
the search for a necessary condition of A is to
narrow, as far as can
be done, the positive analogy between some basic sets of

initially

possible necessary conditions of the property. This is done by adding

new positive instances to those already considered, which
differ
from the latter in as many properties as possible. Or to use a classical
it is done
by 'varying the circumstances' under which A
occurs. According to the
way in which this variation is affected, we
say that we use eliminative induction as a method of observation or

terminology,

as a

method of experimentation.

conceivable that the positive analogy between the basic sets

of one property only. This, of course, does not
mean that use of eliminative induction will actually, sooner or later,
in every case lead to a situation in which all
save one are
It is

might

finally consist

properties

eliminated from the positive analogy. If we are left with more than
one property in the positive analogy, this may be either because we
have not yet 'varied the circumstances* to the utmost, or because the
conditioned property actually has more than one necessary condition

among the members

of the basic

sets.

The latter

alternative

is

known

from

traditional logic of induction under the name of 'Plurality of

Causes'. It is to be observed that there is no answer to the question,

which of these two

alternatives is true, other than the answer which

future experience alone can give after continued recourse to eliminative induction.

Let us, however, suppose that we actually were

65

left

with only one

THE LOGICAL PROBLEM OF INDUCTION

property in the positive analogy. From this fact it cannot be concluded that this remaining property is also the only remaining
not even in the 'realm', so to
possible necessary condition of A
of
the
the
basic
sets.
in
For, we have still to consider
say,
properties
the possibility known by the name of 'Complexity of Causes',* i.e.

no single member of the basic set is a

a
of
condition
A,
necessary
disjunction of two (or more) such members
is a necessary condition of A*
That the disjunction of two properties, say B and C, is a necessary
the possibility that, although

condition of

A means

that

whenever

is

In symbols: (x) [Ax-+Bx v Cx\.

present, then

or

is

Necessary conditions of

present.
the disjunctive

form are no mere 'theoretical possibilities' but

from the practice of science.' Thus, e.g., in order to bring
about a variation in the volume of a gas, it is necessary either to

familiar

vary the pressure, to which the gas is subject, or to vary its temperature. Variation in pressure or temperature is thus a necessary condition of variation in volume.
It might here be suggested that the eliminative method should be
applied also to the properties which can be constructed by forming
all the possible disjunctions of logically independent properties which
are

members of some of the

basic sets.

Actually, an application of the eliminative method to such 'dis'

7
But the resort to elimination is
junction-properties is possible.

here subject to an important limitation in its logical powers. This

is seen from the following considerations:
It is clear that if a property P is a necessary condition of A, then
the disjunction of P and any property is also a necessary condition

of A.

For, that

is

positive instances of

where P is present
Consequently,

it is

if

a necessary condition of

means

that in

all

the property P is present, and of all instances

is
trivially true that P or
present.

'

elimination has left us with the property

possible necessary condition of A, then

as a

us with all
as a constituent, as such

it

has also

left

disjunctions of properties, containing P

possible conditions.
From the above it follows that if all the 'disjunction-properties'
mentioned with the exception of one have been eliminated, i.e.
excluded from the possibility of being necessary conditions of A 9

then this one remaining property must necessarily be the disjunction

66

INDUCTIVE LOGIC
of as

many

logically independent properties as there are in all the

basic sets considered

together.
The relevance of this

applied to possible

peculiarity of the eliminative method,

complex conditions, to that method's

when

power of

attaining certainty will be estimated later.

We

Sufficient Conditions.

proceed next to an examination of the

where we are looking for a
condition of a given
property A.

mechanism of elimination
sufficient

Sufficient

in cases

and necessary conditions are

If the

interdefinable.

presence of A is sufficient for the presence of B, then the absence of

A is necessary for the absence of B, and vice versa. In symbols:
(x) [Ax-*-Bx]zs(x)[~Bx-+~Ax]. If oxygen is a necessary condition
of life, then the absence of
oxygen is sufficient to extinguish life, and
conversely.
It follows

from this that to ascertain the sufficient conditions of a

given property A is equivalent to the task of ascertaining the necessary conditions of the property
(not-^4). And this means that
the same method of -elimination as was described above for
necessary
conditions can be applied ('negatively' or
'inversely', so to speak)
to the search of sufficient conditions. 8

~A

There

is,

however, a

case of the search of sufficient

'typical'

conditions, to which a different canon of elimination is applicable.

In this case we are interested, not in the sufficient conditions 'as
such' of a given property, but in the sufficient conditions of the

property among the properties of a given positive instance of it.

In this case we compare a given positive instance x of the conditioned property A with a (finite) set of negative instances 9 z, ... of
y
A. (By a negative instance of A we mean any individual, of which A
9

can be truly denied.)

To

the positive instance

there answers a set of logically inde-

which can be truly predicated of x. 10

a basic set ofpossible sufficient conditions of A.
To the negative instances j, z
also answer sets Y19 F2
and
etc. of
2
Zi,
properties which can be truly predicated of the
individuals. We call these sets of properties basic sets of not-possible
pendent properties

We call

l9

X*

it

sufficient conditions

of A.

in a negative instance of

(It is clear

that any property which occurs

sufficient condition of A.)

cannot be a
67

THE LOGICAL PROBLEM OF INDUCTION

We

form

the logical

sum of the

basic sets of not-possible sufficient

we form a set consisting of each

occurs
in
that
at
least
of the above sets of not-possible
one
property
sufficient conditions.

conditions of A.

With

That

is

this sum-set, thus

to say:

formed,

we compare

the one basic set of

possible sufficient conditions of A, It is obvious that if a property

belongs to the positive analogy between these two sets, then it is elim-

inated as a possible sufficient condition of A. For, from the fact that

it belongs to this
analogy, it follows per definitionem that at least one
instance exists in which the property in question is present but in
A is lacking. The property mentioned, in other words, though

which

A in x, cannot universally imply A.

reason each increase in the positive analogy mentioned
entails the elimination of some alternative hypothesis as to the
sufficient condition of A among properties of x. It is the purpose
of the eliminative method to increase this positive analogy to the
utmost. This is done by taking for examination new negative instances which agree in as many properties as possible with the one
positive instance x of A which we have examined.
Obviously, the elimination of all properties except one from the

present with

For

this

basic set of possible sufficient conditions, is logically possible. But

here, as in the case where we looked for a necessary condition, we
can never a priori exclude 'Plurality of Causes', i.e. the possibility
that

has not one but several

of this basic

sufficient conditions

among members

set.

Suppose, however, that

except one from the basic

we had actually eliminated all properties

Then the question of possible 'Com-

set.

plexity of Causes' arises. This means that, although no single

is
member of the basic set of possible sufficient conditions of

actually such a condition, nevertheless a conjunction of

such members

is

two

(or

more)

a sufficient condition of A. 11

That the conjunction of two properties, say B and C, is a sufficient

condition of A means that whenever B and C are both present (but
not necessarily when one of them is present), then A is present too.
In symbols: (x) [Bx&Cx-*Ax]. Sufficient conditions of a conjunctive
form are certainly a commonplace in the practice of science." Their
importance to the logical study of induction was (vaguely) recognized

by

Mill. 13
68

INDUCTIVE LOGIC
It is
possible to apply the same canon of elimination also to the
properties which are constructed by forming all the
possible conjunctions of any two, three, etc. members of the basic set of
possible
sufficient conditions. 14
But the use of elimination
such

among

'conjunction-properties' is subject to a limitation, analogous to the

one described above for
necessary conditions.
It is clear that if a
property P is a sufficient condition of A, then
the conjunction of P and
any property is also a sufficient condition of
A. For, that P is a sufficient condition of A means that in
all positive
instances of P A is present, and of all instances where P and P' are

present it is trivially true that they are positive instances of P. Thus,

if elimination has left us with the
property P as a possible sufficient
condition of A, then it has also left us with all
conjunctions of
properties, of which P is a constituent, as such possible conditions.
From the above it follows that
the
//all

'conjunction-properties'

mentioned with the exception of one have been

eliminated, i.e.
excluded from the possibility of
sufficient
conditions
of A, then
being
this one
be the
remaining possibility will
of

necessarily
conjunction
the properties in the basic set of
initially possible sufficient
conditions of A.
all

Our

description of induction by elimination, when used for the

purpose of ascertaining necessary conditions, roughly answers to
Mill's Method of Agreement, and our description of the elimmative
method, when used for the purpose of ascertaining sufficient conditions of a given property in a
given positive instance of it, roughly
corresponds to Mill's Method of Difference. For the purpose of
ascertaining necessary-and-sufficient conditions canons of elimina15
tion may be used which
roughly answer to Mill's Joint Method.
the
above discussion we have assumed that the generaThroughout
lization or law, of which we are in search, is of the
type [1] of the
preceding Section, i.e. of the type that the conditioned and the
conditioning properties are attributes of the same individual. The
description of the mechanism of elimination can without difficulty
be extended so as to apply to generalizations of the
type [2] as well.
We have only to let the basic set of properties answer, not to
(positive

or negative) instances of the conditioned

property itself, but to
instances correlated through some relation F to instances of the
conditioned property. This modification in the determination of
F

69

THE LOGICAL PROBLEM OF INDUCTION

the basic sets
tive

is

altogether inessential to the

functions. 16

way in which

the elimina-

mechanism

Conclusions.

It

was mentioned

at the

end of Section 3 that

if

elimination, as such, without the aid of any general postulates or

assumptions as to the nature of the universe, were to attain certainty
in

any of the various senses in which

to this

'certainty' has

been attributed

method of

effect, in

any

induction, then it should be logically possible to

given case, the elimination of all alternative hypotheses

except one as to a necessary or a sufficient condition of a characteristic.

Examination has taught us that the two sub-methods of
induction by elimination, roughly corresponding to the classical
methods of Agreement and Difference, are both equally efficacious

And

to both of them applies the

of
reaching this aim:
power
to
achieve an elimination of all
it is
Although
logically possible
relative to given basic
conditions
one,
initially possible
except
sets of logically independent properties, this 'ideal' elimination must
always lead to the result that the only remaining possible condition
is of 'maximal
complexity', i.e. either a disjunction or a conjunction
of as many properties as there are logically independent members in
the basic sets of initially possible conditions. This important restriction on the 'direction' of the eliminative process makes pure
elimination less valuable, as a means of finding the only possible true
hypothesis as to a necessary or sufficient condition of a given characin the

attainment of this aim.

following limitation in their

teristic, than it might seem at first sight to be. It would have been
more valuable, if the 'direction' of elimination had been 'free', i.e.
if it had been possible to eliminate from the class of initially possible

hypotheses all but one, without it being possible to know

beforehand (on purely logical grounds) which degree of complexity
must characterize that hypothesis which will finally be the only
rival

17
remaining one.

Let us illustrate the limitations in the power of pure elimination by

an example. This example will to some extent go beyond the basis of
the above strictly formal considerations, but might be useful in
assessing the value and epistemological significance of the classical
ideas about eliminative induction.
70

INDUCTIVE LOGIC

As an example of how the Method of Difference works the

follow-

sometimes mentioned: 18
We observe a yellow band at a characteristic place in the spectrum
of a spirit-flame containing sodium. We wish to establish a 'causal
relationship*, i.e. a relationship of universal implication, between the
yellow band in the spectrum and the presence of sodium in the flame.
To this end we remove the sodium from the flame leaving all other
ing

is

circumstances unchanged.

If then, together with this removal of

flame, the yellow band in the spectrum also
vanishes, we feel inclined to assert that the presence of sodium in the
flame was the 'cause' of the yellow band in the
This

sodium from the

spectrum.

based on the following argument;

Since the yellow band did not occur when the flame did not contain
sodium, it cannot be universally implied by any circumstance present
in this case. But since, on the other hand, this second case differed
only in the absence of sodium from that case where the yellow band
appeared, we conclude that, if there is any cause at all for the occurrence of the yellow band, then this cause must be the
presence of
assertion

is

sodium in the flame.

Already Mill has observed that an argument of this type
clusive in the following respect: 19
Even if it is assumed that the yellow

is

incon-

cause, we cannot
be sodium alone,
since the possibility remains that it were sodium and some other
substance
and neither of them alone which universally implied
the occurrence of the yellow band in the
spectrum. In other words,
the possibility of a 'complex cause" remains. We can only, strictly

from the experiment conclude

band has a

that this cause will

speaking, conclude that, if there is any cause at all for the occurrence of the yellow band, then sodium is at least part of this cause,
that is to say in our terminology, that if the condition is complex then
consists of a conjunction of characteristics,

one of which is sodium.

Mill and later authors, however, have not rightly estimated the
significance of this inconclusiveness in the argument. Actually this
inconclusiveness implies that we cannot draw any general conclusion

it

at all

from the above described experiment.

important to point this truth out, since there appears to be a

strong inclination to overlook it for the following reason: We admit,
following Mill, that we cannot conclude from the experiment that
It is

71

THE LOGICAL PROBLEM OF INDUCTION

when sodium is present in the flame there will appear a yellow
band in the spectrum, since the cause may be 'complex'. But since,
on the other hand, this possible complexity in the cause means that
sodium is at least part of the cause of the yellow band's occurrence,
always

seems that we could conclude conversely that always when there

a yellow band at the characteristic place in the spectrum, then
there must be (at least) sodium present in the flame. This would
amount to asserting that, although we cannot from the experiment
conclude that sodium is a sufficient condition of the yellow band's
occurrence, we can nevertheless conclude that it is (at least) a necessary condition of it.
The suggested conclusion, however, it must be observed, is entirely
unjustified. The method of ascertaining necessary conditions by
elimination is, as was shown above, a method which can be roughly
identified with the classical Method of Agreement. The experiment
described above has not, however, the slightest relevance to the
possible result of applying this method in ascertaining a necessary
condition of the yellow band in the spectrum of the spirit-flame. It
will also be immediately clear on reflexion that there is nothing in the
experiment mentioned to exclude the existence of the characteristic
yellow band in the spectrum in spite of the fact that there were no
it

is

sodium in the spirit-flame.

Thus we can, on the basis of the experiment described above,
neither conclude universally from the presence or absence of such
and such substances in the spirit-flame the presence or absence of
such and such spectral phenomena, nor conversely from the presence
or absence of anything in the spectrum the presence or absence of
any substance in the flame. In other words, there is no universal
implication whatever to be concluded from the experiment. This

important truth
sical inductive

is

clearly revealed to us

methods

is

when

the theory of the clas-

treated as a theory of necessary

and

suffi-

cient conditions.

Although the value of pure elimination as a means of attaining

is limited, the description of the logical mechanism of
elimination is of importance as being the exact and formalized
expression of age-old ideas on the way in which truly 'scientific'*
certainty

induction proceeds, particularly when employed for experimentation.

The nature of elimination makes it clear why we sometimes consider
72

INDUCTIVE LOGIC
the examination of a
single instance, when pursued carefully and
with a certain methodical aim, to contribute
very much more to the
weight of a general proposition than does the verification, regardless
of further circumstances, of even an enormous multitude of instances
21

The method of ascertaining necessary conditions, in

the exact expression of the rule, which has
always been
regarded as one of the leading maxims in the practice of science, viz.
that the true test of a scientific law does not lie in the number of conconfirming
addition,

it,

is

firming instances as such, but in the multitude and variety of different

22
conditions under which the testing has taken
place.

Remarks about the comparative

and Difference.

5.

The

value of the methods of

Agreement

Method of Agreement is, for the purpose of

Method of Difference might
appear highly surprising when we consider the value attributed to the
latter method in the inductive
logic of Mill. As is well known Mill,
and following him most later authors on the subject, regarded the
Method of Difference as being the only method by means of which
result that the

elimination, not less 'effective' than the

we can

reach absolutely certain conclusions as to possible 'causes', 3

and as being much superior for this and other reasons to the method
of Agreement. 4
In order to make clear to what extent our results as to the comparative value of the two methods conflict with those of Mill we
have to make the reader conscious of certain peculiarities in Mill's
description of the inductive methods, peculiarities which are partly
the offspring of serious mistakes and obscurities,

In the logic of Mill the word 'cause' means sufficient condition* (in
For this reason the Method of Agreement, as described by
Mill, can be used as a method of elimination solely for the purpose of
looking for the effect of a given cause and not for the cause of a given
6
effect. This, however, was overlooked by Mill.
After having given
a substantially correct description of how the method works by
elimination for the detection of the sole possible effect of a given
cause 7 he continues: 8
*In a similar manner we may inquire into the cause of a given
time).

73

THE LOGICAL PROBLEM OF INDUCTION

Let a be the effect. ... If we can observe a in two different
combinations, a b c and a d e\ and if we know, or can discover, that
the antecedent circumstances in these cases respectively were
and ADE; we may conclude by a reasoning similar to that in the
9
preceding example, that A is the antecedent connected with the
consequent a by a law of causation. B and C, we may say, cannot be
causes of a, since on its second occurrence they were not present; nor
and E, for they were not present on its first occurrence. A,
are
alone of the five circumstances, was found among the antecedents
'
of a in both instances.
Here Mill obviously has failed to see that the fact that 5, C, D and
E belong to the negative analogy between the two cases has no bearing
whatever upon the question of finding a sufficient condition, i.e. a
'cause* in Mill's sense, for a. Actually none of the characteristics has
been eliminated as a possible sufficient conditon of A\ all we have
achieved with the two instances is that the hypotheses that B, C, D or
E respectively is a sufficient condition of a have been confirmed once,
and the hypothesis that A is such a condition has been confirmed
twice. Thus Mill's Method of Agreement,, when applied to the task of
finding sufficient conditions is not an eliminative method at alt but
simply a kind of induct io per enumerationem simplicem.
This important truth never became clear to Mill. Misled by the
fact that the increase in the negative analogy, when the Method of
Agreement is employed, actually brings about an elimination (of
possible necessary conditions) he believed this method also to be one
of elimination when used in the search for causes of given effects, i.e.
of sufficient conditions. On the other hand the fact
although not
effect.

ABC

that the method, when applied to sufficient

by Mill
conditions, was not one of elimination but one of enumeration, drove
him to the reservation against the Method of Agreement, which he
clearly grasped

expressed by saying that this method can prove a characteristic to be

invariable, but not an unconditional antecedent of a given charac10
teristic.
By this very confused and gravely misleading formulation

an

Mill simply wanted to express that the Method of Agreement cannot

prove the characteristic A to be a sufficient condition of a, even if A
and a are the only properties common to all the examined instances. u
Again, with the aid of the Method of Difference, we might prove
that if a has any sufficient condition in the cases in question, then this
74

INDUCTIVE LOGIC
condition cannot be any other than A or some condition more
complex than A, i.e. some condition part of which is A.
Thus we can understand how Mill, from the false idea that the

Method of Agreement as well

eliminative method when used

as the

Method of

Difference were

an

in the search for sufficient conditions

together with the true insight that the Method of Difference, but
not that of Agreement, can under certain circumstances
one

prove

characteristic to be in a given case the only possible sufficient

condition or part of such sufficient condition of another characteristic,

arrived at the further idea that the

Method of

Difference

was

'superior' to the Method of Agreement in the search for 'causes',

i.e. sufficient conditions. The
absurdity of this comparison becomes

apparent so soon as we realize that the Method of Difference alone

an eliminative method when applied to sufficient conditions.

Apart from this fundamental misconception of the eliminative
power of the Method of Agreement there is another reason which
is

caused Mill to regard the Method of Difference as 'superior* to that

of Agreement. This reason consists in his always assuming, in
describing how the Method of Difference works, that the first
instance lacking the conditioned property agreed with the instance
having the conditioned property in all other characteristics except a
single one, or else that only two instances were needed in this method,

whereas in the method of Agreement we need perhaps an unlimited

number of instances. 12 Such an assumption was intelligible from
the use we actually make of the Method of Difference as a method of
15

scientific experimentation.

But logically' we are equally

justified

making a corresponding assumption in the Method of Agreement,

when used as a method of elimination, viz. that the second positive
instance of the conditioned property differs from the first instance
in all its properties except a single one. (This assumption might also,
as in the case of the other method, be 'practically' justified when the
1
Method of Agreement is used as an experimental method. *) Mill
was prevented from grasping this probably for the simple reason
that he did not realize that the Method of Agreement is an elimina-

in

tive

method when applied

to necessary conditions only*

75

THE LOGICAL PROBLEM OF INDUCTION

6.

The general postulates of induction by

elimination.

In ascertaining the possibilities of eliminative induction to attain

certainty we must distinguish from each other the following two
questions:
(1) Is it logically possible, by elimination alone, to invalidate all
concurrent hypotheses except one as to a
necessary or sufficient
condition?
(2) Is it possible to determine, in a given case, whether
current hypotheses except one have been eliminated?
It was stated above that an affirmative answer to the first

was necessary

if

eliminative induction

was

all

con-

question

to attain
certainty in

any
of the three previously defined senses.
It is, however, to be observed that this affirmative
answer, although
necessary, is not sufficient for the attainment of certainty even in the
first of the three senses mentioned. To this
end, evidently, it must be
possible to answer also the second of the above questions in the
affirmative.
It is
immediately clear that if the number of independent possible
conditioning properties of a given conditioned property were infinite,
then it would never be possible to determine whether,- in a
given
situation, all concurrent hypotheses except one have been eliminated.

For, under such circumstances, we could never know for certain

whether, in the basic sets, the positive and negative analogies of
which it is the business of eliminative induction to increase and
decrease respectively, all relevant
have been included or
properties

whether some property has not, for one reason or another,

escaped
notice or been neglected.
however, not possible for reasons of logic alone to exclude the
possibility that the number of concurrent hypotheses as to a necessary
or sufficient condition are infinite. The exclusion of this
possibility
can be effected only by the introduction of some
It is,

general assumption

as to the constitution of the universe.

Now

has been suggested that the

assumption necessary on this
of eliminative induction of
attaining certainty, were that the number of logically independent
properties of any individual are finite in number. 1 This assumption
we shall call the Postulate of Limited Independent
Variety. It can
it

point, in order to secure the possibilities

76

INDUCTIVE LOGIC
be said to be the basic supposition of the inductive
3
logic of Bacon.
a
form
of
the
Baconian
Recently
developed
postulate has been advocated by Keynes. 3 There appears, furthermore, to be some

prima

facie presumption in favour of the truth of this postulate. 4

It is, however,
extremely important to observe that the assumption
of a finite number of properties of
individual is not sufficient for

any

the purpose of knowing when all concurrent

hypotheses except one
have been eliminated. Suppose that we knew the number of
properties to be finite. Then, even if we have taken into
account, in pursuing
the elimination, any finite number n of concurrent
hypotheses as to a
necessary or sufficient condition, we could never be sure that there

does not exist at least one p roperty which so far has not been reckoned
with, and which therefore may represent also a true hypothesis as to
the condition in question. For from the mere
knowledge that the
number of properties of an individual is finite, it does not follow that
the number of properties (and
consequently also of concurrent
hypotheses), which in a given situation we ought to take into account,
is not
greater than the (always finite) number of properties, or hypo5
theses, which actually has been considered.
Consequently, in order to make possible knowledge as to whether
in a given situation all concurrent
hypotheses except one have been
eliminated, we have to introduce some assumption 'stronger' than
that of Limited Variety. This new
assumption must, generally

speaking, assert that, under certain circumstances, it is possible to

know when we possess complete knowledge of all properties of the
examined instances which are to be taken into account for the purpose

of elimination of concurrent hypotheses. This postulate of the

logic of induction which is to replace the postulate of Limited

Independent Variety we

shall call that

of Completely

Known

In-

stances. 6

For a more

detailed formulation of the postulate of Completely

Instances two principal ways are left open. One of them
might be described as a continuation of the way leading to the
introduction of the postulate of Limited Variety. It would consist

Known

our introducing some more precise assumption as to the number

of possible properties of an individual than simply that it is finite. A
sort of 'minimum assumption' in this direction would be to suppose
the number of possible properties never to be greater than a fixed
in

77

THE LOGICAL PROBLEM OF INDUCTION

number

n.

Under

this

assumption

it

would be possible

at least in

we have in each examined instance discovered

n properties, to know for certain when all concurrent hypo-

those cases where

exactly
theses except one as to a necessary or sufficient condition have been
eliminated.

On the other hand it seems arbitrary, if not absurd, to assume

a priori, of the number of possible properties of an individual, that it
cannot supersede a fixed number n. This way of giving the postulate
of Completely Known Instances a more specific formulation must
therefore be regarded as extremely unplausible

The second way of

make no assumption

and

unsatisfactory.

specifying this postulate is the following:

We

number of possible hypotheses or

not even that it must be finite.
an
of
individual,
possible properties
We assume instead that certain categories of simple properties can be
as to the

out of consideration as being irrelevant to the eliminative method

of induction, and that in each single case we are able to judge
whether the information about the instances, which has been taken

left

into account, represents complete knowledge as to all the remaining

relevant properties or not. If it does then we can enumerate the

relevant properties of each individual and hence also determine

whether or not, after the examination of a number of instances, the
eliminative process has reached the elimination of all concurrent

hypotheses except one.

The defects also of this formulation of the postulate will be obvious
to anybody. How is it possible, it will be asked, to define what is
'relevant' in this connection in any way other than that the definition
involves a reference to the eliminative method itself and hence is
circular? 7 But apart from this obvious defect it cannot be denied
that such a specification of the postulate has some plausibility in itself.
First there is, so to speak, a 'practical plausibility' in favour of it,

we are in any given case usually "practically sure'

about which hypotheses might conceivably be true for a given
phenomenon and which again can at once be dismissed as being
consisting in that

irrelevant.

The number of

the

first

hypotheses, furthermore,

is

as a

rule not very great.

This 'practical plausibility' of the postulate, incidentally, is a fact

of the greatest psychological significance for the possibilities of, the
human mind to detect law and order in the multitude of phenomena
78

INDUCTIVE LOGIC
with which it is confronted. If it were not
possible for us to confine
our attention to a fairly small sector of 'relevant' circumstances in
the multitude of given data, we should have seldom succeeded
in
detecting the uniformities and laws of nature of which we
actually

possess knowledge.

must

also be noted that there seems to be at least one class of

8
'properties' which can generally be excluded as being 'irrelevant'.
This class consists of the characteristics which state the
spatiotemporal position of the instances. It is an old idea that the validity
It

of natural laws cannot be restricted

by time and space as such, but
that if a law is not valid under
spatio-temporal conditions differing
from those under which it has been detected and confirmed, then this

is due to some difference in

circumstances, other than spatio-tem9
This idea seems extremely plausible, and one is
poral.
tempted to
say that its plausibility, which almost
self-evidence, is founded

equals

not upon matters of

but upon some a priori grounds. 10

So much for and against the postulate of Completely Known
Instances as specified above. Irrespective of whether this
postulate
can be upheld with some plausibility or not, we have to observe that
it is
absolutely necessary if the second of the above two questions is
to be answered affirmatively, i.e. if it is to be
possible to determine,
in a given case, whether all concurrent
hypotheses as to the necessary
or sufficient condition of a given characteristic have been eliminated.
Thus only under the Postulate of Completely Known Instances can
induction by elimination reach certainty as to which one of a number
of concurrent hypotheses is the only generalization fitting all the

known
Mill,

12

fact,

data. Actually there are passages in the writings of Bacon, 11

and other authors, 13 from which one gets the impression that
just this kind of certainty that they deemed the eliminative

was
method capable of
it

achieving. The failure to distinguish different

senses of the term 'certainty' from each other has, however, caused
those authors to make apparently contradictory statements in other
14

places.

be useful here to introduce the following definition:

of the elimination in a case when the eliminative method
is
applied we mean all (singular) propositions stating that such and
such individuals, examined for the purpose of the method, possess
such and such properties.
It will

By

the data

79

THE LOGICAL PROBLEM OF INDUCTION

evident that the above postulate of Completely Known
is not sufficient if it is to be possible from the data
of the
elimination to deduce general propositions, that is to say, if eliminative
It

is

Instances

is to reach certainty also in the second of the three senses

mentioned. For even if the result of actual elimination together with
the above postulate had informed us that the only characteristic
which can possibly be connected by general implication with the
given characteristic, say A is the characteristic B, this information
cannot exclude the possibility that the next instance, so far unknown
to us, which exhibits the characteristic supposed to be the implicans

induction

lacks the property supposed to be the implicat, or in other words,

that the general implication between A and B is, after all, false.

Thus a second postulate

is

needed

if

the

method of

eliminative

to result in the deduction of general implications (or

equivalences) from the data of the elimination. This postulate we

induction

is

15
It can be given various,
Assumption.
weaker or stronger, formulations. One such formulation is that every

shall call the Deterministic

property has, in every positive instance of its occurrence, at least

one sufficient condition. From this formulation follows that every
property also has at least one necessary condition, viz. the disjunction of all

its sufficient

conditions. 15

Mill restricted the applicability of the Deterministic Assumption

to properties of individuals succeeding each other in time, thus
getting the more specific form of it which might be called the Univer-

Law

of Causation, 17 and did so on the ground that he regarded

the postulate as being unjustifiable for simultaneously existing
18
Causal Laws, according to Mill, but not Uniformities of
properties.
Co-Existence, get their strength from such a general principle.
Bacon on the other hand seems to have assumed precisely the
opposite to Mill, viz. that for each property or 'nature' of an individual there exists another simultaneous property, called its *form\
10
being a necessary and sufficient condition of it, whilst he does not
sal

state

any corresponding general principle for temporally related

properties.
*

The two

general 'inductive principles called by us the Deterministic

Assumption and the Postulate of Completely Known Instances are
necessary, but also sufficient, if eliminative induction is to attain
certainty in the second of the three senses, i.e. if inductive generaliza60

INDUCTIVE LOGIC
tions shall be deducible from the data of the
elimination. This
deduction, generally speaking, is made so that the Deterministic
Assumption is applied to a situation where the Postulate of Completely Known Instances, together with the actual results of the
elimination, entitles us to the conclusion that all concurrent

hypo-

theses except one have been eliminated.

Any system of conditions which makes it possible to deduce
inductive generalizations from
singular data we shall call a Complete
System of Inductive Logic. In this sense the above two postulates,
together with the logical mechanism of elimination, make up such

a Complete System.

It is,
obviously, logically possible to strengthen the postulates of
our Complete System so that the
system's powers of deduction are
widened. We might, for
example, introduce some more definite
assumption as regards the number of (sufficient or necessary) conditions of a given conditioned
property. Or, we may assume the
existence of some restriction to the
complexity of the possible
conditioning properties. Such stronger assumptions have actually
been suggested. 20 They will, however, not be discussed here.

7.

The justification of the postulates ofeliminative

induction.

If induction by elimination is to reach

certainty in the third of the
previously mentioned senses, that is to say, certainty as to the truth
of a generalization, then it is not sufficient to know
only which
premisses are needed for the purpose of deducing general propo-

We

must, in addition, know that those premisses are true.

postulates of eliminative induction mentioned are all general
propositions, by the aid of which from singular propositions other
sitions.

The

general propositions are deduced. The truth of the singular propositions, i.e. the so-called data of the elimination stating that such and
such properties belong to such and such an individual, is in this

connection unproblematic. This cannot be said of the truth of the

general propositions.
Let us first assume the principles to be true a priori. As was seen
above the sole way to guarantee the truth of a general proposition

a priori

is

to

make

it

analytical.

Now it can be shown that if the postulates are supposed to

81

be true

THE LOGICAL PROBLEM OF INDUCTION

a priori, i.e. analytical, then any general proposition which, with
their aid, is deduced from particular data can be proved true only
in the sense of being analytical. This amounts to the same as that if
with a general analytical proposition and a singular synthetical
proposition as premisses, we draw a general conclusion, then the
conclusion must itself be analytical. One might be inclined to say
that this thesis is 'almost self-evident'. It will, however, be of some
interest to see, how it is related to our previous elucidations of the
notions of analyticity and logical consequence. 2
If the proposition b follows from the proposition a, then the

an analytical proposition and hence 3 necessary.

This relation between logical consequence and necessary implication
cannot be converted. For, if a is an impossible proposition or b a
implication a-+b

is

necessary proposition, then the implication is necessary, irrespective

shall assume, 5 however, that
of whether b follows from a or not. 4
if a-+b is necessary, then b follows from a in every situation when a

We

is

not impossible or b not necessary.

The

desired proof can

now be

given as follows:
If the general proposition

g follows from the particular proposianalytical

proposition a, then the implication a&p-+g
tion/?
is a necessary
proposition. Now, according to a law of modal logic,
if a is necessary and a&p-+g is necessary, then p~+g is necessary too.
and the

on our assumption above, if /?-># is necessary, then it is the case

g follows from/? or that/? is impossible or that g is neces-

But,

either that

We

already know that g, the general proposition, is not a

consequence of/?, the data of the elimination. Nor is it the
case that p is an impossible proposition. Hence the only remaining
sary.

logical

is that g is
necessary (analytical)/
In presenting the scheme of induction by elimination we intended
primarily to inquire whether it could reach inductive propositions
which were synthetical. Now we have seen that if eliminative

alternative

induction
this

is

to reach this then the general postulates necessary for

synthetical. Our next task will be to see what

method must be

bearing this has upon the question as to whether eliminative induction can justify inductive inference or not.
If the postulates are synthetical principles, their truth cannot be

guaranteed a priori* But perhaps it could be established a posteriori,

7
i.e.
generally speaking according to some 'inductive method'.
82

INDUCTIVE LOGIC
It is
immediately clear that this 'inductive method' cannot be
induction by elimination. For this method
presupposes, for the
attainment of truth, the two postulates the truth of which is now to

be proved. Thus to try this way would be circular.

But it is equally clear that the method cannot be enumerative induction. This method is
constantly 'periculo ab instantia contradictoria
and
can
hence
never reach certainty. 8
exponitur\
The possibility then remains that there exists some third inductive
method, beside the eliminative and the enumerative ones, which
could be used for proving the truth of the two
postulates. But this
also
can
be
ruled
out
on
the
possibility
following general grounds:
That a general proposition g can be established by an 'inductive
method' means 9 that g can be inferred from some
experiential
'data'

in conjunction with

some

'rules'

or 'principles' p.

Since

a general and d a singular proposition, it follows

that/? must be
10
If the inductive method has to establish the truth of
general
g,
the truth of d and p must be known. But, how can the truth
ofp be
known? If it is known a priori, then p is analytical. And then it

is

11
If, on the other hand, p is known
analytical too.
a posteriori, then its truth must have been established by means of
some other inductive method using other rules or principles p',

follows that

is

different from/?. Then the questions arises, how/?' is established arid

are driven to consider some further principles p", different 18

we

from both p and /?

Thus we see that we cannot prove the truth, as general synthetical
propositions, of the postulates upon which eliminative induction is
based, without reference to some new inductive method, and it
cannot be proved that this new method establishes the truth of the
'.

postulates without reference to a further inductive method, again

different from the former ones, and so on in infinitum. As the truth

of the two postulates as synthetical propositions was necessary in

order that induction by elimination should lead to true generalizations, not being analytical, we can conclude that the whole idea of an
inductive method reaching general synthetical propositions, known
to be true, has failed as leading to an infinite retrogression. For
from its leading to an infinite retrogression follows that it is never
possible to know
regarded as true.

when and

if

the generalization aimed at can be

83

THE LOGICAL PROBLEM OF INDUCTION

8.

The elimmative method and

the justification

Although Bacon's idea of an inductive

ut instantia contradictoria inveniri
failure,

non

of induction.

logic able to conclude 'ita

potest' is proved to be a

we must not underrate

inductive logic

on the above

the relevance which a system of

principles nevertheless possesses to the

question of justifying induction.

If anybody were to assert that the principles of elimmative induction justify a certain general proposition he is likely to assert, not
that this generalization can be proved to be true, so much as that it

can be proved to follow from premisses of a particular kind. These

are partly singular propositions asserting that certain
individuals have certain properties and that, consequently, certain
generalizations are invalidated; partly general propositions or
'inductive principles' such as the above Deterministic Asumption

premisses

or Postulate of Completely Known Instances. The truth of the

former premisses is, as a rule, unproblematic. In the truth of the
latter premisses, again, we are seldom directly interested. Therefore
also the task of proving their truth is of minor importance in view
of the practical needs of science.
The really important task of a logic of induction is to analyse the
logical mechanism of elimination so that it becomes clear what pure
elimination alone can achieve, and exactly to formulate the content
of the principles needed in order to extract inductive generalizations
from the data of the elimination. 1 When this is done we can determine in the case of given generalizations how far they are based on
known data and how far they go beyond our direct experience. This
knowledge, which gives us elucidation as to the logical relation
between our inductive conclusions and the experiential evidence
on which they are based, can sometimes truly be. said to constitute a
justification

of induction.

84

CHAPTER V

INDUCTION AND PROBABILITY

1.

The hypothetical character of induction.

IN the three preceding chapters we have examined different attempts

to justify induction as a species of reasoning leading to certainty.
We have seen that those attempts are successful or not depending
upon what we expect the justification of induction to be. They
succeed, under appropriate conditions, inter alia, if by 'justification'
we mean any of the following three things:
(1) A proof that a given general proposition is a 'law' for certain
data, i.e. that those particulars can be deduced from the generalization.
(Whewell.)

(2)
proof that a general proposition can be deduced from given
particular propositions by the aid of certain other general propositions, called 'inductive principles'. (Bacon-Mill,)
demonstration that a general proposition, obtained by
(3)

induction,

is

analytical,

i.e.

true per conventionem.

(Convention-

alism.)

But we have also seen that all those attempts fail to justify induction in one very important sense, viz. that of proving predictions from
an inductive generalization to be true. Or to use a different mode of
to be
expression: we cannot prove a synthetical general proposition
true prior to experiential testing.
The truth that inductive propositions, when used for the purpose
of predicting future happenings, are and must always remain hypotheses which coming experience may either confirm or refute, has
been already clearly apprehended by men of science centuries ago.
for instance, by Newton in his Opticks, when he
It was
expressed,
said that 'the arguing

from experiments and observations by induc-

1
The same
'no demonstration of general conclusions'.
a law to
from
of proving predictions
clarity as to the impossibility
another
be true a priori also pervades the works of
great man of
2
science of about the same time, Huyghens.

tion'

is

85

THE LOGICAL PROBLEM OF INDUCTION

In spite of all the various doctrines of synthetical judgments a
priori it can hardly be maintained that any philosopher had explicitly
asserted the possibility of proving predictions of concrete events in
advance of experience. But it is true that philosophers, even those
who have devoted much ingenuity and work to the theory of induc-

have until recently paid only slight attention to the fact that
inductive propositions, the consequences of which are deduced and
successively tested, are hypotheses.* The significance of this truth

tion,

has usually been entirely minimized by the stress that those philosophers have laid upon the element of certainty which is also inherent
in the inductive

The

mode

of reasoning. 4

philosopher to have clearly apprehended and separately

emphasized the epistemological significance of the hypothetical
first

of the impossibility of proving the truth of

this is the chief merit of his
predictions,
the
of
induction.
Today
opinions of Jevons in these
philosophy
matters may seem almost trivial, but the way in which they were
misunderstood and contested in contemporary philosophy is the
best proof that they represented a real step forward in the theory of

element in induction,
is

i.e.

Jevons. 5

To have done

induction. 7

2.

Hypothetical induction and probable knowledge.

One

of the chief aims of science

predictions.

Does the

is

to provide a basis for successful

knowledge used for predictions is

the possibilities of science to achieve this aim?
fact that

hypothetical affect
Is the impossibility of guaranteeing the truth of predictions prior
to testing a 'catastrophe to science', does it mean that all prediction
is
simply haphazard guessing, and that we are left in 'complete
uncertitude' as to the future course of nature?

It is obvious that if the results of our

previous investigations make
such questions seem justified, then the results have been misinterpreted. Because those results, as has already been pointed out
several times, are 'granxmaticar in their nature, i,e. concern the use
of certain words, whereas the above questions seem to protest
against some absurd consequence of our results for 'matters of fact'.
The questions arise out of the same fundamental mistake as that

86

INDUCTION AND PROBABILITY

which makes Berkeley's treatment of the existence of things and
causation appear absurd and
the
catastrophic
confusion between the clarification of thought and the discovery of

Hume's theory of
facts.

The above

questions, therefore, could be said to express a vain

about
the
worry
implications of our investigations. But if we are
asked to show in detail why we need not worry about these imagined
consequences, we are soon confronted with the most perplexing

problems.

The first answer which suggests itself as settling our anxiety as to

the 'catastrophe of science' is roughly the following:
From the fact that we cannot prove what is going to happen it
does not follow that we could not estimate, prior to testing, the degree
of reliability possessed by a prediction. Such estimations as a matter
of fact take place, since certain predictions are actually regarded as
very reliable ones, and others again as less or in a very small degree
reliable. These different degrees of belief in predictions we also
express by saying that predictions are more or less probable. Inductive

knowledge, in so far as

it is
hypothetical, is probable knowledge.
of
the
guaranteeing the truth of predictions
why
impossibility
is no
to
and
science',
why it is overhasty to say that we
'catastrophe
are left in 'complete uncertitude' as to the future.

This

is

answer were satisfactory it would imply, that the fact

that certain predictions are believed more, and others again
less strongly, were all the justification of inductive predictions that
If this

itself

we need.
To this, however,
tion.

From the fact,

there appears to be a strong prima facie objecwill be said, that one prediction or one genera-

it

regarded as more reliable, i.e. believed more strongly than

does not" follow that this prediction or generalization
really is more reliable than the other one. That is to say, we might
have been mistaken in our judgments about probabilities. The
reference to degrees of belief, therefore, is unsatisfactory as a justi-

lization

is

another,

it

fication of induction unless we can justify the beliefs themselves,

that is to say can guarantee with certainty, or at least with probability,
that we are not mistaken in our estimations of the probability of

inductive propositions. 1
Thus we have arrived at the idea that the justification of hypotheti87

THE LOGICAL PROBLEM OF INDUCTION

cal induction does not lie in the fact, as such, that we estimate
degrees of probability, but is to be found in some 'mechanism of

probability' underlying these estimations and guaranteeing their

validity. To the examination of this idea, which has played, and still
plays, a profound role in discussions and philosophical controversies
about induction, we shall devote the following chapters.

3.

The

scheme for

the treatment

of inductive probability.

idea of a

'mechanism of probability' underlying our estimations of reliability in inductive

propositions gains support from the
observations:
following
These estimations, it appears, are not made on 'intuition'
only but
take place in conformity with certain rules. Such rules are for
instance: The probability of a generalization increases with the
number of verified instances of it; the verification of an unexpected
or surprising instance of a law contributes more to its
reliability
than the verification of an instance of a
type with which we are
familiar; the probability of an induction is, somehow, proportionate
to the scope of the generalization.
It will

be our task in Chapter VI to give a formal analysis of the

rules of inductive
probability.

We

will inquire into the conditions

under which the above-mentioned rules and certain others are

provable, that is to say, what assumptions need to be made about
probabilities in order that those rules shall become logically necessary. That this analysis is purely 'formal' also implies that it is

pursued without any presuppositions as to the 'meaning' of probability or as to how probability-values are empirically determined.
The formal treatment of inductive probability will show that the
rules mentioned are all deducible from a common set of

simple
assumptions. This is important as it proves that those rules, frequently mentioned in works on induction but seldom analysed into
their formal interconnections, form
part of a coherent system of
inductive probability.
Moreover the analysis

is
peculiar inasmuch as that it shows the
formal structure of the probability-concept of this
system of inductive probability to be the same as the formal structure of that
proba-

88

INDUCTION AND PROBABILITY

bility-concept which is treated in the branch of mathematics known
to us under the name of the ('classical') calculus of
probability.
'
This shows that the idea of two kinds of probability
'mathematical

probability and 'philosophical' probability, the latter being essentially

the probability of inductions
is
unnecessary at least in so far as

the formal nature of inductive probability is concerned.

The formal analysis of the rules of inductive probability cannot
in itself determine the relevance of those rules to the
justification of
induction.

In Chapter VII

we return

to the

problem of justification.

We shajl

try to show, why any justification of induction with probability,

intended to refute Hume's scepticism as regards the possibility of

guaranteeing anything about the future course of nature,

is

doomed

to failure irrespective of how we interpret the

probability-concept.
From this will follow that no dichotomy into different kinds of

probability

terms be

is

'mathematical' and 'philosophical' or whatever the

helpful towards a solution of Hume's Problem.

89

CHAPTER

VI

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

1.

The Abstract Calculus of Probability.

Historically, the mathematical study of probability was developed

basis of a study of certain mathematical models of the concept.

on the

The

models is provided by the well-known definition

of probability as a ratio of cases or possibilities, 'favourable' and
'unfavourable' to a certain event or to the outcome of a certain
experiment. This model was suggested by considerations pertaining
to games of chance. Another model is provided by the definition of
probability as the relative frequency of a characteristic or an event
within a class ('population')- Probability-mathematics, when developed on the basis of the first model, was primarily a branch of the
theory of combinations and permutations. Probability-mathematics
developed from the frequency or statistical model, may be termed a
oldest 1 of these

class-ratio arithmetic. 2

In the two models are reflected different opinions as regards the

meaning of probability and about the relation of the mathematical

theory to empirical reality. 'Philosophically', these opinions are

highly divergent. We shall not here be interested in the question,
which of them is right or whether some of them can be 'reconciled'
3
by virtue of the fact that they fit different concepts of probability.
The logico-mathematical nature of the models themselves will be
somewhat more closely scrutinized in the next section.
It is a most important fact that the theories of probability, which
can be developed on the basis of the two models mentioned, though
differing in their conception of the 'meaning' of their fundamental
notion, yet agree, by and large, in their logical structure. This fact
suggests the possibility of creating an Abstract Calculus of Probability, i.e. a deductive theory which is 'neutral' with regard to
conflicting opinions about the meaning of probability and studies
only the mathematical laws which this notion obeys. Within a
theory of this abstract kind 'probability' figures as an undefined
90

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

term, for which certain 'axioms' or 'postulates' are laid down.
The axioms are sometimes said to constitute an implicit definition of
probability.
4
Several abstract calculi of probability have been
suggested
some have been developed in detail. They fall into two groups.

them the

and

We

and the logistic calculi.

An Abstract Calculus of Probability on a set-function basis has
been developed by the Russian mathematician A. Kolmogorov in
an important publication from the year 1933. In Kolmogorov's
calculus probability figures as a function
of sets. The theory has
received much favour among mathematicians and is
perhaps the
most satisfactory mathematical treatment of probability which has
shall call

set-function calculi

been presented. It incorporates probability mathematics within the

general theory of measurable sets of points.
The theory of J. M. Keynes from the year 1921 may be regarded as
the first attempt on a large scale at the development of an abstract
calculus of the logistic type. Another, more consciously 'formalist*,
system of the same type is that of Hans Reichenbach. It was first
8
presented in 1932. Axiomatic systems, similar to that of Keynes,
are due to S. Mazurkiewicz and H. Jeffreys. 9
Systems of probability of the type here called logistic' form a
rather heterogenous group. Common to all members of the group is
that they conceive of probability as a 'logical relation' between two
entities. The entities may be propositions, as in the
systems of
Keynes and Jeffreys, or they may be attributes (propositional10
functions, properties, classes) as in the theory of Reichenbach.
They might with a common name be called proposition-like entities.
Their 'proposition-likeness' consists, vaguely speaking, in the fact
that they can all be manipulated with the aid of the so-called truth7

connectives: negation, conjunction, disjunction, etc. 11

The logistic calculi of probability thus stress the relative (or relational) nature of probability.

for

its

value upon the

'field

Probability is a quantity depending

of measurement' in which it is deter-

mined. 12
For our purposes a logistic calculus of probability is better suited
than a set-function one. We shall in this section outline a logistic
Abstract Calculus of Probability. In order to simplify the treatment
we have made as many omissions of points of a. formal nature as has
91

THE LOGICAL PROBLEM OF INDUCTION

seemed

to us possible without seriously damaging the logical rigour

of the arguments. The purpose of our inquiry, it will be remem13
bered, is to indicate, how certain popular ideas about the probability
of inductions may be assigned a place within the common framework
of probability mathematics.
We introduce a symbol 'P(a/hy which we call probability-functor.
It can be read: the probability of a relative to h. Instead of 'relative
to' we may also say 'on data' or 'given' or simply 'in'.
It may be asked, whether P(a/hy makes sense for any two propositions, a and /z, i.e., whether it makes sense to say that any given
proposition possesses, relative to any other given proposition a
(known or unknown) probability. It is prima facie plausible to think
that a and h would need to be somehow 'materially related' in order
to determine a probability-relation. What sense could it make to
speak of the probability of a proposition of, say theoretical physics
6

relative to a proposition of history? It is further doubtful, whether a

proposition can have a probability relative to a self-contradictory

or for other reasons logically impossible proposition. 14

It is thus reasonable to think that the propositions (propositionlike entities) a and h must satisfy certain conditions in order to
determine a probability-relation.
It may further be asked, whether 'P(a/h)\ when significant,
necessarily signifies a number, i.e. whether the probability of a
proposition relative to another proposition is necessarily a numerical magnitude. This is a serious question, not least from the point of

view of a theory of induction. For it is sometimes said that probaan attribute of inductive conclusions, though a legitimate
concept, differs from 'ordinary' probability, among other things,
in being non-numerical 15 Of probability which is considered as
non-numerical the question may be raised, to what extent it is
comparative, i.e. obeys laws about 'greater', 'equal', and less'."

bility as

We shall, however, in this chapter deliberately ignore the problems

about

the existence

of the probability-relation and about the numerical

or non-numerical nature of (inductive) probability. In constructing

our Abstract Calculus we simply proceed on the assumption (con-

vention) that with such pairs of proposition-like entities as may enter

our arguments there may be co-ordinated a unique, non-negative
real number subject to the following axioms or postulates:
92

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

AL

P(a/h)+P (~alh)=L The sum of the probability of a

proposition and the probability of its negation, relative to one and
the same datum, is 1
A2. P(a&blh)=P(a[h)xP(b/h&d). The probability,
given A, of a
of
two
a
and
is
the
b,
conjunction
propositions,
probability of a on A
as datum multiplied by the probability of b on h and a as datum.
.

We

call this the Multiplication

Principle.

A3. Ifh

is self-consistent,
thenP(/z//z)=l.
the basis of these axioms, and a few principles of subordinate
nature, the whole fabric of what might be called 'classical' probability

On

mathematics can be erected.

The deduction of theorems from the axioms can be completely
'formalized', i.e. subjected to explicitly stated rules for the manipulations of formulae. We shall not burden the exposition
by enumerating
the rules of inference. We only mention that the deduction, by
and large, proceeds through 'substitutions of identities*, and that of
such substitutions there are two kinds, viz.
external substitutions, of numerically identical' expressions, in
(i)
the equations,
(ii)

and

internal substitutions, of logically identical propositions, in

the probability-functors.
(Examples of the two types of substitution of identities are indicated below.)
mention a number of elementary theorems.

We

Tl.

O^.P(a/K)^L Probability

is

a magnitude between

and

1,

inclusive the limits.

That probability is not-negative, i.e. equal to or greater than O,

was established by convention. (P. 92.) From this convention and
the postulate (Al) stating that the probabilities of a proposition and
of its negation, on one and the same datum, add up to 1, it immediately follows that probability must be equal to or smaller than 1
72. If h is self-consistent and entails a, then P(a/h) = 1
.

Proof:
17
logically identical with /z.
is logically identical with h, then by internal substitu-

(1) If

h entails

(2) If

a&h

a,

then

a&h

is

(3)

= P(a&h\h\
P(a&h/h) = P(h/h)xP(a/h&h).

(4)

h&h is

tion P(h/K)

logically identical

with
93

(From A2.)
h.

THE LOGICAL PROBLEM OF INDUCTION

Thus, by internal substitution, P(ajh&K) = P(a/h).
(6) Hence, by external substitution in (3), P(a&h/Ii)
P(h/h)xP(alh).
= L (A3.)
(7) If h is self-consistent, then P(h/h)
(5)

Hence, by external substitution in

(8)

identity established in
proof of T2.

T3.

we

(2),

P(alh)=P
P(b/h&a)+P(~b/h&a)

(1)

(2) Since a probability

P(a/h).
P(alh)

is,

get P(a/h)

=L

(6),

and considering the

1.

This completes the

(From A I.)

by convention, a unique value,

we have

- P(a/h) x [P(blh&a)+P(~b/h&a)].

(From (1) and (2).)

P(a/h) =P(alh)xP(blh&a)+P(alh)xP(~blh&a). (From (3).)
P(a/h) ^P(a&blh)+P(a&~blh). (From (4) with the aid of

P(a/h)

(3)
(4)
(5)

A2.)

We shall call T3

the Division Principle.

T4. P(avblh) ^P(alli)+P(blh)-P(a&blh).

P(avblh) =P((avb)<&lh)+P((avb)&~blh). (From T3.)

(avb)&b is logically identical with b alone, and (av&)&~6

(1)

(2)

logically identical with

a&~b. Thus by internal substitution

in (1)

is

we

obtain
(3) P(avb/ti)

=P(bjh)+P(a&~blh\

= P(alh)-P(a&blK).

(From T3.)
(From

(4)

P(a&~b/h)

(5)

P(avb/h)=P(alh)+P(blhy-P(a&blh).

external substitution.)
T4 will be called the Addition Principle.
That two propositions are mutually exclusive

(3)

and

means

(4)

by

that the

negation of either proposition logically follows from the other

proposition. Thus, if a and b are mutually exclusive, ~b follows

from
T5.

a.

If

P(avb/h)
It

a and b are mutually exclusive and consistent with

- P(a/K)+P(b/h).

may

easily

be proved

that, if

consistent with A, then P(a&b/h)

exercise to the reader.

A,

then

a and b are mutually exclusive and

= O. We leave the proof as an

We shall call T5 the Special Addition Principle.

We say that a is independent of b (for probability)
P(alh&b).
94

in A, if P(ajh)

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

T6. If a is independent of b in h or
then P(a&blh} = P(a/h)xP(blh).
(1)

independent of a in A,

is

P(a&b/h)=P(a!h)xP(b!h&a)^P(blh)xP(alh&b). (From
a is independent or 6, then P(0/A) = P(alh&b). If 6

(2) If

independent of

from

if

(1)

then P(6/A)

a,

by external

= P(b/h&a).

In either case

we

is

get

substitution:

= P(alh)xP(b/h).

(3)

P(a&bjh)

We

call T<5 the Special

Multiplication Principle.

For the higher development of the

calculus, the notion of inde-

of the greatest importance. 18

pendence
If extreme probabilities (the values
and
is

1)

prove the following theorems:

T7. If a is independent of b in h, then b
a in A.
T8.

~b

If

is

independent of b in

A,

then a

are excluded,

we can

is

also independent of

is

also independent

of

in A.

79.

If

b and of

We

<2

is

independent of b in A, then

^a

~& in A.

is

also independent of

leave the proof of these elementary theorems to the reader.

Consider an event such

We symbolize the

a coin.
non-occur-

as, e.g., getting 'head' in tossing

occurrence of the event by

'" and

its

rence by <-v/T.

Consider further a sequence of occasions, on which the event may

occur or fail to occur, e.g., a sequence of tosses with a coin. We
symbolize the sequence of occasions by *Xi\ x z \
*Exn means that the event has occurred on the nth occasion, and
'^Exn* means that the event has failed to occur on the nth oce

casion.

We shall say that the occurrence (and non-occurrence)

occasions

of E on the

are independent for probability in A, if the

probability, relative to A, of any proposition Exn is independent for
has occurred
probability in A of any proposition A to the effect that
i9

-x 2 ,

'

or failed to occur on some other occasions than xn (The proposition

A' is thus itself a conjunction of propositions of the form Ext and
.

~Exi.)

sequence of occasions of independent occurrences (and non95

THE LOGICAL PROBLEM OF INDUCTION

occurrences) of an event are said to constitute an independencerealm. 19

An

independence-realm will be called normal,

on

for the occurrence of the event

if the
probabilities
the respective occasions are

nor 1.
normal independence-realm

neither

will be called Bernoullian, if the

for
the
occurrence
of the event on the respective
probabilities
occasions are all equal If the probability of the occurrence of the

event

is
p (the Bernoullian probability), then the probability of its
non-occurrence is 1 p.
The notions of the various independence-realms can easily be
generalized so as to apply to sequences of occasions for the occurrence
and non-occurrence, not of one event E only, but of any number n
of events Ei
For our purpose, however, it will suffice to
., En
consider the simplest case, when there is one event E
only.
For Bernoullian independence-realms can be proved a very
9

important theorem, usually known as Bernoulli's Theorem. It is

convenient to divide its content into two parts or stages.
We ask for the probability that, on n occasions, E will occur m
n
times and fail to occur n-m times. This can
happen in Cm different
20
ways, all of which are mutually exclusive. The probability that E
will occur and fail to occur in a given one of these n Cm
ways is, by
repeated use of the Special Multiplication Principle, found to be
m
m
p x (1 -pY~ where p is the Bernoullian probability. The proban
bility again that E will occur and fail to occur in any one of the Cm
ways is, by repeated use of the Special Addition Principle, found to be
,

21

It can be proved
that, for given p and n, the calculated probability
has its maximum value for that value of m:n which is closest to
p. In
other words: the most probable value of the event's relative
frequency
on n occasions is the value which is closest to the event's Bernoullian

probability p.

This, which

may

also be called the Direct

Law

of

Maximum

Probability (for Bernoullian Independence-Realms),

constitutes the first stage in the proof of Bernoulli's Theorem.

We next ask for the probability that, on n occasions, E will occur

times and fail to occur n-m times, m now
being a variable which

runs through
interval

e.

values (integers) for which the ratio m:n falls in the

(e is an arbitrary quantity which may be as small as

all

96

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

we

please.) This probability is calculated with the aid of a further

application of the Special Addition Principle and

2 of the values

Cm xp m x(l-~~p) n - m when m
Of

interval just mentioned.

given values of p and

this

sum Z

it

is
simply the sum
runs through the

can be proved 22 that, for

greater, the greater n is. As n approaches

I
1 as a limit. In other words:
infinity,
approaches
if the Bernoullian
the
event
is
then
the
probability of
p,
probability that the event's
e, it is

relative frequency on n occasions mil deviate

from p by less than an
amount s, however small approaches as a limit the maximum value 1
as n is indefinitely increased. This, which may also be called the Direct
9

Law

of Great Numbers (for Bernoullian Independence-Realms),

and final stage in the proof of Bernoulli's
theorem.

constitutes the second

Thus, loosely speaking, the first part of Bernoulli's Theorem tells

us that the most likely relative frequency of an event is that which is
indicated by its probability, and the second part that 'in the long
run' it becomes infinitely probable that the event's relative frequency
will equal its probability. The danger of using this loose mode of

speech is that

it

mention the, rather sweeping, assumpwhich are essential to a correct proof of the

leaves without

tions of independence

theorem.

2.

The

It is

interpretation

offormal probability.

usual to distinguish between two types of interpretation or

abstract calculi. 1 In the one type of interpretation it is

model of

assumed that certain empirical objects 'satisfy' the postulates

(axioms) which the calculus lays down for the undefined concepts.
In the other type of interpretation it improved of some entities that they
conform to the postulates.
An example of an interpretation of the first type is when we
conceive of Euclidean geometry as a physical theory of space ('lightray geometry'). As is well known Euclidean geometry, as such a
theory, is supposed to have been falsified by some experiments which
are relevant to the acceptance of Einstein's Theory of Relativity.
An example of interpretations of the second type is Descartes 's
invention of analytic geometry. In this geometry the system of
97

THE LOGICAL PROBLEM OF INDUCTION

5

'modelled in the realm of numbers, the axioms and

theorems of abstract Euclidean geometry becoming provable propositions within another branch of mathematics.
It is of some importance to note that all known interpretations of
abstract probability are interpretations of the second type, i.e.
logico-mathematical models. For this reason one ought not, as is
Euclid

is

sometimes done, 2 to regard the relation between abstract and interpreted probability as presenting a close analogy to the relation between axiomatic geometry and the physical theory of space.
There are many models of abstract probability, but all of them
which are known to be important fall within one of two main
3

categories.

We

range-models.

shall call these categories

Since the

members of

frequency-models and

the respective catagories are

also speak of the categories

closely similar to each other, we may

themselves as the frequency- and the range-model of abstract probaalso be called a statistical, and the
bility. The frequency-model may

range-model a modal interpretation of probability

'possibility

'-

interpretation).

Any model of the statistical type preof

the probability-relation are, not proposithat
terms
the
supposes
but
tions,
propositional-functions. It will suffice for our purpose to
The Frequency-Model

consider propositional-functions of the simplest kind only, viz.

propositional-functions of one variable. We shall further make the
simplifying assumption that the propositional-functions which enter

one and the same probability-relation are functions of

one and the same variable. Under these simplifying assumptions
we can speak of the probability-relation as a relation between some
as terms of

attributes or properties or classes.

the frequency view, the probability of a given h is the relative
frequency of such values of the variable which statisfy the proposi-

On

values of the variable which satisfy the

(Instead of Value of the variable' we may
propositional-function
say 'individual'.) Popularly speaking, the probability of a given h is
tional-function a

among

all

h.

the relative frequency with which an 'event' of the nature a takes

place when the 'conditions' h are fulfilled. Or differently again: the
probability of a given h is the proportion of A's which are a's.

Since the relative frequency in question thus

98

is

a proportion of true

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

propositions within a class of propositions, it may also be called a
truth-frequency*
The notion of a relative frequency or a proportion is straightforward and presents no difficulties, if the number of values of the
variable which satisfy h is restricted to a finite number n. It is
easy to

show

that such a Finite Frequency-Model satisfies our

postulates of

abstract probability. 5
The Finite Frequency-Model

is, for reasons which we need not

consider here, thought very unsatisfactory from the point of view of
accounting of the 'meaning of probability.
If the number of values of the variable which satisfy h is potentially
5

infinite, the proportion of A's which are a's is the limiting value of a
relative frequency in a sequence. 6 The notion of a
limiting-frequency
must not be regarded as 'meaningless' or in any other way logically
unsatisfactory. But it is important to observe that the notion makes

sense only relative to a

way ofordering the values of the variable which

By re-ordering the sequence we may alter or even destroy

the limiting value. 8
The Frequency-Limit-Model can be shown to be also a valid interpretation of the Abstract Calculus of Probability.
satisfy h.

Against the Frequency-Limit-Model too, as a suggested analysis

of probability, many grave objections can be levelled. One is that
empirical propositions about proportions in infinite 'populations'
can be neither verified nor falsified by statistical observation. Another
is that a
probability is (usually) not thought to depend upon a way of
ordering the members of a population. And a third is that use of
probability calculations for statistical predictions

is

by no means

always tied up with beliefs in limiting-frequencies in nature.

problem which has been very much discussed in connection with
the Frequency-Limit-Model of probability concerns the manner in
which those values of the variable which satisfy a are distributed
among the values which satisfy h. It has been thought that unless this

distribution satisfies some conditions as to irregularity or randomness,

the frequency-model cannot give an adequate account of what we

mean by

a probability.

The

definition of

random

distribution,

10

however, constitutes a difficulty.

It should be observed that none of the

difficulties

mentioned

impairs the logico-mathematical correctness of the Frequency-Model

99

THE LOGICAL PROBLEM OF INDUCTION

(for finite or infinite populations) of abstract probability. And this
correctness of the frequency-view is all that concerns us in the
present
11

investigation.

The Range-Model 12 A model of this type can be worked out both

on the assumption that the terms of the
probability-relation are
propositions and on the assumption that they are attributes. 13 We
here adopt the former alternative, which
appears to be the more usual
one. 14

The range-theory of

probability, in its simplest form,

as
follows:
loosely explained
'analyse' the evidence-proposition h into a number,

can be

We

say n, of
h n which are mutually exclusive and such that
some of them, say m, entail the proposition a and the rest entail the
15
In conformity with a traditional
proposition ~#.
terminology, we
shall call the alternatives, which entail <z, 'favourable' alternatives
and the alternatives, which entail ~a, 'unfavourable' alternatives.
By the probability of a given h we now understand the ratio of

alternatives, hi,

'favourable' alternatives to
alternatives 'favourable' to a

all

alternatives, or the

among

all alternatives

proportion of

which

fall

under

h.

We may call this the 'classical' form of the range-definition

of probability. Omitting an
important qualification to be mentioned
presently, it answers approximately to the definition of
probability

proposed by Laplace

16

and current

in

books on the subject up to the

present day.

The
of

it

can be generalized
exact in the following
way:

'classical' definition

made more

We

and the description

consider some set v of

propositions, such that a and h are
truth-functions 17 of some members of o-. 18
thereupon consider
the disjunctive normal forms of a&h and of h in terms of all the

We

members of a. Let us assume that the normal form of a&Ji is mtermed and that the normal form of h is ^-termed, i.e. let us assume
that the normal forms are
disjunctions of m and of n conjunctions
respectively. By the probability of a given h we now mean the ratio
m:w. 19

By

the a-range of a
proposition we shall
in terms of the members of cr. 20

normal form

100

mean

its
disjunctive
(In this definition it

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

presupposed that the proposition in question is a truth-function
of the propositions in or.) The conjunctions in the normal form we

is

shall call unit-alternatives

The numbers m and n above are measures of the o-ranges of the

propositions a&h and h respectively. These measures are obtained
simply by counting the number of units in the ranges of the propositions. But we may also
adopt some other method of measurement
which

not an equal, but an unequal 'weight' to these units.

arrive at a generalized notion of the measure of a
We
introduce
the symbol mr a for 'measure of
range.
cr-range
On the basis of this generalized notion of a range-measure, we
In this

assigns,

way we

'

introduce the definition P(alK)

= df
J

It may be shown that for

any choice of a and mr a
subject only
to a few restrictions of a very general nature 23 this ratio of measures
of ranges satisfies the postulates of abstract probability. The above
'classical' range-definition and its
generalized form may be regarded
as special types of such ratios.
It is characteristic of this definition that if,
given a and h, we ask
what is the value of P(ajh) the answer will depend upon the choice

of a and the choice of rar 24 Since, from the point of view of the
applications of probability, not all choices are equally good, we are
led to consider the problem of adequacy in the choice.
The answer given to this problem in the 'classical' theory of
probability can, in our terminology, be stated as follows:
The choice of a ought to be such that each unit in the ranges can
be given an equal weight. Or, popularly speaking, the data ought to
be analysable into a number of equipossible alternatives.
When this condition is added to what we called above the 'classical'
range-definition we get the following: The probability of a given h
is the
proportion of alternatives, which are 'favourable' to a among
a number of equally possible alternatives which fall under h.
But how can we be sure that an analysis of the data leads to equiCT ,

possible unit-alternatives?
proposed as an answer the

To

this question the

'classical'

theory

famous Principle of Insufficient Reason,

also called the Principle of Equal Distribution of Ignorance or the
Principle of Indifference. The applicability and formulation of this
101

THE LOGICAL PROBLEM OF INDUCTION

26
principle have been the object of much discussion and controversy.
In the original formulation, given to it by James Bernoulli, the

principle states that alternatives should be held equally possible, when

is known why one of them rather than another should

no reason

come about. 26
One other answer to the problem of the adequate choices of or
and mra should be mentioned before we leave this topic:
The choice of mr a ought to be such that for a given a, the
//

ratio

jp

~2j

~\

equals the value of P(ajh) on the frequency-interpre-

tation. 37

To

accept this answer is to let the frequency-interpretation function

on the adequacy of any particular range-interpretation of
the concept of probability. This may be said to ignore the problem
of adequacy for the frequency-interpretation. But it is nevertheless
as a check

noteworthy that one of the difficulties confronting the range-theory

of probability is how it can be used to account adequately for
probability-values without taking refuge, so to speak, in the
28

frequency-theory.

In this inquiry we are, however, not interested in the difficulties

and problems confronting the range-theory as a proposed analysis
of the meaning of probability. For our purposes it will suffice to note
that a definition of probability in terms of range-measures
just as
is
a definition of the concept in terms of relative frequencies
our
a
model
of
Abstract
correct
and
mathematically
gives
possible
Calculus.

3.

The doctrine of Inverse Probability.

In the Theorem of Bernoulli, one might say, we argue from

probabilities to probable values of relative frequencies or proportions. Can this argument be reversed or inverted? Is there a theorem

of the calculus which enables us to conclude from knowledge of

relative frequencies to probable values of probabilities? If this
theorem were analogous to the Theorem of Bernoulli, it would,
speaking approximately, tell us that the most likely value of an
event's probability

is

that indicated
102

by

its

actual relative frequency,

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

and that
of

its

it becomes
infinitely probable that the
which an event occurs gives us the true value

the long run'

'in

relative frequency with

(hitherto

unknown)

probability.

a noteworthy fact that James Bernoulli, in proving the

Direct Law of Great Numbers, evidently believed himself to have
proved also the inverse of it, i.e. that knowledge of actual frequencies
1
entitles us to probable conclusions about
Later
probabilities.
authors again sometimes believed the inverted theorem to follow as a
matter of course from the direct one. 2 The reasoning underlying
such an idea has an apparent plausibility in its favour, but is nevertheIt

is

thoroughly fallacious. Since the question is of a certain interest to

the problem of induction, we shall examine in some detail the way to
a correct proof of the Inverse Laws of Maximum Probability and of
less

Great Numbers.

The

inversion essentially relies upon an elementary formula of

probability theory which is not needed for the proof of the direct

We shall call

principles.
Its

it,

3
following Keynes, the Inverse Principle.

proof is as follows:

(l)P(blh&a)

=pMfo'

(From ^2 provided P(

= P(blh)xP(a!h&b).

(From A2.)
(3)P(alfy=P(aMlfy+P(a&~b{h). (73.)
(4) P(a/K) ^P(blh)xP(a/h&b)+P(~blh)xP(alh&~b). (From
(3) and A2.)
(2)

P(a&b\K)

PthiiAA;

Tin

(From

__

P(blh)xP(aIh&b)

P(b/h)

xP(alh&b+P(~blh) xP(a/M~6).

(1), (2), (4).)

We next conof
it.
generalizations
b s be s mutually exclusive and jointly exhaustive
Let bi
alternatives. Then we have P(a/h)
P(a&(b :v
vbJ/K)
This

sider

is

the simplest form of the Inverse Principle.

two

P(a&b l

v ...

va&bJK)

P(b l /h)xP(alh&b 1)+.

If,

in T10,

we

=
.

substitute the last expression for the

103

denominator

THE LOGICAL PROBLEM OF INDUCTION

and

bi for b,

Principle:
TIL If bi

we

reach the following generalization of the Inverse

one of s mutually exclusive and jointly exhaustive

is

alternatives b,

n o ,
then Pfa/h&d)
t

b,,

=-

P(bilh)xP(alh&bi)
^

^P(bilh} xP(a/h&bi)

From Til we

immediately reach one further generalization of the

viz..

principle,
T12. If bilt

.,

bik

v ... vb ik lh&d)

k of some s mutually

are

exhaustive alternatives

61.

exclusive

and

jointly

b s , then

In speaking of the Inverse Formula we shall usually mean the

formula given in TIL In conformity with traditional terminology
we shall speak of the probabilities P(bijh) as the initial or a priori
probabilities, of the probabilities P(a/h&bi) as the likelihoods, and
of the probabilities P(bilh&d) as the a posteriori probabilities.
If all a priori probabilities are equal, the values P(bt/K) cancel out
in the right-hand member of the Inverse Formula. Then it is readily
seen that the a posteriori probability P(btlh&d) has its maximum

when

the corresponding likelihood P(a/h&bi)

From the Inverse Principle we arrive

Maximum Probability and <3reat Numbers

is

greatest.

at the Inverse

Laws of

in the following principal

steps:

Let there be
Bernoullian independence-realms for the event E.
let the Bernoullian probabilities bej^i
ps We assume, for
the sake of simplicity, that these probabilities are all different from
each other and that^i<
<ps
The reader may think of these s independence-realms as s different
sets of conditions under which E may occur. Each time the conditions
are satisfied we have an occasion for jB's occurrence. On each occasion E may either occur or fail to occur. That the conditions constitute an independence-realm for E means that the probability of E's
5-

And

104

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

occurrence on a given occasion is not affected by E's occurrence and
non-occurrence on other occasions. And that the independencerealm is Bernoullian, finally, means that the probability of E's
occurrence remains the same on each occasion.
The 'classical' illustrations are from games of chance. An independence-realm could be the potentially infinite sequence of drawings,
with replacement, of balls from an urn. If there are s different urns,
there are s such independence-realms. The event E could be the
drawing of a black ball. Under 'normal' conditions of drawing we
regard the results as independent for probability of each other.
The Bernoullian character of the independence-realms, again, is
guaranteed by the stipulation that the ball should be replaced after
each drawing.
In each of these independence-realms we can calculate the probatimes on n occasions. In the
bility that E will happen exactly
realm with the Bernoullian probability/?! this value is

Let qi be the probability that (a random set of) n occasions for "s
occurrence belong to the independence-realm with the Bernoullian
and qs the probability that they belong to the
probability p
realm with the Bernoullian probability p s
For example: #1 is the probability that n drawings, with replacement, are from an urn in which the probability af drawing a black
ball is pi. It is assumed that the n drawings are from one and the
.

same urn.

We can now, given that E occurs exactly m times on n occasions, use

Formula of Til to calculate the probability that those
occasions belong to an independence-realm with the Bernoullian
probability/?/. (!<<$.) This value is:
the Inverse

<A\
(A)

use the Inverse Formula of T12 to calculate the

to an independenceprobability that those occasions belong either
or to an indepen...
realm with the Bernoullian probability p^ or
This
value is:
dence-realm with the Bernoullian probability/?,^.
Further,

we may

105

THE LOGICAL PROBLEM OF INDUCTION

k

OB)

For example: Let there be n drawings, with replacement, from one

and the same urn. On exactly m of those n occasions a black ball is
drawn. (A) now gives us the probability that the drawings were from
an urn, in which the probability of drawing a black ball is p
(fi)
were
either
from
that
the
an
in
which
urn,
they
gives
probability
this probability is /?, 1? or ... or from an urn in which it is pik
Let us assume that all ^-values are equal. Then they cancel out
from 04) and 03) above and we get the simplified expressions:
t .

(A')

/=!

and

p
2rt.

tj

mx
v/'/
(i

_p^)

n-^\ n -- n

f=l

For given values n and m, the denominators of the four expressions

have a constant value. The maginitude of the four expressions themselves is then directly proportionate to the value of their numerators.
This value again varies with the choice of i or of the set i l9
., fo.
Of pi m x(lp!) n - m it can be proved 5 that it reaches its maximum
for that value of i which is nearest to the ratio m:n. This means that
(A'} reaches its maximum for that value of i too. The result can be
expressed in words as follows:
If it is initially or a priori equally probable that a set ofn occasions
for the occurrence of an event E belongs to any given one ofs Bernoullian
independence-realms for E's occurrence, then it is a posteriori, i.e.
given the information that E occurs on m of those n occasions, most
probable that the set of occasions belongs to an independence-realm,
.

106

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

which the Bernouttian
probability
frequency m:n.

in

ofE

is

closed to

actual relative

its

We shall call this the Inverse Law of Maximum Probability (for

Bernoullian independence-realms).
For example: If it is initially equally probable that a set of n
drawings
are from any given one of s urns, then it is a
posteriori most probable
that the drawings are from an urn, in which the
probability of
of
drawing a black ball is closest to the actual relative

frequency

black balls drawn.

Consider an interval

round the ratio m:n. Let us assume that

the
t
p-values which happen to fall in this interval,
and
all the
corresponding ^-values qil
q equal O.
On these assumptions it can be proved 6 that the value of the numerator in (E) increases with n and
approaches the value of the denomina/

p\ k are
that not

all

fjt

tor as n

is

indefinitely increased. Thus the value of (B) increases with

1 as a limit. The result can be
in words as

n and approaches

expressed

follows:

If it is not initially or a priori infinitely improbable, i.e. probable to

degree O, that a set of n occasions for the occurrence of an event E
belongs to some Bernoullian independence-realm for E's occurrence,
in which the Bernoullian
at most, an
probability of E differs
by,
arbitrary
the actual relative
frequency m:n ofE, then the probability that the set of occasions belongs to some s\ich independencerealm increases with the number n of occasions and
the

amount sfrom

approaches

maximum

value

as n

is

indefinitely increased.

We shall call this the Inverse Law of Great

Numbers

(for Bernoul-

lian independence-realms).

For example: Unless it is initially infinitely improbable that the

drawings are from an urn, in which the probability of drawing a
black ball differs by, at most, the value s from the actual relative
frequency of black balls in those drawings, then the probability that
the drawings are from some such urn increases with the number of
drawings and approaches the maximum value 1 as a limit.
The Inverse Laws of Maximum Probability and Great Numbers
can be given a particularly elegant formulation, if we use integration.
Then we have to replace the assumption that there is a limited number
s of Bernoullian independence-realms for the event
by the assumption that there is a not-denumerable infinity of such
independence*
107

THE LOGICAL PROBLEM OF INDUCTION

"s Bernoullian
realms, one for each of the possible values of
values
is the whole
of
the
The
range
possible
range from
probability.

to

inclusive.

The a

or

prforf-probability

#-value,

associated with

Bernoullian probability or ^p-value p, we denote by qp

(A) and (B) above now become formulae
n ~m
m

mX
(Jf

a given

The formulae

xqp

p x(l-p)

jp

m
pY~ X qp dp

and
(m:/7)-fe

p x(lp)

n-

xqp

dp

(D)

x (1 p) n ~m x qp dp
o

If all ^-values are equal, the values

expressions and we
and (5') above

qp cancel out from these

get simpler expressions corresponding to (^4')

-r^

(C')

n~

n -m

n -m

p x(lp)

dp

and

p x(lp)

f T\ f \

(L>

(m:)

dp

E
.

j p x(l-p)

dp

(C") has

its

value for max. p m x(lp) n - m which is

(The Inverse Law of Maximum Probability.)

maximum

= m:n.

reached when/?
If qp is not
for all values of p in the interval m;ne, then (D)
increases with n and approaches 1 as a limit. (The Inverse Law of
Great Numbers.)
The inversion of Bernoulli's Theorem is also known under the
name of Bayes's Theorem.
The first part of Bayes's Theorem or the Inverse Law of Maximum
7

Probability depends on an assumption of equal a priori probabilities.

108

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

The second part of Bayes's Theorem or the Inverse Law of Great
Numbers requires only the much weaker assumption that it is not
a priori infinitely improbable that the event's Bernoullian probability
deviates by at most s from its actual relative frequency. One might
say that this assumption means that the increase in a posteriori
probability towards I is 'practically independent' of a priori
8

probabilities.

Thus, loosely speaking, the first part of Bayes's Theorem tells us

that, if all values of an event's probability are a priori equally likely
then it is a posteriori most likely that the event's probability is as
indicated by its relative frequency. And the second part tells us that,
'practically independently' of a priori probabilities, it becomes 'in
the long run' infinitely probable that the event's probability will
equal its actual relative frequency. The danger of using this loose
mode of speech is, among other things, that it leaves without mention the assumptions of independence needed in order to warrant the
Bernoullian character of the event's probability.

From Bayes's Theorem we easily reach another famous principle of

Inverse Probability known as Laplace's Law of Succession. 9 We
raise the following question:
If an event has occurred on all of n occasions, what is the
probability that the event will occur as well on the next occasion?
It is taken for granted that the formula (C) above can be used for
all
calculating the probability that, if the event has occurred on
The
will
be
n
of
occasions, its Bernoullian probability
Special
p.
Multiplication Principle is then used for calculating the probability

that the Bernoullian probability is p and the event occurs on the

next occasion as well. Finally, use of the Special Addition Principle
to 1, all
is made to 'add up*, p passing through all values from
these conjunctive probabilities. Thus we get for the calculated

probability the value

(E)
n

J p xqp dp
o

109

THE LOGICAL PROBLEM OF INDUCTION

On the

assumption that all the initial probabilities qp are equal,

the
get
simplified expression

we

jp
(')

n+2

dp

*.

pdp
By

integration

we obtain

the probability that,

sion,

it

will occur

if

11 -{- 7

for

(') the 'famous' value

for

the event has occiu.ed n times in succes-

on the next occasion as

well. 10

Mention should be made of a version of the Law of Succession

which is independent of the assumption of
equality in the initial
probabilities and obtainable without the use of
integration.

Let there be s Bernoullian independence-realms for a certain event.

Let the Bernoullian probabilities be p t
.../, and the corresponding
initial probabilities c^
qs We assume that q,<
<qs
Now suppose that the event has occurred n times in succession.
We choose an arbitrary value s such that at least one of the Ber.

noullian probabilities falls in the interval 1-s. Formula

(5) may
be used for calculating ths probability that the n occurrences
belong
to some independence-realm in which the Bernoullian
probability
of the event falls in the interval 1-s. It follows from the Inverse Law
of Great Numbers that the calculated
probability is the greater, the
greater is n (provided that not all the initial probabilities, which
correspond to the Bernoullian probabilities in the interval 1-s, are

For a

sufficiently large n it is more probable that the occurrences

to
an independence-realm, in which the Bernoullian
belong
probaof
the
event falls inside this interval, than that
to a
bility
0).

they belong
realm in which the Bernoullian
probability falls outside the interval.
And 'in the long run' it becomes infinitely probable that the n occurrences belong to some such
independence-realm.

We raise the question: What is the probability that the event, which

has occurred n times in succession, will occur on the next occasion

as well? This value we
may calculate from (A) with the aid of the
and
Addition Principles in a manner exactly
Special Multiplication
similar to the derivation of the formula
(E) from the formula (C).
We get the formula:

no

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

/=!

This formula can be expanded as follows:

.

w
^
Wehave/?x<

we can always choose e in such a way that /us the

in the interval 1-

e, it

follows from the Inverse

.<z?5 Since
.

only/-value which falls

Law of Great Numbers

ps xq

that

increases with n

(provided qs

is

not

0).

For a

sufl&ciently large

is

greater than any other term

From arithmetical considerations of an elementary nature now follows

11
that (F) increases with
In other words, it has been
increasing n.
proved that, relative to the assumption mentioned, the probability
that an event which has occurred n times in succession will occur on

the next occasion as well, increases with

Non-Numerical Law of Succession. 12
On the assumption that all initial

n.

We may

call this the

probabilities are equal, (F)

reduces to

(F)

Let us assume that p =

and ps = 1 and that the difference
between any two successive ^-values is always the same. The interval
l

from

to

1,

in other words,

intervals of equal length.

is

On

sl

divided by the ^-values in

these assumptions it may be

in

sub-

shown

THE LOGICAL PROBLEM OF INDUCTION

that,

with increasing

as a limit. Thus,

the value given

s,

the value of (F

approaches the value

on the assumption of equal probabilities a priori,

by the Non-Numerical Law of Succession and

obtained without the use of integration approaches as a limit the

Law of Succession, which is obtained

value given by the Numerical

13
by the use of integration.

4.

Criticism of Inverse Probability.

The doctrine of Inverse Probability

is

also

known

as a doctrine of

'inductive' probability or of the probability of 'hypotheses'

'causes'. It is not difficult to see the reasons for these names:

Consider

the Inverse Principle or T10-T12 above. It

first

is

or

some-

times natural to speak of the mutually exclusive and jointly exhaustive

b s as 'causes' of the 'event' a, and of the assumpalternatives b
tion that an occurrence of a is due to a specific one of these 'causes'
.

On each 'hypothesis' the event possesses a

'hypothesis'.
certain likelihood, and each 'hypothesis' has itself a certain initial
or a priori probability of being true. Given these likelihoods and

as

an

we use the Inverse Principle to calculate the

a
that
given occurrence of a is to be explained by the
probability
it is due to a
If any one 'cause'
that
specific 'cause' bi.
'hypothesis'
initial probabilities

('hypothesis') has the same initial probability of being operative

(true), then the most probable 'cause' ('hypothesis') is the one which
2
gives to the event the greatest probability.
In a similar manner one may speak of 'cause'

and 'hypothesis'

in connection with Bayes's Theorem. 3 The 'cause' is here the event's

membership of a certain Independence-Realm, i.e. the 'cause'

consists in the presence of certain conditions under which the event

will occur with a certain (Bernoullian) probability. The 'hypothesis'

again is that those conditions are satisfied, i.e. that the occasions for
the event's occurrences belong to a certain Independence-Realm.
Very often one has talked of the 'hypotheses' involved in Bayes's
Theorem as propositions to the effect that the event possesses a
certain (Bernoullian) probability.

seriously misleading. (Vide infra.)

112

This loose

mode

of speech

is

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

The Law of Succession was

traditionally regarded as a rule for

the
of
future
events relative to past experience.
estimating
probability
It has been said that no formula in the
alchemy of logic has exercised

a more powerful fascination over the human mind. 4 Not only was it
uncritically accepted by those, who followed closely in the footsteps
of Laplace. Many respectable authors of the nineteenth century on
the subject of induction regarded it as an altogether sound formula,
whereby to judge with probability of the future course of events/
Some put it to the wildest uses, such as calculating that the sun will
rise tomorrow or that it will continue to rise regularly for the next
1000 years. 8 The formula still finds favour with authors, who regard
the doctrine of inverse probability as being, with due qualifications,
tenable. 7

Criticism of Inverse Probability in general and of the Law of

Succession in particular is historically connected with the criticism,
mainly by early proponents of the frequency view, of the Laplacean
definition of probability and of uncritical use of the Principle of
Indifference. Among the early critics Boole, Peirce, and Venn should
be mentioned. 8 Keynes took a guarded and in many respects sound
view of Inverse Probability, but in his criticism of the Law of Succession he went too far when charging the law with a contradiction. 9

In recent times Inverse Probability has been severely criticized by

one of the champions of modern

statistical science,

R. A. Fisher.

He rejects

the entire doctrine as theoretically unsound and as useless

in practice. 10 Fisher's criticism of the theoretical foundation, however,
is

not in every way clear and convincing. The subject

is still

open to

11

controversy.

Here we

shall

have to content ourselves with a number of

critical

observations which no attempted rehabilitation of Inverse Probability

will be able to 'get round'.
(1) In the Inverse Principle (T10-T12) three factors may be said to
be involved. These are the individual propositions a and h and the
set of propositions bi. It is convenient to speak of a as a proposition
to the effect that a certain event has occurred, and of each proposition bi as some condition, under which the event will take place with
a certain probability. We may, for present purposes, ignore h.
That the probability of the event relative to conditions bi is pt
is a
probability-proposition.
113

THE LOGICAL PROBLEM OF INDUCTION

which is calculated by means of the Inverse Formula, cannot
properly
be described as the probability of a

probability-proposition ('second
order probability'). The calculated
probability is the probability
that the conditions, under which an occurrence of the event took
place, were just the conditions 6/. The unknown of the problem,
therefore, is not a probability. It is the presence of certain conditions,
relative to which the event has a
probability. The 'inverse problem'
in all its variations can be described as a
problem of re-identification

of the conditions under which an event has occurred, these conditions

constituting a 'field of measurement' (data, information) of the
event's probability.
When from the Inverse Principle we pass on to Bayes's Theorem
we conceive of the event a of the Inverse Principle as a
complex
occurrences of an event
event, consisting of
on n occasions.
The complex event a is, as in the Inverse Principle, supposed to have

taken place under some condition(s) A/. This Z>/ is,

moreover,
supposed to be a Bernoullian independence-realm for the occurrence
of
In other words: given that the condition 6/ is satisfied, the
event E will occur with a certain Bernoullian
probability/)/. On this
supposition it is then possible to calculate also the probability that,
given &/, the complex event a will occur. The 'inverse problem'
.

now

consists in determining the

probability that the conditions,

under which an occurrence of the complex event a took

place, were
It is thus a
of
exactly the conditions b
problem
'identifying', with
probability, an independence-realm for E.
(2) What has so far been said goes a long way towards
explaining
certain limitations in the
applicability of inverse probability to
concrete cases. Let us first consider to what
type of situation inverse
formulae may be applied:
Let there be a number of urns
containing black and white balls.
With each urn we associate a certain probability of
drawing a black
ball. (How we have come to associate these
probabilities with the
t.

respective urns is for present purposes quite irrelevant: it might

have been on the basis of knowledge of the
proportions of black and
white balls in the various urns, or it might have been on the basis of
the results in long series of
drawings from the urns.) Now n drawings,
with replacement, are made from one of the urns and we
exactly
m balls. We do not know which probability of drawing aget
black ball
114

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

has been associated with this
particular urn. The inverse problem
before us is now to use the information obtained from the
drawing to
'identify' the urn as being one associated with a certain Bernoullian
probability p of drawing a black ball.
Any legitimate application of inverse probability has to be essent

tially analogous to this case as regards the problem of identifying

the independence-realm. And from it follows at once that such uses
of inverse probability as those of
determining the probability that
the sun will rise tomorrow or that the next raven to be observed will
be black are illegitimate. The
possibility of identifying birds as
ravens, independently of observations concerning their colours,
makes the suggested application of the Law of Succession lose its
point For, it is a part of the data on which this law would rest, when
applied to birds and their colourings, that being a raven is associated
with a certain (Bernoullian) probability of
p being black. This value
is
the
that
random
p
bird, which is known to be a
probability
any

raven, will be black

and hence also that the next raven to be

observed will be black. There is, however, a
question to which the
Law of Succession could be quite sensibly applied here, although this
question would hardly arise in practice. It is the following: What is
the probability that the next member to be observed in a set of
(one
and the same) unknown species of bird will be black, given that all
the n members of the set which have been so far observed have been
found to be black? In calculating an answer to this question, using
the Law of Succession, we would have to
rely on probabilities
PI, pz, etc. that a random individual of the species of bird, b l9 b t
etc. are black. And
among these probabilities would also be the
probability that a random individual of the species of raven is black.
(3) We have so far said nothing of the a priori probabilities
traditionally considered the crux of the doctrine of inverse probability.
We have been concerned to show that, independently of the problems
connected with the initial probabilities, the doctrine, if applicable at
all, is so only to situations of a very peculiar nature. When initial
,

probabilities are considered too, further severe restrictions to the

applicability of the formulae

become

apparent.

The a

some

priori probabilities are the probabilities, relative to

piece of information A, that the respective conditions &/,

under

which the event a may or may not occur, are

of the

115

satisfied.

If h is

THE LOGICAL PROBLEM OF INDUCTION

form hv~h, the

initial probabilities

may be

said to

be 'eminently'

no particular information.
To many of the notorious uses of the Inverse Formula for
determining the probability of some 'causes' or 'hypotheses', and
to any use of the Inverse Law of Maximum Probability and the
Numerical Law of Succession, it is essential that the a priori probaa priori,

bilities

i.e.

subsisting relative to

involved in the problem under consideration should be

equal.

Traditionally, the needed equality of the a priori probabilities was

regarded as a consequence of a Principle of Indifference. In short:
ignorance of the initial probabilities was considered a sufficient

condition of their equality. 12 We shall not criticize this deduction of

knowledge from ignorance here. It seems to us that a Principle of
Indifference may be legitimately invoked as a ground for framing a

hypothesis about the equality of certain initial probabilities, but that

use of this principle can never amount to a proof of the equality in
13

question.

how any assumption about the a priori probaabout their equality or inequality or exact numerical
could be anything but a hypothesis for the correction of
values
which future experience about the case under consideration may
constitute a reason. This hypothetical nature of the initial probabiliIt is difficult to see,

bilities

be

it

already destroys the faith which the 'classical doctrine put in

the power of inverse probability to justify induction.

ties

(4)

In order to reach that level of the doctrine of Inverse Probability,

where Hayes's Theorem and the Law of Succession belong, it is

necessary to assume that the conditions under which the event in
question may occur or fail to occur constitute Bernoullian independence-realms for the event. These assumptions of independence are of
a sweeping nature. Their problematic character was practically

never noticed in the traditional doctrine. It is difficult to see, how

the truth of these assumptions could be established on a priori
grounds. The assumptions of independence may, with the assumptions about initial probabilities, be said to form part of the hypothetical framework of any significant use of Bayes's Theorem and
the Law of Succession.
(5) In the case of those formulae of inverse probability which

employ

integration,

an additional

difficulty enters.
116

They presuppose

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

that the conditions under which the event
may occur or not, constitute not only an infinite, but a not-denumerable manifold. This

presupposition can never be empirically satisfied. It is an Idealizaat least in the

tion', the legitimacy of which
opinion of the present
writer
remains problematic even in those cases where all the other
conditions for a legitimate use of the inverse
principles are fulfilled.
We do not think that the question of the
of this 'idealilegitimacy

zation' can be settled simply

by reference to an analogy with other

applications of mathematical formulae involving integration to

cases in nature. For it is not clear to what extent use of
integration
in the doctrine of inverse
probability is, from the point of view of
application, peculiar and to what extent it is analogous to such other
cases.
It is not a correct
presentation of the nature of the case to say
5
that the 'idealizing assumption is the
assumption that the possible
values of a certain event's (Bernoullian)
probability continuously
cover the range from
to 1 inclusive. What we have to
say is that
the alternative conditions under which an event
occur
the
may
e.g.

from which the drawing of balls takes place

form
a manifold within which all possible values in the interval from to 1
of a certain (Bernoullian) probability are
represented. It is not
even as an 'idealization*
prima facie obvious that this assumption
alternative urns

makes any

5.

sense at

all.

Confirmation and probability.

Let a be some proposition and h some logical consequence of it.

Deductive logic studies the relation of entailment between a and h.
Inductive logic, one might say, studies the degree of confirmation or
support which h gives to a. (We do not mean to say that this is the
only task of inductive logic.)
1
By the hypothetical method or use of hypothesis is often meant
the deduction and subsequent verification of consequences of an

assumed (usually

general) proposition. It is a major task of inductive

logic to study the way in which verified consequences 'inductively'
affect the hypothesis from which they deductively follow.

By Confirmation-Theory we
i

shall here
117

understand the theory

THE LOGICAL PROBLEM OF INDUCTION

how

of

in the

the probability of a given propositon

form of propositions which are

is

affected

by evidence

logical consequences of

it.

case of particular importance to this theory is when the given

proposition is a generalization and the evidence for it is some of its
Verified instances confirm the generalization. It is a
of Confirmation-Theory to evaluate, in terms of
task
primary
the
confirming effect of the instances on the generalizaprobability,
instances.

The notion of an 'instance' of a generalization will have to be

somewhat more in detail later. For present purposes it
suffices to lay down merely that an 'instance' of a generalization is a

tion.

discussed

logical consequence of

it.

The doctrine of Inverse Probability, which we have examined in

the two preceding sections, is not a Confirmation-Theory in the
sense here understood. The same is true, moreover, of some recent
investigations which call themselves theories of confirmation,*
The creation of a Confirmation-Theory for inductive generalizais of
comparatively recent date. The theory was founded by
C. D. Broad (1918) and J. M. Keynes (1921). After Keynes there has
been very little further development of the theory, but a certain

tions

amount of dicussion of

We

the significance of his achievement.

Lemma which is of basic

both
to
and to the Theory of
Confirmation-Theory
importance
Scope to be discussed later:
Lemma. The probability of a proposition., on given data, is smaller
than or at most equal to the probability , on those same data, of any of
prove the following elementary

its

logical consequences.
Proof: Let h be the datum, and let a entail

b.

In virtue of the

we have P(a&b/K)

General Multiplication Principle (A2)

= P(b/K) x

P(a/h&b). Since a entails b, a&b is logically identical with a. Thus

we haveP(a//z)
P(b/K)xP(ajh&b). Since probabilities are in the
interval from
to 1 inclusive, it follows that P(a//z)<P(Z?//i).
Let g be a generalization. Let z"i
in
be confirming instances
of it. Let In be the conjunction ii
&in of the first n instances of
g. Let e, finally, be some piece of evidence or information, relative to

&

which we may estimate the probability of g and of /.

Let p be the probability of g given e. Thus we have P(g[e) =p.
We shall callp the initial or a priori probability of the generalization.
Let pi be the probability of z\ given e p z the probability of i,
;

118

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

and h
etc.
Thus we have P(inle&In ^ =p n We shall
values p n the eductive
probabilities of the instances of the

given e
call the

generalization.
Given the eductive probabilities we can use the General
Multiplication Principle to calculate the
probability of ! given e, i.e. the

probability of n successive confirmations of the generalization. Let

ttn be the product l x
p
xp n Then we have P(In /e) //.
Let qn be the
Thus we have
probability of g given e and In
shall call the values qa the
P(g/e&In)
qn
probabilities a
posteriori of the generalization.
.

From

We

the General Multiplication Principle follows P(g&IJe)

^
F

Since the generalization entails (the

conjunction of)
it follows from our Lemma above that

instances,

From this immediately Mows

And if IIn >0 we can transform the

that,

P(g/e<Mn
with

)^^^

if

(g&In

Le.toqn^^-^

is

confirming

p^IIn
p>0, then

last identity

its

IIn >0.

above to

logically

identical

alone).

Now

compare qn and

dent that,
bilities

if

p>0 and

#, +1 ,

i.e.

//n >JTn+1 ,

compare -^- and =^-. It is evilln

//+ 1
then qn <qn +i. But, since proba-

from
and only ifpn +i<l.

are in the interval

//>//+!,

if

to

inclusive,

it

follows that

Herewith has been proved that, ttP(g/e)>0 and P(in+i/e&I )<l,

thQnP(g/e&In+1)>P(g/e&In). In words:
If the initial probability of a generalization is not minimal, its a
tt

posteriori probability increases with each new confirmation which is

not maximally probable relative to the previous
confirmations.
(It should be observed that in this formulation of the theorem in

words no mention
is

is

made of

e.

presently,
shall call this the Principal

We

Consider the difference jf---

formyrr

--

/ 1

the value

The choice of

e,

as

we

shall see

crucial to the
'meaning* of the theorem.)

\
1

j.

n+1

Theorem of Confirmation.

lln

It

can also be expressed in the

It is

seen that

it is

inversely proportionate to

ofpn+1 Thus we have the theorem:

.

119

THE LOGICAL PROBLEM OF INDUCTION

The smaller the eductive probability of the confirming instance,
the greater is its contribution to the increase in the a posteriori probability of the generalization.

The

two theorems, viz. the ideas that the

of
a
generalization increases with the number of conprobability
that it increases the more, the more improbable,
instances
and
firming
and
i.e.
unexpected, are the confirmations, can truly be
surprising
as
regarded
belonging to the 'classical' ideas of the theory of induction

ideas contained in these

and

probability.

The next question

to be raised is, whether the increasing probaa

of
generalization tends towards a limit and, particularly,
bility
whether this limit is the maximum value 1
On this question divergent opinions have been expressed. Keynes
wanted to show that the increasing probability approaches 1. His
proof makes use of extra-logical assumptions about the constitution
of the universe. 4 Nicod tried to show that Keynes 's use of his
assumptions was based on an error, and that the increasing
5
probability cannot be proved to approach I.
.

The condition of approach

stated.
if

From

the formula qn

to

maximum

=77-^
Jin

ws

probability

is

easily

that q* approaches

1,

IIn approaches p. In words:

The probability a posteriori of a

generalization approaches 1, if the

of the generalization
confirmations
n
successive
of
probability
a
its
probability priori.
approaches
P(~g/e)xP(In/e&~g) =
By virtue of A2 we have P(~g&In/e)
1
n
Al
we
virtue
have
of
p and
P(~g/e)
P(In/e)xP(~g!e&I ). By

P(~gle&In}

= 7-

-r

Lin

We also have P(/*/e) =//.

Substituting these

values in the equation P(~g[e) xP(//ecfc~) =P(//e)

we

get

P(Lle&~g)

P on
9

condition that 1

1p =

xP(~gfe&L\

p>0.

0, then/>
/, which
(This last condition is trivial. For, if
means that the generalization is already a priori maximally probable.
And in this case its probability can no longer be increased by confir-

mation.)
If

now

IIn

approaches p, then P(L/e&r^g) approaches

120

0,

and

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

conversely.

Thus we

can be formulated

in

get a

new

condition of convergence, which

words as follows:

The

probability a posteriori of a generalization approaches 1, if the

probability of n successive confirmations, on the assumption that the

generalization

is false,

O.

-approaches

The thought

that the probability of n successive confirmations

as
a
limit the probability a
approaches
priori of the generalization
is
if
we
think
of
the
intuitively plausible,
generalization as a con-

junction of its confimiing instances. If the generalization

i.e.

unrestricted, this

numerically
infinite. In symbols:
(1)

ii&

&i*&

conjunction

ad

inf.

is

is

'genuine',

(at least 'potentially')

= Df g.

A shorter way of expressing the same thought symbolically is

(2)

Urn
n >oo

L = Df g.

Here we have a case of a sequence of propositions approaching

another proposition as a limit. 7 (The sequence is the sequence of
conjunctions /.) It is 'natural' to think that, under such circumstances, the probabilities associated with the members of the sequence
of propositions will converge towards the probability associated with
the limiting-proposition. The probabilities associated with the
members of the sequence are the values IL. The probability of the
limiting-proposition is/?. In view of (2) it is therefore 'natural' to
think that we also have:
(3)

limlln=p.
n >oe

The 'naturalness' of this thought, moreover, has nothing to do with

assumptions about the structure of the universe. It is of a logicomathematical character and connected with the conception of the
generalization as an infinite conjunction. But we cannot prove this
idea from the axiom system of probability, presented in the first
section of the present chapter. In order to prove it we have to add
to the system a new axiom to the effect that, if a sequence of propositions approaches a given proposition as a limit, then the probabilities
associated with the members of the sequence approach as a limit the
probability associated with the given proposition. (It is assumed
121

THE LOGICAL PROBLEM OF INDUCTION

that the probabilities are throughout taken relative to the same
data.) In symbols:

^ = a,

A4. If Urn

n+

then lim P(*//z)

= P(tf//2).

n >

=o

axiom and the identity (2) above we can prove that the
of
n successive confirmations of a generalization approbability
a
as
limit the probability a priori of the generalization
proaches

From

this

the probability a posteriori of the generalization

the maximum value 1.
as
a
limit
approaches
The above argument, we believe, is the right and conclusive

and that hence

answer to the Keynes-Nicod dispute over the question, whether or

not the increasing probability of a confirmed generalization approaches maximum probability as a limit.
It is important to stress that the convergence towards maximum
the fact alone that the generalizaprobability does not follow from
tion is confirmed in an indefinite number of instances ('indefinitely
confirmed'). It must also be the case that the confirmations cover
the -whole range of instances of the generalization, and not only some
of this range. From the sequence l9 ...,/,...,
consist of all instances of g 9 we may select every second

infinite sub-class

assumed to

member

l9

zs

fs,

etc.

and thus obtain another

infinite

sequence of

confirming instances of g. But there can be no assurance (of a logicomathematical nature) that indefinite confirmation of the generalization through these instances will
9
eralization approach 1 as a limit.

make

the probability of the gen-

of our Principal Theorem

Lastly, it should be noted that the proof
does not require the use of principles of Inverse Probability.
6.

The Paradoxes of Confirmation.

It is

a major task of Confirmation-Theory to make the notion of a

(confirming) instance of a generalization precise. Of this notion we

have so far only said that the confirming instance must be a logical
consequence of the generalization. The further clarification of the

concept turns out to involve

difficulties.

We

shall here briefly consider some of these difficulties. Attention

will be confined to the simplest case only, viz. to Universal Generaliza-

tions of the

form

(x)

(Ax-^Bx).
122

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

First a certain ambiguity in the term Instance' should be noted.
to refer only to instances which confirm the

The term can be used

generalization, or it can be used to refer to instances which either

confirm or disconfirm it. Only if 'instance' means 'confirming

instance'

is it

true that the instance

is

entailed

by the

generalization.

In speaking about instances we have so far always meant confirming instances. For present purposes, however, it is convenient to take
Instance' in the broader sense and distinguish between instances

which are confirming and instances which are disconfirming of the

generalization.

Anything which

is

but

is

not

B disconfirms (falsifies) the general

(Ax-^Bx).
By a disconfirming instance of
(Ax~*Bx) we shall therefore understand any proposition of the
form Ax&~Bx. And we shall say of the thing x that it 'affords or

proposition

(x)

(x)

'constitutes' the disconfirming instance (disconfirmation).

The notion of a disconfirming instance raises no problem.

the notion of a confirming instance.

several possibilities are open.

One

possibility

is

As

regards

its

Not

so

definition

to define a confirming instance as that which is

Or, more precisely, as the contra-

not a disconfirming instance.

dictory of a disconfirming instance. In the case of (x) (Ax-*Bx), the

general form of the confirming instance would thus be ~(Ax&~Bx).

But ^(Ax&^Bx) is the same proposition as Ax-*Bx and also the

same as Ax&Bx v ~Ax&Bx v ~Ax&~Bx. Thus, if the notions of
a disconfirming and a confirming instance are contradictories, then

we shall have to say that the generalization that all A are B is confirmed by () anything which is both A and B, (ii) anything which
is not A but is B, and (Hi)
anything which is neither A nor B,
This way to define a confirming instance, however, leads to two
'paradoxes'. These axe variants of the well-known Paradoxes of
Implication. The first 'paradox* is that anything which is not A
will constitute a confirming instance of (x) (Ax-^Bx), irrespective of

whether

it is J5

or not.

And

the second 'paradox'

is

that anything

is B will likewise constitute a confirming instance,

irrespective
of whether it is A or not. Thus for example any swan, whether white

which

or not, would serve to confirm the proposition that all ravens are
black. And the same holds good for any black object, whether a
raven or not. Such consequences as these plainly conflict with our
123

THE LOGICAL PROBLEM OF INDUCTION

intuitive notions of

what

it

is

for a state of affairs to 'confirm' a

generalization.

Another possibility
firming instance.

To

and

which are both

is

to look for a narrower definition of a con-

this

end one might suggest that only things

B are

really confirmative

of the generalization

The general form of the confirming instance of

thus on this proposal be Ax&Bx. This idea is
would
(x) (Ax~>Bx)
sometimes called the Nicod Criterion. 2

that

all

are B.

a 'paradox'.
form of a confirming instance of (x) (Ax-^Bx) is
Ax&Bx, then, by substituting ~B for A and ~A for B, it follows
that the general form of a confirming instance of (x) (~Bx-+~Ax)
must be ~Bx&~Ax which is the same as ~Ax&~Bx. This piece
of reasoning seems quite unobjectionable.
Now (x) (Ax~*Bx} and (x) (~Bx-+~Ax) are logically equivalent.
It is highly plausible to think that anything which counts as a confirming instance of a certain proposition should also count as a
confirming instance of any propositon which is logically equivalent
with the first. This idea may be called the Equivalence-Criterion

But

this definition, too, leads to

If the general

(-Condition).

we should thus have to

accept the Equivalence-Criterion,
is the sole general form of a conAx&Bx
that
the
proposal
reject
should have to recognize that
firming instance of (x) (Ax-^Bx).
If

we

We
~Ax&~Bx is another form of confirming instance.

On the other hand, to regard a proposition of the form ~Ax&~Bx

B seems not to

as a confirmation of the generalization that all A are

accord very well with our 'intuitions' in the matter.

We

would not

normally regard the fact that this particular object, say a swan,
is neither a raven nor black as a confirming instance of the proposition that all ravens are black. And we would admit that the fact
in question confirms the generalization that all things which are
not black are not ravens.
This conflict between two "intuitions', the obvious plausibility

of the Equivalence-Criterion on the one hand and the reluctance to

Ax&^Bx as a confirming instance of (x) (Ax-+Bx) on the
accept
other hand, has been called a Paradox of Confirmation. 4
Of those 'paradoxes' which are special cases of the Paradoxes of
Implication

it is

easy to

show

that they are 'harmless' in the sense

124

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

that paradoxical confirmations of a generalization cannot influence
(increase) the generalization's probability by virtue of the Principal
Theorem- of Confirmation. The fact that this particular animal is -a

swan cannot effect the probability that all ravens are black. This is
seen from the following considerations:
The proposition Axn -*Bxn logically follows from the proposition
~Axn and also from the proposition Bxn. If therefore, in estimating
the probability of (x) (Ax-^Bx) relative to verified propositions of
the form Axn-^Bxn, it is part of our data that an object xn does
not possess the property A or that it possesses the property 5 5 then

and hence also maximally probable that this object will

the
verify
proposition Axn-*Bxn. Expressed in our symbolism above:
If e contains either
>Axn or Bx, i.e. if e is identical with e'&~Axn
it is

certain

<

Axn-*Bxn and L-i

for

& ... & (Axn-r+Bx^, we have by virtue of T2 (p.

= /. And this means that /, i.e., Axn-^Bx does

93)

or e'&Bxn, then, writing

for

not satisfy the

condition which is necessary if its verification is to contribute to
increase in the generalization's probability.
'Paradoxical' confirmations of a universal implication, afforded by
things which are either known not to satisfy the antecedent or
n,

known to

satisfy the consequent, are thus 'harmless', i.e. they do not

affect the probability of the general proposition. Instead of saying
that they are 'harmless' one may also say that they are 'valueless'

for confirmation, have no 'confirmatory value'.

It seems to us that the solution to the 'paradox' which results
conflict between the Nicod- and Equivalence-Criteria has to
be sought along the following lines:
We should start by questioning the validity of the Nicod-Criterion.
Is it necessarily the case that anything which is both A and B genuinely
affords a confirmation of the law that all A are 5? It seems to us that
the answer is quite certainly negative.
Whether the fact that a thing which is both A and B is a genuine
or a paradoxical confirmation of the law that all A are B depends
upon the way in which this fact becomes known to us. If we know that
the thing is A but do not know whether it is or is not B, then it will be
of interest, from the point of view of testing the law in question, to
find out whether it is B or not. If the thing is found not to be B,
the law is falsified, and therefore, if the thing is found to be B, the

from the

125

THE LOGICAL PROBLEM OF INDUCTION

'Genuinely confirmed', one might say, means
from falsification after having stood the risk'.
But if we know that the thing is B and do not know whether it is or is
not A> then it will be of no interest, from the point of view of confirmation, to find out whether it is A or not. For in neither case would
the law be falsified and therefore not (genuinely) confirmed either.
'Paradoxically confirmed', one could say, means 'confirmed under
circumstances which involve no risk of falsification'.
Thus the fact that a thing is both A and 5, genuinely confirms the
law that all A are B only if, on first knowing that the thing is A, we

law
the

is

confirmed.

same

as 'saved

subsequently verify that it is

Next we raise the following question: Is it necessarily the case that
nothing which is neither A nor B genuinely affords a confirmation
of the law that all A are 5? It seems to us obvious that the answer
.

we know that a thing is not B

not A, then showing that it is not
A saves the law that all not-5 are not-^4 and thus also the law that all
A are B from falsification, and can therefore be truly regarded as
confirmatory of either law. But if we know that a thing is not A but
do not know whether it is or is not 5, then showing that it is not B
saves neither the law that all not-5 are not-A, nor the law that all A
are B from falsification and therefore does not genuinely confirm
to this question, too,

but do not

is

If

negative.

know whether it

is

or

is

either of them.

Thus not any confirmation of the law that all ravens are black
through an object which is neither black nor raven can be rightly
called 'paradoxical'. It is paradoxical only if, prior to knowing
anything about the object's colour, we know that it is not a raven.
Then the object is valueless for the purposes of confirming that all

ravens are black and also for confirming that all not-black things are
not-ravens. But if, prior to knowing anything about the object's
membership of a certain species of bird, we know that it is not black,

then there is no 'paradox*. The object is then of value as affording a

potential confirmation both of the law that all ravens are black and
of the law that all not-black things are not-ravens.

The conclusion

that the Equivalence-Criterion is sound and that

the 'paradox' arises from the Nicod-Criterion alone. It is a mistake
and B constitutes a genuine
to believe that everything which is both
is

confirmation of the law that all

are B.

126

And

it is

equally a mistajce

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

to believe that nothing which is neither A nor B could constitute
a genuine confirmation of the law that all A are B. Whether something which is both A and B genuinely confirms the law or not,
pends upon whether the fact that this thing is B, is or is not known
prior to knowledge of the fact that it is both A and B. Similarly,
whether something which is neither A nor B paradoxically confirms
the law or not, depends upon whether the fact that this thing is not A,
is or is not known prior to
knowledge of the fact that it is neither A nor
B. But anything which genuinely confirms the law that all A are B
also genuinely confirms the law that all not-J? are not"A. And
anything which paradoxically confirms the first law also paradoxically
confirms the second. For they are one and the same law.

7.

Confirmation and elimination.

the Principal Theorem of Confirmation mean that induction

the
by
multiplication of instances, sometimes also called Pure Induction, possesses a value independently of induction by elimination?

Does

Keynes tried to show that the condition P(in+ife&I)<l mentioned in the Theorem, is satisfied only if the thing affording the
n+lth confirming instance differs in at least one property from all
9

the previous things affording confirmations of the generalization.

In his argument he makes a rather dubious use of the principle known
as the Identity of Indiscernibles. He interpreted his result as meaning
that the contribution of confirming instances to the probability of
laws really is rooted in their contribution to the elimination of

concurrent possibilities. 1
In opposition to the view of Keynes, Nicod made an attempt to
defend the value of Pure Induction independently of elimination. 2
He pointed out some errors and insufficiencies in Keynes 's arguments,
but he cannot be said to have been successful in vindicating his own
3

position.
It is hardly possible to settle the Keynes-Nicod controversy
over the value of Pure Induction without resort to a model of the
abstract notion of probability as it occurs in the Principal Theorem.

An

adequate model is provided in the following way:

Let g be the Universal Generalization (x) (Ax-+Bx). We replace J5
by a variable X. Thus we get a propositional function (x) (Ax~*Xx).
4

127

THE LOGICAL PROBLEM OF INDUCTION

It is satisfied

By O we

shall

by any property which

understand the class of

a necessary condition of A.
necessary conditions of A.
such a condition. Thus g can

is

all

generalization g states that B is

be expressed symbolically by G> (B).
Let e be of the form Y (5), i.e. let e be a proposition to the
that the property B is one of a certain class of properties Y.

The

also

The confirming instances

Axi-*Bx l9 Ax z->Bx z etc.

z'

i'i,

2,

etc.

effect

of g are the propositions

That a property

mean

that

it is

co-present with the property ^4 in a thing is to

not the case that A is present but the property in
is

question absent in this thing.

Consider the prepositional function Ax l-^Xxi which is obtained
from z\ by replacing B by the variable X. By 93. we shall understand
the class of all properties which satisfy this prepositional function,
9

properties which are co-present with A in the

thing
Similarly, we define /p 2 <p s , etc. By <D we understand the
&9. Thus On is the
conjunction (logical product) of classes 9i&

i.e.

the class of

all

XL

which are co-present with A in all (every one of)

the things x l and
and jc.
or P(<&CB)M#) ) is the probability that B will be a
necessary
P(gje)
condition of A, given that .# is a member of the class of properties y.
That this probability has the value p means in the FrequencyModel: the proportion of necessary conditions of A among all
members of vp is p.

class of properties

P(fn/e<fe/n_0

or P(<?4B)lv(B)&<S>r - l (B)')
t

is

the probability that

be co-present with A in x* given that B is a member of y and

B has been co-present with A in every one of the things #1
jcn_!
That this probability has the value p* means in the FrequencyModel: p is the proportion, among all member of y which are
and x- ly of properties co-present
co-present with A in x^ and
will

with

in

x.

P(L/e) or P(<fc<jB)M5) ) is the probability that B will be copresent with A in the first n things, affording confirming instances of
g, given that B is a member of y. That this probability has the value

X ... xp means

in the

Frequency-Model: this probability is the

of properties co-present with
proportion, among all members of
A in Xj, multiplied by the proportion, among members of y copresent with A in x l9 of properties co-present with A in x, multiplied

^i

128

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

by
with
.

among members of y co-present

of properties co-present with A in x

multiplied by the proportion,

in

X L and

and x

tt

-i,

tt .

the probability that B will

P((B)jv(B)&Q
be a necessary condition of A, given that B belongs to a class of
properties y <?/zrf that B has been co-present with A in every one of the
first n things which have afforded confirmations of the law that B is
such a condition. That this probability is qn means in the

P(gle&In) or

is

n (B))

Frequency-

Model that
members of
qn

the proportion of necessary conditions of

<y which are co-present with A in x l and
.

That qn equals the

ratio

~~- means

A among
.

and xn

that the proportion just

is

men-

tioned equals the proportion of necessary conditions of A among all

members of y divided by the proportion, among members of y, of
properties which are co-present with

That 9+!

in

Xj.

and

and xn

greater than qn means in the Frequency-Model that

the proportion of necessary conditions of A among members of y
which are co-present with A in x : and
and Xi +1 is greater than the
of
of
conditions
A
necessary
proportion
among members of vy which
is

are co-present with

in x^

and

and xn

in the Frequency-Model, of the ratios

Considering the meaning,

-=~- and w~ ^
*

llpn+ 1

fU ws

that

tlpn

two conditions must be

fulfilled if qn -i is to be greater than q

viz:
the
or
of
of
A
conditions
the
all
(i) p 9
proportion
necessary
among
members of y, must not be minimal, i.e. must be greater than 0, arid
(ii) pn+i, or the proportion, among all members of \y which are coand xn of properties co-present with A
present with A in x^ and
in #11+1, must not be maximal, i.e. must be smaller than 1.
If the class of properties y is finite, then (z) simply means that there
must exist at least one necessary condition of A in vy, and (ii) simply
means that, if 'xn +i is an instance of A then xn+l must lack at least
one property which is co-present with A in x l and
and xn And
the latter again means that the n+l:ih instance must exclude at
least one member of y, which has not already been excluded, from
the possibility of being a necessary condition of A.
If the class vy is infinite, then (z) means that a 'perceptible', i.e. notminimal, proportion of members of the class must be necessary
conditions of A and (ii) means that the n + Jf :th instance must exclude
a 'perceptible' proportion of members of the class, which are cotty

129

THE LOGICAL PROBLEM OF INDUCTION

present with A in the n first things affording confirming instances,
from the possibility of being necessary conditions of A.
The condition that p or the a priori probability of the generalizais thus, in the Frequency-Model,
must be greater than'
tantamount to a Condition of Determinism. And the condition that
p n +i or the eductive probability of the confirming instance must be
smaller than 1 is tantamount to a Condition of Elimination,
The above considerations started from replacing B in (x) (Ax~*Bx)
by a variable X. We might instead have replaced A by a variable.
Then we should have got the prepositional function (x) (Xx-*Bx).
It is satisfied by any property which is a sufficient condition of B.
The Universal Generalization states that A is such a condition.
The generalization can also be expressed symbolically by 3>(A).
Let e be of the form y(A).

tion

Consider the prepositional function Xxi-+Bxi.

the class of

properties which

all

By 9 we now mean
x

satisfy this prepositional function.

member 91 is thus a property, of which it is not true that it is

Of such a property we shall say that
present in x if B is absent in x
lf

ly

it is

co-absent with

Similarly,

&y n

we

B in (from)

the thing in question.'

define 9*, 9,, etc.

(Ox thus

the

By

<D*

we understand

the product

same

as 91.)
If, in our previous interpretations in frequency terms of the a
of the a posteriori
priori probability, of the eductive probabilities, and
9i<&

is

of B'
probability, we substitute the phrase 'sufficient condition(s)
for the phrase 'necessary condition(s) of A ', and the phrase *cothen we get
absent with B for the phrase 'co-present with
9

A\

another interpretation in terms of frequency of the magnitudes in

question. The two conditions of increase in probability now assume
the following shape:
of B
(*') p, or the proportion of sufficient conditions

members of

vy,

among

all

the

must not be minimal, and

(") pn +i, of the proportion, among all members of y which are

and Jcn , of properties co-absent with B
co-absent with B in *,. and
in xn +i, must not be maximal.
As before, the condition p>Q is tantamount to a Condition of
Determinism, and the condition />+i<l is tantamount to a Condition
of Elimination.
It is clear that, on the interpretation <D(B) of g, elimination Can
.

130

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

is
only if *n +1 has the property A. For it is only when
that
present
any other property could be denied to be co-present with
it.
And similarly it is clear that, on the interpretation (A) of g,
elimination can take place only if x +1 lacks the
B. For it

take, place

property
only where B is absent that any other property could be denied
to be co-absent with it.
This is in harmony with the two basic facts of
Elimination-Theory,
viz. that when we are in search of a
necessary condition of a given
property we ought to compare with each other positive instances of
that property, whereas when we are in search of a sufficient condition
of a given property the elimination is promoted by
negative instances
of that property. (Cf. above Ch. IV, 4.)
is

We now

also see how the fact that the

probabilifying effect of a
instance
is
confinning
inversely proportional to the eductive probaIt
bility of the instance, is reflected in the

Frequency-Model.

simply

means

that the probabilifying effect is

inversely proportional to
the eliminative effect. The eliminative effect, in its turn, is the
instance is as
greater the more unlike the new

confirming
compared
with the previous confirming instances. Thus, we increase the
probability of a generalization by confirmation, the more effectively,
the more we succeed in varying the circumstances under which the
generalization is put to successful test.
If the eductive probability of an instance

is 1, its

eliminative effect

This sheds light upon the Paradoxes of Confirmation.

thing
which has the property B cannot eliminate anything from the
possibility of being a sufficient condition of B. And a thing which
lacks the property A cannot eliminate anything from the
possibility
of being a necessary condition of A. Such things are therefore
necessarily ineffective from the point of view of elimination. This
is the
counterpart, in the Frequency-Model, to the fact that such
things are ineffective from the point of view of producing an increase
in probability. But by being necessarily ineffective, the
'paradoxical'
confirmations are also 'harmless', they do not 'genuinely' or 'really'
confirm the generalization in question at all.
The abstract notion of probability, which figures in the Principal
Theorem of Confirmation, can thus be given a model which makes
the logical mechanism of the theorem reflect the working of the
logical mechanism of induction by elimination.
is nil.

131

THE LOGICAL PROBLEM OF INDUCTION

The possibility of this model may be said to support the view,
which Keynes put forward but supported with dubious or false
arguments, that the multiplication of instances in induction has a
far as it has an
probabilifying effect on the conclusion only in so
eliminative effect within a field of concurrent possibilities.
Can the model be said definitely to settle the dispute over the

The answer to this question depends upon

whether some other model of the Principal Theorem can be worked
out which would establish the independence (in that model) of the
from their eliminative effect.
probabilifying effect of the instances
No such model is known. And it seems to us doubtful, whether such
a model could be given without the introduction of highly arbitrary
7
assumptions as regards the way in which probabilities are measured.
It would involve no contradiction, if there existed a model which
would establish the independence of the probabilifying effect from
the eliminative effect. We should then only have to say that, abstract
in one way, the Principal Theorem
probability being interpreted
makes the probability of inductions dependent of elimination, and
abstract probability being interpreted in another way, the theorem
makes confirmation independent of elimination.
In any case the possibility of a model, which makes the probabilifyis of
great
ing effect of confirmation mirror a process of elimination,
interest. It establishes a not-trivial logical connection between the
two main branches of the formal study of induction, viz. Eliminationvalue of Pure Induction?

Theory and Confirmation-Theory. And in doing this it lends

shared by some of the ablest
support to an epistemological attitude,
thinkers in the field of induction (Bacon, Mill, Keynes), as regards
the logical nature of inductive reasoning.

Reasoning from
Probability, scope and simplicity.
Mathematical and philosophical probability.

8.

analogy.

Besides the point of view of confirmation, there are at least two

other points of view from which the probability of inductive generalizations may be studied. One is the point of view of scope of generalizations. The other is the point of view of simplicity of generalizations.
J.

M.

Keynes, with C. D. Broad the founder of Confirmatidti132

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

Theory, also

made a first attempt to

study the probability of generali-

from the point of view of their scope. 1 The theory of scope

turns out to bear relevantly on one 'classical' type of argument
intimately connected with induction, viz. reasoning from analogy.
zations

We

need not here give a general definition of the notion of the

of
a generalization. The idea is connected with some difficulties,
scope
which are related to the difficulties which arise in connection with the
notion of a confirming instance. (See this chapter 6.)
We shall content ourselves with a notion of relative magnitudes of
scope in propositions (not necessarily generalizations). We shall say
that, if one proposition entails another., then the scope of the first

smaller than, or at most equal to, the scope of the second.

with
our previous Lemma (p. 118), according to which the
Together
of
a
probability
proposition, on given data, is smaller than, or at
most equal to, the probability of its logical consequences, we immediately obtain the following result:
The probability of a proposition., on given data, is directly proportion-

proposition

ate to

its

is

scope.

must be understood that

formulation is a shorthand for

data, a propostion of given
scope cannot have a smaller probability than a proposition of smaller
It

saying that,

relative

to

the

this

same

scope.

a Universal Generalization of the simple form

(Ax-+Bx).
The scope of this generalization is increased, if (z) to the antecedent
conjoined a new term, or (if) to the consequent is alternated a
Consider

(x)

is

new

term.

For example: (x)(Ax-+Bx)

(x)(Ax-+Bx v Cx).

entails

(x)(Ax8tCx-+Bx) and also

Similarly, the scope of a generalization

is alternated a new term, or

antecedent

is
(if)

decreased, if (/) to the

to the consequent is

conjoined a new term.

For example: (x)(Ax v Cx-*Bx)- and also (x)(Ax-*Bx&Cx) entail

From the definition of relative magnitudes of scope of propositions

follows that, in general, a conjunction has a smaller scope and a
disjunction a greater scope than its single members. This result is

it

now easily
K

generalized to the following:

133

THE LOGICAL PROBLEM OF INDUCTION

The scope of a universal implication is directly proportionate to
the scope of its consequent and inversely proportionate to the scope
of its antecedent.
Or, considering the proportionality between scope and probability:
The probability of a universal implication is in direct proportion to
the scope of its consequent and inverse proportion to the scope of its
antecedent.

We may call this the Principal Theorem on the Scope of Generalizations.

*

Reasoning from analogy

following character:
From the fact that

two

is,

roughly speaking, an argument of the

and y resemble each other

in a
a further feature B
which is characteristic of x will also be characteristic of j.
This sort of argument is, in general, considered to be stronger (its
conclusion more probable) the greater the number n is of common
characteristics of the two things. (This is not to say that the strength
of the argument depended only on the number of common charac-

number of features, AI

things,

:c

A n we conclude that
,

teristics.)

What

is

the logical foundation of this belief?

Why do we

think

that the degree of likeness between two things is relevant to the

question whether a certain property, known to belong to one of the
things, will also belong to the other?
It is difficult to see, how the argument

from analogy could appear

we suspect a connecA*, and the

between the presence of the properties Ai
of
the
former
is
The
a
B
in
properties
thing.
presence
property
thought of as somehow 'responsible' for the presence of the latter
convincing at

all, if it

were not for the

tion

fact that

property.
connection, which satisfies this requirement of causal 'responsibility', is the connection of sufficient (or necessary-andA* contains
sufficient) conditionship. If the set of properties Ai

among themselves a sufficient condition of B, i.e. if one or other of

the properties individually or some of them taken in conjunction
are sufficient to produce B, then the presence of all of them in a thing
will be accompanied by the presence of B. Now the meaning of
reasoning from analogy becomes this: The probability that n proper134

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

An, contain among themselves a sufficient condition
greater, the greater n is.

ties,

Ai

of B

is

validity of this argument may now be examined within a

of
the scope of generalizations. For the argument amounts to
theory
&A n
saying that, in general, the probability of (x) (A&&
will be greater than the probability of (x) (A : x&
.&A nx
(It is understood that both probabilities are taken relative to the

The

same

data.)

From

our Principal Theorem in fact follows that

&A nx^Bx)/h)^ P( (x)(AiX&
&A H +ix-+Bx)lh).
P((x)(A>x&
It is of some interest to examine when < holds and when = holds
.

between the two

probabilities.

&AnX~*Bx) is the same proposition as the conjunc(A,x& ...& A nx&~A n+1x-*Bx) & (x) (A Ix& ...& A*+&-*Bx).

(x) (A!.X&

tion (x)
By virtue of the General Multiplication Principle it is easily shown
that the two probabilities under consideration are equal if, and

&~A

n +iX-+Bx)lh&(x) (A^x&...
only if, P( (x) (Ax&. .& A nx
L Conversely, the first probability is smaller than
&An+*x-+-Bx) )
the second, if this last (third) probability is smaller than L
This means: If the addition of a new common property n +i to
A n is to strengthen the
the previous common properties
.

argument by analogy under consideration, then it must not be

A n ~A n +i contains a suffimaximally probable that the set AT,
A n +i contains a sufficient condition of
given that the set Ai
.

cient condition of B.

Considering reasoning from analogy, these conditions of equality

intuitively most plausible. For, assume that both
contained a sufficient condition of B. Then it would

and inequality are

sets actually

follow that

Ai&

&A n &^A n ^

and

And from

A&
t

&A n ^

are both

again would follow that

ScA n is a sufficient condition of B. In other words: if both
sets contained a sufficient condition of 5, then it would be certain
contained a sufficient
that already the smaller set
condition of B. And if this were the case, then the discovery of any
further resemblance between things, which already agree in all the
sufficient conditions

A&

of B.

this

A&

&A

An, would be worthless as a contribution to the

properties Ai
logical force of reasoning from analogy. Conversely we may say
n+i is to be of value
that, z/the discovery of a further resemblance
.

135

THE LOGICAL PROBLEM OF INDUCTION

to the argument, then there must be some chance that a sufficient
condition of B is to be found in the set A l
A n +i rather than in the
set AI
An. And this is precisely what we should say there is not,
if the assumption that there were a sufficient condition of B in the set
.

Ai

An+i would make

the set Ai

A ~A n +i
n9

it

maximally probable that there

is

one

in

as well.

Thus reasoning from analogy may be shown to depend, for its

logical force, on simple ideas concerning the proportionality of
scope and probability in generalizations.
3

The

idea of relating the probability of generalizations to their

'simplicity' can truly be said to be among the classical ideas of
4

Simplex sigillum veri. The notion of

oh
generalizations has often been compared to the notion
simplicity
of simplicity in curves (and their algebraic expressions); and the

scientific

methodology.

problem of generalizing from particular data has been compared to

the problem of tracing the simplest curve through a number of
5
Sometimes ideas on simplicity have been
points in a diagram.

on the scope of generalizations. 8

of many efforts, no satisfactory theory of the

related to ideas

In spite
relation
of simplicity to the probability of inductions has as yet been developed.
The subject largely remains a virgin field of inductive logic. 7 In this
work we shall not make an attempt to penetrate into it. 8

It is sometimes alleged that probability,

relation to simplicity of curves and of laws,

from the probability-concept which

is

when contemplated

in

of a different nature
'implicitly defined' in a set of
is

postulates such as ours for abstract probability and 'explicitly

defined' either in a Frequency- or a Range-Model of the abstract

This probability of a different nature was traditionally

called philosophical probability and contrasted with the notion of the
calculus.

calculus which

was

called mathematical probability. 9

appears, however, that the suggested dichotomy is at least not

very helpful from the point of view of & formal examination, such as
It

136

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

the one undertaken in this chapter, of ideas concerning the probability
of inductive conclusions. It is noteworthy that all achievements so
far in the formal clarification of these ideas have
either with the aid of the abstract calculus or

been reached
its models

some of

or with the aid of some weaker calculus such as the various systems
of so-called comparative probability. We must not, of course,
exclude on a priori grounds the possibility of a formalism of probability

which would be

significantly different

from the

'classical'

ones and which might be successfully used for analysing some such
ideas as those relating to simplicity of inductions. 10 But it is initially
difficult to see how such a formalism, if invented for the ad hoc
purpose of dealing with an obscure corner of inductive theory,
could be of much interest either to the logician or to the philosopher.

For any

contribution, it would seem, becomes significant only if it

succeeds in assigning to some of our natural, though notoriously
obscure and vague, ideas of inductive probability a place within the
common framework of all other significant uses of probability.
But irrespective of the question of plausibility and intrinsic
interest of other formalisms of probability, any formalism of what-

ever structure would, so far as concerns its power of justifying

induction, be subject to the same general conditions as our abstract
calculus and its various models. Which these conditions are wiU be

discussed in the next chapter.

137

CHAPTER Vn

PROBABILITY
1.

AND THE

JUSTIFICATION OF INDUCTION

Probability and degrees of belief.

our intention in this chapter to answer the following question:

In what sense and under what circumstances can the assertion either
that a proposition is probable to degree p, or that one proposition
is more
probable than another proposition, be said to justify induc-

IT

is

tion*

very use in ordinary language the word 'probability' is

to ideas such as those of 'possibility', 'degree of
related
closely
or
'degree of rational belief. This connection naturally
certainty',
causes us to think of probability as justifying induction in the sense

By

that

its

it

were

'better'

or 'safer ', in

'practical life', taking precautions

and, in
making predictions
to prefer the more probable

propositions to the less probable ones. In other words, this connec1

tion suggests the idea of probability as being 'the guide of life',
i.e. the safest finger-post to follow in the search for truth.
In order to assess the significance of this idea, it will first be necessary to discuss the relation between abstract probability and the

notion of a partial belief.

By an actual degree of belief vis mean a state of mind, a psychological fact expressing our attitude towards something (usually outside
the sphere of our direct knowledge). In order to get a clearer idea of
this psychological fact we must consider how degrees of belief might
be measured, i.e. determined empirically.

We

can, in the first place, regard the psychological facts called

degrees of belief as feelings of different intensity. It is, theoretically,

conceivable that a psychometrical method could be invented for

comparing intensities in feelings and for assigning to those intensities
numerical magnitudes. Nevertheless, it is for several reasons obvious
that this way of evaluating degrees of belief is wholly inappropriate
if we wish to relate degrees of belief to degrees of probability. We
138

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

need only consider the fact that our belief in

things which we habitually take for granted is often accompanied by practically no feeling

at all. 2

In the second place we might measure

degrees of belief as follows:
say that our degree of belief in a proposition is p 9 when we
believe that the event which the
proposition asserts will occur *in

We

average (or in the long run') in a proportion/? of

its

all

occasions for

occurrence.

This definition

is peculiar in that it defines

one
degree of belief
reference to belief, as such, in another
proposition,

proposition by
'Belief in connection with this latter
proposition, again means the
psychological fact which is expressed in saying that we 'believe* a
certain proposition to be true (as
opposed to certain other propositions

which are

'believed* to

We

be

false).

can, of course, go further and ask what degree of belief we

have in the latter proposition believed to be true. This
is then

degree

defined

reference to a further
proposition believed to be true.
It seems to us that this second
way of measuring degrees of belief
is a true
of
the
old-established
idea of measuring a man's
analysis
belief by proposing a bet, and also of the
philosophical doctrine that
degrees of belief were causal properties of our beliefs, that is to

by

say
of our preparedness to act on our beliefs. 3
Supposing a method of measuring actual degrees of belief to be
given, we turn our attention to the interpretation-problem of formal
probability. It

measurement
belief,

would

is

now

to be observed that, whatever this

method of

be, the definition of probability as actual degrees of

make the axioms and the theorems of the probability-

calculus general synthetical propositions, i.e. a kind of

psychological
laws for our distribution of beliefs. The
interpretation would thus

unlike the frequency- and range-interpretations

be of the same
type as the interpretation of Euclidean geometry as a theory of
light-rays.

Consider for example the Special Multiplication Principle. If

means degree of belief, then this principle says that my
degree of belief in the proposition a & b (relative to some datum
K) is the product of my degree of belief in a and my degree of belief
in b, assuming that knowledge of one of these propositions does not
influence my belief in the other proposition. This assertion,
obviously
probability

139

THE LOGICAL PROBLEM OF INDUCTION

synthetical in the sense that from
to degree/? and the
proposition b to
I shall believe
is

my

believing the proposition a

it does not
follow that
There is nothing in the nature

degree/

a&bto degree p xp'.

of things to exclude that
my belief in a

& b, on the above premisses

isnotpxp'.*
The assumption that an axiom or theorem of the
probabilitycalculus were false under this
interpretation is, furthermore, not
only possible but also likely to be true. 5 Thus the calculus of
probability as a theory for the distribution of actual beliefs would
presumably soon break down as being a false theory.
It is, however, evident that the adherents of the
'psychological'
theory of probability have not intended the propositions of the
probability-calculus to be synthetical propositions
concerning the
distribution of actual beliefs.
They appear, on the contrary, to have
assumed that the axioms of
probability were a kind of standard of
correctness in actual beliefs. TTie calculus of
probability does not
tell us how we believe cs a matter
offact, but how we ought to believe.*
'

If we
accept this regulative function of formal probability, it is
necessary to abandon the idea of probability being defined or interpreted in terms of beliefs as psychological states of mind. 7 The
exact nature, however, of this
regulative function of the formalism,
as also the true connection between
probability and belief, still
remains to be determined.

We

will characterize the

regulative function

calculus

by saying that

of the probability-

provides a standard of rationality in degrees

of belief. What does this characterization
convey?
It is first to be observed that the
standard of rationality cannot
be defined as a standard
of consistency. This is important. Not only
does the calculus of
probability not tell us how we actually believe
but it does not even tell us that,
if OUT actual beliefs in certain simple
cases are distributed in such and such a
way, then in order to be consistent

it

we

ought to distribute our beliefs in certain other cases

compounded of those simple ones -in a determinate way. 8 For
actual degrees of belief are
psychological facts and can, as actual
facts, never contradict one another. This has been
illustrated
already

in the example above of the

Special Multiplication Principle.
It thus
appears that 'rationality' in
of belief is not a

degrees

property of the

way

in

which actual
140

beliefs are related,

formal
but that

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

what may be termed a 'material* characteristic

between
certain actual beliefs and others.
discriminating
This material characteristic determining certain
degrees of belief
is

'rationality'

is, evidently, the same as that referred to when we

say that
'rational' to prefer the more probable to the less
probable, implyas was already observed above
that probability is
ing with this
'the guide of life' or the best finger-post to follow in the search for

as rational

it is

rational degree of belief, in other words, is a

degree of
induction.
It
remains
to
be
probability justifying
analysed what this
truth.

'rationality' in beliefs really

2.

Rationality of beliefs

means.

and success

in

predictions.

Let us, throughout this paragraph, assume that the proposition

9
probable to degree p implies the proposition 'belief of degree
p is rational', and conversely. We ask the following question:
How is it to be determined whether or not it is rational to entertain
belief of degree/? in a given proposition?
As a first answer to the above question we suggest the following:
Whether a certain degree of belief in a proposition is rational or not,
is determined
by the knowledge relative to which we consider the
truth or falsehood of this proposition. By knowledge we mean here
'it is

any analytical proposition and any

to

be

true.

inductive

is

synthetical proposition known

(That is to say, no synthetical proposition which is
included under the term 'knowledge'. 1)

For example:
coin will

that the coin

made under
it

may

come down
is

call it rational to believe to

degree

J-

that this

on the knowledge
symmetrical and homogeneous and that the toss is
'heads' in the next toss-up

certain determinate conditions. 2

Suppose that on this way of determining rationality in beliefs

were rational under the circumstances C to believe to degree p a

certain proposition asserting the event E. It follows from the initial

assumption of this paragraph that it is rational under the same

circumstances to believe the negation of this proposition to degree

1

p.

We suppose further that p>lp.

Under
beliefs

is

must be

these suppositions we can conclude that, if rationality in

to be of relevance to the justification of induction, then it
rational, in considering

whether under the circumstances

141

THE LOGICAL PROBLEM OF INDUCTION

the event

will occur or not, to

prefer the prediction of E to the
of
not-j as a 'guide of conduct*.
prediction
Let us assume that, in predicting the event E n times under the
conditions C, the prediction turned out to be true
times and false

m'

and that m\n<m \n. If the process of

predicting is continued so that n becomes very large, and if the
proportion of true
and false predictions shows a marked
tendency to cluster round the
same ratios m\n and m'\n respectively, this
may make plausible the
times,

further assumption either that, in the

long run, the events
occur under the circumstances C in the
respectively, or at least that notIt is to be observed that there is

E and not-

proportions m\n and m'\n

occurs more frequently than E.

nothing in the previous suppositions

E which could preclude any of the
frequency of E from being true.
This implies that the suggested way of
determining rationality in
beliefs may lead to a situation of the
following somewhat paradoxical
as to the rational degree of belief in
above assumptions as to the relative

character:

We

call

it,

for determinate reasons, rational to

prefer the predicunder the circumstances
to the prediction of

tion of the event

not~E. But
(i.e.

that

we

we

assume, nevertheless, that we shall be less successful

shaU arrive at the truth in a smaller number of
in
cases)

under these circumstances than in

predicting not-.
We know, furthermore, that this assumption may well be true.
From the possibility of such a situation it follows that the
suggested
way of determining rationality in beliefs cannot provide a satisfactory
justification of induction. By this we do not wish to maintain that
it were not
possible to define rationality in beliefs without regard
to success in
predictions, and not even that one might not call this
kind of rationality a 'justification* of the predictions which we
actually
make. One must simply be clear that this
'justification' is not of the
predicting

slightest relevance to the

'sceptical'

results

of

Hume

as to the

4
impossibility of foretelling the future.

degrees of probability qua degrees of rational

from Humean scepticism, then the answer
to the question as to whether or not it is rational to
entertain, in a
given proposition, a certain degree of belief must involve some reference to success in using the proposition concerned for

Consequently,

if

beliefs are to 'relieve' us

We

predictions.*

must, in other words, be able to give some kind of guarantee

142

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

that, in preferring the more probable to the less

probable,
more successful than in making the
opposite preference.

we

shall

be

In considering what the

'guarantee' mentioned could possibly
be, the following will
If, in the example

be instantly clear to

us:

given above, it is rational under the circumstances C to prefer the

prediction of the event E to the prediction of not-2f,
we cannot with this assertion of
wish to exclude the
'rationality'

possibility

that,

those circumstances

being

realized,

the

more

probable prediction will after all turn out to be false, the less probable
6
again to be true. Nor, of the assertion of 'rationality', do we demand that, of all predictions of E on the conditions C which will
actually ever be made, a majority is going to be true and a minority
false. We only demand that, if it is rational to
prefer the prediction
of E to the prediction of not-jE , and it nevertheless
happens that only
a minority of actual predictions of E are true, then this distribution
of truth and falsehood on the
predictions must be regarded as
a
'chance-event'
which
cannot be excluded, but which
representing
1

we

think, in the long run, will give place to another distribution where
7
the true predictions are in
The statement that 'chancemajority.
events', such as the above 'abnormal' distribution of true and false

predictions of the event E9 will be cancelled out or eliminated in the

long run so that ultimately the frequencies tend to be proportionate
to the probabilities, we shall call the statement on the
Cancelling-out of

Chance CAusgleich des Zufalls'). 8

It can thus be stated that the
guarantee of success needed in
predictions, if induction is to be justified with reference to rationality
in beliefs, concerns the statement that the
Cancelling-out of Chance
will take place for the
degrees of probability corresponding to the

degrees of rational belief.

It will
immediately occur to us that there is one way of securing
off-hand the Cancelling-out of Chance for
every probability. This
consists

in

frequencies.

interpreting formal

Under

this

probability in terms of limitingthe statement that the

interpretation

Cancelling-out of Chance will take place would become analytical

If this method be resorted to, the assertion that it is rational to

E would mean that this event

entertain belief of
degree/? in an event
occurs in a proportion/? of all occasions.

hand, that the

relative

The statement, on the other

frequency of an event on
143

all

occasions of

its

THE LOGICAL PROBLEM OF INDUCTION

occurrence

is/?, is

a general synthetical
proposition of the type called
It is plain that if the truth of such Statistical

Statistical Induction.

Inductions be

made the criterion for

the truth of statements concerning rationality in beliefs, then any argument which tried to justify
induction by reference to degrees of
probability as representing
rational degrees of belief would be circular.
The state of our problem is now the following;

We have shown that if rationality in beliefs is to justify induction,

must be possible to give some kind of 'guarantee'

Cancelling-out of Chance will take place for degrees of

then

it

that the

probability
have further seen that
representing rational degrees of belief.
this guarantee can be given if we resort to the
frequency-interpretation of formal probability, but that this
way of securing the Cancelling-out of Chance at the same time vitiates
from

We

any argument

rationality in beliefs to the justification of induction. Consequently,

if we are able to show
of
convincingly that the

only possible

way

guaranteeing the Cancelling-out of Chance is to interpret formal

probability in terms of limiting-frequencies, then it will have been
5
demonstrated that the idea of
compensating the 'sceptical arguments of Hume with a theory of inductive
probability is vain.

The Cancelling-out of Chance and the theorem

of Bernoulli.
The theorem of Bernoulli, speaking
tells us that

3.

if
approximately,
are concerned with propositions the
of
which
all
probabilities
have the value/?, then it is infinitely probable that 'in the
long run',
exactly the proportion p of those propositions are found to be true.

we

Thus,

if

we have two

propositions with the respective probabilities p'

and/?", and/?' is greater than/?", then we know that it is infinitely
probable that the former proposition will be true on a greater number
of occasions than the latter. In
spite of this it may of course happen
that, of the propositions which we
have
those with
actually

tested,

the probability/)" have more

frequently been true than those with
the probability/?'. This, however, has been due to 'chance'. In the
long run we know, according to the theorem, that it is infinitely
probable that such chance-events will be eliminated
'cancelled
out'

so that finally the proportions of true

propositions in the
respective classes become as indicated by the probabilities.

When

stated in this

way

it

looks as though the theorem of Ber144

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

were closely related to that which we have called the

Cancellingout of Chance. Actually it is an old idea that this theorem amounted
to a proof that
although irregularities and chance-events may
upset our calculations when applied only to a narrow sector of the
world
in the course of nature as a whole
law and order
noulli

regularity,

prevail

It is therefore

intelligible that the proof

intellectual achievement of the

of the theorem was

regarded as an
greatest philosophical
2
In the nineteenth
significance.
century mathematicians and philosophers still spoke with the deepest awe about the wonderful philosophical implications of this theorem.

deeper insight into the logical nature of probability, however,

in our days to a common abandonment of the

hashed

philosophical
aspirations originally connected with the theorem of Bernoulli and
the Laws of Great Numbers in
general. Nevertheless, the idea of the
relatedness between the formal theorem of the
probability-calculus
and the statement on the course of nature which we have termed the

Cancelling-out of Chance, possesses high philosophical value as an

of a fallacy of thought, which not
only underlies the
main 'classical' misuses of the
principles of probability for philosophical considerations concerning induction, but also is the source
of various erroneous ideas about inductive
probability which still
play a prominent role in philosophical discussion. We shall therefore examine in some detail the
fallacy made in relating the theorem
3
of Bernoulli to the
Cancelling-out of Chance.
The apparent relatedness between the theorem of Bernoulli and
the Cancelling-out of Chance has its root in the fact that in the
illustration

theorem mentioned two probabilities are involved. Of these

probabilities one remains constant
throughout the course of considerations,

whereas

maximum

value

acquires

the

1.

the

other

increases

It is this
probability

appearance

probabilities of the first order

of

and

approaches the

of the second

order that

between
and corresponding frequencies This
providing

bridge

occurs through an unconscious

interpretation of the empirical
implications of an increasing probability.
It

has been already observed

bility

is,

by

its

(p. 138) that the concept of probause

in
very
ordinary language, related to the concepts

145

THE LOGICAL PROBLEM OF INDUCTION

of certainty and of possibility.

It seems
plausible to say that prob4
measures
The more
ability
degrees of certainty or of possibility:
a
the
smaller
the
probable
proposition,
possibility that this proposition will turn out to be false. If the
is
probability of a

proposition

infinitely close to the maximum value 1, it means that its possibility

of being false is infinitely small, or in other words that the

proposi-

tion itself

is

'almost certain'.

The

interpretation of probability as a magnitude of possibility

natural, particularly when we have to do with a variable probaFor whereas the grounds for
bility-value.
interpreting a fixed
as
a
of
must
be
in
some way or other
probability
degree
possibility
in
the
sense
in
which
in
'objective', (e.g.
games of chance certain
is

properties of symmetry, being physical properties, provide us with a

number of equally possible alternatives), the statement, on the other
hand, that a probability varies seems by its very nature to mean a
statement about the altered possibilities of a certain
thing being true.

For

this reason it happens that, in the theorem of Bernoulli, we

might
speak about the constant probability of the first order without
attaching to it any special 'interpretation', but nevertheless take as a
matter of course the variable probability of the second order to
'mean' an increasing degree of possibility or
certainty.
Thus the first step towards the use of the Bernoullian theorem as a
bridge from the realm of probability to the realm of empirical
frequencies consists in giving the theorem the following content: If
the probability in certain propositions is
p, then it becomes in the
long run almost certain that among those propositions a proportion

are true.

At this point the following remark will take the reasoning further.
Let us suppose that we have demonstrated, with the aid, for
example,
of the theorem of Bernoulli, that one
possibility is greater than
another, or that one is very great, another again very small. Is there
anything in this which will preclude, even in the long run, the small
possibility from being realized very frequently, the great possibility
again very seldom or perhaps even not at all?
It is obvious that if there were
nothing to preclude this, then a
degree of possibility, even at its maximum, could not serve as a
bridge to frequencies, since then the statement that a certain frequency
very, or even 'infinitely', possible would not tell us anything

is

146

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

about the way in which that frequency will be realized. Therefore,

the second step in using the theorem of Bernoulli as a bridge from
probabilities to frequencies consists in our tacitly assuming the
increasing degrees of possibility, involved in the theorem, to have the
following implication: If the probability in certain propositions is p,
then it will in the long run almost always happen that among those
5
propositions a proportion p are true.
On the other hand there is nothing in the proof of the theorem of
Bernoulli which would exclude the highly possible from happening
very rarely/ If we want to effect the necessary exclusion we must
therefore turn our attention to the way in which degrees of
possibility are measured empirically.

A well-known type of empirical

determination of degrees of posthe

cases
where
certain physical attributes,
represented by
as properties of symmetry, afford a basis for calling alterna-

sibility is

known

tives equally possible.

geneous coin.

Consider, for example, the case of a homoare inclined to call the

As circumstances stand we

occurrence of 'heads' an event equally as possible as the occurrence

of 'tails'. Suppose, however, that in actual trials, 'heads' occurred
more frequently than 'tails'. Would this under any circumstances
affect the judgement, passed on the basis of the
properties of symmetry, that the alternatives are equally possible?
To this question two answers can be given. First, we may say that
the judgment mentioned is not under any circumstances affected by
what is true of the proportion of 'heads' and 'tails*. Secondly, we
might' say that if it were true that the proportions of 'heads* and
'tails were unequal, even in the long run, then we were mistaken in
calling the alternatives equally possible. This does not imply that
statistical frequencies must be used as the measure of equal and
possibilities, as the observations of frequencies may represent chance-events, but it implies that any comparison between

unequal

degrees of possibility is "checked up* by a comparison between

proportions, so that we cannot imagine the relative frequency of the
event E' to be greater in the long run, than that of E" without also
assuming that E' is more possible than E'\ and conversely.
It is clear that if the first answer be accepted then again we could

not exclude that which was necessary if the theorem of Bernoulli was
to. serve as a bridge to frequencies, viz. that the small possibility will,
147

THE LOGICAL PROBLEM OF INDUCTION

even in the long run, occur frequently, and the great
possibility
seldom or never. Therefore we must resort to the second answer,
which moreover seems to accord much better than the first answer
with the way in which in practice we would judge the situation, and
which actually has been suggested by several supporters of the
theory
that probability is to be defined in terms of
quantified possibilities
or 'Spielraume'. 7 But the acceptance of this answer has a remarkable
consequence which the supporters of the theory mentioned have as a
rule not observed. 8
If a statement that two
possibilities are equal is checked up by a
statement about proportions, then the 'grounds' on which the former
statement was made
properties of symmetry or whatever these
be
cannot
be defining criteria of equal possibilities.
grounds may
Rather than criteria they are only symptoms of this equality and
inequality respectively. For a proposition about proportions, which
is a
general synthetical statement, plainly cannot be a logical consequence from those (singular) propositions which lay down the
observable content of the circumstances under which an event takes
9
Since, on the other hand, a statement about proportions is
place.
'checking up' the assertion about possibilities based on the latter
propositions, it follows that the truth of these latter propositions can
never imply the truth of the assertion about possibilities. It
might
always be the case that the assertion is false although the propositions
in question are true.
have now arrived at the following general conclusion: In

We

whatever way degrees of possibility may be measured 10 it is not

possible with this measurement to exclude that a great possibility
will, even in the long run, be realized extremely seldom and a small
again very often, unless the grounds for measuring
involve the assumption that a proposition will, on
repetition, be true in the long run in a proportion of cases proportional to its degree of possibility.
possibility

possibilities

With this we have shown that the theorem of Bernoulli provides

a bridge from the realm of probabilities into the realm of
empirical
frequencies solely under the condition that the probability of the
second order, involved in this theorem, is given an
interpretation
which implies that a proposition will, on repetition, be true in the
long

run, in a multitude of cases proportional to

148

its

probability.

But

this

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

implication is nothing but that the Canceliing-out of Chance will

take place. Consequently, that which makes it appear as though the

theorem of Bernoulli were of relevance for the Canceliing-out of

Chance in the case of the probabilities of the first order involved in

it,

nothing but the tacit or unconscious assumption that the Canceliingout of Chance is already established as regards the probabilities of
the second order. 11
The idea of the theorem of Bernoulli being a proof of the Cancelling-out of Chance is thus circular. Bernoulli's theorem can be used

is

for proving inductive predictions concerning future frequencies only

on assumptions which are themselves inductive. This important
truth, incidentally, was pointed out already by Leibniz in his polemics
against the uncritical use of the theorem which James Bernoulli

himself suggested. 12

The idea of probable success 9

l

4.

The above examination of

Bernoulli's theorem

was pursued

mainly in order to point out an interesting fallacy of thought which

will be relevant also to the following discussion concerning probability and the justification of induction. It is, however, clear for

and thus independently of the

general epistemological reasons

analysis in the preceding section

that neither Bernoulli's theorem,

other deductive chain of thought, could ever provide a

nor any
proof of the Canceliing-out of Chance relevant to the question of
justifying induction. This is seen from the following considerations:
The statement that the Canceliing-out of Chance will take place
for a given probability is a general proposition. It is such because it
concerns truth-frequencies 1 'in the long run', i.e. in an infinity of
cases.

The only way of guaranteeing

tion

to

is

make

it

the truth of a general proposi-

analytical.

That the statement on the Canceliing-out of Chance for a given

probability, say p, is analytical means the following: If it is true
probable to degree^, then it is logically necessary
that this proposition will be true in a proportion^ of all occasions. Or
in other words: that the Canceliing-out of Chance is analytical
means that the frequency-interpretation is accepted as a model of
that a proposition

is

formal probability.
L

149

THE LOGICAL PROBLEM OF INDUCTION

On the other hand it was seen above

that if the frequency-interpretation be accepted as a model of formal probability, then any statement that a proposition is probable to such and such a degree
i.e. an inductive
becomes a general synthetical assertion
proposifor which reason it
tion of the form called Statistical Generalizations
would then be circular to justify induction by reference to probability.
We can therefore conclude that the guarantee of the statement on

the Cancelling-out of Chance, necessary for the justification of

induction with probability, applies to this statement as a general
4
Since it is
synthetical proposition concerning truth-frequencies,

impossible to guarantee a priori the truth of a general synthetical

proposition, it follows further that the 'guarantee' in question
cannot be a proof that the statement on the Cancelling-out of Chance
will

be certainly

true.

On this point the following idea suggests itself: Perhaps the Cancelling-out of

synthetical statement could be secured or

not with certainty, at least with some degree of

Chance as a
if

guaranteed,

probability in

its

favour.

This idea, according to which the justification of induction with

success' in predictions,
probability consists in a proof of 'probable
5
does not seem unplausible at first sight. If we are asked whether
the statement that the proposition a is more probable than the
proposition b can justify induction, we should be likely to give
roughly the following answer: Of course we cannot be certain that
in predicting the proposition a we shall be 'on the whole' or 'in
the long run' more successful than in predicting the proposition &,
but it is very likely, if the proposition a is asserted to be more probable
than the proposition b, that the former proposition will be true in a
greater proportion of occasions than the latter.
This answer is very suggestive. It gains its suggestiveness partly
from the undetermined way in which the word 'likely' or 'probable' is used in speaking about probabilities. This makes it appear as
though we had to do with two kinds of probability, the one, used in
saying that a is more probable than Z>, being 'mathematical' probain
bility, i.e. a quantity treated in formal calculations, the other, used
the
quantitative
judging the relevance of those calculations with
concept to matters of fact, being 'philosophical' or 'inductive'
6

probability.
150

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

This distinction between two kinds of

probability seems the more
plausible, as it would make it possible to interpret 'mathematical'
probability in terms of limiting-frequencies, i.e. to make statements
such as "a is more probable than b 9 general synthetical
propositions
concerning proportions, without nevertheless depriving, with this
interpretation, propositions on probabilities of their power of
7
justifying induction. The justification in question would be a judgment in terms of 'philosophical probability about the inductive
propositions of 'mathematical' probability.
It is, however, to be observed that,
irrespective of whether we want
5

to distinguish between different kinds of

probability or to keep to the
same interpretation of the concept throughout, the idea that success
in predictions could be guaranteed with
probability is of no value for
the justification of induction unless we can compare the magnitudes

of the probabilities which are to guarantee

this success, (at least) in

the following way:

If,

in preferring

on a given occasion the prediction of the more

probable proposition a to the

less

probable proposition

b,

we wish

to justify this preference by stating that it will

probably lead to
success, then this statement must imply that it is more probable that
we shall arrive at the truth in preferring a to b than in making the

opposite preference. For if the statement on 'probable success' did

not imply this, then any assertion according to which it were 'probable' that a certain proposition will be true on a greater number of
c
occasions than another proposition would simply mean that we do
not know, but perhaps' the truth-frequency of the former proposition will be greater than the truth-frequency of the latter, and this is
exactly what we can say in any case of a general synthetical proposition concerning proportions. It would, under such circumstances,
be possible to assert equally as well that I shall probably succeed
in preferring the more probable to the less probable as that I shall
probably succeed in preferring the less probable to the more probable, without saying anything as to which of these two alternatives
is
preferable. But in this case I have not justified the choice which I
actually make between them.
If, therefore, philosphical probability is to justify induction, that is
to say if it is to be a guide as to which opinion we ought to follow in
the search for truth, then this probability must be capable of quanti151

THE LOGICAL PROBLEM OF INDUCTION

tative evaluation in the sense determined

above/ This truth, which

obvious even upon the slightest reflexion and is
wholly independent
of how we wish to interpret probability, will soon be seen to
possess
the most remarkable consequences.
It has been shown above that if induction is to be
justified with
probability, i.e. if it is to be 'rational' or 'better' or 'safer' to
is

the

more probable

prefer

must be possible to
that
the
more
'in
the
run'
is realized on a
guarantee
probable
long
number
of
occasions
than
the
less
more
greater
probable, or
the
that
of
Chance
will
take
for
the
exactly
Cancelling-out
place
under
consideration.
On
the
hand
other
we
have
seen
probabilities
that if the Cancelling-out of Chance is to be
guaranteed with probathis
must
that
one
of
two ways of possible
bility,
guarantee
imply
success is more probable than another. From this it follows that we
must be able to guarantee the Cancelling-out of Chance also for
to the less probable, then

it

the 'philosophical' probability guaranteeing the

Cancelling-out of
Chance for any 'mathematical' probabilities. From this

important
conclusion the reasoning easily proceeds as follows:
The Cancelling-out of Chance cannot be guaranteed with certainty if we wish to justify induction with probability. The remaining
alternative is that it could be guaranteed with
probability. Thus, we
might introduce the idea of 'philosophical' probability for a second
time. If this new probability is to
express anything more than
uncertainty in general, it must be capable of 'quantification* in the
sense described above which tells us that the one of two alternatives is
more probable than the other. This again leads to a justification of
induction only if we can guarantee the
Cancelling-out of Chance for
these last two probabilities. In this way we are involved in an infinite retrogression. 10

The crucial point of the whole discussion concerning probability

and the justification of induction is to see that any statement,
according to which something is 'probable', is relevant to what is
going to happen, only if it implies that this 'something* is going to
happen in a proportion of cases proportionate to its probability. 11
For this reason the idea that success in predictions could be guaranteed with 'probability' is dependent, in its
of
induction,

power
justifying
on the possibility of guaranteeing that of those predictions

a determinate proportion are true (the

Cancelling-out of Chance)
152

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

and consequently any attempt

to the former

to guarantee the latter with reference

circular.

is

thus

impossible to substitute, for the demand of a guarantee

with 'certainty' that the Cancelling-out of Chance will take
place,
2
the 'weaker' demand that this is to be guaranteed with
'probability
The apparent possibility of making this substitute arises from an
unconscious use of the word 'probability* so as to imply truthIt is

is to
say from the same fallacy of thought which
of
the
classical
idea that Bernoulli's theorem amounted
origin
to a proof for uniformity and order in the course of nature. 13 The
only way to justify this use of probability, however, is to interpret
the concept in terms of truth-frequencies, and as soon as this is done
it is
for
easily seen that any argument which invokes

frequencies, that

was the

probability

guaranteeing success in predictions has become circular.

It

deserves mention that David

Hume who was the first to

see that

general synthetical propositions cannot be proved true a priori, also

clearly apprehended that this result of the impossibility of foretelling
the future cannot be 'evaded* or 'minimized' by reference to
proba14

bility.

He was

introduction of

aware of the

infinite retrogression 15 to which the

probabilities in this connection leads and also of the

necessity of interpreting probability as a statistical concept if it is to

be of relevance to statements on future events. 16 This clarity, in our

opinion, gives the highest possible credit to the philosophical genius

of Hume and strikingly contrasts him with those numberless critics
of his ideas who have in the realm of probabilities found an escape

from the

'scepticism*

which he taught.

Logical and psychological, absolute

induction with probability.

5.

and

relative justification

of

The results of the analysis in the preceding sections can be said to

have taken us back to the same point from which
in Chapter V
we started our investigations concerning inductive probability. No
'mechanism of probability*, whatever be its formal structure and
whatever its interpretation, is in itself a better guide to the truth than
the mere fact, as such, that
reliable,

We

we

regard certain propositions as

more probable, than other

more

propositions.
previously refused to accept this fact alone, as & justification of
153

THE LOGICAL PROBLEM OF INDUCTION

induction. It

is

now

appropriate to consider this point afresh.

The

following will then occur to us:

In trying to justify inductions as general synthetical propositions

guarantee that certain things are going to happen in the
future. Such a guarantee can only be given if it is relative to some

we need a

other assumptions as to the future, 1 and this leads to an infinite

retrogression. If, in spite of this, we break the chain of superimposed

assumptions as to the future and declare ourselves content with the

'guarantee' given in the last of them, the sole justification of this
behaviour, to which we can refer, is the fact that we regarded this
last assumption as being so highly 'probable' as not to need a
'guarantee'

itself.

not inappropriate to call the mere fact that

deemed more, others again less 'probable',
i.e. reliable, a justification of induction. It is not a justification in the
sense of its being & proof or: guarantee about what is going to happen

Thus

it is,

after all,

certain propositions are

in the future, it is simply an expression for the attitude which we

use certain inductions for
take to conjectures on future events.

We

making predictions and taking precautions, and discard others for

the same purpose. Our use of inductive arguments is, in other
words, guided by estimations of reliability which as a matter of fact

we perform.
The different weights attached to inductive conclusions might be
characterized as different degrees of actual belief. It recommends
itself to say that degrees of actual belief provide a psychological
with probability.
justification of induction

The essence of a logical justification of induction with probability

consists in a proof or guarantee that certain inductions are better
guides to the truth than others. If such a proof is to be given without
interpreting probability statistically, i.e. without the introduction
of any assumptions concerning the truth-frequencies in classes of

we might call the justification absolute.

Such a justification, we have seen, is not forthcoming.

propositions, then

This,

however, does not imply that our formal analysis and proofs of
inductive probability, have been altogether useless. We might also
speak of a relative justification of induction with probability, meaning
proofs of the calculus to the effect that a proposition is a guide to the
154

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

truth, reliable in proportion to its probability, provided that certain

other propositions are also such guides to the truth. Such a 'relative
justification' is obtained, if the probability of propositions is inter-

preted in terms of frequencies.

It remains to be considered to what extent

it is
possible to apply
the frequency-interpretation to inductive propositions, especially
to generalizations.

In the Frequency-Model, the probability of a given proposition

means a truth-frequency within a class of propositions. This class is

determined by those values of a variable which satisfy a certain
propositional-function. The truth-frequency is the proportion of
those values of the variable which also satisfy a certain other propositional-function. And the proposition, in the probability of which
we are interested, is one of the propositions which this second
propositional-function yields, when for the variable is substituted a
constant.

The problem of how to define the two propositional-functions

needed for measuring the truth-frequency, presents no particular
difficulties when we are dealing with singular propositions. The
reason for this is that a singular proposition contains no existential or universal operator. We need only substitute variables for
some constant parts of the proposition and we get a propositional
function. Any other propositional function with at least the same
number of variables (of appropriate logical type) then constitutes
a possible 'collectivity' for measuring the probability of the singular
proposition in question.
When we preceed to general propositions, i.e. to propositions
containing existential or universal operators, the situation is different.
It does not seem natural to speak of an 'occasion' on which a general
proposition might 'occur', or of a general proposition being true on
certain 'occasions' and false on others. But, this being the case, is it
then at all possible to speak of the probability of a general proposi-

tion, if probability has to

be interpreted statistically?
in the affirmative. It was
This question has already been answered
shownin Chapter VI, section? that the Principal Theorem of Confirmation can be given a frequency-model, in which the logical mechanism
of the theorem 'mirrors' the logical mechanism of induction by
elimination.

The

desired interpretation in frequency terms

155

was

THE LOGICAL PROBLEM OF INDUCTION

obtained by regarding Universal Generalizations as instances of
9
prepositional functions of the type 'X is a sufficient condition of A
a
or 'X is a necessary condition of A\ where A is a constant and
variable property. The probability of the Universal Generalization
becomes, on this interpretation, a proportion of actual sufficient (or
necessary) conditions of a given characteristic within a class of
of it.
possible sufficient (or necessary) conditions

If the class in question

is infinite,

the proportion in question will

have to be a limiting-frequency. The notion of a limiting-frequency

the class. The
again presupposes an ordering of the members of
of the
necessity of this presupposition challenges the adequacy
as an analysis of the 'meaning' of probafrequency-interpretation
not impair the logico-mathematical correctness
bility. But it does
of the model. 2
It is an old idea that the 'probability' of a generalization depends
on whether it belongs to a class of good or less good inductions. 3
We shall show that this idea can be worked out to a formally correct
4
definition in frequency-terms of the probability of natural laws.
Consider the two propositions 'all ravens are black' and 'phosmelts at 44 C'. They are both general implications of the

phorus
form
(1)

If

(jc)

we made

tions were

(Ax-Bx).

a statement according to which one of these generalizathe other, it is likely that we would

more probable than

be expressing something such as the following: The generalizations

the proportion
represent two classes (kinds, types) of natural laws,
of true generalizations among all generalizations of the one class
being greater than the corresponding proportion in the other class.
It would presumably not be immediately clear how those classes of
but it
generalizations to which we refer are to be characterized,
of
class
seems plausible to suggest, for example, that they are the
constant
a
hypotheses attributing respectively to a species of bird
combination of colour, and to a chemical substance one and only
is a less
probable generali44
if
C' there are more species of
zation than 'phosphorus melts at
birds, the individuals of which vary in colour, than there are chemical
substances with more than one melting-point.

one melting-point.

'All ravens are black'

156

PROBABILITY

AND THE

JUSTIFICATION OF INDUCTION

Let Si denote the class of all species of birds, and S z the class of
species of combinations of colours in a bird. The statement that
all individuals of the
have the same colouring Y
species of birds
is then a
the
of
form
proposition
all

(2)

we

If

& (X) & S (Y) & (Ex) (Xx & KC)-KJC) (Xx-+Yx).
Z

take

X and 7 to be real variables (2) becomes

a prepositional
of
constant
values
and
7
when subwhich
Xi
t
Any pair
stituted for the real variables in the prepositional function Si (X)
S 2 (Y) (Ex) (Xx Yx), make this a true proposition can be said
to constitute an 'occasion' for any hypothesis of the form (x) (Xx->
Yx). The proportion of times on which the prepositional function
function.

&

&

&

(x) (Xx-+Yx) turns into a true proposition on all such 'occasions*

constitutes the truth-frequency in the class of hypotheses as to

constancy of colour in species of birds.

In an analogous way we may, in the form of a prepositional
function, define a class of 'occasions' for a statistical interpretation
of the probability of the hypothesis that phosphorus melts at 44 C.
Against the idea of defining a 'collectivity' for the purpose of
measuring the probability of a general proposition the following
objection has been made:
proposition of the form (jc) (Xx-*Yx) is unverifiable. Consequently it is not possible to determine whether such a proposition
is true on a
given 'occasion' of it, nor to count the truth-frequencies
even among a finite number of hypotheses. Or as Popper expresses
it 'Dieser Yersuch scheitert
daran, dass wir von einer Wahrheitshaufigkeit innerhalb einer Hypothesenfolge schon deshalb nicht

sprechen konnen, weil wir ja Hypothesen zugestandenermassen nicht

"wahr" kennzeichnen konnen. Denn konnten wir das wozu
brauchen wir dann noch den Begriff der Hypothesenwahrschein*

als

lichkeit?' 5

This remark, however,

is

beside the point.

It is

a fact that Universal

Generalizations cannot be verified, but there is no logical objection

to the assumption that, in a class of such generalizations, a certain

proportion

is true.

Leaving for a

moment

all

considerations as to

'probability', it is surely in no way absurd to assume a certain thing

to be true of, say, the proportion p of all species of birds, although
such an assumption, strictly speaking, cannot be 'verified* even for
157

THE LOGICAL PROBLEM OF INDUCTION

a single species. It is a reason for regarding this assumption as
p of all species are known to
plausible if roughly the proportion 1
which knowledge can be obtained
lack the property in question
and if continued
since Universal Generalizations can be falsified
observation and long experience had confirmed the hypothesis as to
the presence of this property in individuals of the remaining species.
Propositions concerning the probability of Universal Generalizations,

meaning truth-frequencies in

classes of hypotheses, therefore

cannot be verified, but nevertheless can be, so to speak 'recommended' or 'discountenanced' by the records of experience. The
logical difference, consequently, between singular propositions and
Universal Generalizations, consisting in the fact that the former
may be verifiable and the latter not, is irrelevant to the question of
whether or not it be possible to interpret the probability of general
propositions in terms of frequencies.
There remains the further problem whether it be possible to deter-

mine the propositional functions

('collectivities') for

measuring the

a way that the loose estimaprobability of generalizations in such

tions of reliability, with which we usually content ourselves, can be
made into exact numerical evaluations. We shall not embark upon a
discussion of this problem here. For several reasons we are inclined
6
to take a sceptical attitude to the possibility in question.

158

CHAPTER

INDUCTION AS A SELF-CORRECTING OPERATION

1.

Induction the best

mode of reasoning about

the unknown.

The

ideas ofPeirce.

WE

say that

we employ

the inductive

mode

of reasoning or an

'inductive policy' when we make inferences to the unknown on the

principle that future experience will be in conformity with the past.

For instance: We conclude from the fact that all observed A 's have
been B that the unknown A's will also be B, or we infer from the
of the observed A's have been B that the
We have seen that we cannot
guarantee, prior to testing, the truth of any such inference as a

fact that a proportion

same proportion of

all

A's are B.

synthetical proposition, nor can we show one generalization from a

of data to be more probable than another in the sense that it were

set

a better guide to the

inferences to the

truth.

unknown

We might, in other words, have made the

any other than the 'inductive* way and

But in spite of these 'negative' results
that the inductive method as such is, in a

in

nevertheless been successful.

seems possible to assert

certain sense., the best way

it

of making conjectures about the unknown.

method seems to have a certain advantage possessed by no
other method of conjecture. 1
The methods of reasoning about the unknown could be compared
to the methods by means of which we find our way out from a

For

this

complicated labyrinth. This last can be accomplished in many

different ways. We might just run ahead and guess the right course
at each turn. Or we might determine the course to be chosen according to a fixed rule. One of such rules holds a peculiar position. It is
the determination consistently to keep to the same hand, either to the
right or to the left. Of this rule it can be proved that it must finally
lead us out of the maze. It is very likely that the employment of
some other rule, or even mere guessing, will lead us more quickly
out of the labyrinth, but the employment of any such method may
also never attain this end. The determination to keep to the same
159

THE LOGICAL PROBLEM OF INDUCTION

hand

is

the only

consistently,

it

method of which we can be sure that,

if

persisted in

will lead to the goal.

This power of the method mentioned of finding one's

a maze is due to its being, by its very definition, a

way through

self-correcting

method. That

is to
say, it follows from the way in which the method
defined that, even if it momentarily takes us
away from the exit
of the labyrinth, this deviation from the goal will be
automatically
corrected until the exit is reached and we leave the maze.

is

Something corresponding to this seems to be true also of induction.

classes A and B intersect and we want to
generalize as to the
of
the
one
class
falls
which
within
the
other. This can
proportion
be done either by guessing or 'methodically', i.e.
according to some
rule. One such rule is
the
inductive
method. It is not
provided by

Two

impossible that we might arrive at the true generalization more quickly

by employing some method other than the last-mentioned one, or even
by guessing. But, on the other hand, none of these methods may lead
to the truth. The inductive method on the other hand

may momentarily

give results deviating very much from the true value of the proportion, but it follows from the way in which this method is used that
any such deviation is in time corrected by experience's own indica-

This process of correction continues until the true

proportion
Thus induction, like the method mentioned of finding
one's way out from the labyrinth, is by its very nature a selfcorrecting operation, and as such is the only method of making
inference about the unknown of which it can be
proved that it must,
when consistently employed, finally lead to a true generalization.
This property of the inductive method might plausibly be
regarded
as & justification of our use of induction.
This argument about induction as 'the best mode of reasoning
about the unknown', which is related to the idea that our
tion.
is

reached.

experiences
samples' from a larger totality, was expressed for the first
time by Peirce. 2 Peirce speaks of 'the constant tendency of the
inductive process to correct itself as the 'essence and the 'marvel'
of induction. 3 He also says that 'the validity of an inductive
argument consists, then, in the fact that it pursues a method which, if
duly persisted in, must, in the very nature of things, lead to a result
4
indefinitely approximating to the truth in the long run'.
The validity of the argument, naturally, depends upon the
are

'fair

assump-

160

INDUCTION AS A SELF-CORRECTING OPERATION

tion that in the respective cases the
proportion as to which we
For if, of the two classes A and B, no definite
first is included in the second, then
obviously any
method of generalizing as to the proportion would be
equally vain.
This condition, however, was not stated
by Peirce. It is likely that
he assumed it to be tautologous, and hence
to formulate
generalize exists.
proportion of the

unnecessary

explicitly. But in fact the condition is not tautoiogous.

dwell on this point for a moment, as there is

We

shall

psychologically a most

when two

classes

proportion of the

first is

interesting origin for the supposition that always

intersect it is a tautology that a definite

included in the second.

We denote each member of the class A that is also a member of
the class B by 1, and each member of A that is not a member of B
by
0. The observed A's can
always be pictured in a series of a definite
number of alternating 1 's and O's, such as this

110100011111011.

(1)

In any such series there is a definite

proportion of A 's which are B,
This follows per definitionem from the way in which such a series
constructed.

is

something

And now we

fancy the picture of all A's to be

like this

(2)

110100011111011...,

where the dots indicate that the series of 1's and O's perhaps goes
on indefinitely,
This picture, however, is most fallacious. It causes us to think ofthe
class A as in some way
'resembling' finite collections of the class,
picturable in rows such as (1), chiefly on account of its being a
'very long' row of this kind. (It is the same type of fallacy wMch
occurs over and over again in the philosophy of mathematical conceptions. We know it, for
example, from the controversies about the
infinitesimals or the transfinite, controversies of which the
deepest
root lies in the inclination to think of infinitesimal quantities as something 'very small' and of transfinite magnitudes as something Very
great'.)

Actually the only way to state the fact that a definite proportion
A is included in the class B is to assert the proposition

of the class

161

THE LOGICAL PROBLEM OF INDUCTION

N
(3)

(J5//0

0) (x)

(Ex.)

Axt&B

n>m & =
:

some proposition equivalent

or

to

But the statement

it.

(3) is

no

5
tautology, for, as was seen above it is conceivable that it were false.
And hence the assumption that when two classes intersect there must

be a definite proportion of the one included in the other is itself an

inductive assumption. 6
This fact can hardly be said to intrude upon the validity of the
argument about induction being the best mode of reasoning about
the unknown, as this argument makes sense only under the assumption that the proportion as to' which we generalize exists. It imposes,
however, a serious restriction upon the applicability of the argument
to single cases. For as the statement that a definite proportion of
one class is included in another is itself inductive, it follows that we
can never guarantee that in this case such a proportion exists and
hence that induction here will be the best mode of reasoning.
On the other hand knowledge that the existence of a proportion
is a non-tautologous fact, expressed in a statement of the form (3),

an unexpected possibility to strengthen Peirce's argument.

This has been pointed out by Reichenbach, whose system of an
'inductive logic' is very closely related to the idea about induction
gives us

as a self-correcting operation. 7
It can be proved that if the statement (3) is true, i.e. if a definite
proportion/? of the A's are 5, then also the statement
i

N
n>m-

(4)

i
i

=
=

N
is

true for this

from the m'th

i.e.

then there exists a

e.

such that

which are

B remains

number

finite

A onwards the proportion of A

in the interval/?

& B xt

(The series of relative frequencies corresponding

162

INDUCTION AS A SELF-CORRECTING OPERATION

to the observations

that

in other words,
convergent.)

is,

But

means

this

two

classes intersect so that a definite

proportion of the first is
included in the second and we want to
generalize as to this proporif

tion, then the inductive

number of steps,

finite

method can be proved to attain this goal

after a finite number of corrections. 9

in

i.e.

It is clear that for

'empirical' classes it is not possible to calculate
the exact value of the ordinal
from which point onwards (for a
given s) the observed proportions will remain within the fixed interval.
Thus it is never possible to tell, in a given situation where we are
using induction, whether the corrections can still be expected to
continue, or whether the true generalization has already been reached.
But perhaps in spite of this it would be
possible to devise some
could
'technique' by means of which the approach to this important

be accelerated. Such a
'technique' has actually been suggested by
Reichenbach. We shall therefore examine his ideas as
regards this

somewhat more

topic

2.

closely.

Reichenbach's Method of Correction.

In this section we shall

attempt to give a simplified description of
Reichenbach's idea which should make it easier to assess its
episte1
mological significance.
Let the problem be to find 'by induction what
proportion of
members of a certain, potentially infinite, sequence 5
possess a
certain characteristic A.
We have observed an initial segment of n members of S. The
recorded proportion of A among them is
p. We generalize, following
1

2
the inductive principle that 'the future will resemble the
past', that
p is the limiting frequency of A in S. This generalization we call a
posit ( 'Setzung*} of the fast order*

For

We

we can find an appraisal ('Beurteilung') which

a correction of the posit. The procedure is as follows:

this posit

result in

consider s different, potentially

infinite,

sequences

may
.

Ss

assume that our sequence S above is one of them.) We

observe initial segments of n members of each
sequence and record

(We

shall

the proportion of
are /?i
p9
.

in these segments.

The recorded proportions

Following the inductive principles we assume that p*

163

.'/vare

THE LOGICAL PROBLEM OF INDUCTION

the limiting-frequencies of
posits of the first order.

Some

We

of the s p-values

assume that there are

in the s sequences. Thus,

we

get in all s

may be identical (or 'nearly identical').

/-different /7-values. (/"<.)
call them

We

We

record the relative frequency of /7-values which are lf (If

q
such ^-values, the recorded relative-frequency is ra:s.)
Similarly, we record the relative frequency of ^-values which are q z ,
etc.
In this way we get in all r relative-frequencies /(^)
f(q r).
Some, or even all, of them may be identical. Their sum, of course,

there are

isl.

Following the inductive principle we assume that f(q^

f(qr)
are the limiting-values, which those r recorded
will
frequencies
approach, when the segments of n members of the s sequences are
indefinitely increased. Thus we get r inductive generalizations.
.

We

call

them posits of the second

order.

We raise the following arithmetical problem:

If the limiting-frequency of A in a sequence is qf , what is the
limiting-frequency, among all sets of n members of this sequence, of
such sets in which the proportion of A is qk 1

In order to be able to answer this question, certain assumptions

about the 'inner structure' of the sequences in question will have
to be made. 5 For purposes of our
simplified description, however, we

We imagine the question to have been

answered for all possible pairs q and qk (Since there are r ^-values,
there are r 2 such pairs.) The calculated
limiting-frequency, for given
ignore these assumptions.

qt

and qk we symbolize by/(^, qk).

,

We

can now use the values /(#/) and /(#i, qk) for calculating an
answer to the following question, which may be termed the 'inverse'
of the question just answered:

What

is

the limiting-frequency,

any of the

among

all sets

of n members from

sequences, of such sets as satisfy the following

two

conditions:

A in the set is qk , and

the limiting-frequency of A in the
sequence, from which the set

(0 the proportion of
(ff)

is

selection,

is

qi ?*

The

calculated limiting-frequency
(/(?0; ffabQk) ) or shorter F&.
164

we

symbolize by

INDUCTION AS A SELF-CORRECTING OPERATION

Now, for a fixed i, the value of F/,* will, in general, be different

for different values of k. Its maximum value we call Fimax.
Since z can take r different values in all, we have
altogether r such
maximal values F\rnax
Frmax.
.

The value of

Ft,t

we

call the appraisal

(inductive generalization) that qt

of the first order posit

the limiting-frequency of
in
segment of n members has been

is

a sequence, from which an initial

found to contain A in the proportion qt.
The appraisals are used for the purpose of correcting the posits
according to the following rule:
Iff},/ equals Fimax, no correction is needed. If F& is less than
the

maximum, then we

correct the posit by assuming that the

of
A
in
the sequence is, not qf , but a value q?
limiting-frequency
such that Fi r equals Fimax. 7
This correcting procedure, which even under the above
simplified
description of its mathematical mechanism may appear involved and
cumbrous, has in fact a very simple and clear-cut meaning. It
t

amounts

to this:

We

should posit that value as the limiting-frequency of A in the

sequence S, which is most frequently the limiting-frequency of A in a
sequence, of which a set with the recorded relative frequency of A
is a selection. 8
It must now be observed that the
posits of the second order upon
which the correction essentially depends, are themselves generalizations about proportions, and may as such be in need of correction.
To this end we may derive an appraisal of the second order for
each posit of the second order and then use the appraisal for possible
corrections, all in a way analogous to the one described above for
first order
posits. The corrected values of the posits of the second
order could then be used for correcting our previous calculations of
appraisals for the posits of the first order, and may thus ultimately
lead to further corrections in the first order posits.

the

The correction of the posits of the second order would in their

turn depend on certain posits of the third order, which again are
capable of appraisal and correction. Thus we get an infinite hierarchy
of superimposed posits and appraisals,

We

are now in a position to tell in what sense the Method of

Correction means an accelerated approach to the points of conver-

165

THE LOGICAL PROBLEM OF INDUCTION

gence in sequences of relative frequencies, and to assess the
significance of this to the problem of induction.
On the assumption that the recorded relative frequencies possess
limiting values in the sequences under consideration, we can from
what was said above (in section 1 of the present chapter) on the
nature of Statistical Generalizations, draw the following conclusion:
For any given value of e, however small, there exists some value
of H, call it /2i, from which onwards all the posits of the first order are
And similarly, there exists some value of
true within the limits
all the posits of the second order are
onwards
which
from
it
n
call
z
H,
.

true within the limits

e.

order is true means that the limitingin the sequence under consideration
characteristics
the
of
frequency
is as indicated by the relative frequency of this characteristic in the
observed initial segment of the sequence. That a posit of the second
order is true means that the proportion, among all sequences under
consideration, of sequences with a certain limiting-frequency of the
characteristic, is as indicated by the observed proportion of initial
of the
segments of sequences showing this relative frequency

That a posit of the

first

characteristic.
it is clear that if all the posits of
the posits of the second order are
true also. But it is equally clear that this proposition cannot be
converted. For all the posits of the second order may be true, and
first order false. It follows from this
yet some of the posits of the
that 72 2 </2i. And this means that all posits of the second order will

From considering these meanings

the

first

become
with,

all

order are true, then

all

true not later than, i.e. either sooner than or simultaneously

posits of the first order.

Generalizing,
necessarily

we can prove

become

that

all

posits of

a higher order

will

true not later than all posits of inferior orders.

the sense in which the building up of the hierarchy of posits

and appraisals and its use for corrections may be said to 'accelerate'

This

is

or 'speed up' the inductive approach to truth.

It is important to observe that the 'acceleration' in question does
not amount to a proof that we shall, with our posits of the first
order, sooner reach the point of convergence if we resort to the
Method of Correction, than if we generalize on the basis of the
alone. 9 And it is also evident from the way in
inductive
principle

166

INDUCTION AS A SELF-CORRECTING OPERATION

which the hierarchy of posits and appraisals is constructed that we
can never determine, after any number of corrections, whether the
true value of the proportion as to which we are
generalizing has been
reached or not, nor can we ever know how many corrections, if any,
still remain to be made. And we cannot even exclude the
possibility
that the corrections will, for any length of time, take us further away
from the true value of the proportion instead of letting us approach
it.

10

For these reasons it seems to us that Reichenbach's Method of

Correction cannot be said to add anything epistemologically significant to the idea that induction is an indefinite and self-correcting
approximation to the

truth.

The goodness of inductive policies

3.

reconsidered.

We

return to the problem, whether induction can be justified as

being, in some sense, the best policy for making conjectures about the
unknown. The superiority of induction, we have seen, was thought
to be in its self-correcting nature and in its alleged indefinite approximation to the truth. In view of what has been said above of these
attributes of induction, it is at least doubtful whether they can really
be said to constitute a 'superiority' of inductive over alternative
1
We shaU now consider a somewhat different way, related
policies.

we

believe to the Peircean approach, of establishing the superiority

of induction.
It is useful

here to

make a rough

distinction

between prediction

A prediction, we shall say, is about a single case

('event') or about a finite number of cases. A prediction should be,
in principle, verifiable and falsifiable. A generalization is about an
and

generalization.

unlimited number of cases. 2

Accordingly, we shall say that a method or policy for reasoning

about the unknown can be either & prediction policy or a generalization

policy.

A prediction policy will be called inductive, if

to a

maxim

() If

all

it

proceeds according
of one of the following schematic types:
observed A are 5, then predict that the n next A are B.
167

THE LOGICAL PROBLEM OF INDUCTION

observed A are j3, then predict that a
proportion (as near)/? (as possible) of the n next A are B. (>1.)
Similarly, a generalization policy will be called inductive, if it
proceeds according to a rule of one of these types:
(i) If all observed A are B, then generalize that all A are B.
(ii) If a proportion/? of all members of a sequence S are 5, then
generalize that the limiting-frequency of B in S is p.
This characterization of inductive policies is only a rough first
approximation. It would probably be sensible to count as 'inductive'
also policies which employ rules somewhat 'laxer' than those above,
but resembling them in essential features. We need not discuss this
(ii)

If a proportion

question here.
(In view of the

p of

all

among others, that generalization policies of

the
ordering of the members of a class into a
type (if) presuppose
it seems to us doubtful whether they play any great role
sequence,
fact,

might be suggested that there is a more important

(and embracing) type of policy, viz.)
of all observed A are B, then generalize that
(ii)' If a proportion/?

in science. 3

It

the probability that any given A is B is p.

shall not, however, here discuss inductive policies of the

We

probability-type.)
shall say that a policy

We

is

truth-producing (possesses a 'truth-

producing virtue') if its predictions or generalizations always, or at

shall also say that
least in a great majority of cases, are true.

We

verified predictions confirm a policy.

shall now briefly consider policies

which are, or may be

We
claimed to be, not-inductive. Not-inductive policies are probably
worth a closer scrutiny than is given to them here and elsewhere in
the literature

on

induction.

and 1
Let n be an integer and /? a value in the interval between
inclusive. f(p,ri) is to be a function of/? and n which satisfies the
following three requirements: (a) for any given /? and n we can calculate a unique value of /(/?,); (b) the value of /(/?,) is in the interval
between and 1 inclusive; (c) the value of/(/?,rc) is different from/?.
Consider a prediction policy of the following type:
If a proportion/? of the n last A which have been observed are J5,
then predict that a proportion (as near) /(/?,) (as possible) of the
n next A are B.
168

INDUCTION AS A SELF-CORRECTING OPERATION

policy of this type resembles induction in that it is guided by

experience. What we anticipate according to this policy is rigorously
determined by what we have recorded. The policy is thus
selfcorrecting. (Being, in this sense, self-correcting, is therefore no
4
privilege of the inductive method.)

The policy differs from induction in that it proceeds, not on

the principle that 'the future will resemble the past', but on the
principle that the future will, in a characteristic way, be different
from the past. We shall call a policy which proceeds on this prin5
more rigorous definition of such policies
ciple, counter-inductive.

not be attempted here. The counter-inductive policies are a

sub-class of not-inductive policies.
As a prediction policy the above not-inductive method may be
superior to an inductive prediction policy in a very palpable sense.
Consider the following situation:

will

The property B becomes

rarer

and rarer among instances of A.

If

predict
frequency in a set of n new instances of A following
an inductive policy we shall, on the whole, predict a too high fre-

we

its

quency. But iff(p,ri) is a function of p and n which corresponds to

the 'rate of diminution' in the frequency, then we shall with the
aid of the counter-inductive policy, on the whole, predict the right
frequencies.

Consider next a generalization policy according to which the

limiting-frequency of a characteristic in a sequence is consistently
assumed to differ in a determinate way from the recorded relative
frequency.

be

Use of such a policy would,

in the following sense, always

futile:

Either the relative frequency of B in S has a limiting-value, or it

has not. In the first case we shall, foEowing the inductive policy,
in a finite number of steps reach the point of convergence which
answers to an arbitrary value of e. This entails that by following
the counter-inductive policy we shall in a finite number of steps
reach a point from which onwards we make only false generalizations. In the second case again, no generalization policy about
limiting-frequencies will approximate to the truth.
As we have said repeatedly before, 6 it is not necessary that the
relative frequency of a characteristic in a sequence should have any
limiting value at all. Beside approximating to a limit there is also the
169

THE LOGICAL PROBLEM OF INDUCTION

behaviour of relative frequencies which
between two extremes. 7 Special policies

is

best termed

an

oscillation

may be devised for predicting

and for generalizing about such oscillating behaviour of

frequencies.
Such policies may be either inductive or counter-inductive or notinductive. We shall not stop to examine them here. It is a lacuna
in the literature on induction that, as far as we
know,
oscillating

have never received systematic attention. 8

Of inductive and counter-inductive policies we may,
comparing
them with each other, say that they use the same premisses but draw
different conclusions from them. We now turn our attention to
not-inductive policies which do not use the same
premisses as inducrelative frequencies

arguments (but may reach the same conclusions).

noteworthy that it is difficult to give any uniform characterization of such not-inductive policies or to illustrate them
by an example
which seems worthy of being seriously considered as a
policy or
method at all. (This, by the way, throws light on the
proposition
that induction is the best method for
making conjectures.)
As a crude example of such a not-inductive policy we may take the
consulting of an 'oracle' for purposes of prediction and generalization. We shall not attempt to
explain what should be the other
characteristics of an 'oracle' besides the
negative one that we must
not be able to calculate the oracle's answer from
experiential
premisses according to some known rule. If the 'oracle' did not
possess this characteristic, consulting it would be equivalent to
tive

It is

adopting a counter-inductive policy. Since this negative feature must

be common to all not-inductive policies, which are not counterinductive, we shall call such policies oracle-policies.
It is clear that we cannot exclude the
possibility that an oraclepolicy would be superior to induction in the sense that its predictions
and generalizations were more often true than those made in accordance with inductive (or counter-inductive) policies. 9 And it is
noteworthy that there is no limitation to an oracle-policy's powers
of competing successfully with an inductive policy, when we are
generalizing about proportions, which would correspond to the
limitation in a counter-inductive
policy's capacity.

Having thus made clear in what respect not-inductive policies

may and may not be superior to inductive policies in the search for
truth, we raise the question: Could there ever be a ground or reason
170

INDUCTION AS A SELF-CORRECTING OPERATION

for adopting a not-inductive policy, and what would such a reason

look like?
If asked

why we adopt a certain policy for predicting or generalizthe

'reason'
ing,
given is often some fact about our beliefs and (other)
attitudes in the matter. Why, for
example, did we predict B with a
lesser frequency among the last 100 A than the recorded
frequency
among previous Al Because we believe that the occurrence of B will
become rarer. Why did we consult an 'oracle' about tomorrow's
weather and act according to its prediction? The answer could be
we regard the oracle as the representative of a God, whose
powers of foretelling the future we trust, or whose wrath for not
having taken his advice we fear.
'
If belief is called a 'reason for adopting a policy, it should be borne
in mind that a 'reason' of this sort is quite without relevance to the
question of justifying the choice of policy, i.e. to the question of
that

objectively appraising the policy's truth-producing capacity. The

same is true of any 'reason' which consists in our attitude to our

source of information about the future

such as an attitude of love
or trust or fear of a divine power. We shall therefore, for the sake
of clarity, distinguish between reason and motive and say that a belief
or other attitude may be a motive for adopting a certain policy, but

not a reason for doing so. 10 By a reason for adopting a policy we

shall understand a reason for belief in the truth-producing virtue of
reason in this sense, moreover, should be some known
the policy.
fact about the 'world', i.e. about something which exists independently of the predicting or generalizing subjects. (This last excludes

beliefs

and

attitudes

from being reasons for

beliefs.)

The above is a rough characterization only, but it will have to suffice

for present purposes. Be it remarked, however, that the logic of the
case

is

further complicated by the fact that an attitude (other than

a source of information, is sometimes a motive both for

belief) to

adopting a certain policy and for believing it, and sometimes a

motive only for adopting but not for believing a policy. Thus, for
example, fear of punishment for not having taken the oracle's advice
could be a motive for adopting a policy in which we do not believe.
In such a case it is difficult to see how anything which counts as a
reason for our attitude to the source of information could ever be

a reason for belief in the policy.

171

If,

however, the attitude to the

THE LOGICAL PROBLEM OF INDUCTION

source of information is also a motive for belief, then a reason for
may but need not be at the same time a reason for the

the attitude
belief.

In addition the following point should be noted about our

concepwhich constitutes the reason should

tion of 'reason'. Since the fact

be known, we cannot
on our terminology count as reasons
facts which, if known, would have been reasons for
adopting
a certain policy. Thus, for example, we must not
say of the fact,
that B actually became rarer and rarer
among A's from a certain
moment on, that this was already before it became known a reason
for adopting a counter-inductive
policy for predicting B.
After these preliminaries we raise the
question: Of what kind
would the facts about the world have to be in order to
qualify, if
known, as reasons for adopting a certain 0Mnductive prediction or

unknown

generalization policy?
first consider counter-inductive
policies.
What should we, for example, consider a reason for predicting that
the relative frequency of B among the next 100 A will be 45

We

per cent,
although the relative frequency of B among, say, the first 500 A which
we have observed was 48 per cent? As a reason we might count the
observation, i.e. known fact, that among the first 100 A the
frequency
of B was 50 per cent, among the next 100 it was 49
per cent, among
the next 100 again 1 per cent less, and so on. Or the reason
be

some other,
of

might
about the fluctuation of the
frequency
could be the observation,
say, that the

similar observation

B among

the A.

Or

it

A n has sunk
frequency of B among certain other properties A^
from 50 per cent among the first 100 instances to 45
per cent among
the sixth 100 instances.
This is only a very rough indication of
possible examples. And in
this place it is
important to warn against a misunderstanding. The
above illustration of reasons must not be taken to mean that
any
observation of the kind indicated would
ipso facto constitute a
reason for the prediction
under discussion. Whether the
.

policy
observation will be a reason or not for
adopting the policy depends
upon its relation to all the other facts which are known about the
case. Thus the 'weight' of some of the observations mentioned in
favour of the counter-inductive
policy may be, so to speak, 'counteracted' by some other observation
speaking in favour of a different,
172

INDUCTION AS A SELF-CORRECTING OPERATION

e.g. of an inductive prediction policy. What the examples were
intended to show was only what kind of facts might count as reasons
for adopting a counter-inductive policy.

The

which we indicated agree in an important feature, which

may already have discerned. The reasons for adopting
a counter-inductive policy were thought to be confirmative of some
inductive prediction policy which produces the same
predictions as
facts

the reader

the counter-inductive policy.

first 100 A the proportion of

The observed

facts that

cent,

was 50 per

among the
among the next

100 49 per cent, etc. confirm the inductive policy, always to predict
B among the 100 next A with 1 per cent less frequency than among
the 100 last A. This policy is inductive in as much as that it may be
said to proceed on the principle that the future will resemble the past
with regard to the 'rate of diminution' for the relative frequency of
B among A. The policy, moreover, conforms to the prediction-

scheme
for 'B

(i)

we

preceding

on p. 167 above,

if

for 'A'

substitute "showing
set

of 100

we substitute

per cent

less

of 100 A and
9

'sets

B than the immediately

9
.

Thus from the view which we took of the possible reasons it

follows that a reasoned counter-inductive policy is equivalent to
some inductive prediction policy.
similar argument may be

conducted for counter-inductive generalization policies.

It may be asked: How can we be sure that any known facts which
constitute reasons for a counter-inductive policy must be confirmative
of some inductive policy? The answer is that, in a sense, we cannot
be sure of this at all. The only indubitable certainty about it which
we could reach would arise from a decision not to call any other
facts 'reasons' for a counter-inductive policy. But no such decision

would strengthen the case which we are here pleading. If the argument which we have presented carries any weight, it must be because
it is difficult to see what other known facts (excepting beliefs and
other 'subjective' states of affairs which are admittedly incapable
of justifying a choice of policy) could conceivably be counted as
reasons.

We next turn our

attention to oracle-policies.

constitute a reason for believing an oracle? As

already observed, the reason cannot consist in the authority which
the oracle enjoys among those who consult it, or, which means the

What would

173

THE LOGICAL PROBLEM OF INDUCTION

same:

may

it

be

cannot be the attitude

of love or trust or fear, as the case

which the oracle's consultants take to

it.

For

facts

concerning attitude or authority are not 'objective' in the sense in

which we required that facts providing reasons, as opposed to
motives, should be objective.
reasons and motives in mind,

But, with this distinction between

not the sole reason which we can

is

imagine for believing an oracle, knowledge of the fact that it had

proved to be a 'good guesser' in the past or, more precisely, proved
to be a better 'guide to the truth' than alternative policies?
Granting an affirmative answer to the last question, a reason for
adopting an oracle-policy is thus something which is confirmative of
a certain inductive policy. This inductive policy proceeds on the
the past with regard to the
principle that the future will resemble
oracle''s truth-producing powers.

A certain ambiguity in the use of the term 'induction' or

'inductive

us assume that all A so far

policy' should be noted here. Let
observed have been B, but that an oracle tells us that the next A will
not be B. If, in this situation, we actually believe that the next A will
not be jB, it might be natural to say that we here believe the oracle
rather than induction. Now, what would be a reason for believing the
oracle rather than induction? If we accept the account of the concept
*
of a 'reason which we have given, it could only be our past experience that inductive inferences to the next case from facts that all Z's
so far observed have been 7, have more often broken down than the
oracle's predictions in similar situations. But this, of course, is
oracle rather than
equivalent to saying that a reason for believing the
induction is tantamount to a reason for believing the induction from
past experience of the oracle's predicting-powers rather than the
induction from past experience of certain regularities in nature. In a
sense, therefore, it is misleading to say that we have a reason for
believing an oracle rather than induction. What we have a reason for
doing is to trust one inductive policy rather than another.
The conclusion which emerges from the above considerations is
that a reasoned policy for purposes of prediction and generalization
is necessarily equivalent to an inductive policy. If we wish to call
reasoned policies better than not-reasoned ones, it follows further
that induction is of necessity the best way of foretelling the future,
The suggested use would certainly be a sensible use of 'better'.
174

INDUCTION AS A SELF-CORRECTING OPERATION

But

not the only sensible use of the attribute in connection with

prediction and generalization. A policy might also be called good
in proportion to its
successfulness. And under this use of 'good' it
may well be the case that a reasoned policy turns out to be inferior
to a not-reasoned one. Neither with
certainty nor even with 'probacan
this
be
excluded.
bility*
possibility
With our term 'reasoned' may be compared the words 'reasonit is

able'

and

'rational'.

to say, a double face.

It is

noteworthy that the two

latter have, so

One face looks

to the past, another to the future.

'rational' as attributes of a
policy for prediction

'Reasonable' and
and generalization may mean 'reasoned' in our sense, i.e. grounded
on past experience. But they may also mean the policy which will hold
good in future.
Thus, depending upon which use of 'reasonable' and 'rational'

we

contemplate, we are entitled or not to say that induction is ipso

facto rational. But there is no way of securing the rationality of induction under every sensible use of 'rational'.
Our examination of policies for predicting and generalizing has
thus led us to the conclusion that the truth contained in the idea that

mode of

a
reasoning about the unknown is
not that the inductive method possesses
some features, besides being inductive, which give it a superiority
over other policies. Its superiority is rooted in the fact that the
inductive character of a policy is the very criterion by means of which
we judge its goodness. The superiority of induction, in other words,
is concealed in the
meaning of a policy's goodness.
Our argument, it will be remembered, hinges on the assumption that the only things which count as reasons for a belief about the
future are known facts which are confirmatory of some inductive
policy. We shall not dispute that this assumption may be successfully challenged, although we see no way of doing it. But it is our
thesis that with a changed conception of a 'reason' we should have
to give up or modify the idea that the justification of induction consists in the superiority of inductive over rival
policies.

induction

is

the best

disguised tautology.

It is

175

CHAPTER IX

SUMMARY AND CONCLUSIONS

1.

The

thesis

of the 'impossibility* ofjustifying induction.

THE

subject-matter of this treatise has been an investigation into

the logical nature of the relation in which so-called inductive inferences or inductive conclusions stand to the data or grounds on which

they are established. This investigation has been pursued with the
chief purpose of answering the question as to whether or not it be
to find a justification
possible in the logic of the relation mentioned
for induction as an operation on
and in every-day life, has to rely.

We

which reasoning, both

have seen that the demand for a

in science

justification of induction

in ordinary language as well as in the records of the history

covers
not one but several distinct ideas, and that conseof philosophy

quently the question mentioned can be answered in the negative or

in the affirmative depending upon what we expect a justification of
induction to be. We have, moreover, seen that there is one sense in
which any attempt at such a justification necessarily fails. This
failure consists, roughly speaking, in the impossibility of guaranteeing
(with certainty or even with probability) the truth of any synthetical
assertion concerning things outside the domain of our actual or
recorded experience. This impossibility was first pointed out by
Hume, and from the anxiety over its alleged philosophical implications has originated the 'problem of Hume* or the 'problem of
induction' par preference.
Before leaving our topic we must scrutinize the logical nature of
the statement concerning the sense in which any attempt to justify

induction fails. This statement will in what follows be referred to as

the thesis on the impossibility of justifying induction. With regard
to this thesis the problem of induction has been called 'the despair
of philosophy' 1 and the failure to justify inductive inference has been
deemed a scandal to philosophical thinking. 2 From such statements
it
might appear that philosophy, in attacking the inductive problem,
176

SUMMARY AND CONCLUSIONS

has undertaken a task too
mighty for its faculties and that the thesis
under consideration was the
acknowledgement of ultimate defeat
in this task.
Actually, however, these statements are the offspring
of certain typical
misinterpretations of Hume's results and of the
failure clearly to
apprehend the logical character of the question
to
us
presented
by the demand for a 'justification' of induction.
When this is understood it is seen why the thesis on the
impossibility
of justifying induction is, not a scandal to
philosophy, but a philosophical achievement of great importance.
The typical way of
is the
misinterpreting the thesis in
following:

Our

question

thesis states that in a certain

and furthermore, as

it

seems, very important sense our efforts to justify induction have been
all in vain, since such a
justification is not forthcoming. Induction,
in other words, is in a certain sense an
unjustifiable operation. We
cannot 'prove', with certainty or even
that the sun will
probability,

rise
all,

tomorrow.

It is

easy to take this as implying that it were, after

worthy of a rational man to take a sceptical attitude towards this

conjecture.
On tho other

hand such 'sceptical* consequences of a theory of

induction are repugnant to sound
judgment. We revolt against
ideas such as that it is not
'probable' that the sun will rise tomorrow, and as this idea was the logical outcome of a certain philosophy of induction we revolt against this philosophy as well And
we maintain with a deep moral awe that induction must be
justifiable

which we are here concerned, even if such a

justification has not hitherto been found.
Of such an attitude towards induction there are good representaalso in the sense with

tives also in recent

philosophy.

The following passages may be

quoted from a distinguished contemporary philosopher as

it.

He says:
The most

illustrating

important postulate of science is induction. This may

be formulated in various ways, but, however formulated, it must
yield the result that a correlation which has been found true in a
number of cases, and has never been found false, has at least a
3
certain assignable degree of
*I
probability of being always true.'
am convinced that induction must have validity of some kind in some
degree, but the problem of showing how and why it is valid remains
unsolved
Until it is solved, the rational man will doubt whether
.

177

THE LOGICAL PROBLEM OF INDUCTION

food will nourish him, and whether the sun will rise tomorrow.' 4
This typical misinterpretation of the thesis on the
impossibility
of justifying induction
originates from a confusion with which we are

his

already familiar from earlier portions of the present treatise. This

confusion, which is deeply rooted in the
philosophical inclinations of
man and which is one of the fundamental sources of
philosophy as
such, consists in the failure to separate from one another
questions
of language and questions of fact. In almost
any situation where
there is alleged to be a conflict between
'philosophy' and 'common
sense' the conflict can be shown to be the
offspring o this confusion.
It will

be our

the above thesis

final task, therefore, to show

why and in what sense
on the impossibility of justifying induction is when

rightly understood

grammatical in its nature, and, as such, free from

'sceptical' implications. We shall do this by applying a characteristic
'technique of thought' to the thesis mentioned, the essential of
all

this

'technique' being to demonstrate that the truth in the thesis

disguised tautology.

The

2.

logical nature

Let us give to the

ing

more

It is

of Hume's 'scepticism

'thesis'

is

7
.

which we are here examining the follow-

specific formulation:

impossible to guarantee, with certainty or with probability,

an unknown instance of the

property A will also exhibit the
property B, if A and Bare different properties.
(This formulation must not be taken to represent, in
itself, an
important result of philosophical thinking. The philosophical
achievement which it has been the chief
purpose of this treatise to
expound does not consist in the above thesis itself but in a certain
interpretation of it.)
The thesis states that a certain thing is
'impossible". It is important
4
to observe that the
phrase it is impossible* here means the same as it
is
contradictory'. There is also another interpretation of the phrase
which suggests itself, viz. 'there does not exist'. This last
interpretation need not be false if used in the
appropriate way, but it should,
however, be avoided for the reason that it is misleading. For, if we
that

take

*it is

impossible to justify induction' as meaning 'there does

justification of induction*, then this suggests that we

not exist a

178

SUMMARY AND CONCLUSIONS

know what

the desired justification ought to be,

although upon
investigation we have not been able to find it. But this gives an entirely
wrong picture of the logical situation. Actually the failure of the
attempts to justify induction was caused by the fact that we had no
clear idea as to what exactly we were
seeking. Now we maintain that

meaning of 'justification' we find that the reason why

was impossible to justify induction in a certain sense is that this
'sense' was a hidden contradiction.
The next task, therefore, is to show that that which the thesis asserts
to be impossible is a contradiction. This is done
by summarizing
the chief results of the foregoing
chapters into an analysis of the
in clarifying the
it

constituents of the thesis.

was shown

VII that a guarantee for something with

which is going to happen only if it
'probability'
means that this 'something' is going to be true in a certain proportion
of cases. The difference, therefore, between a guarantee with
'certainty' and one with 'probability' is that the former concerns the
truth of a single statement, the latter the truth of a statement on the
It

is

in Chapter

relevant to that

truth-frequency in a class of statements. The problem of 'guaranteeing' something as to the future is thus fundamentally the same in
both cases, viz. that of assuring that a proposition, about the truth of

which we are uncertain,

will

be

true.

We can, consequently, without

mentioned on the impossibility of

justifying induction, omit from its formulation above the qualification 'with certainty or with probability* added to the word 'guaranaltering the content of the thesis

tee'.

(This, it must be noted, does not mean that it is the same thing
to guarantee that A will be
with certainty and to guarantee it with
will
probability.
merely maintain that to guarantee that

We
be B means

to guarantee the truth of another proposition

probably
itself of the same 'inductive* kind as this one. The
guarantee of the
truth of the second proposition can again be demanded either with
certainty or with probability.)
We next turn our attention to the condition that the properties
A and B ought to be 'different'. The meaning of this was analysed
in Chapter II, section 2, where we distinguished between 'psychological* and logical* difference, the latter alone being of relevance to the
problem of Hume. That two properties are logically different meant
179

THE LOGICAL PROBLEM OF INDUCTION

that the presence (or absence) of one of the
properties does not
logically follow from the presence (or absence) of the other property.
If one property entails another, then its
presence, in a given
situation, is a standard or criterion for the presence of this other

property. (The absence of the second property again is a standard

for the absence of the first.) That two
properties are (logically)
different therefore means that neither of them is a standard for the

presence or absence of the other.

The phrase that the instance of the property A ought to be 'unknown' then remains to be interpreted. The first interpretation to
suggest itself is the following:

An

A is 'unknown' so long as we do not know of any

it is
which
property
going to possess except A, which it has per
That
the
instance of A is 'unknown' thus
definitionem.
implies that
we do not know whether it will have the property B or not.
To 'know that an instance of the property A will have the property
B can mean different things. But, as was shown in Chapter II, section
5, unless it means that the presence of A is taken as a standard for the
instance of

'

presence of

'know' that

for the absence of A), then to

(or the absence of
will be
is not relevant to the
question whether

B
be B or not.

'really' is going to
It follows that not to

know whether or not A will be B must

imply that the presence of A is not taken as a standard for the presence
of B. Otherwise the interpretation of the phrase that the instance of
A must be 'unknown' would contradict the condition that A and B
are logically different.
It is,

we

however, easy to see that even

get a contradiction.

The

thesis

if

the phrase

on the

is

thus interpreted

impossibility of justifying

induction would then imply that it is impossible to guarantee that A

will be B if the presence of A is not a standard for the
presence of B.

According to the analysis in Chapter II a 'guarantee' that A will be

several things, but unless it means that we make the
assertion 'A will be 5' analytical, the 'guarantee' is not relevant to the
question whether or not the property B really will be present in A.
On the other hand, if it is analytical that A is B then the presence of
A is a standard for the presence of B and the demand for a 'guarantee
becomes contradictory.
On this point the following 'objection* is likely to be suggested:

B can mean

'

180

SUMMARY AND CONCLUSIONS

The above

interpretation of the qualification that the instance of

to
be
'unknown' is evidently beside the point, since on
ought
reflection it is clear that under it the demand for a
of

justification

induction becomes a

demand

for knowledge about something of

which, according to our own premisses, we can know nothing. But
surely, in demanding a guarantee that certain things are going to be
such and such in the future, we do not demand this. Therefore, an
interpretation of our demand as such a self-contradictory wish does
not do justice to that for which we are really asking. The true
interpretation seems to be something of this kind:
In speaking of an instance of
as being 'unknown' we mean that

certain reasons for judging the presence of

in the instance are as yet not available.

(and other properties)

These reasons, sense-

perception or whatever they may be, we shall call the experiential

grounds on which a proposition concerning the presence or absence
of properties is Verified' or 'tested'. Induction, speaking generally,
is the
anticipation of the results to which experiential tests will

subsequently lead.

To

grounds or reasons

justify induction is to provide

us call them inductive grounds

some other

let

which are

somehow

to 'rationalize' this process of anticipation. Since these

'inductive' grounds are to be different from the above 'experiential*
be
ones, the demand for a justification of induction may

perhaps

impossible to

satisfy,

but

surely not self-contradictory.

easy to reply:
is

To this objection it is
What would be the logical relation, we ask, between the
grounds and the

'inductive'

ones for judging the truth of propositions on future events? Let us suppose that the former were criteria
of the results to which an anticipation of the testimony of the latter
will lead. This would mean that if I had 'inductive' reasons for
anticipating that A will be B then I am bound to interpret any
subsequent experiential information so as to accord with the anticipation. But then the proposition 'A will be "B is analytical, and the
presence of A is a standard for the presence of 5, and the absence of
B a standard for the absence of A. This again contradicts the condition that A and B ought to be different.
Consequently in demanding a justification of induction we cannot
demand criteria of the truth of inductions. The remaining possibility
is that we demand something which can be conveniently called
181
N
'experiential'

THE LOGICAL PROBLEM OF INDUCTION

symptoms of the experience anticipated in inductive inferences.
Such 'symptoms' for the anticipation of truth may be obtained in
various ways, but irrespective of how they are obtained we must

know something

of their reliability if they are to be relevant to the

statement concerning their reliability, again,
is either an assumption that those 'symptoms' will lead to true
inductions, and as such is itself in need of justification, or it is a
guarantee that the indications of the inductive grounds will accord
with the testimonies of the experiential grounds. On the other hand
we know that the sole way of guaranteeing that those indications and
testimonies are going to give concordant results is to make the statement on their concordance analytical. This means that the indications of the inductive grounds are standards or criteria for the
truth of inductions.

testimonies of experience, and this again contradicts the assumption

that the former are only 'symptoms' of the latter.

We

have thus seen that the same contradiction, which, according

to the above 'objection' it was unfair to attribute to the demand for
a justification of induction, is innate also in the 'objection' itself.
It is the logical peculiarity of this demand that, although it need not
be interpreted as self-contradictory, any interpretation which evades
the contradiction is not relevant to the demand in that sense of it,
1
in which it is to be satisfied by a solution of 'Hume's problem'.
But if the demand for a justification of induction is self-contrathen the above thesis
dictory, when taken in that particular sense,

on the 'impossibility' of justifying induction is a tautology.

The view that Hume's 'sceptical' result as to the justification
of induction is a consequence, not of the constitution of the world
but of the use of language, can truly be said to constitute the 'solution' of the problem put to philosophy by Hume. To Hume the
failure to justify induction seemed the discovery of a serious limita-

We, in realizing what this

'failure' means, also understand that from the very meaning of words
it follows that we can never imagine these faculties, in the aspect of
them under consideration, to be greater than they are. When this is
of induction in the
clearly apprehended, the demand for a justification
from itself as
it
vanishes
to
Humean sense is 'satisfied', that is
say
tion in

man's

intellectual faculties.

being devoid of object.

182

SUMMARY AND CONCLUSIONS

3.

The

The

critical

and the

clarification of

constructive task

of inductive philosophy

language leading to a solution of Hume's

problem can be said to constitute the critical task of inductive

philosophy. With this can be contrasted what we shall call the
constructive task of a
theory of induction. The latter, unlike the
former, is not concerned with the meaning and use of words, but
with the formal peculiarities of given conceptual structures. It consists in the
application of formal logic and mathematics to the
analysis of inductive propositions.

As regards this constructive task the treatment in the

work has been far from exhaustive. Our contributions have

present
to some

extent been in the nature of first sketches which it will be the task of
others to work out in fuller detail.
finally mention some points
on which a continuation of the task undertaken here would seem
to be worth while.

We

In Chapter IV we showed that the idea of a

logic of induction',
form of it which was invented by Bacon and later developed
by Mill, can be profitably treated as a formal theory of necessary
and sufficient conditions. This treatment reveals unexpected asymmetries and other logical peculiarities in the fundamental
types of
method of scientific inquiry. The examination, as pursued by us,
in that

applied specifically only to inductive propositions of certain very

simple structures. It would be of interest to investigate, inter alia,
whether the theory of necessary and sufficient conditions can be extended also to relational and quantitative laws of nature, and whether
there is some analogy to such a theory among Statistical Inductions.
In Chapter VI we analysed the formal nature and interrelatedness

of certain ideas concerning the probability of inductions. We

endeavoured to show that those ideas can be formalized and made
exact within the 'ordinary' probability-calculus.

This system re-

mains to be embellished, especially we think by a fuller analysis of

the ideas of simplicity and scope in relation to the probability of
inductions.

Philosophy of induction has, at least since the days of Hume,

been seriously hampered in its progress by an unwholesome confusion of the two tasks of inductive theory, called by us the critical
and the constructive tasks respectively. The constructive value of
183

THE LOGICAL PROBLEM OF INDUCTION

system of inductive logic or inductive
minimized
been
has
by the fact that those constructions
probability
have been undertaken for the vain purpose of solving Hume's
problem. On the other hand most critical treatments of inductive
of showing
philosophy have contented themselves with the easy task
that those constructive efforts have failed in that they did not lead
to a solution of the problem mentioned, in which case one has not

most

efforts to develop a

been able to estimate the value of the constructions in

spite

of their

'failure'.

It

appears to us that

we have now

arrived at a point where the

completion of the

clarification of philosophical ideas has led to a

of inductive theory, and from which the constructive

task can be pursued with a clear purpose unhampered by false
from all misleading
philosophical pretensions and disentangled

critical task

expectations.

184

NOTES

NOTE

In the notes any work is usually referred to under the author's name, or
there are several works quoted by the same author, under the author's name
followed by a number in brackets. The names of the works themselves are found in
if

the Bibliography. The references and quotations apply, of course, always to the
edition mentioned in the Bibliography, but the Author has in certain cases tried, by
instead of those of pages, to make them
giving the number of chapters and sections
to other editions of the work also. If no special edition of any work is mentioned,

apply

the references should apply to

all

editions of

it.

186

CHAPTER

INTRODUCTORY REMARKS ON INDUCTION

i.

Inductive inference

1.
1

and

the

problem of induction.

211 and 239.

2
See Mill, bk. II, chap, iv, 2 and 3.
3
Aristotle [1], 1565; Mill, bk. Ill, chap, n, 1.
4
Mill, bk. Ill, chap, m, 1.
5
To this rule there are indeed important exceptions. The way of applying the principles of probability to induction, which is characteristic of Laplace and his followers,
marks an important epoch in the development of the logical problem of induction.
(See below chaps, vi and vn.) Ajnong works, typical of the psychologizing tendency
of French writers on induction, may be mentioned: The methodological works of the
great physiologist Claude Bernard, Introduction a la m?decine experimental and La
science experimentale, Lalande's Les theories de r induction et de V experimentation,
Naville's La logique de Vhypothese, and Picard's Essai sur la logique de r invention
dans les sciences and Essai sur les conditions positives de r invention dans les sciences.
6
Mill, bk. Ill, chap, i, 2: 'An analysis of the process by which general truths are
arrived "at, is virtually an analysis of all induction whatever' and 'we shall fall into no
error, then, if in treating of induction, we limit our attention to the establishment of
Reichenbach [6], p. 265f.: 'Das erkenntnistheoretische
general propositions'.
Problem
liegt gar nicht in der Unendlichkeit der Folgen, sondern darin, dass die
Folgen sich stets, iiber vergangene Ereignisse hinaus, auf zukiinftige Ereignisse erstrecken
Eben darin liegt das eigentiimliche Problem der Induktion; und dieses Problem
wird nicht im geringsten dadurch erleichtert, dass man die Zahl der zukiinftigen
See,

e. g.

Jevons

[1], p.

Ereignisse
2.

finitisiert.

Different forms of inductive generalizations.

The use of logical symbols and formulas in this book assumes that the reader is to
some extent famiSar with the symbolic language of modern logic. For any reader, not
acquainted with logistics, the following elementary elucidations will be added:
Let a and b be propositions. The conjunctive proposition a&b ('# and &') is true
only if both a and b are true. The disjunctive proposition avb ('a or 6*) is false only if
a and b are both false. The implicative proposition a-*b is false only if a is true and b
+b is true when a and b have both the same
false. The equivalence-proposition a<
truth-value ('true' or 'false'). The negative proposition r^/a(*not-a') is true when a is
false, and false when a is true.
If the truth-value of a proposition depends (uniquely) on the truth-value of certain
other propositions, the former is said to be a truth-function of the latter, a&b, avb, a-+b>
and a+*b are, according to this definition, truth-functions of the propositions a and
b. The proposition
>a, again, is a truth-function of a.
The proposition a&b, avb, a-*b, and a* >b are called molecular relative to their constituent propositions, a and b, which again are called atomic relative to the above com<

pound propositions.
Of two propositions, a and b, and then* negations we can form four conjunctions:
a&b, a&> >b, r^a&b, and (^a&<-^b. In a similar manner we can of n propositions and

form 2 n conjunctions. Any proposition, which is a truth-function of the

2 n conjunctions. This
is
equivalent to a disjunction of some of these
called the disjunctive normal form of the proposition.

their negations

n propositions,
disjunction

is

187

THE LOGICAL PROBLEM OF INDUCTION

Ax means: the individual x has the property A. R(x,y) means: the pair of individuals,
x and y, stand to one another in the relation R.
x in Ax may be regarded as a constant or as a variable. If x is a constant, Ax is a
proposition. If x is a variable, /lx is called a propositional-functlon. The propositionalfunction 'becomes' a proposition when for the variable is substituted a constant. The
constants which may be substituted for a variable are called the values of the variable.
value is said to satisfy the propositional-function when the resultant proposition is

true.

(Ex)Ax means: there exists an x which is A or, in other words, at least one x has the
property A. (Ex) is called an existential operator. (x}Ax means: all x*s are A. (x) is a
universal operator.

The symbols used by us differ from those of Principia Mathematica inasmuch as that
as a sign for conjunction, the arrow instead
we, following Hilbert, use '&' instead of
of the horseshoe as a sign for implication, the double-arrow instead of *=' as a sign for
equivalence, and brackets instead of dots for combining and separating the respective
parts of a symbolic sentence.
It is of some importance to observe that phrases such as the proposition a\ 'the
property A ', the individual x\ when they occur in this book, are not about the symbols
but about the symbolized entities. Similarly, 'a is false*, 'every A is B\
V, *A\' and
'x is A are about entities and not about symbols.
On the other hand, a phrase such as 'Ax means' or '(Ex) is called' is about symbols

and not about things symbolized.

Usually, it is clear from the context, whether we are talking about the symbols or
about their meanings. Sometimes, however, when we
we enclose the symbols within single quotation-marks
'

this

same

are talking about the symbols,

(as, for example, on line 13 on

'

page).

A good modern introduction to logic is Cooley, A Primer ofFormal Logic (New York,
1946).

An

excellent text-book of a

more advanced character

is

Hilbert- Ackermann,

Grundzuge der theoretischen Logik (3rd edn., Berlin, 1949).

2
Cf. Keynes, p. 220.
8
Keynes (p. 220) calls them Inductive Correlations. Mill's Approximate Generalizations correspond roughly to our Statistical Inductions. Mill pays very slight attention
to this type of inductive inference. See bk. Ill, chap. xx.
4
The contrast in question is perhaps more appropriately described as one between
Causal Laws and Probability-Laws, than between Universal and Statistical Generalizations. It might be maintained that induction from statistical data usually aims at the
establishment of Probability-Laws, and that Statistical Generalizations accordingly
shall not discuss the question here. Some
are relatively unimportant in science.
writers on induction (Broad [7] and Kneale [1], 48), have tried to argue that Probability-

We

Laws presuppose the existence of causal (nomic) connections in nature. Their arguments
do not seem to us convincing.
Characteristic of the elements of a series are said to be intensionally given when they
a rule. Characteristics which cannot be calculated from a rule
but have to be determined by empirical observation from case to case, are called
5

may be calculated from

extensionally given.

For
7

this notation cf.

Reichenbach

[4], p.

81 and 347.

Cf.

Popper [2], p. 128ff.

Observe for example the

where the
series 01010011000011110000000011111111
proportion of 1's is 'oscillating' between the limits 1/2 and 1/3.
9
From the above expression (4) follows a proposition asserting the existence, for any
.

188

NOTES

such that from xm on the frequency of A's which are B remains

e, of a number
within the interval p
s.
See below chap, vm, 1.
10
The existence of empirical propositions which involve both universal and existential
operators and hence are neither verifiable nor falsifiable originally presented a difficult
puzzle to adherents of the so-called verificationist theory of meaning. Later, however, the
criterion of meaningfulness, advocated
by the logical positivists, has been gradually
widened, so as finally to include also propositions with any number of both universal
and existential operators in any combination. See Carnap [1 ].
11
which contains an infinite number
Consider, e.g. the series 100100010000100000
of both 1 *s and O's, but where nevertheless the
limiting ratio of the number of times
when 1 occurs to the total number of members in the series is zero. Sequences of this
structure may occur in nature (the order of 1's and O's
reflecting, say, the temporal
succession of results in repeated experiments or observations). It seems to us, therefore,
that R. B. Braithwaite is mistaken, when he writes ([6],
p. 152): 'all general statements
are in fact probability statements, since to say that all A *s are 2?'s is the same
thing as
to say that 100 per cent of A's are 's, which is (on
my thesis) the same thing as to say
that the probability of an A being a B is 1. Similarly, to say that no A'$ are B's is to
say that
per cent of A 's are 5*s and to say that the probability of an A being a B is 0.
Universal generalizations, whether affirmative or negative, are special cases of
probability statements.' He argues (loc. cit.) that 'within a science, ascriptions of zero
are
taken
to
be
from
universal
probability
indistinguishable
negative
generalizations'.
This, it would seem, is a confusion between what is usually the case (as a matter of fact)
and what must necessarily be the case (as a matter of principle).
12
Darwin, On the Various Contrivances by which Orchids are Fertilized, (London
.

1862), chap. v.

'

Remarks about

various usages of the term 'induction*. Induction and eduction.

relevant passages are Topics bk. I, chap. XTI and bk. VIII, chap, i-n; Prior Analytics bk. II, chap, xxm; Posterior Analytics bk. I, chap, i and xvm and bk. II, chap. xix.
It is usually said (cf. Keynes, p. 274 and Kneale [1],
pp. 24-37) that Aristotle used
'induction* in two senses, viz. to mean either summative induction (Prior Analytics) or
3.

The

intuitive induction (Posterior Analytics). It

*
'induction in the Topics must be counted as

by the example quoted in the text, which

Cf. Lalande, p. 3 and 6.
2

is

seems to us, however, that the use of

a third sense. This opinion is confirmed

a clear case of so-called ampliative induction.

105*12.
105 a 13-15.
[1], 165* 5.
For a discussion of Aristotle's account of induction in relation to the syllogism see
[1],

[1],

4
5

Whewell

[4], p. 449fT.

p], 68* 19-24.

Cf. Kneale

PL

10
11
12
l

14
15

[1], p. 25.

68* 29.

[31,71*8.
[3], 100*12.
PL 81*6.

Johnson [1], vol. II, chap, ix,

Kneale [1], p- 30.
Johnson [1], vol. II, chap, vm,

1.

Peirce, vol. II, especially par. 680

and

vol. Ill, chap, n.

and 709; Lalande,

189

p. 6;

Kneale

[1], p. 44.

THE LOGICAL PROBLEM OF INDUCTION

1S
17

Johnson [1], vol. II, chap. vra.

For a good account of recursive induction see Kneale

l8

SeeMach

[3], p.

[1],

10.

306.

19
1:
Mill, bk. Ill, chap, in,
'every induction may be thrown into the form of a
syllogism by supplying a major premiss'.
20
Johnson [1], vol. II, chap. x.
21
For significant examples of the use of summative induction see Mach [3], p. 305f.

and Lalande,

p. 8ff.
Mill, bk. HI, chap. n.

22
23

Johnson

24

[1], vol. Ill,

Mill, bk. II, chap,

Mill, bk. II, chap,

2*

M Kneale

[1], (p.

The chapter
chap.

m,
m,

is

called 'Of Inductions Improperly So-Called'.

rv.

and

7.

5.

45) argues against Mill that there can be

no

'inference

from the

observed to the unobserved without at least tacit reliance on laws*. It is not quite clear,
how this shall be understood. If 'inference from the observed to the unobserved* is
meant to include generalization, then Kneale's statement must be rejected as false.
For all generalizations cannot be said to 'rely on laws ', i.e. rely on some other generalizations. If again 'inference from the observed to the unobserved' means eduction
to us unduly dogmatic. It is hardly possible to
only, then Kneale's statement seems
deny that cases of genuine eduction occur. At most one might say that the rationale of
an eductive inference is a hypothetical general truth. And if this is what is meant by
saying that inference from the observed to the unobserved must rely, tacitly at least, on
laws, then Kneale's position in the matter would seem to coincide with Mill's.
27
Catnap [11], 44s and 110, especially p. 208 and p. 574f.
38

Carnap [11], 110F.

Cf. von Wright [10],

*9

CHAPTER
2.

n.

Hume's

p. 364.

INDUCTION AND SYNTHETICAL JUDGMENTS A PRIORI

theory of causation.

See above ch. I, 2.

2
His definition of 'cause', however, sometimes
conditions, See Hume [3], sect. VII, pt. 2.

is

given so as to include also necessary

IV, pt. 1, and Hume [2], p. llf.

bk. I, pt. m, 12: 'There is nothing in any object, considered in itself,
which can afford us a reason for drawing a conclusion beyond it.*
6
Ibid.: 'Even after the observation of the frequent or constant conjunction of objects,
3

Hume
Hume

[3], sect.
[1],

we have no reason to draw any inference concerning any object beyond those of which
we have had experience.
*
Hume [3], sect. 2: 'When we entertain, therefore, any suspicion that a philosophical
term is employed without any meaning ... we need but enquire, from what impression is
'

that supposed idea derivedT

7

Hume

[1],

bk.

I,

pt m,

14.

In another part of the Treatise on

discussion of this problem. See Hume

Hume

10

[1],

bk.

I,

pt. in,

Human
[1 ],

bk.

Nature there
I,

pt. iv,

is,

however, a most acute

2.

6.

has been a matter of philosophical controversy whether the relation between

cause and effect is a relation between 'objects' or between 'events'. This controversy is
wholly irrelevant to our discussion of the causal relation and so also to Hume's.
11
Cf. the resemblance between Hume's thesis and the following proposition from
Wittgenstein: 'Auf keine Weise kann aus dem Bestehen irgend einer Sachlage auf das
It

190

NOTES
*
Bestehen einer, von ihr ganzlich verschiedenen
Sachlage geschlossen werden. (Wittgenstein 5, 135.) Also here we are entitled to ask what is meant
two
by
'Sachiagen' being

'ganzlich verschieden.' Note the qualification 'ganzlich'!

12
ought rather to have said: In the case of the billiard-balls, as we want to Conceive of it here, and as it was conceived of
by Hume, logical and psychological difference
are concomitant properties. For as will be seen later (ch. m,
1) it is also possible to
conceive of the case in such a way that in it cause and effect become
logically connected.
13
It is not
necessary here to embark upon a discussion of the exact nature of the
concepts of entailment (the 'follows from') and deducibility. The concepts are open to
debate. It is sufficient for our
purposes to emphasize the 'formal' or 'logical* (as distinct
from 'material' or 'physical') character of the notions concerned.
14
Cf. Braithwaite [2],
pt. I, p. 467.
15
Outbursts about the absurdity of Hume's
theory, based upon typical misunderstanding of the nature of his arguments, are to be found everywhere even among recent
authors.
good example is offered by Ewing's criticism of the Humean theory or the
so-called 'regularity view' of
causality. See e.g. Ewing, p. Ill: If the regularity view
were the whole truth, all practical life would become sheer nonsense.'
Similar
exclamations are abundant in the works of Reichenbach. See
e.g. Reichenbach [9], p.
344ff. (Note the phrase 'intellectual suicide*, ibid.,
p. 344.)
16
do not think it necessary here to introduce a definition of the

We

We

or 'logically necessary'. The following three remarks will

which we in this treatise make of the term 'analytical':

suffice to

concept 'analytical'
make clear the use,

proposition which is true per definitionem, is analytical.

the falsehood of which has been excluded on the ground that
would be contrary to the (correct) use of language to say the proposition is false,
analytical. It is clear that (2) includes (1).
(1)

Any

(2)

A proposition,

Any analytical proposition

Cassirer, E., vol. II, p. 262
Natorp, p. 127-163.

(3)
17

18

it

is

true.

is

(necessarily)
- See
also Church,

R, W., p. 210ff.

10
See Revue de Metaphysique et de Morale 40, 1933, suppl. /, p. 15f., where William
of Auvergne is quoted, and the possibility that he had influenced Hume is mentioned.

20
21

Hobbes, p. 15ff.
Malebranche, Quinzieme fclarcissement. Malebranche

also uses the

example of

the billiard-balls.
22
Leibniz [2], 26.
23

24
25

Leibniz [1], vol. IV, p. 161f.

Kaila [6], p. 1 12. Cf. Mill's use of the term 'empirical law*, bk.
Mill, bk. Ill, chap, xn,

2*

Ill,

ch. xvi.

1.

'Wie sind synthetische Urtheile a priori moglich? So lautet das

es allgemein auffasst.' Cf. Hobart, pt. i, p. 284. Hume
himself, however, regarded causality as the only relation by means of which, as he says,
'we can go beyond the evidence of our memory and senses*. (Hume [3], sect, rv, pt. 1.)
Apelt

[2], p. 53:

Humesche Problem, wenn man

3.
1

Kant and Hume.

The development of Kant's views

as to these problems,

which may be said to

constitute the central question in the Kritik der reinen Vernunft, can be followed in the
Reflexionen.
2
Kant [1], 291, 292, and 726.
*

See

Kant

e.g. the definition

18.

of 'Wahrnehmungsurtheile' and 'Erfahrungsurtheile' in

[3],

191

THE LOGICAL PROBLEM OF INDUCTION

4

See

e.g.

Kant

[3

],

17.

A slightly different form of the same question is 'Wie ist Natur

selbst moglich?'. (Kant [3], 36). Cf. ibid.: 'Die Moglichkeit der Erfahrung uberhaupt
ist also
zugleich das allgemeine Gesetz der Natur, und die Grundsatze der erstern sind
*

Gesetze der letztera. In a certain sense, therefore, we are with Kant entitled
to use the word 'Natur' as synonymous with 'Erfahrung'.
5
Another important factor in this change in Kant's opinion was that he always seems
to have held mathematical judgments to be synthetical. (Kant [1 ], 496.) When this
had to be reconciled with the aprioristic nature of those judgments he turned to the
doctrine of synthetical judgments a priori.
*
Kant [3], Vorwort.

selbst die

Cf. Cassirer, E., vol.

II, p. 526fT.

and 534.

Kant [2], p. 232ff.

9
Whitehead [2], p. 34.
10
Kant [2], p. 237f.
11
For a more detailed analysis of Kant's causal theory of time see Mehlberg, especially
the importance of reversibility and irreversibility, respectively, in
pt. i, pp. 135-158. For
series of sensations as regards the logical

'Aufbau' of the physical world, see Kaila

[4],

p. 29-63.
12

are, however, not causal laws according to the terminology followed in this
(See chap, i, 2 and chap, n, 2.)
13
The idea that the so-called physical world is a system of invariances or laws prewell stressed by Kaila in several of his writings.
vailing in the phenomenal world is very

They

treatise.

and Kaila [6], p. 89.

*Die Natur ist ... der InbegruT ihrer allgemeinen
Gesetze. Helmholtz [2], p. 39: 'Das Gesetzmassige ist daher die wesentliche Voraussetzung fur den Character des Wirldichen.'
16
See Kaila [6], p. 228.
18
Leibniz [3], vol. VII, p. 3I9ff.
17
Actually all that could possibly follow from the 'deduction' as outlined above is
that causal laws must exist in order for time to be possible. But this is not the same as
must be true, which is a much 'stronger'
stating that the Universal Law of Causation
assumption. Now this peculiarity of the 'deduction' seems, in a way, to be attached also
to Kant's own arguments, and is thus not to be attributed to an incompleteness in our

See Kaila
14

Kaila

[3], p. 81ff.,

Cassirer, E., vol.

[4], p. 14ff.,

II, p. 530f.:

'

restatement of them.

Kant [2], 1st edn., p. 189. In the second edition (p. 232), we read only: 'Alle Veran'
derungen geschehen nach dem Gesetze der Verkmipfung der Ursache und Wirkung.
18

19

20
81

4.
1

Kant
Kant
Kant

[2], p. 102ff.

[3],

21.

[2], p. 102ff.

Kant and the

application-problem.

For the following

synthetical or as

it is

irrelevant

whether we have tried to establish

this

law as a

an

analytical principle. Actually several philosophers, in insisting upon

the aprioristic nature of the Universal Law of Causation, have regarded it as an analytical principle. According to Mach causal relatedness is a kind of functional relatedness.

So long as the class of functions, which are to connect cause and effect, is not specified,
the Universal Law of Causation becomes tautological. See Jourdain and Mach [3],
p. 270-281. See also Helmholz [1 ], vol. HI, p. 26ff. Helmholz's view is an intermediate
between Kant's and Meyerson's and illustrates very well how the theory about the
192

NOTES
aprioristic nature of causality

is driven to conventionalism, i.e. to

accept the a priori
principles as analytical. Cf. later chap, n, 6 and chap, in, 2.
2
Cf. Cassirer, E., vol. Ill,
p. 92f.: 'Der allgemeine Satz der Ursachlichkeit enthjUt
kein Merkmal und gibt kein Kriterium an, kraft dessen wir die besooderen Falle seiner
Anwendbarkeiterkennen und ihm subsumieren konnen. Ich weiss aus dem Grundsatze

zwar, dass Objekte der Erfahrung uberhaupt in Kausalverbinding xniteinander gedacht

werden miissen, keineswegs aber, dass eben diese Objekte es sein miissen, die in diesem
*
Verhaltnisse stehen.
3

Kant

[2], p. 165.

Cassirer, E., vol. Ill, p. 93, observes acutely

Problem steht vor neuem vor uns'.

on

this

point that 'das Humesche

Cf. Maimon, p. 190f.: 'Kant nimmt den wirklichen Gebrauch der

Kategorien von
*
empirischen Objekten als ein unbezweifeltes Faktum an. This, in our opinion, hits the
nail on the head. The application-problem never
seriously bothered Kant.
6

Kant
Kant

19,

[3],

18

20,

and

22, 29.
22. Cassirer, E., vol. II,
p. 525:

'Audi das Erfahrangsurteil als

solches enthalt eine eigentumliche "Notwendigkeit".' As to Kant's use of the term
'Notwendig*, see Reinach, Kant [2], p. 279ff., and Kant [3], 19.
8

[3],

Maimon,

p. 190ff.

Maimon uses Kant's example from the Prolegomena

tion between the radiation from, the sun

of the rela-

and the

increase in temperature in a stone

touched by the rays. See also Maimon, p. 420: 'Daraus, dass Objekte
uberhaupt z.B.
im Verhaltnisse von Ursache und Wirkung gedacht werden miissen, wenn eine Erfahrung uberhaupt moglich sein soil, lasst sick noch nicht begreiflich machen, warum z.B.
eben das Feuer und die Warme in diesem Verhaltnisse stehen miissen? *
9

Cassirer,

W.

H., p. 110. Cassirer emphasizes that Kant's views as to the application-

problem had undergone a radical change from the Kritik der reinen Vernunft and the
Prolegomena to the Kritik der Urtheilskraft.
10
Kant [4], Einleitung, IV, and Kant [6], p. 22ff.
11
Kant speaks of such principles under the names 'lex parsimoniae*, 'lex continui in
natura' and others. All these are modifications of a more general principle, which he
calls the
principle of 'die Zweckmassigkeit der Natur'. Kant [4], Einleitung, V, and

Kant [6], p. 22ff.

12
Kant [4], Einleitung,
5.

The

IV, and Cassirer,

inductive problem in the school

W.

H., p. 109.

of Fries.

Kastil, p. 29: 'Dieser vielgepriesene "transzendentale Beweis" . . lauft also eigent*

lich darauf hinaus, die synthetischen Urteile a priori zu analytischen zu machen.
2
3

4
8

Kastil, p. 296.
Cf. Popper [2], p. 52.
Fries [1], vol. I, p. 27

Fries [1], vol.

Fries [1], vol.

I,

and

35ff.

p. 28.

I, p. 21 and Kastil, p. 3 Iff. For the theory of synthetical

in its Neo-Friesian form see especially Nelson [1] and [2].
Kastil, p. 31.

judgments

a priori
7

8
Cf. Nelson [2], p. 532: 'Die metaphysische Erkenntnis ist eine Erkenntnis allgemeiner Gesetze, und allgemeine Gesetze werden a priori erkannt. Die Erkenntnis der
allgemeinen Gesetze ist aber nicht selbst wieder ein allgemeines Gesetz, sondern ein
individuelles Faktum. Individuelle Fakta aber werden a posteriori erkannt. Also wird
auch das Faktum der unmittelbaren Metaphysischen Erkenntnis nicht a priori sondern
*
a posteriori . , erkannt.
.

193

THE LOGICAL PROBLEM OF INDUCTION

theory of synthetical judgments a priori very similar to that of Fries and the
Neo-Friesians has been held by the Oxford philosopher Cook- Wilson and his adherents.
fl

10

Apelt [1], p. 44 and p. 56ff.

Apelt [1], p. 73 and p. 91ff.
12
sind allgemeine und notwendige Wahrheiten d.h.
Apelt [11, p. 92: 'Gesetze
Wahrheiten a priori.* Ibid., p. 106: 'In der Natur sind die physischen Gesetze von gieicher
*
Notwendigkeit wie die mathematischen und philosophischen.
13
74f.:
'Die
Induction
also
nur
die
des theoretischen
Untersatze
bringt
Apelt [1], p.
Lehrgebaudes. In der vollendeten Theorie miissen diese Untersatze auf doppelte
Weise festgestellt werden: einmal
deductiv, d.i. als Lehrsatze, die durch systematische
Ableitung aus ihren Principien folgen, das anderemal inductiv als Erfahrungssatze.
11

. .

'

"Apelt [1], p. 72.

15
Apelt [1], p. 56: "Die Induction ist

der

Weg zu der Verbindung der notwendigen

Wahrheiten mit den zufalligen Wahrheiten.'

"Apelt
17

18
19

[1], p. 101.
77ff.

Apeit [1], p.
Apelt [l],p.

95f.

observe that Maimon, after having shown that Kant's synthetical

judgments a priori are not sufficient to establish the truth of single inductions, (Maimon,
It is of interest to

p. 382), tried to 'complete' the theory of

Kant in roughly the same way

as the Friesians,

by determining the synthetical principles a priori so that special inductions follow

from them. Thereby he also approached the conventionalistic attitude. For a confirmation of this interpretation of Maimon 's view see, e.g. the way hi which he tries to prove
that iron necessarily must be attracted by a magnet. This seems in some way to follow
from the very definition of what a magnet is. Maimon, p. 255: 'In diesem Urtheile z.B.:
der Magnet zeiht das Eisen an sich, wird das Eisenziehen ... als etwas eingesehen

i.e.

zu dessen Bewusstsein wir nicht eher gelangen, als wir zum Bewusstsein des Magnetes
an sich gelangt sind; und so ist es mit alien Objekten der Erfahrung der Fall, deren
Subjekte* gegeben, und deren Pradikate nach und nach durch Abstraktion gefunden
werden.
6.
1

Some
Cf.

other theories of causation.

Whitehead

[1], p. 55: 'It is

impossible to over-emphasize the point that the key

to the process of induction, as used either in science or in our ordinary life, is to be

found in the right understanding of the immediate occasion of knowledge in its full

concreteness.
3

67,
3

Kerby-Miller, p. 177, Whitehead [3], p. 26 and p. 251, Kelly, p. 22, Meyerson

Bosanquet [1], vol. I p. 135, Bradley, p. 546f.

Whitehead

*Hutne

[2], p.

[2], p. 39.

[3], sect. 4, pt.

I.

Kaila

For a theory of causality slightly approaching that one of Whitehead see Russell
For a criticism of the theory about causal perception see Ayer's rejoinder to

[5].

[5], p. 27ff.

Russell's paper, especially p. 274.

7
do not maintain that the interpretation in question answered to the intentions
of Whitehead himself. See also Robson and Gross.

We

8
Meyerson [2], p. 136: 'Nous avons explique le phenomene, le changement, en
deduisant le consequent de Tantecedent, en montrant que le consequent etait necessairement tel qu'il a ete, r&pouvait pas Stre different de ce qu'il a ete, parce qu'il se trouvait
deja implicitement contenu dans cet antecedent.*

194

NOTES
9

Kelly, p. 24f.:

The

explicated, brought out

here

effect is identical

from the fields, but

with the cause, for after

'

itself

unchanged.

all it is

The meaning of

the cause
'identity*

complicated by the fact that Meyerson often also considers the causal relation
quantitative equivalence (identity) between cause and effect.
One of the reasons why we speak of cause and effect as 'different' from each other,
obviously, is that they occur at different time-points. The time-factor involved in causality
has presented some difficulties for the doctrine that cause and effect are 'identical'. See
Meyerson [1 L the chapter called L'eKmination du temps. Also Renouvier, p. 26 and
is

from the point of view of

Bosanquet
10

[1], vol. !, p.

Cf. Joseph, p. 409:

258.

Tor the causal relation which connects a with x connects a cause

of the nature a with an effect of the nature x. The connection is between them as a and
and therefore must hold between any a and x, if they really are a and x
respectively/
11
Cf. Bosanquet [1], vol. I,
p. 174: *If, in an alleged causal nexus, the alleged effect
;*:,

sometimes absent while the alleged cause is present, ceteris paribus, it is

impossible
'
that the alleged cause should be the real cause of the effect in
question. See also ibid.,
*
similar view upon causality was
p. 255: 'Same effect, in the same form, same cause.
advocated by Lotze.
is

12

Blumberg [2], especially p. 76ff.

Cf. Bosanquet [1], vol. II, p. 2.

13

14

Mill's definition of a natural kind is this: 'By a Kind ... we mean one of those classes
which are distinguished from all others not by one or a few definite properties, but by
an -unknown multitude of them: the combination of properties on which the class is
grounded, being a mere index to an indefinite number of other distinctive attributes/
(Mill, bk. IV, chap, vi, 4.) The doctrine of Kinds as it occurs with Mill has nothing to
do with synthetical judgments a priori. Cf. Mill, bk. Ill, chap, xxn, 7. For the connection between Mill's theory of Kinds and the theory of Concrete Universals see Acton.
With the theory of Kinds and Concrete Universals are connected Lachelier's ideas on
induction and final causes. The best statement of the theory of Natural Kinds and its
relevance to induction is found in Broad [1 ], pt. i.
15

[3], p. 8.

Bosanquet
7.
1

General remarks about synthetical judgments a priori.

what follows immediately is that there does not exist an

A which is not B. As
the equivalence between this proposition and 'all A's are R* has been a
matter of controversy among modern logicians. This controversy, however, is of no
relevance to the context to which our reasoning applies.
is

I.e.

well

known
f

'Notwendig heisst einmal das, dessen Gegenteil einen inneren Widerspruch enthalt . . . Es gibt aber noch eine andere . . Notwendigkeit; namlich dann, wenn
das Gegenteil einer anderen, sonst schonfeststehenden Wahrheit widerstreitet . . . Urteile
Kastil, p! 248:

im ersten, logischen, Sinne nocwendig sind, sind analytische; Urteile, die im zweiten
Sinne notwendig sind, sind synthetische. Denn was in diesem zweiten Sinne des wortes
dessen Gegenteil ist logisch moglich.
notwendig ist
die

'

CHAPTER m.
1.

The way

CONVENTIONALISM AND THE INDUCTIVE PROBLEM

which conventions enter into inductive investigations.

Cf. Mill, bk. Ill, chap, x, 2.
in

Some

examples,

Cf. Schuppe, p. 242.

This is why we do not want to call the use of the word 'phosphorus' ambiguous in
the strict sense of this term. We have not used the word to mean two different things.
*

195

THE LOGICAL PROBLEM OF INDUCTION

We have not expressed
and thus

it

opinion as to the 'real' meaning of that word at all,

not clear what the word actually is intended to cover.
treatment of induction and conventionalism.

any

comes about

definite

that

it is

Cf. Britten's (p. 183ff.)

4
Cf. Cornelius [1], p. 291.
5

Cf. Poincare [3], p. 189. Poincare, however, does not see that this leads to a point
is relevant also to the conventionalistic
arguments.

which
6

Cf. the following observation of Jevons in speaking about classification (Jevons [2], p.

675): 'Now in forming this class of alkaline metals, we have done more than merely
have arrived at a discovery of certain emselect a convenient order of statement.
a metal which exhibits
pirical laws of nature, the probability being very considerable that

We

'

some of the properties of alkaline metals will also possess the others.
7
The example of the melting-point of phosphorus was used for the purpose of

illus-

trating conventionalistic lines of thought for the first time by Milhaud (p. 280fT.). The
same example is mentioned by Le Roy ([2], p. 517) and discussed by Poincare in several
Poincare [3], p. 189f. Cornelius uses for the
places. See Poincare [2], p. 235ff. and

same purpose a slightly different example, also from chemistry. See Cornelius [1], p.
289ff. and Cornelius [2], p. 211.
8
This idea is expressed clearly by Schuppe, p. 240: *Sind die Bedingungen eines
natiirlich in aller Vollzahligkeit und ohne
Ereignisses erst voUstandig erkannt
behinderndeandereUmstande
dasEreigniss mussunter alien Umstanden eintreten.'
.

2.

Conventionalism as an 'elimination' of the inductive problem.

Mill, bk. Ill, chap, x, 5 and chap, xi,

p. 14 and Berlin, p. 90.
2
Whewell [3], p. 36: 'The question really

Whewell

1.

[4], p.

453.

how the Conception

See also Fowler,

shall be

understood
'The business of
definition is part of the business of discovery. Ibid., p. 70: 'Induction is ... the process
of a true Colligation of Facts by means of an exact and appropriate Conception. Ibid.,
made by Induction, there is introduced some General
p. 73: 'Thus in each inference
*
Conception, which is given, not by the phenomena, but by the mind. See also Whewell

and defined

in order that the Proposition

is,

may

be

true.'

Ibid., p. 39:

'

'

[4], p. 253ff. Although Whewell 's ideas can in part be interpreted conventionalistically,
he himself was of the opinion that the truth to which induction leads, in so far as it is
absolute, is a kind of synthetical truth a priori. For Whewell's ideas about synthetical
judgments a priori see the very lucid account in Whewell [1], vol. I, p. 53-75. (Also
Whewell [2], vol. I, p. 57-76.) It is most interesting to see, how easily Whewell's 'fundamental ideas', i.e. the general synthetical and a priori principles, can be understood in a
without the use of the terms analytical
conventionalistic way. This was pointed out
or conventionalistic
already by Boole, in an extremely interesting passage in The Laws
of Thought. (Boole [1], p. 406.). About the general idea of order and uniformity in nature
Boole, quoting Cournot [2], says (ibid.) that 'it carries within itself its own justification
or its own control, the very trustworthiness of our faculties being judged by the conformity of their results to an order which satisfies the reason.'
3
Bacon [2], lib. I, aph. CV: 'Atque huius inductionis auxilio, non solum ad axiomata
invenienda, verum etiam ad notiones terminandas, utendum est. Atque in hac certe

Induction
4

Jevons

spes

maxima

[2], p.

675.

sita est.*

see also Mill, bk. IV, chap. vn.

Kinds. See above chap, n, 6.
5

Mach

Sigwart, vol.

Cf. Ellis [1], p. 37.

For the question about

The question

[3], p. 307ff.
II, p.

451.

196

and convention
connected with that of the existence of

classification, induction,
is

NOTES
7

Broad

[1], pt. n, especially

pp. 17f., 32f., and 34f.

The

discussion

is

of Natural

Kinds.
8

Mach

[3], p.

307.

See above chap. I,

10
See e.g. Poincare

and Keynes, p. 274.

HOff. and 135ff. for conventions in the foundations of
92ff. for convention and
geometry. For a further development of
3

[1], p.

and ibid., p.
some of the ideas of Poincare

physics,

see Lenzen, p. 259ff.

11

That Poincare regarded the essential problem of induction as a question upon

which conventionalism had no bearing is seen from several statements of his.
(E.g.
Poincare [I], p. 6. and p. 167ff.)
12
The arguments given here do not pretend to be identical throughout with the
all the
philosophers, whom we call 'radical conventionalists*. They are
to characterize a certain attitude towards the
problem of induction, and the
philosophers separately mentioned may in minor points differ from the attitude which

opinions of

meant
is

here
13

'typified'.

With

this might be compared the ideas of J. R.

Weinberg ([1], p. 108 and p, 157ff.).
Weinberg 's arguments seem to aim at something similar to the lines of thought developed by us here, but are somewhat obscured by his reference to the 'neo-positivistic'

idea that general propositions are not

'propositions,' but 'schemes for the construction
of propositions*. For this doctrine of general propositions and its
bearing on the
inductive problem see also Blumberg [1J, p. 581, Ramsay,
237ff., and Schlick [2],
p.

p. 151.
14

Poincare was aware of the perplexing features in the not always easily
perceptible
from synthetical to analytical, which takes place when conventions originate.
(Poincare [1], p. 11 Off. said passim.)
15
The experiential instances, which recommend the adoption of a new convention and
the dismissal of a previous one might be called renegade instances. For this name and its
use see Aldrich.
transition

16

See Le

criticism of
17

Roy

[1

Le Roy

is

and

[2] for

a fuller account of his ideas.

to be found in Poincare

For the relatedness of the

[2], p.

(partly very good)

213-247.

ideas of Ajdukiewicz to those of Le

Roy see Ajdukiewicz,

p. 260.
18

Schuppe's theory of what he calls 'rationale Induktion' (Schuppe, p, 31 Off.) is a

beautiful example of the view that induction in so far as it is to have
scientific value must
lead to absolutely true and consequently analytical propositions. Induction which does
not lead to absolute truth,

i.e. all

induction which attains synthetical generalizations* he

calls 'nichtig'. (Ibid.).

ia

Cornelius expounds his system of radical conventionalism in Cornelius [1 ], p. 277299 and Cornelius [2], p. 21 Off.
20
See Dingier [1], p. 6; [2], p. 135ff.; [3], p. 178ff. (here on p. ISO it is very clearly
stated that the semper et ubique of the natural laws is a tautologous property of theirs):
[4], p. 25; [5], passim; and [6], p. 340ff. For a criticism of Dingier *s ideas see H.

Weinberg and v. Aster- Vogel.

21
Dingier and his adherents do not, for reasons which are connected with this peculiarity of his system, want to call it conventionalistic at all, but give it the name 'DezerniV
mus'. (See Krampf, p. 45ff. where the difference between the 'Dezernismus' and
conventionalism in the usual sense is stressed.) Nevertheless what is said here about
conventionalism and the inductive problem applies also to Dingier 's theory of induction.
Cf. H. Weinberg, p. 40.

197

THE LOGICAL PROBLEM OF INDUCTION

Conventionalism and prediction.
Poincare was quite clear as to the importance of this in the question of whether
conventionalism can justify induction or not. Laws of nature can always be kept true
if we
regard them as definitions, but that these laws are also used for making predictions
is not
justified by this. (Poincare [2], p. 236.) I may for instance regard Galileo 's law
concerning falling bodies as analytical, but this does not contribute anything to the
reliability of my predictions as to how a certain body in a given case is going to fall.
The law which gives reliability to such predictions is no convention. *I1 ne me servirait
a rien d 'avoir donne le nom de chute libre aux chutes qui se produisent conformement
a la loi de Galilee, si je ne savais d'autre part que, dans telles circonstances, la chute sera
probablement libfe ou a pen pres libre. Cela alors * est une loi qui peut etre vraie ou
fausse, mais qui ne se reduit plus a une convention.
Ibid., p. 237f.)
2
Poincare [2], p. 238.
*
Roughly the same objection against radical conventionalism as the one put forward by
us, is the following which has been made by several philosophers. We can guarantee the
absolute truth of inductive generalizations by making them definitions. But how do we
3.

that definitions derived from past experiences will be suitable for new experiences
This is a question which must be decided by the new experiences themselves.
These experiences may 'correct' our definitions or suggest the adoption of new ones
instead of the old. Nevertheless we know that this process of altering the definitions

know

also?

does not take place from case to case, but that the definitions which we employ have a
certain 'stability*. To account for this stability is beyond the power of conventionalism.
It is the inductive problem recurring. For these ideas see Sigwart, vol. II, p. 45 Iff., Mach
[3], p. 139f., Meinong [1], p. 637ff., and Lenzen, p. 262. For a conventionalist's way of
dealing with these objections see Cornelius [1], p. 292ff.

4. Conventionalism

and the justification of induction.

Cf. H. Weinberg, p. 40: 'An der Tatsache, das wir "prophezeien", Naturereignisse
'
voraussagen konnen, scheitert aller (Konventionalismus und) Dezernismus.
*
This observation underlies the following very good remark of H. Weinberg (p. 40)
in his criticism of Dingier 's theory of induction: *Gewiss, wenn eine auf Minute und
Sekunde vorausgesagte Sonnenfinsternis nicht oder nicht piinktlich eintritt, so werden
wir diese "Stoning" wahrscheinlich "exhaurieren'V ('Exhaurieren' is Dingier 's
expression for the process of completing the formulations of inductive laws by considering new circumstances.) 'Aber es war doch schon unzahligemal moglich eine derartige,
sich bestatigende Vorausbestimmung zu treffen.
3
Tne word 'compelled' is used here to mean logical and not psychological compulsion.
*
It is, for instance, very interesting to note that Whewell who, as was shown above,
closely approaches conventionalistic lines of thought in his doctrine of inductive truth,
at the same time underlines the importance of testing inductions, by making predictions
from them. *It is a test of true theories not only to account for, but to predict phenomena.' (Whewell [3], p. 70.)
6
This point was emphasized also by Poincare. See e.g. Poincare [2], p. 239: 'Quand
une loi a rec.u une confirmation suffisante de Inexperience ... on peut 1'eriger snprincipe,
'
en adoptant des conventions telles que la proposition soit certainement vraie.
'

Cf.

We

above

p. 45.

are here touching upon a point which is at the same time one of the deepest
sources of certain metaphysical ideas about language and knowledge. Of these ideas
the metaphysical systems of Bergson-and Le Roy, for example, are good exponents.

198

NOTES
CHAPTER

rv.

INDUCTIVE LOGIC

Justification a posteriori of induction.

1.
x

Cf. above chap, n,

1.

A full proof of this

statement would take us into general considerations about the

nature of logic, which it does not seem
appropriate to introduce in this treatise.

and discovery. Induction

Mill, bk. Ill, chap, i, 2.

2. Induction
1

2
8

and deduction as

inverse operations.

Jevons [2], p. 122ff.

Ibid., p. 139. Jevons 's idea that there is

no other method of discovering the function

in question is, however, false. Actually the function can be written down
directly on the
basis of the given truth-possibilities
to a rule.
according

Already before Jevons, Tissot had mentioned induction and deduction as inverse
operations. (Tissot, p. 248.) Jevons 's account of induction has been criticized by
various authors, thus

Those

Venn

([2], p. 361),

Meinong

([1], p. 656),

and Erdmann (p. 710ff.).

however, tend to overlook that Jevons, in calling induction the inverse of

deduction, did not intend to attribute to inductive reasoning syllogistic powers. The
lengthy criticism of Erdmann is based on a complete misinterpretation of Jevons *s
critics,

opinion on this point.

5

Cf.

Whewell

64 and

[3], p.

[4], p.

456 and Liebig,

p.

20ff.

See also Popper

[2],

p. 207ff.
6

Whewell

[3], p. 103.

pp. 142-9 and 189-204; Mill, bk. Ill, chap, n, 4; Whewell [3], p. 75.
For the 'Okonoinie des Denkens* see Mach [1], p. 452tT. For induction and the
economic nature of thinking see also Kaila [6], p. !8ff.
9
For this resemblance see Couturat [1], p. 265ff.
10
For a further description of this method see Mach [3], p. 252ff. The invention of
the method is attributed to Plato.
11
Cf. Cassirer, E., vol. I, p. 136H It appears from this context that the description of
the method of science given by Zabarella were almost identical with Galileo's descrip-

Apelt

[1],

tion of his 'resolutive* method.

12

Galileo, vol.

II, p.

21.

13

See Couturat

14

The best ? xount of Whewell's inductive logic is to be found in Whewell [3], bk.
v and *i. Ideas similar to those of Whewell are expounded by Sigwart (vol.

[1], p. 265ff.

chap,

p. 434ff.) in his talk of induction as

II,

II,

a 'Reduktionsverfahren*. Cf. also Trendelenburg's

account of induction. (Trendelenburg,

vol. II, p. 316ff.)

15

Whewell [3], p. 105.

For a description of the inductive tables see ibid.,
astronomy and another of optics is given ibid, after bk.
16

17

Ibid,, p. 114.

18

Ibid., p. 115.

19

Ibid., p. 75.

20

Whewell

21

(ibid., p. 64)

Ibid., p. 98,

p. 100.
II,

chap.

An

inductive table of

ix.

speaks of discoveries of science as 'happy Guesses*.

explicitly says that his system of inductive logic justifies

Whewell

induction.
22

This most important feature of certain common types of inductive logic has been
by Kaila ([6], p. 97). Cf. also above chap, i, 2,
23
2 and 4. Although Whewell *s philosophy of induction
Mill, bk. Ill, chap, n,
on the whole gives a clearer picture of the actual procedure of science than the induc-

stressed

199

THE LOGICAL PROBLEM OF INDUCTION

one can hardly deny that Mill in his criticism of Whewell was right
as regards most points of importance.
24
This follows from the definition of (ampliative) induction given above (chap, i, 1).
25
Cf. Mill, bk. Ill, chap, n, 3. The only real induction concerned in the case,
consisted in inferring that because the observed places of Mars were correctly represented

tive logic of Mill,

by points in an imaginary ellipse, therefore Mars would continue to revolve in that same
that the position of the planet during the time which
and in concluding
intervened between two observations, must have coincided with the intermediate points
.

ellipse;

of the curve.
26

'

As

Stoll rightly points out (p. 91) Whewell never realized the significance of this.
seems to have assumed that the verification, which shows that the given data follow
from the law, was sufficient to establish the truth also of the law as a general proposition.

He

is explicitly stated in Whewell [4], p. 454, where he says of an inductively obtained

but adds parenthetically to this 'except
proposition 'no one doubts its universal truth',
when disturbing causes intervene'. The last clause indicates why Whewell regarded
.
the conclusion as to the universal truth of the induction as justified: this truth was to be
guaranteed by a convention that if there was a fact apparently contradicting the law,
then it was to be 'explained away' in some way. That this was the opinion of Whewell
is confirmed from Whewell [3], p. 234ff., where he at some length discusses the use of
induction for predicting new facts. The law, it seems, can never be 'falsified', but it can

This
,

under circumstances be 'corrected', i.e. formulated more accurately. (Ibid., p. 235.)

These passages clearly show the above-mentiond relatcdness of Wliewell's inductive
See also
philosophy to conventionalistic lines of thought. (See above chap, m, 2.).
Whewell 's criticism of Newton's view of induction (Whewell [4], p. 196ff.), where he
are never secure of excepagainst Newton, who maintained that inductive propositions
tion (Newton [1], lib. Ill, reg. phil. iv), says that *to judge thus would be to underrate
the stability and generality of scientific truths*.

3.

The idea and aim of induction by

Bacon

Ibid.

elimination.

[1], vol. I, p. 137.

3
Cf. Mill, bk. Ill, chap, m, 3: 'Why is a single instance, in some cases, sufficient for
a complete induction, while in others, myriads of concurring instances, without a single
a
exception known or presumed, go such a very little way towards establishing universal
has solved the problem of inducproposition? Whoever can answer this question
.

tion.

This distinction is equivalent to that between induction by confirmation and induction

See Nicod, p. 219ff. and p. 222.
5
Keynes (chap, xxxm) has made a first attempt to extend the general principles of the

by

invalidation.

eliminative method to Statistical Generalizations.

6
For the following considerations it is irrelevant whether the individuals are objects
or events. Cf. chap, n, 2, fn. 10.
7

Cf.

It is to

above chap. I, 2.
be observed that the main task of the inductive logic of Bacon was to devise
a method for the discovery of necessary and sufficient conditions of given characteristics.
According to Bacon the business of induction was to find the 'form* of a given 'nature'.
That the 'form' is a necessary and sufficient condition of the 'nature* is seen from the
following statement (Bacon
est ut, ea posita, natura data

[2], lib. II,

aph.

infallibiliter

iv.):

'Etenim

Forma naturae

sequatur. Itaqueadest perpetuo

200

alicujus talis

quando natura

NOTES
adest
Eadem Forma tails est ut, ea amota, natura data infallibiliter
fugiat.
Itaque abest perpetuo quando natura illabest*. That is to say: the form implies universally
the nature and
conversely.
ilia

D. Broad, in [8], was the first author to deal with the

logic of elirninative
induction within the framework of a
Logic of Conditions. Broad's paper marks an
important advance in the logical study of induction.
10

Bacon [1], vol. I, p. 137. Cf. Mis [1], p. 23: 'Absolute

certainty is ... one of the
distinguishing characters of the Baconian induction.' See also Keynes, p. 267.
11
Bacon [5], vol. Ill, p. 618. Mill (see bk. Ill, chap, in, 3;
chap, rv, 3; and chap, ix,
6) also attributed absolute certainty to his inductive method. As wiU be seen later,
however, it is not clear in which of the three senses mentioned Mill spoke of inductive
conclusions as being 'certain*.
4.

The mechanism of elimination.

We do not wish to commit ourselves on the question, whether the totality of properties of a
given individual can be said to determine a set. We must not speak of the set
1

X X

. . as the set of
z
19
properties of x (beside A). For the description of the mechanism
of elimination given in this section it suffices to assume that
v 2 ...isa set properties
of x.
2
This explanation of the meaning of 'independent' is somewhat
but
.

X X

imprecise

our present purpose. Cf. von Wright [11], p. 42f.

For the notions of positive and negative analogy see Keynes, p.

suffices for
3

223ff. If we consider
the three sets of properties A, By C; A, B, D; A, C, E, then,
according to the definition
given, the positive analogy between the sets is (the set, the only member of which is)
the property A, and the negative analogy is (the set, to which belong) the properties
4

Nicod, p. 229. Nicod is the first author, who has clearly seen the seriousness of the
restrictions to the logical power of the elirninative method which follows from
possible

Complexity of Conditions.
5
The possibility that the necessary condition of A is a conjunction of properties does
not concern us. If the presence of B and C is necessary for the presence of A, then the
presence of B is necessary for A and the presence of C is necessary for A. In symbols:

&

[Ax-+Bx&Cx]=~(x) [Ax-*Bx]
(x) \Ax~+Cx]>
conjunctive necessary condition
thus a case of Plurality and not of Complexity of Conditions.
6
Mill did not distinguish between Plurality and Complexity of Conditions (Causes).
From failure to make this distinction arise several mistakes in his account of inductionIt is
noteworthy that at least in one place (bk. Ill, chap, x, 3) he considers the possibility
(in our terminology) of a disjunctive necessary condition. He, however, mistakenly
describes it as a case of Plurality of Conditions. Cf. von Wright [1 1 ], p. 161f.
7
detailed description of the working of the logical mechanism of elimination hi this
case is given in von Wright [11], pp. 102-16.
(x)
is

Cf. von Wright [11], p. 94f.

The corresponding typical* case never occurs in the search of necessary conditions.
For it is a characteristic logical difference between sufficient and necessary conditions of
9

a given phenomenon A that the presence of

entails the presence of all its necessary
conditions, but that A may very well occur in the absence of some (or maybe even all)
of its sufficient conditions. For a more detailed account of the Logic of Conditions see

von Wright [11], pp. 66-77.

10
See above p. 64 and the present
11

The possibility that

section fn.

1.

the sufficient condition of A

201

is

disjunction of properties does

THE LOGICAL PROBLEM OF INDUCTION

not concern us. If the presence of B or C, 'no matter which one', is sufficient for the
presence of A, then the presence of B is sufficient for the presence of A and the presence
of C sufficient for the presence of A. In symbols: (x) [Bx v Cx-*Ax\=(x) [Bx-*Ax] &
\x) [Cx-+Ax]. A disjunctive sufficient condition is thus a case of Plurality and not of
Complexity of Conditions. Cf. above fn. 5.
12

Cf. Nicod, p. 229.

Mill's recognition of complexity in sufficient conditions (causes) enters in the form
of the reservation which he (sometimes but not always) makes when, speaking of his
Method of Difference, he says that the circumstance, in which alone the instances differ,
13

is

the cause or a part of the cause of the investigated phenomenon. See Mill, bk.

Ill,

2.

chap, vra,
14

A detailed description

16

See Mill, bk.

is
given in von Wright [11 J, pp. 116-19.
chap, vra, 4 and von Wright [11], pp. 97-102 and 119-26. Mill
also describes two further methods, called by him those of Residues and of Concomitant
Variations. These two methods, however, do not contribute anything new to the logical
mechanism, as such, of eliminative induction. (Cf Nicod, p. 220 fn. and von Wright [11],

Ill,

methods was anticipated in Herschel's Discourse on

Study of Natural Philosophy which appeared thirteen years before Mill's Logic.
Mill's description of his

p. 160f.)

the

(See especially Herschel, p. 151ff.) As Herschel's greatest contribution to the theory of

induction, however, must be regarded the emphasis which he laid on what he called
'residual phenomena', i.e. phenomena not accounted for by known laws of a given
context. (Ibid., p. 156ff.) AJI important class of such residual phenomena are so-called
'counteracting causes'. The theory of residual phenomena is connected with conven-

tionalism. (See above chap, m,

see also Jevons [2], p. 558ff.
16
Cf. Broad [8], p. 311.
17

It is

and

For

2.)

residual

phenomena

hi inductive logic

noteworthy that in the search of necessary-and-sufficient conditions (use of the

Method) there is no corresponding restriction to the 'direction of elimination*

among possible complex conditions. See von Wright [11], p. 123.

Joint
18

See Kaila [6], p. 97ff.

Mill, bk. Ill, chap, vra,

19

2. Cf. above fn. 13.

Lachelier (p. 31) speaks of '1 'induction scientifique' as opposed to 'rinduction
vulgaire* intending, as far as we can judge, precisely the distinction between eliminative
and enumerative induction.
21
Cf. Mill, bk. Ill, chap, x, 2: *A single instance eliminating some antecedent which
existed in all the other cases, is of more value than the greatest multitude of instances
20

which are reckoned by their number alone.

22
Cf. Keynes p. 217. Keynes makes the acute observation (p. 218f.) that Hume
apparently on this point misrepresented the nature of inductive argument in his criticism
of it. See also Stocks, p. 202.

Remarks about

5.

the comparative value

Mill, bk. Ill, chap, vra,

See

e.g.

2 and

of the methods of Agreement and Difference.

3.

Fowler, p. 157f.

Mill, bk. Ill, chap, vra, 3: 'It thus appears to be by the Method of Difference alone
that we can ever, in the way of direct experience, arrive with certainty at causes.'
4

Ibid.

See

ditions

is

e.g. ibid., bk. Ill,

.

which being

the cause of B,

if

is the sum total of the conconsequent invariably follows/ That is to say:

in time and universally implies it.

chap, v, 3:
realized, the

precedes

'The cause, then ...

202

NOTES
6
7

Ibid., bk.
Ibid., bk.

Ibid.

This

Ill,

Ill,

chap, vni,
chap, vm,

and

3.

1.

the point were the fallacy enters. Italics mine.

'
'unconditional antecedent would
be 'necessary condition in time'. This
is
excluded
both
however,
interpretation,
by the
use which Mill otherwise makes of the term 'cause* (cf. above fn. 5) and on the
ground
that the method of Agreement can
prove an antecedent to be unconditional in the sense
of being the (only possible) necessary condition. Thus under this 'natural'
interpretation Mill's statement would be
simply false. It must, therefore, be interpreted as is done
by us in the text.
11
Mill himself (ibid.), erroneously, connected the formulation mentioned with 'the
is the only immediate
impossibility of assuring ourselves' (in the given example) 'that
antecedent common to both the instances'.
12
MH1, bk. Ill, chap, vm, 3.
13
Mill, bk. Ill, chap, vin, 3 and Fowler, p. 158.
10

is

Ibid.

The most natural interpretation of the term

14

above

Cf.

p. 65.

The general postulates of induction by elimination.

See Ellis [2], p. 84, where the postulate of Limited Variety is called The fundamental
principle in virtue of which alone a method of exclusions can necessarily lead to a
6.

positive result.
2

Bacon

243 and Bacon [4], vol. Ill, p. 357. Cf. Ellis [1], p. 28: The
view
that it is possible to reduce all the phenomena of the universe to combinations
of a limited number of simple elements is the central point of Bacon's whole system/ It
is, however, not quite clear what the principle of Limited Variety, as it occurs in Bacon 's
philosophy, really means and to what extent it is covered by the idea that the number
of different properties of an individual is finite. For a lucid discussion of the Baconian
idea of limited variety and different possible meanings of it see Broad [6], p. 35ff. Bacon's
ideas on this point are probably related to another favourite idea of his, viz. that of a
complete and definite collection of all human knowledge. (See Spedding, voL I, p. 369ff.)
All these ideas might be regarded as kindred to the Mathesis-Universalis-idesi, characteristic of the systems of Descartes, Leibniz and other philosophers of the seventeenth
century. The relatedness between the scheme of an inductive logic and the Mathesis
Universalis becomes quite apparent in Robert Hooke's treatise A General Scheme of
the present state of Natural Philosophy, which was published posthumously in 1705.
Hooke tried to develop the general method of Bacon, without, however, contributing
anything essentially new to it, into a 'Philosophical Algebra' (note the mathematical
analogy !) which makes it 'very easy to proceed in any Natural Inquiry, regularly and
.

[3], vol. Ill, p.

certainly'.

(See Hooke, p.

6f.)

Keynes, p. 251. Keynes introduces his postulate for the purpose of securing a finite
a/>rz0n-prebability hi favour of each one of all concurrent hypotheses as to a conditioned
property of a given characteristic. See below chap, vr, 5.
4
E.g. if the properties are sense-qualities. Cf. Kaila [6], p. 203.
5
similar objection to the use of the postulate of Limited Variety for the purpose of
assigning finite a /?jior/-probabilities to the concurrent hypotheses was made by Nicod
against Keynes. See Nicod, p. 278 fn.
e
Whereas Bacon resorts to the postulate of Limited Variety, the inductive logic of
Mill may be said to be based on the postulate of Completely Known Instances, although

203

THE LOGICAL PROBLEM OF INDUCTION

never explicitly stated by him. (Keynes, p. 270f.) From the tacit assumption of this
Method of Difference obtains part of its illusory strength. (Cf. Jevons
[3], p. 295ff. and Kaila [6], p. 99.)
7
See Kaila [6], p. 98ff.
8
It is uncertain whether the term 'property here is not used to denote a relation.

it is

postulate Mill's

'

For this idea

10
11

I.e. it is

Bacon

see Maxwell, chap, i (the end), Schlick [2], p. 147f.,

conceivable that it were an analytical truth.

[3], vol. Ill, p.

Hempel

[1], p. 31.

242.

12

Mill, bk. Ill, chap. xix. Mill also says (bk. Ill, chap, rx, 6) with reference to
Whewell, that his inductive methods 'are methods of discovery'. But at the same time
he adds that they are also methods of 'proof. It remains, however, uncertain whether

'proof here means a demonstration that a certain general proposition is true or a

demonstration that a certain general proposition is, in a given context, the only possible
generalization.

Hibben, p. 166. The author states that we may by the inductive method detect that
has been the cause of B in given instances, but that from this it does not necessarily
follow that we can generalize as to the constant conjunction of A and B. In this statement we find ourselves confronted with the source of much ambiguity in discussions
about the aim and power of the inductive logic, viz. the failure to distinguish sharply
between the following two meanings of the word 'cause': (1) A characteristic with the
*power of producing' another characteristic universally, i.e. always when certain conditions are fulfilled, and (2) A characteristic which is the only one that in given cases has
been conjoined with another characteristic.
14
Cf. the above quotation from Bacon (p. 63) and Mill, bk. Ill, chap, in, 3: 'That all
swans are white, cannot have been a good induction, since the conclusion has turned out
13

erroneous.

15

For this term see Nicod, p. 223. Nicod, however, states the principle only in the
weaker form that every characteristic has a sufficient condition. In Keynes again there
226) a corresponding postulate of eliminative induction called the principle of
Uniformity of Nature. (Cf. Broad [1 ], pt. n, p. 13ff.) Keynes has not observed that, since
the method of elimination which he describes is roughly that of Agreement, his principle

is (p.

applies only to necessary conditions.

16
For this formulation of the Deterministic

Assumption and its implications see

pp. 72-7 and 131-5. If the law is of the form (2)
of 3 there must be added some qualification as regards the correlating function F.
Those possible qualifications will not be considered here.

Broad

[8], p.

307 and von Wright [11

],

It must furthermore be assumed that Mill's Universal Law of Causation applies

only to sufficient conditions, to judge from the definition of cause given by him. Cf.
above chap, rv, 5, fn. 5.
18
Mill, bk. Ill, chap, xxn, 4: To overlook this
was, as it seems to me, the capital
error in Bacon's view of inductive philosophy.*
17

19
The Baconian 'forms* and 'natures' have, however, also peculiarities other than
those of being simultaneously existing necessary and sufficient conditions of each other.
Bacon has rightly seen that other peculiarities must be demanded in order to distinguish
the characteristics as forms and natures respectively. Thus he requires the 'form' to
be something of the kind of a differentia specifca of a genus proxirnum (Cf. Bacon [2],
lib. II, aph. iv and xv). This connects his doctrine of induction with the Aristotelian
doctrine of definition (Kotarbinski, p. 11 Iff). Bacon also seems to have thought, at
least at a certain stage in his development, that the 'form' ought to be an external
feature (Kotarbinski, p.
physical property, the 'nature* again a

phenomenological
204

NOTES
I13ff).

Therewith his doctrine of forms and natures becomes connected with the

distinction between 'primary*

20

See Nicod, p.

and 'secondary'

qualities (Ellis [1], p. 28ff).

229ff.

The justification of the postulates of eliminative

7.
1

induction.

Above

chap, n, 7.
2
Cf. above chap, n, 2, fns. 13 and 16.
3
See chap, n, 2, fn. 16.
4
These findings constitute the so-called 'paradoxes of strict (necessary) implication'.
These and other principles of modal logic, mentioned here, are explained in LewisLangford, Symbolic Logic (New York, 1932) or in the Author's An Essay in Modal

Logic (Amsterdam, 1951).

6
No reason in support of this assumption will be given here. The idea appears highly
plausible and it has, as far as we know, never been contested.
6

For a

similar argument see Broad [12], pt. i, p. 23f.

See above chap, iv, 1.
One might make the objection to the use of enumerative induction that it cannot even
confirm the Deterministic Assumption, and correspondingly Mill's Universal Law of
Causation. For the assertion that A has a sufficient or necessary condition is a universal
proposition and hence unverifiable (since it would be circular to suppose it to have been
verified through the eliminative process itself). This important point seems to have
escaped Mill's notice. Mill constantly speaks about the confirmation of the Universal
Law of Causation through enumerative induction, as if it were actually possible to
7

verify single causal uniformities.

9

Cf. above chap,

10
11
12

rv,

1.

The

metalogical proof of this will be omitted here.

From the above argument on p. 82.
This qualification is necessary in order to avoid circularity.

8.

The eliminative method and the justification of induction.

This task the Author has tried to accomplish in his book

Probability (London, 1951), especially chaps, iv

CHAPTER
1.

v.

and

A Treatise on Induction and

v.

INDUCTION AND PROBABILITY

The hypothetical character of induction.

1
Newton
Newton [1],

[2], p.
lib. III.

See also the fourth of Newton's Regulae Philosophandi in

This must not be confused with Newton's own use of the term

31.

'hypothesis'. Cf, Lalande, p. 123ff.

z
Huyghens, vol. XIX, p. 454. See also Lalande, p. 146.
3
use and of
were not
If
incidentally, of what

what

interest

would

it

hypotheses,
they
then be to draw the conclusions and make the tests?
4
This is particularly true of Whewell. Cf. Stoll, p. 91
5
See Jevons [2], p. 152 and p. 218ff. Jevons uses 'hypothetical* and hypothesis' in
several senses which are not always clearly kept apart. (.Cf. Johnson [1], vol. Ill, chap,
fact that
12.) By the 'hypothetical* character of induction we here mean simply the
n,
When
demonstrative
however,
not
Jevons,
does
induction
certainty.
yield
ampliative
the first stage in the inductive process of thought,
says ([2], p. 265fT.) that 'hypothesis' is
the second and third stage being deduction and verification respectively, what he has in
in the sense of the so-called hypothetico-deductive method. See
mind is

Note

'hypothesis'
at the end of this section.

205

THE LOGICAL PROBLEM OF INDUCTION

6

For a 'typical' misunderstanding see Fowler's criticism of Jevons. (Fowler, p. 9ff.)

Fowler is wholly unable to see the significance of the fact that inductive generalizations,
used for predictions, are hypothetical. The following quotation (ibid., p. lOf.) is interesting as it illustrates the very difficulty on this point which Fowler entirely over-looks:
'I maintain as
against Mr. Jevons that many of our inductive inferences have all the
certainty of which human knowledge is capable. Is the law of gravitation one whit less
certain than the conclusion of the 47th Proposition of the First Book of Euclid? Or is
the proposition that animal and vegetable life cannot exist without moisture one whit
less certain than the truths of the multiplication table? Both these physical generalizations are established by the Method of Difference, and as actual Laws of Nature' (cf.
above chap, iv, 6, fn. 13), 'admit, I conceive, of no doubt. But it may be asked if they
will always continue to be Laws of Nature? I reply that, unless the constitution of
the Universe shall be changed to an extent which I cannot now even conceive, they will
so continue, and that no reasonable man has any practical doubt as to their continuance.
*

Note on the role of Induction and Hypothesis in Science.

1
'Hypothesis and 'hypothetical* are used in many different senses in connection with
The
induction.
hypothetical character of induction as understood above and as emphasized by Jevons ought not to be confused with the use of the hypothetical or hypothetico7

deductive method in the inductive sciences.

Some modern authors (K. Popper, J. O. Wisdom) emphasize, against induction, the
role of hypothesis and the hypothetico-deductive method. Wisdom ([2], p. 7) goes as
that there is no inducfar as saying 'that induction plays no part whatever in science

method and that nothing approximating to inductive inference is used*. Novelty is

claimed (Wisdom [2], p. 49) for the approach of Popper's, who is said to be the first to
in metascience'.
give to the hypothetico-deductive method 'serious attention
In face of these modern exaggerations it may be useful to remember:
That the 'metascientific' appreciation of the hypothetico-deductive method is clearly
(i)
manifested in the remarks on scientific method which we find in the works of many of
the champions of modern natural science such as Galileo, Pascal, Huyghens, or Leibniz
(see above chap, iv, 2 and Lalande, pp. 83-109 and 146-71), and
(ii) That the role of hypothesis in science and the relation between induction and the
hypothetico-deductive method has been ably and extensively studied by authors on
scientific method in the 19th century, foremostly by William Whewell and E. F. Apelt.
(See Whewell [3], bk. II, chap, rv-vn and bk. Ill, chap, v, vi and rx; Whewell [4]; Apelt
[1 ], especially pp. 56-64.) It is particularly regrettable that the writings of Whewell on
the philosophy of science have largely fallen into oblivion among modern authors,
probably under the influence of a text-book tradition in inductive logic which has been
nourished mainly by Mill's theory of the canons of elimination.
Anyone who maintains that 'induction plays no part whatever in science' is advised
to study the examples of the use of inductive methods given in Mill's Logic (particularly
in bk. Ill, chap. rx). Against these examples it could conceivably be objected that they
are all of a rather 'primitive' kind. This rejoinder should not be met by assuming offhand that examples of a more 'advanced' nature could be added. It is worth considering, whether the nature of the examples, which are found in Mill's or Bain's works on
inductive logic and similar books, do not indicate some essential limitation to the use
of induction as a 'method' in scientific investigation. To this extent the 'critics of
induction' may be right. Here the following observation suggests itself (see also above

tive

chap, iv,

2):

Situations, in which generalizations are framed,

206

may be

divided into

two

types.

In

NOTES
situations of the first type it

is clear or
'practically clear' which are the features (characterproperties) that possibly lend themselves to generalization. Of this type are many,
or perhaps most, of the situations in which we look for the cause
(condition) of a
phenomenon among a number of antecedent or coexistent phenomena. (Perhaps the
best examples are found in the branch of medicine called
aitiology.) In sucli cases the

istics,

generalization emerges from the observation of a regular concomitance in the occurrence (or variations) of two features (or
groups of features). The 'method of generalization is either induction
by simple enumeration or induction after the elimination of
concurrent possibilities.
That such situations are frequent in
every-day thinking is obvious and that they
occur in 'science' too can hardly be disputed. But it would be a mistake to believe that
all cases of generalization are of this
type. We know of no author on inductive logic, who
had held this belief; but some authors may be said to have
overemphasized the importance of this kind of situation in science. (Cf. Mill, bk. Ill,
4-7 on* the
xiv,
'

chap,
hypothetical method.)
In situations of the second type, the
possible generalizations cannot be 'read off' by
merely inspecting some experiential data. The introduction of a new concept is required
which, as Whewell says, 'colligates' the facts, gives them a uniting feature. As a prototype
of such cases, may be regarded the tracing of a curve
through "a number of points, the
coordinates of which (in a diagram) are given
by observation. Here the colligating
concept is the curve (or rather, its mathematical law')- It is, usually, first introduced
'as a 'hypothesis' which the points are
subsequently shown to fit. In this Verification
procedure deduction plays an essential part. The 'method' involved is thus: hypothesis
plus deduction.

What has been

said so far, however, is

only one aspect of the 'hypothetical method*.
consists in the conjectural character of the hypotheses.
They ought
prediction possible. On this the classics of scientific method agree (with the

Another aspect
to

make

reservation, however, that the idea of conjectural hypotheses seems to be peculiar to

Western science; it is not prominent in Ancient science). Whewell says ([3], p. 85f.):

Thus the hypotheses which we accept ought to explain phenomena which we have
observed. But they ought to do more than this: our hypotheses ought to foretell phenomena which have not yet been observed.' And the fact that a hypothesis has been
verified to fit given data is no guarantee that it will not be falsified when
predictions
from it are confronted with future observations. (Cf. chap, rv, 2, fn. 26.)
Thus hypothesis in science frequently has a double function. It introduces a new
concept or idea to account for observed data. And it makes conjectures about the
unobserved. The first function presupposes that an invention or discovery has been
made. And inventions are, as Whewell said, *happy guesses* or leaps which are out
of the reach of method'. (See above chap, rv, 2.) To fulfil the second function, is to
reason inductively. Thus induction enters as an ingredient in the hypothetico-deductive
method itself. Whether, in this connection, induction should be called a 'method* or
not, is a matter of nomenclature.
The difference between the two types of situation, just described, may be used for
distinguishing between 'primitive' and 'advanced* generalizations in science. And one
may suggest that use of induction as a 'method' is confined to the 'primitive' cases. But
even if it were true, which it is not, that the hypothetico-deductive procedure were *the
actual method of science* (Wisdom [2], p. 46), this would not, in view of the conjectural
character of scientific hypotheses, minimize the seriousness of the problem of the
justification of induction in science.
These remarks do not claim to give an exhaustive account of the typical uses of
207

THE LOGICAL PROBLEM OF INDUCTION

hypothesis in science. Not all 'advanced' generalizations can appropriately be put
under the title of 'colligation of facts', and not every use of hypothesis for conjecturing
be termed 'inductive'. We have
is
'anticipation from experience' or can appropriately
only against certain *anti-inductivist' claims, wanted to show that theory of induction
cannot, in the name of the hypothetico-deductive method, be banished from holding a
prominent place within the methodology of science.
On the notion of hypothesis and the hypothetical method the following works, now
Die Bedeutung der
largely forgotten, may also be profitably consulted: Biedermann,
Hypothese (Dresden, 1894); Hillebrand, ZurLehre von der Hypothesenbildung (Sitzungsberichte der Wiener Akademie, Philosophisch-historische Classe, Bd. 134, Wien 1896);
Mach, Erkenntnis und Irrtum (Leipzig, 1905); Naville, La Logique de V Hypothese
(Paris, 1880). The best historical survey, known to the Author, of the problems discussed

note

in this

is

Lalande, Les Theories de

V Induction

ei

de

V Experimentation

(Paris, 1929).

Hypothetical induction and probable knowledge.

Reichenbach [13], p. 98: 'Belief can be the motive of action, but belief as such
can never justify an action; only & justified belief can do that.'
2.

Cf.

CHAPTER
1.

vi.

FORMAL ANALYSIS OF INDUCTIVE PROBABILITY

The Abstract Calculus of Probability.

definition of probability, the so-called frequency-view would seem to be the

ancient. It goes back to Aristotle. But the frequency-definition was not used as a
basis for the mathematical study of probability until the nineteenth century. See
1

As a

most

chap, vi, 2, fn. 11.

2
Braithwaite [6], p. 118.
3
This, roughly, is the view taken by Carnap, who distinguishes two (main) concepts
of probability, viz. probability! which is a ratio of possibilities (strictly speaking, of
measures of ranges, see below p. 101) and probability 2 which is a frequency. This distinction does not seem to us a very happy one. The fact, on which it is based, is the
existence of two models of mathematical probability of an identical or at least closely
similar structure. The question, how these models are related to the actual use (applications) of mathematical probability, is very complicated. It would be an oversimplification to think that probability is sometimes used to 'mean* a ratio of range-measures
and sometimes to 'mean' a relative frequency, and that these two usages of it can be

sharply separated.
4
As first attempts in this direction may be regarded Bohlmann (1901) and Bernstein
(1917). Neither paper is mentioned in Keynes's Bibliography.
5
Similar systems for probability have been proposed by Tornier (in [1 ] and [2]) and

by Cramer (in [1]).

6
Cf. Cramer [2],
7

The

p. 151.

Keynes's system along with more recent abstract theories of

probability presents some difficulties. It dates from a period when the general ideas of
axiomatic and formalized systems were much less developed than nowadays. Keynes
presents it, not as an abstract calculus, which is supposed to ]be 'neutral* with
regard to various interpretations of it, but as a form of what is sometimes also
classification of

a belief-theory of probability. (See chap, vn, 1.)

further developed in Reichenbach [4].
This does not exhaust the list. An abstract calculus, in

called
8

And

208

line

with Reichenbach 's

NOTES
rather than Keynes's, was
proposed by the Author in [7] and further developed in [11].
See also below fn. 16.
10
In Reichenbach [4] the probability-relation is first
(p. 56) said to be between events
(Ereignisse) and then (p. 57) said to be between propositions (Satze). In the further
development of the system the relation is actually one between prepositional-functions.
On the question whether probability is appropriately attributed to events or to propositions see Ancillon, p. 4, Boole [1], p. 247f., and Reichenbach [11],
p. 57ff. See also

below
11

fn. 11.

To

these entities may also be counted their linguistic

counterparts', i.e. sentences
as counterparts of propositions and (some kind of) names as
counterparts of attributes.

The

'
interpretation of probability in terms of such 'linguistic counterparts will not be

considered here.
12
That probability is always relative to some evidence has been energetically stressed
by Keynes, who rightly says (p. 6) that 'a great deal of confusion and error has arisen
out of a failure to take due account of this relational
aspect of probability'.
18
See above chap, v, 3.
14
In Keynes 's system it is explicitly assumed that the
datum-proposition h must not

be self-contradictory. (See Keynes, p. 116ff.)

15
This was maintained, e.g., by adherents hi the nineteenth century of the
dichotomy
between mathematical and philosophical probability. See later, this chapter 8, fn. 9.
16
took
the
view
that
there are non-numerical (as distinct from
Keynes (chap, in)
unknown numerical) probabilities and that not all non-numerical probabilities are
comparable. An axiom system of comparative probability has been given by Koopman
79-85 and [15] and Shimony.
(in [1]). See also Carnap [11],
17
See above p. 16.
18
For a fuller treatment of the notion of independence the reader is referred to von

Wright

[11], p. 193f.
fuller treatment see

19

For a

20

Cm

is

von Wright

[11], p. 199ff.

the so-called binomial coefficient. Another symbol for it

ni

for the value

n! again

means the product

Ix2x3x

...

is

m j.

It

stands

xn.

21

The proof is not given here as it is of no relevance to the epistemological problems

under discussion. The proof makes no further use of principles of probability, but relies
on considerations of a purely arithmetical nature. It may be found in any text-book
on probability-mathematics.
22
See m. 21. It was this second part of the theorem which, save for a minor difference,
was proved by James Bernoulli ([1 ], pp. 236-8). An elegant proof both of the maximum
principle and of the limit theorem, which uses, only elementary means and closely
follows Bernoulli's own deductions, is found in Kneale [1], 28 and 29.

2.

The

interpretation

offormal probability.

See von Wright [1], p. 6f.

See Reichenbach [3], p. 404.
3
As an example of a model which falls under neither category may be mentioned the
geometrical model given in Reichenbach [2], p. 588ff.
4
The term truth-frequency appears originally to have been suggested by Whitehead
(see Keynes, p. 101). It is used also by Carnap, Reichenbach, and others.
2

209

THE LOGICAL PROBLEM OF INDUCTION

5

See Reichenbach

Model
6

On

For a thorough presentation of the Finite Frequency-

[7].

the notion of a limiting frequency or proportion see above chap,

later chap,
7

18.

[4],

see also Russell

vm,

i,

2.

Also

1.

This important point,

it

seems to

us,

has not received

sufficient attention either

from von Mises or from Reichenbach, not to speak of earlier proponents of the frequency-theory of probability. It is emphasized by Braithwaite C[6], p. 125) and put
forward by him as an objection against the Frequency-Limit-Model as a proposed
analysis of the meaning of probability.
8
See von Wright [11], p. 80f.
9
See Reichenbach [4], 18.
10

Important mathematical contributions to the problem of random distribution are

found in the papers of Church, Copeland [2], Feller, and Wald. For a discussion of the epistemological aspects of the notion of randomness the reader is referred to

to be

von Wright [2], [7] and [1 ].

11
As mentioned above (chap, vi, 1, fn. 1) the frequency-view of probability goes
back to Aristotle. A probability, says Aristotle in [2], 70 a4,is 'what men know to happen
or not to happen, to be or not to be, for the most part thus and thus '. (See also Aristotle
a
[4], 1357 .) A similar view of probability was taken by writers of the seventeenth and
1

eighteenth centuries, other than those

who

studied probability in connection with

games of chance. Thus Locke (bk. IV, chap, xv, 1) says that 'probability is nothing but
but is, or
whose connection is not constant
the appearance of such an agreement
appears, for the most part to be so'. (See also below chap, vu, 2, fn. 5.) In connection
with a mathematical theory of probability the frequency-view made its first appearance
from the same year 1843, viz. Cournot [1], Ellis [3], and Mill.
(See von Wright [6].) Mill, however, in later editions of his Logic (see bk. Ill, chap,
xvm, 1) 'recanted' his earlier criticism of the view of probability taken 'by Laplace
and by mathematicians generally' and withdrew from the frequency position. None of
the three authors mentioned attempted a rigorous construction of the mathematical
calculus on the basis of a frequency-model. The first to do this was Venn in 1866.
Venn also was the first to make use of the notion of a limiting frequency. An improved
version of the frequency-theory was presented by von Mises in 1919. The modern form
of the theory is best studied in von Mises [2] and in Reichenbach [4]. A very good
semi-popular account is found in Nagel [5].

in three publications

12

of the German Spielraum

'range' may be regarded as a translati
in probability theory by J. von Kreis (1886).
13
It is usually not clear which alternative is intended by those authors (particularly
of an earlier epoch), who speak of probability as an attribute of events.

The English word

v/hich

was introduced

14
A range model, in which the terms are attributes, is given in Kneale [1]. In the
most elaborate account of the range-theory which exists, viz. Camap [11], the terms

(arguments of Carnap's probability-function) are sentences. The choice of sentences

rather than propositions as terms has certain technical advantages, but fundamentally
there is no great difference between the two possibilities. See Carnap [11],
10 and
52.
15
I.e. we substitute for h a
logically identical proposition h which overtly has the
form of a disjunction of n propositions of the required nature. That h' overtly has this
form should mean that the sentence h'\ expressing the proposition #', is a disjunctionsentence of n atomic sentences, each of which expresses a proposition which entails
either a or r^a and no two of which express
compatible propositions. If h itself overtly
has the required disjunctive form no analysis is needed.
'

210

NOTES
"

Laplace's view of the philosophical nature of probability is best studied in [6].

For the notion of a truth-function and of a normal form see
chap, i, 2, fn. 1 or
consult any modern text-book on logic.
18
In the limiting case, the set o- may consist of the
propositions a and h themselves
and no other propositions.
19
This essentially answers to the definition of probability within a
theory of truthfunctions proposed by Wittgenstein (5.15ff.). It is also substantially the same as Bolzano's (147, 161 and 167) definition of
as 'relative
(relative
17

Giiltigkeit).

Our

probability

definition

validity'

might therefore be called the Bolzano-Wittgenstein

defini-

tion.

In the theories of Bolzano and Wittgenstein the set a is identified with the set of
atomic propositions, of which a and h are overtly truth-functions. This is the set of
and Vz' are
propositions expressed by the atomic sentences of which the sentences
molecular complexes. (See above fn. 15.) Our definition is thus somewhat more general
than the definitions actually proposed by Bolzano and by Wittgenstein.
As specialized forms of the Bolzano-Wittgenstein definition may be regarded the

on the basis of hypothetico-disjunctive judgments, given by

and Czuber ([3], vol. I p. 5); the definition on the basis of
disjunctive judgments, given by Lange (p. 99ff.) and Stumpf; the definition on the basis
of hypothetical judgments, given by Pick (p. 12ff.); and finally also the definition given
by Mendelssohn (vol. II, p. 248) and by Hailperin.
20
This essentially answers to Carnap's concept of range in [11], 18 D. Our set oplays a role corresponding to Carnap's choice of a 'language'. For a brief account of
the essentials of Carnap's theory see von Wright [10].
21
These units essentially answer to Carnap's state-descriptions. ([11], 18A.)

definition of probability
Sigwart (vol. II, p. 314)

22

This definition was, neglecting notational differences, first proposed by Waismann

essentially answers to Carnap's definition in [11], 55A of a regular confirmation
function. It may therefore be called the Waismann-Carnap definition of probability.
23
See Waismann, p. 236.
24
In Carnap's terminology: on the choice of a language and a regular measure func-

and

tion.
25

The

best discussion of the principle

See also Kneale

[1],

31,

34 and

is

probably the one found in Keynes, chap.

iv.

35.

26
J. Bernoulli [1],
p. 224: 'nulla perspicitur ratio cur haec vel ilia potius exire
debeat quam quaelibet alia*.
27
See Waismann, p. 242.
28
The re-birth of the frequency-theory in the nineteenth century (see fn. 11 above)
was ultimately connected with a criticism of the idea of equipossibility in the theory of
Laplace. It was alleged that the equality of alternative possibilities must ultimately
consist in the equal frequencies of their realization 'in the long run'. See Ellis [3] and

Mill, bk. Ill, chap,

xvm,

1.

The doctrine of Inverse Probability.

See Bernoulli [1], p. 224ff., where the author is discussing the determination of
and use of the inverse Law of
probabilities, as he puts it, *a posteriori*, i.e. the nature
Great Numbers. Of this inverted form of the theorem he says, not only that it is
provable (ibid., p. 226 and Bernoulli [2], p. 2f.), but also that he himself had proved it
after twenty years of effort and was going to give the proof in the Ars Conjectandi
(Bernoulli [1], p. 227). Thereupon he gives the proof of the direct theorem (ibid., p.
236ff,), and here the book suddenly ends. It remains uncertain, whether Bernoulli
3.

211

THE LOGICAL PROBLEM OF INDUCTION

regarded the proof given as a proof of the inverted form of the theorem which he had
mentioned before, or whether he intended the real proof to follow in a later chapter. It
is the true
interpretation of
appears to us (cf. ibid., p. 239) that the former alternative
Bernoulli's own opinion. See also Todhunter, p. 73 and M. Cantor [2], vol. Ill, p. 334f.
Curiously enough Cantor seems to believe that Bernoulli really proved the inverted
form of his theorem.

See e.g. De Moivre, p. 251: 'As, upon the Supposition of a certain determinate Law
according to which any Event is to happen, we demonstrate that the Ratio of Happenor Observations are
ings will continually approach to that Law, as the Experiments
Observations we find the Ratio of the
multiplied: so, conversely, if from numberless
Events to converge to a determinate quantity, as to the Ratio of P Q\ then we cojiclude that this Ratio expresses the determinate Law according to which the Event is to
happen. For let that Law be expressed not by the Ratio P Q, but by some other, as
R:S; then would the Ratio of the Events converge to this last, not to the former: which
'
contradicts our Hypothesis. From this interesting quotation is clearly seen that it is the"
confusion of maximum probability with certainty that makes the inversion of Bernoulli's theorem appear self-evident. The conversion would be self-evident if we had
proved by Bernoulli that, supposing an event's probability to be p, the proportion of
that event's happening on all occasions is certainly p. But what we really have proved
is that the probability that the proportion is p on some n occasions, approaches 1 as n
2

For a similar confusion see G. Cantor, p. 362.

approaches oo
3
Keynes, p. 148f. Keynes's Inverse Principle is of a somewhat more general content
than our 770.
4
It should be noted that the binomial coefficient n Cm cancels out.
.

adaptation of the arithmetical considerations underlying the Direct

Probability. Cf. chap, vi, 1, fn. 21.
an adaptation of the arithmetical considerations underlying the Direct

By an

Law

of

Law

of

Maximum
6

By

Great Numbers.
7

The name

used for what

'Bayes 's Theorem' is used very ambiguously in literature. Sometimes it is

the Inverse Principle, i.e. for T10 or some of its generaliza-

we have called

(See Kolmogorov, chap, i, 4. and Nagel [5], p. 29.) Sometimes it is used to

denote the Inverse Law of Great Numbers. (See von Mises [2], p. 147f.) Actually,
Bayes's chief achievement consists in having proved the theorem which we here call
the Inverse Law of Maximum Probability. To this end he had, of course, first to prove
a form of the Inverse Principle. Bayes's considerations apply to 'continuous probabilities', i.e.
they assume that the Bernoullian probabilities of the problem cover the whole
to 1 inclusive. Bayes, however, did not use integration but relied on
range from
'geometrical' considerations. (See Bayes [1], p. 388ff.) The first to prove the formulae
involved in the problem by the use of integration was Laplace in a Memoire of the
year 1774. It appears that, in 1774, Laplace was not acquainted with the essay of Bayes.
Concerning the history of Bayes's Theorem see the next fn. and fn. 3 of the next section.
8
The asymptotic property of the expressions (Z>) and (/)'), i.e. the Inverse Law of
Great Numbers, was not known to Bayes. It was noted by Price, who communicated
Bayes's paper for publication. (See Bayes [1], p. 418 fn.) It was proved for (>') i.e. on
the assumption of equal initial probabilities, by Laplace in the Memoire of 1774. The
corresponding proof for (>), which is 'practically independent' of initial probabilities,
is of later date. (See Bachelier [1], vol. I,
p. 472f. and p. 488; Edgeworth, p. 228; von
Mises [2], p. 148f.) We have not been able to discover, who was the first to produce it
and thus to complete the proof of that part of Bayes's Theorem or the inversion of
Bernoulli's Theorem which we here call the Inverse Law of Great Numbers.
tions.

212

NOTES
9

The name 'Rule of Succession'

appears originally to have been suggested by Venn,
(See Venn [1], p. 190) The use of the name in literature is
Often it refers
ambiguous.

only to the formula

It

^r^-

seems to us useful to
distinguish between a numerical and a

non-numerical form of the law.

10
See Laplace [1], vol. VIII, p.

30f., [3], vol. X, p. 325fF., [6], vol. VII,

p. 16ff. Prior
to Laplace, Price (see
Bayes [1], p. 405ff.) had used a somewhat different formula for
the determination of 'the
probability of future events'. See also Mendelssohn, vol. II,

where

given the formula

for the
probability that if two events have been
j
observed n times in conjunction, they will be
constantly conjoined,
The problem behind the Law of Succession can be
put quite generally as follows:
If an event has occurred
times on n occasions, what is the
probability that the
event will occur i times on the k next occasions? On the same
assumptions as those
made in proving Bayes 's Theorem, we get the value:

p. 264ft.,

is

i
J

On

the additional assumption that

all

the initial probabilities q are

p
equal

we

get,

after integration, the expression:

k! x (m+i)f
i!

If

i-sks-j, we obtain from

will

m = n,

on the next
11

and

This

we

--~

value

n+2

for the r
probability
J that the event

has occurred

if it

^or

77-

(n+1)!

IV x mix (nrm)!

t ^ie

times in n occasions.

it

+7n s xAfs <m

assumed

von Wright

If,

probability that the event will occur

has occurred on every one of some n occasions.

but a consequence of the following arithmetical truth: If 1
2 <.
Af's and -WS
then m 1
+Ms==M'l +M'2 +
S,

AT-values are
Cf.

+ k-m-i)!x

w <w

is

obtain the value

occasion, if

M^M^r

XM +
12

this the

occur on the next occasion,

further

x(n

(fc-i)/x(/H-& -h

xM +m2XM

<M'

a -f.

to be in the interval" from

[11], p. 21 3f. for

fuller

-fws xM's
to

,<m s

xM +m
1

The m- and M- and

1 inclusive.

treatment of the problem.

The Non-

Law

of Succession, naturally, can be proved in a more general form which

deals, not with the question of the probability of the event's occurrence on the next
occasion, but with the question of the probability of its occurrence on the k next occa-

Numerical

which can be deduced from either the non-numerical or

and which has much impressed the philosophical
imagination, that the greater this number k is, the smaller is the calculated probability.
See^Quetelet [2], p. 21 and de Morgan [2], p. 21 3f. It was on similar considerations
about diminishing probabilities that Craig based his notorious calculations concerning
the dying-out of belief in Christianity, on the supposition that this belief was based solely
on oral or written tradition. See Craig, pp. 21 and 24.

sions.

It is

a mathematical

fact,

the numerical version of the law

213

THE LOGICAL PROBLEM OF INDUCTION

18

we know, has first been shown by Broad. See Broad

and von Wright [11], p. 215.

This, as far as

[12], p. 196f.,

pp 4-9 and

[7],

Criticism of Inverse Probability.

4.
I
3

For good

illustrations of this idea see

Kneale

A characteristic use

[1],

27 and Nagel

[5], p. 30f.

of the Inverse Principle (Formula), was for

determining, with
probability, whether a phenomenon was due to a cause or to chance. This meant a
probability-judgment on the respective alternatives that the probability of the phenomenon, on a given condition, was 1 and that it was, on the same condition,
Kirchhoff
determined the probability that the occurence of the 60 dark
rays, observed by him at
characteristic places in the spectrum of the sun, were
by chance coinciding with the rays
in the spectrum of iron by this use of inverse
79.) Hartprobability. (See Kirchhoff,
.

mann

p.

has offered abhorrent instances of the uncritical use of the formulae

for determining the probability of causes,
calculating inter alia the probability for a
non-material cause operating in a given case to 0.9999985! For formal treatment of
the formulae for estimating the likelihood of causes and critical remarks about their
(p. 24-35)

applicability see Bertrand, p. 142ff.

3
Bayes himself never spoke of his

theorem as one of the probability of causes. His

treatment of the problem is purely mathematical and free from
philosophical
aspirations. Price, however, who communicated Bayes 's paper to the Royal Society,
regarded the formula as being relevant to the estimation of probable causes and to
inductive conclusions in general. (See Bayes [1],
p. 402fT.) With Laplace's abovementioned Memoire (see especially vol. VIII, p. 29ff.) the use of the formula
being an
instrument for determining the probability of causes was
ultimately established. The
literature, predominantly from the late eighteenth century, illustrating this use of it is
very extensive. As examples, apart from the writings of Laplace, may be mentioned,
Quetelet [1], p. 123ff. and [2], p. 24fL, the paper of Lhuilier and Prevost called 'Sur
1'art d'estimer la
probabilrte des causes par les effets' (1799), and Trembley's paper
*De probabilitate causarum ab effectibus oriunda' (1795).

own

Keynes,
B

and

p. 82.

For various usages and acceptances of the formula

[2], p. 18;

de Morgan

[2], p. 213ff.;

Jevons

see, e.g.

[2], p. 257ff.;

Quetelet [1], p. 128

[1], p. 425: Bobelc,

Lotze

p. 207f.; Whittaker, p. 163ff. ;

Edgeworth, p. 234.
See Keynes, p. 383 and Bobek, p. 207f.
Among such authors Jeffreys and Carnap are most prominent. Carnap 's Quantitative System of Inductive
Logic ([11], 110) may be regarded as a revival, with modifications and further developments, of essential
aspects of the classical doctrine of
6

inverse probability.
8

See Boole [1], p. 368ff. for very acute criticisms. See also Bryant and Broad
393-400 and [7], p. 19-23.

[1],

pt. I, pp.
*

Keynes,

removed

if

p. 377f.

probabilities of the
10

The

alleged inconsistency which Keynes sees in the formula is

is
paid to the data relative to which the various

due consideration
problem

See especially Fishex

exist.

[2], p.

10 and

[7], p. 6f.

the Bibliography.

and the minor publications

listed in

II

For an attempt at a logical and

epistemological clarification of the problems of
inverse probability see von
Wright [7], pp. 60-6 and [11], chap, x. This attempt, we
feel, still stands in need of improvement.
12

Laplace

[2], p,

419: 'Lorsqu'on n'a aucune donnee a

priori sur la possibility d'un

214

NOTES
evenement,

il

faut supposer toutes les

1'unite egalement
possibilites, depuis zero jusqu'a
^
&

probables.'
13
For the idea of 'equal distribution of
ignorance' see Boole [1], p.370 and Donkin
p. 354. In the first edition of the present work
(p. 121) an attempt was made to account
tor the
as a
of
equality of a
priori probabilities
what we mean by 'equal
consequence
For an appraisal of this
attempt see Broad [12], p. 119. Edgeworth and
Pearson ([!], p. 365rT. and [2],
p. 143fL) tried to justify the assumption of equality on
an empirical basis. All the
about the value of a

ignorance'.

possible hypotheses
proportion (probaare initially
equally probable, because experience is alleged to show that all
proportions (probability-values) as a matter of fact occur equally often in nature. As
Edgeworth (p. 230) put it: 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'. For a criticism of this idea see
Keynes, p. 381ff. and von Wright
[11], p. 283.
bility)

Confirmation and probability.

See above chap, v, 1,
especially fn. 7.
2
For example the. theory developed
by Rudolf Carnap.
degree of confirmation
means with Carnap the same as a degree of
probability, defined as a relative measure of
ranges. (See above 2 of the present chapter.) The proposition
conferring a degree of
probability or confirmation upon another proposition need not be entailed
by the
latter, nor need it be, in any obvious sense of the word, an 'instance of it. The
probabilified (confirmed)
proposition again need by no means be a generalization. It is, oa
the contrary, characteristic of
Carnap 's theory that any (numerically unrestricted)
Universal Generalization possesses a
zero-probability, relative to any (finite) number of
confirming instances of it. (See Carnap [11], 110f.) In an important sense this theory
is
incapable of evaluating the bearing of individual instances
conclusions
5.

'

upon general
already for this reason, no Confirmation-Theory at all in our sense of the word.
This incapacity, in our opinion, must be considered a serious defect of
Camap's treatment of induction, irrespective of whether one wishes to restrict the use
of the term
'Confirmation-Theory' to the treatment of converse entailment-relations (as is done by
us), or use it in some wider sense (as is done
by Carnap). See also above chap, i, 3.
3
For the idea that induction by simple enumeration,
although unable to reach certainty, yet contributes to an increase of probability in a generalization see

and

is,

Huyghens,
XIX, p. 454, Bayes [1], p. 406, Mendelssohn, vol. ii, p. 267f., Poisson, p. 161f.,
Lotze [2], p. 70. For the idea that the
strength of support which a confirmation affords
to a law is inversely
see
proportionate to the confirmation's initial
vol.

p. 170f,,
4
5

Broad

probability,

[1], pt.

I,

p, 402, Russell [4], p. 194f.,

and Kaik

Herschel,

[6], p. 105f.

Cf. Keynes, p. 237ff.

Cf. Nicod, p. 248 and 2524.

e
This is the form in which the condition of
convergence towards maximum probability
was stated by Keynes (p. 236f.) and Nicod (p. 276). Keynes, moreover, substitutes for
it a somewhat
stronger condition (op. cit., p. 238). The reason, why the authors mentioned do not discuss the condition in the
simple form, first introduced in the text, is
probably that their proof of the Principal Theorem is more complicated than the simple

proof given here.

7

For a

precise definition of this type of the notion of a limit see

54f.

215

von Wright

[11], p.

THE LOGICAL PROBLEM OF INDUCTION

s

For a statement of

this

axiom

in a

more developed symbolism

see

von Wright

[1 1

],

I76f.

p.

6.

Cf.

von Wright

p. 248.

[1 1 ],

The Paradoxes of Confirmation

For discussions of the notion of a confirming instance the reader is referred to

86-8.
Hempel [6] and Carnap [11],
2
Hempel [6], p. 9ff. and Carnap [11], 87. See also Nicod, p. 219.
3
Hempel [6], p. 124 and Carnap [11], 87.
4
The Paradoxes of Confirmation were first hinted at by Hempel in [3] and discussed
by Hosiasson-Lindenbaum in [3] and by Hempel in [6]. Hempel, however, does not
make a clear distinction between the paradox which results from the clash between the
Nicod- and Equivalence-Criteria, and the paradoxes which are special cases of the
familiar Paradoxes of Implication. The discussion with Hempel and Carnap mainly
centres round the first type of paradox. In von Wright [1 1 ], pp. 254-6 the second type of
paradoxes
7.
1

2
3

is

discussed.

Confirmation and elimination.

Cf. Keynes, p. 226, 234 and 236.
Cf. Nicod, pp. 249-65 and pp. 269-73.
In the first edition of the present work the

Author sided with Nicod against Keynes

both on the question, whether the increasing probability of a generalization approaches

1
as a limit, and on the question, whether confirmation contributes to probability
von Wright [7],
independently of elimination. That this was a mistake was shown in
3 and 4 and chap, x, 5.
3 and 4 and von Wright [11], chap, rx,
chap, nr,
4
For a more detailed development of this idea see von Wright [11], chap, ix, 4,

5
\ B is co-present
'Co-presence with A thus means "not-absence in the presence of A
with A in x, if either A and B are both present in x, or A is absent and B present, or A
and B are both absent. This definition of co-presence is made necessary by the fact that
we regard anything which satisfies the prepositional function Ax >Bx as affording a
'genuine' or 'paradoxical') of the law (jc) (Ax-+Bx).
confirming instance (either
1
6
'Co-absence with B thus means 'not-presence in the absence of B\ A is co-absent
with B in x, if either A and B are both present in x, or A is absent and B present, or both
A and B are absent. We notice that the two phrases "B is co-present with A' and 'A is
co-absent with B' mean exactly the same. Cf. fn. 5 above.
7
The Author has also worked out a Range-Model of the Principal Theorem, using
the general theory of measuring ranges which is developed in Carnap [11], It turns out
'

<

that in this model the condition /?^ + i

1, which in the Frequency-Model means eliminais tantamount to certain measures of
ranges becoming zero. No conditions for the

tion,

existence of such zero-measures can be deduced within the model. But it becomes
reasonable to assume that the ranges in question acquire a zero-measure ('become
extinct') because of the incompatibility of a new confirming instance fm + 1 with some or

number of alternatives* covered by the evidence e, to the generalization g.

this supports the idea that increase in probability is, in the case of the RangeModel too, effected by elimination.

other of a

And

8.

Probability\ scope,

and

simplicity.

Reasoning from analogy.

Mathematical and

philosophical probability.
1
Cf. Keynes, p. 224ff. Keynes 's reasoning in this place
every detail correct.

216

is

neither very clear

nor in

NOTES
2

This must not be confused with Keynes's use of the term

'analogy' or with our use
above in chap, iv, 4.
'For a fuller examination of the argument from
analogy within a theory of scopes see
von Wright [1 1 ], chap, ix, 7. In Carnap [1 1 ], 1 10 D there is an outline of an analysis
of reasoning from analogy within a Range-Model of probability. Carnap 's
analysis
resembles ours in that it relates analogy to scope of propositions (in Carnap 's
terminology
'width' of properties). The main difference seems to lie in the fact that
Carnap does
not relate reasoning from analogy to considerations about nomic connections between
characteristics. (Cf. our idea that the analogy contains some factor or
conjunction of
factors which is 'causally responsible' for the occurrence of the property, whose
presence
we know in one thing and conjecture in another thing.) The outline given in Carnap
[1 1 ] is too sketchy to make possible a more detailed comparison of the two attempts to
clarify the logical nature of reasoning from analogy. Although analogy may be said to
belong to the traditional topics of inductive theory and the methodology of science,
there are evidently very few attempts at a formal treatment of the
subject. The only
formal examination, beside Carnap 's and ours, known to us, is in Hosiasson-Lindenbaum [4] and J. R. Weinberg [2].
of

it

4
forceful early expression of this idea is Leibniz's comparison between
discovering
laws of nature and rinding the key to a cryptogram. See Couturat [1], p. 254f. and [2],
175
and
232.
Kaila
See
also
[6], p. 103ff., where an interesting suggestion is made,
p.
p.
relating the idea of the cryptogram and of simplicity of curves to ideas of inverse probaLeibniz's thought on the topic calls for a more thorough examination than has
bility.

been given to them in

5

Cournot

literature.

saw

in the

comparison of curves with regard to their

and in curve-fitting the basic type
He says (ibid.): 'En general, une
theorie scientifique quelconque
peut etre assimilee la courbe que Ton trace d'apres
une definition mathematique, en s'imposant la condition de la faire passer par un
certain nombre de points donnes d 'avance.
6
In the first edition of the present work and in von Wright [1 1 ] the theory of scope
([2], vol. I, p. 82)

simplicity the basis of all probability of inductions,

of all making of theories and hypotheses in science.
.

'

and probability was regarded

For similar ideas

probability.

Kaila

as a special case of a general theory of simplicity

see Bolzano, vol. II, 151, Broad [1], pt. i, p. 402,

and
and

[2], p. 139f.

For contributions to a clarification of the notion of

see Goodman [3] and
simplicity
[4], Kemeny [4], and Lindsay. For discussions of probability in relation to simplicity
of curves see Cournot [2], vol. I, p. 82, Weyl, p. 155f., Jeffreys [1], p. 43ff. and [2],
Braithwaite [3], Popper [2], p. 87fT., and Kaila [6], p. 103ff.
8
In the first edition of the present work a treatment of probability and simplicity of
curves was attempted. The treatment, however, was most unsatisfactory and contained
a bad error. For a conclusive criticism of it with some interesting positive suggestions
see Broad [12], pt. m, p. 199ff.
9
The distinction between mathematical and philosophical probability was, as far as
we know, made for the first time by Fries in his System der Logik (1811). It was later
developed by the same author in the Kritik der Prinzipien der Wahrscheinlichkeitsrechnung (1842). Fries regarded philosophical probability essentially as an attribute of
inductions. (See Fries [3], p. 16ff.) The distinction between the two kinds of probability
was current with many authors on induction, probability, and scientific method in the
nineteenth century. See Apelf [1], p. 38ff.; Beneke, vol. II, p. lOltT.; Cournot [1], p.
440 and [2], vol. I, p. Tiff, and vol. II, p. 386; Drobisch, p. 177; Grelling, p. 459ff. and
in relation to the notion of simplicity
p. 478. For the notion of philosophical probability
217

THE LOGICAL PROBLEM OF INDUCTION

see especially Cournot [2], vol. I, p. 71ff.; Poirier, p. 107ff.; and Picard [2], p. 436. For
a distinction resembling that between mathematical and philosophical probability see
also Peirce, vol. II, p. 416ff. In recent times the distinction between probability as an
attribute of inductive conclusions and as an attribute of 'events' finds favour with many

Some prefer not to call the former 'probability' at all. Thus Carnap, Hempel,
Hosiasson-Lindenbaum, Popper, and others call the probability of inductions (hypothe36) speaks of acceptability, Braithwaite([6],
ses) degree of confirmation', Kneale ([1],
of reasonableness. These notions with some modern
p. 120f. and pp. 354-60) speaks
authors.

authors cannot, however, all be equated with the notion of philosophical probability
with the authors mentioned of the nineteenth century (or Grelling). Kneale 's concept
of acceptability comes very near the classical notion of philosophical probability. But
his followers
it is
important to observe that the distinction made by Carnap and
'
between 'probability/ (degree of confirmation) and 'probability 2 is not directly
comparable with the classical distinction between philosophical and mathematical
within a Range-Model (probaprobability. Carnap 's distinction is between probability
a Frequency-Model (probability 3) of abstract probability.
bility!) and probability within
To call probabilitVj by the name of 'degree of confirmation' is, in our opinion, somewhat misleading. (See the present chapter 1, fn. 3 and 5, fn. 2.) Mention should also
be made of the distinction which Russell makes in [7] between credibility and mathematical probability. By the former he means the probability of an individual event on
all relevant information as data. Russell's notion of credibility, it would seem, is
thus a limiting case of the generic notion of mathematical probability. (Cf. von Wright
[ll],p.302f.)
10

The

a
possibility of such

formalism

is

mentioned by Popper

([2], p. 245)

with a

'reference to Hosiasson-Lindenbaum.

CHAPTER
1.
1
2

Probability

PROBABILITY

AND THE JUSTIFICATION OF INDUCTION

and degrees of belief.

This well-known saying comes from Bishop Butler. (See Butler, p. 3 and passim.)

Ramsey,

p. 169.

This doctrine

is

expounded by Ramsey,

p. 170ff.

This must not be confused with the fact that, on the above second way of defining
degrees of beliefs, if my belief hi a to degree/? and in b to degree q are true beliefs, then
belief in a&b to degree/? x q would be true too.
5
Consider the well-known psychological phenomena underlying the arguments from
'maturity of odds'. If I toss ten successive times 'heads' with a homogeneous coin I
am likely to expect 'tails' rather than 'heads' in the following toss. Various less successful attempts have been made to
justify such arguments on the ground of their not having
a purely psychological foundation. See Maibe, Sterzinger, and Kammerer. For a
criticism of these attempts see v. Mises [2], p. 166-72.
6

That

a true interpretation of the view taken by the adherents of the 'psycholoconfirmed from the following statements of De Morgan ([2], p. 172f.):
'By degree of probability we ... ought to mean degree of belief ... I ... consider the
word (sc. "probability") as meaning the state of the mind with respect to an assertion
on which absolute knowledge does not exist. *Tt is more probable than improbable"
means ... "I believe that it will happen more than I believe that it will not happen".
Or rather 'I ought to believe &c. '.
this

is

gical* theory is

'

'

213

NOTES
From the quotation in the preceding footnote it is apparent that
not realize this consequence of the regulative function of formal
7

De Morgan

did

probability.
Donkin (p. 354), after mentioning the principle of 'equal distribution of
ignorance'
as an expression of how beliefs are distributed on a set of alternative
propositions, says:
This being admitted as an account of the way in which we actually do distribute our
belief in simple cases, the whole of the
subsequent theory follows as a deduction of the
cases
we would be consistent. This
way in which we must distribute it in
8

'

complex

statement,

it

must be noted,

is

false.

It

if

appears uncertain to the Author whether

when he described

(see especially p. 180, 182 and 188) clearly apprehended this

the laws of probability as laws of
consistency for partial beliefs.

Ramsey

2.

Rationality of beliefs

and success

in

predictions.

This answer, interpreted as here is done, seems to be in exact accordance with

Keynes's opinion as to the basis on which rational degrees of belief are determined.
See Keynes, p. 17: 'I assume then that only true propositions can be known . . and
that a probable degree of rational belief cannot arise
out of the
directly but only .
knowledge ... of a ... probability-relation in which the object of the belief stands to
.

some known
2

'

proposition.

might be suggested that a statement concerning symmetry and homogeneity in a

coin involves an inductive element. This we are not going to dispute. Our argument
only requires the not unplausible/cfww that the question as to the properties mentioned
were settled by reference to known (i.e. not-inductive) data concerning weight, shape and
centre of gravity of the coin.
8
According to A^ on p. 93.
It

4
Keynes explicitly expresses the opinion that the criterion of rationality in beliefs is
altogether independent of any reference to success in predictions. (See Keynes, p. 107
and p. 322f.) In spite of this he strongly insists that degrees of probability, as rational

degrees of belief, justify induction. The

validity of the inductive method does not
*
depend on the success of its predictions. (P. 221.) 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".' (P. 323.) As already observed there is no objection to such statements as
these, if one is clear about their philosophical implications. Otherwise they might be

thoroughly misleading.
5
This important truth has been clearly apprehended and expressed in Venn [1], p.
150f.; Peirce, vol. II, p. 394; and Ramsey, p. 188, 196 and 202. (Ramsey, p. 199:
*Reasonable degree of belief = proportion of cases in which habit leads to truth. )
The idea of probability as a *guide of life* being a 'guide to success' is contained already
in Locke's opinions on probability. (See especially Locke, bk. IV, chap, xv, 1 and 4.)
Probability, according to Locke, is to supply the (defect of our knowledge, and to guide
the proof being such as for the most part carries truth with it.'
us where that fails
5

Cf. Kahle, p. 22 and Price, p. 410,

Cf. Jevdns [2], p. 261: 'All that the calculus of probability pretends to give, is the
result in the long run, as it is called, and this really means in an infinity of cases. During
*
any finite experience, however long, chances may be against us. See also Ramsey, p, 207.
8
What we call the Cancelling-out of Chance is roughly equivalent to the basic
7

Kaila ([1], p. 9, p. 41 and passim.) calls 'Kontinprinciple of inductive arguments which

aller moglichen Ffille'.
genzprinzip' and Bruns (p. 13) 'gleichmassige Erschopfung

219

THE LOGICAL PROBLEM OF INDUCTION

The Cancelling-out of Chance and the theorem of Bernoulli.
For the idea that the Laws of Great Numbers were laws of nature assuring order
and uniformity, see e.g. Laplace [4], p. 360f., [6], p. 47; Poisson, p. 7; Quetelet, [3],
261. The natural philosophy of the school of Laplace was
p. 15, [4], p. 38f.; Lacroix, p.
much inspired by Hume's philosophy of the causal relation and the similar theories
of Condillac. (See e.g. Lacroix, p. 3ff.; also Ellis [3], p. 1 and G. Cantor, p. 366.) To
the philosophers of the school mentioned the principles of probability served as a kind
of 'substitute* for the uncritical belief in the uniformity of nature, which was 'destroyed
by Hume. A much later, and naive example, of the use of probability as a weapon
3.

'

late president of the Republic

against Hume is offered by the little book by Masaryk
of Czechoslovakia entitled David Hume'sSkepsis nnddie Wahrscheinlichkeitsrechnung.

(See especially Masaryk, p.

De Moivre

14f.).

in the Laws of Great Numbers a proof of the prevalence

of 'Intelligence and Design* in nature. Kant ([5], vol. VIII, p. 17) found them to have
some bearing on the problem of the freedom of the will, so also Quetelet ([3], p. 70).
8
The error in relating the theorem of Bernoulli to the fact called the Cancelling-out
of Chance consists, in its gravest form, in that a statement about a probability is believed,
as a consequence of the theorem mentioned, to imply a statement about a proportion.
have been guilty of this error. Some
Distinguished mathematicians and philosophers
114f. and [2],
p. 184). In
very significant examples are offered by De Morgan ([1], p.
the latter place the author says that 'it is a remote, but certain, conclusion from the
that events will, in the long run, happen in numbers proportional to the
theory
322.
probabilities '. See also Lambert, vol. II, p.
2

(p. 252f.)

saw

*
For a detailed analysis of the relation of probability to possibility see Meinong [1].
James Bernoulli ([1], p. 212) calls probability a degree of certainty. He does not speak
n this point the account given in Kneale
of degree of belief, as does later Laplace,

[1], p.
B

We

124

is

in error.

that the gravest error in relating the theorem of Berin the interpretation of the relation of
maximum probability as one of implication.
might now say that the next-gravest
error consists in uncritically giving to the theorem the interpretation just mentioned in

have already said

(fn. 3)

noulli to the Cancelling-out of

Chance consisted

We

the text, according to which the increasing probability is taken to be a frequencyOf this error Czuber ([3], vol. I, p. 154) presents a nice example. He says:
4
Der Sinn des BernoulHschen Theorems ist dahin zu verstehen, das mit wachsender
Versuchszahl die absolute Differenz zwischen der relativen Haufigkeit und der Wahrsch*
im attgemeinen abnimmt. It is to be noted that Czuber does not interpret
einlichkeit
but
in terms of 'Spielraume*. One must accuse
in
terms
of
probability
frequencies

constant.

Keynes of a similar error to that made by Czuber, although Keynes on the whole adopts
a very guarded attitude to the use of Bernoulli's theorem for predicting averages. See
Keynes, p. 109, p. 337, p. 344 and passim. Cf. also Cournot [2], vol. I, p. 61 and v.
Kries, p. 81.
*

might be suggested that one of the reasons why we are apt tacitly to assume a great
in the sense in which it occurs in the theorem of Bernoulli
to be realized
frequently, and again a small possibility very seldom, is connected with the following
mathematical 'picture*. Take two events such as 'heads* and 'tails* in tossing with a
coin. In each toss there are two possibilities
we shall call them possibilities of the first
order
one of which will be realized. We can denote them with 1 ( == 'head') and
= 'tail*). In two tosses there are four such possibilities of the first order, 11, 10, 01,
(
00. By & possibility of the second order, again, we shall mean the class of all possibilities
'
of the first order, containing the same number of digits, which contain, 'heads* and 'tails
It

possibility

220

NOTES
in a given proportion. (E.g. the class [10, 01] is such a
possibility of the second order.)
Continuing the construction of such rows of digits, each one corresponding to a certain
possibility of the first order for the occurrence of 'heads' and 'tails', we shaU find that
the longer the rows become, the greater is the
proportion (among rows with a given
number of "digits) of rows which contain 'heads' and 'tails' in roughly the same proportion, whereas rows where the distribution of 'heads' and 'tails' differs
considerably
from the value |- become the rarer, the longer the rows are. Thus to each
possibility of
the second order there corresponds a relative
for
the
occurrence
of its classfrequency

members within

the class of all possibilities of the first order with the same number of
frequencies, furthermore, are found to equal the magnitudes of
those possibilities for the occurrence of 'heads and 'tails * in a
given proportion, which
are determined by the probabilities of the second order in the theorem of Bernoulli.
1
Consequently this mathematical picture of the possible ways in which 'heads and 'tails'
may be realized contains a frequency-interpretation of the possibilities determined by
the theorem of Bernoulli, and it seems
highly plausible to assume that this frequencyinterpretation will influence our inclination to assume that also the actual realization of
those possibilities will take place with frequencies proportionate to their relative
frequencies in the 'picture'. Cf. Ellis [3], p. 4.
digits.

These

relative

'

See e.g. Drobisch, p. 191; Czuber, [3], vol. I, p. 210; Waismann, p. 242. Note also
the remark by Quetelet ([2,], p. 12) that if a coin fell regularity twice on one face, while
it fell but once on the other, the
tossing of the coin might be considered as presenting
three possibilities, two of them in favour of one face, and one in favour of the other.
8

See

e.g. v. Kries, p. 5fL

and compare

it

with the analysis which follows in the text.

Or, if general propositions are also included in the class of propositions mentioned,
a statement about proportions cannot follow from them unless one at least of those
general propositions is itself a statement about proportions. This trivial truth is of farreaching importance. From it follows, for example, that from the knowledge of the
physical constitution of a die, together with the known or assumed laws of Newtonian
mechanics we can never deduce that the proportion of, say, aces will be such and such.
This would be possible only if this bulk of knowledge itself contained some assumption
about proportions, e.g. the assumption that certain causes, operating according to the
rules of classical mechanics, are divided into different categories in determinate proportions. (See also v. Mises [2], p. 90fl)
10
It is to be observed that the suggested measurement on the basis of properties of
symmetry merely served as an illustration. What is said here applies to all conceivable
ways of measuring degrees of possibility.
11
On the other hand the theorem of Bernoulli, when probability is interpreted statistithe
cally, ought not to be taken as asserting the Cancelling-out of Chance itself. For
Cancelling-out of Chance only implies that events will be realized in the long run with

frequencies proportionate to their probabilities, whereas the theorem of Bernoulli

contains an assertion about the frequencies with which frequencies of the event, in
finite series of occasions, will be realized, supposing that the Cancelling-out of Chance
takes place. This difference is sometimes overlooked. (See e.g. Poisson, p. 7 and
v.
Charpentier, p. 35 for such confusions. For a clarification of this important point see

Mises

[2], p. 129f.

and

p. 136.)

79-98 and M. Cantor [2], vol. Ill, p. 339f. Leibniz (in

the letter from 3. XII, 1703) in opposition to Bernoulli's idea that one could determine
the probable value of human life with ever-increasing probability on a statistical basis,
makes the very acute observation that 'novi morbi inundant subinde humanum genus,
rerum limites
quodsi ergo de mortibus quotcunque experimenta feceris> nan idea naturae
221
12

See Leibniz

[4], vol. Ill, p.

THE LOGICAL PROBLEM OF INDUCTION

posuisti,

ut

pro future variare nonpossit*. Bernoulli (the

letter

from

20. IV, 1704)

was

wholly unable to grasp the epistemological significance of this ingenuous remark of

Leibniz.
It is interesting to observe that also mathematicians of the school of Laplace were not
of making certain inductive assumptions if the
altogether unaware of the necessity
calculus of probabilities were to be applicable to proofs concerning future events,
although they never realized the epistemological significance of this necessity. (See

e.g.

Laplace

[6],

pp. 14, 48, and 53f.,

and Condorcet,

p. 10.)

The knowledge that the theorem of Bernoulli and other

propositions of the probabilitycalculus can be applied to inductive predictions solely on the condition that we have
already made some assumptions as to the future, is related to the well-known statement
that 'probability presupposes causality'. (See Mach [3], p. 283, and Struik, p. 51 and
65f.) For analyses of the causal conditions for this applicability see Poincare [3],
p,

p. 64-94
4.

and Hopf.

The idea of 'probable success

9
.

the truth-frequency of a proposition we mean the proportion of values of a

variable, which satisfy a certain propositional-/crto, among all values of the variable
1

By

in question.

(See above chap,

vi,

2, p. 98f.)

See chap, ir, 7.

8
See above p. 143f.
4
Zilsel is of the opinion that the fact which impresses us as the 'Ausgteich desZufalls'
obtains its character of an unquestionable truth from a convention. (See Zilsel, especially
to us essentially right but, as Kaila ([1], p. 69fF.)
p. 123.) Zilsel's analysis appears
acutely observes, the statement on the Cancelling-out of Chance, when used as a basis
for induction, must be taken as synthetical. It deserves mention that Ellis, who gave the
first substantially correct detailed criticism of Bernoulli's theorem as a bridge from
of Chance to be a
probabilities to empirical frequencies, believed the Cancelling-out
synthetical truth a priori roughly in a Kantian sense. See Ellis [3] and also Fick, p. 2

and
*

p. 46.
Cf. E. J. Nelson, p. 580:

'No "justification" is worthy of respect unless it is based

principles the application of which will in theory probably lead to success.'
Cf. chap, vi, 8, p. 136 and fn. 9, p. 217.
Keynes' chief objection against the frequency-interpretation is that it makes proba-

upon
6
7

bility-statements inductive and consequently deprives probability of the power of justifying induction. 'The Calculus of Probabilities', he says (p. 96 fn.), 'thus interpreted, is

no guide by itself as to which opinion we ought to follow.* Therefore, according to

Keynes (p. 95) the statistical interpretation can at most only cover part of what we mean
by probability, the other sense in which probability is used being that justifying induction. 'It is, in my opinion', he continues (p. 96), 'this other sense alone which has
importance'. See also Broad [11], p. 487 for the suggestion that, since probabilitystatements in frequency-interpretation are inductive, another kind of probability might
'
be needed for judging the likelihood of those statements truth.
8

supporters of 'philosophical' probability have clearly

and Cournot ([2], vol. II, p. 386) decisively
state that philosophical probability cannot be estimated numerically. The latter author
for example says (ibid.) that 'la probabilite phttosophique repugne tout a fait a une
evaluation numerique\ It is, however, plausible to assume that these authors wished to
deny only a metrical, and not a topological quantification of philosophical probability,
although they did not explicitly state the necessity of the latter.
It is

questionable whether

apprehended

this.

Both Fries

all

([3], p. 18)

222

NOTES
9

10

Chap, vii, 2.
Cf. Popper [2],

p.

4 and Feigl

[2], p. 25.

11

Probability taken as a 'GrundbegrhT, in other words, cannot justify induction as

and
leading to 'probable success'. It is questionable whether authors such as

Keynes

who

regard probability as a kind of undefinable fundamental idea, have clearly

apprehended how 'empty' of empirical content is this undefinable probability-concept.
12
One might object to our view that it moves in the grooves of the 'classical' twovalued logic, and that the nature of inductive argument cannot be
grasped until we have
left this logic for the more
comprehensive system of a 'probability-logic \ (Cf. Reichenbach [4], p, 360 and p. 377.) To this objection the following rejoinder will be sufficient;
It is possible to
develop a formal system, treating degrees of probability, which exhibits
certain analogies to the formalized systems of two-valued logic. This
system we might
then call 'probability-logic'. (For the development of such a system see e.g. Reiehenbach [4], p. 379ff. and [5].) But from the mere fact that such a formalism can be
developed nothing follows as to the justification of induction. For its capacity to justify
induction the probability-/^fc is dependent on the same conditions as any formalism
of probability, and those conditions, we have tried to show, are such as to make any
argument that probability were the 'guide to success' circular. (See also Tarski, p.

Jeffreys,

174ff.
13

and Hertz.)

our opinion, evident that this fallacy of thought underlies the whole of
as to probability and the justification of induction. Consider, for
example, the following passage: Keynes states (p. 309) that although it is not certain
It is, in

Keynes 's reasoning

that

we

shall ultimately succeed in preferring the

more probable

to the less probable,

this
nevertheless be probable. That success is 'probable' implies,
seems to be Keynes *s opinion here that we shall 'generally* or 'on the whole* succeed
in preferring the more probable to the less probable. Concerning this implication as to

the success

may

future frequencies he states that it need not be certain. It is, evidently, sufficient only to
assume it to be itself 'probable*. On this point Keynes breaks the hierarchy of superimposed probabilities. This is fatal, because his argument then becomes supported by a
new statement on frequencies which remains unformulated but is nevertheless implicitly

contained in the reasoning.

14
Hume [2], p. 15: 'Nay, I wUl go further, and assert, that he could not so much as
All
prove by any probable arguments that the future must be conformable to the past.
betwixt
probable arguments are built on the supposition, that there is this conformity
the future and the past, and therefore can never prove it. This conformity is a matter of
*
See also Hume [1 ], pt. iv, 1 and E. Cassirer, vol. II, p. 264.
fact.
15
Hume [1 ], bk. I, pt. iv, 1 s As demonstration is subject to the control of probability so
a reflex act of the mind . Here then arises a
is
probability liable to a new correction by
'
. and so on ad
of probability to correct and regulate the first
new
infinitum.
:

species

16

5.

Hume

[2], p. 15.

and relative justification of induction with probaLogical and psychological* absolute

bility.

This truth is usually expressed by saying that all induction proceeds upon the
will resemble the past, and that this principle must be taken for
principle that the future
of all proofs concerning the future. Cf.
granted without proof, since it is itself the basis
1

Hume

[2], p. 15, Poirier, p.

216 and Olzelt-Newin.

Cf. chap, vr, 2.

8
See e.g. Mill, bk. Ill, chap, rv, 2: 'Experience testifies, that among the uniformities
which it exhibits or seems to exhibit, some are more to be relied on than others; and

223

THE LOGICAL PROBLEM OF INDUCTION

number of instances, with a
uniformity, therefore, may be presumed, from any given
as the case belongs to a class in which the
in
of
assurance,
proportion
greater degree
uniformities have hitherto been found more uniform.'
4
For attempts in the same direction see Poirier, p. 94ff., and Reichenbach [4], 71,
[7], p. 274ff., and [9], p. 348-73.
5

Popper [21 p. 192.

For an argument in support of
Nagel [5], pp. 60-75.
6

CHAPTER

viii.

this attitude see

von Wright

[11], P. 243f.

See also

INDUCTION AS A SELF-CORRECTING OPERATION

Induction the best mode of reasoning about the unknown. The ideas ofPeirce.
The characterization of the use of induction as being the adoption of SL policy seems
to be of recent origin. It is used by Kneale (especially [1], pt. iv) and Braithwaite [5]
and [6]. With the authors mentioned and some other recent authors (Reichenbach,
J. O. Wisdom) the problem of the justification of induction may be said to have under1.

gone a transformation. From having been a problem of ascertaining the conditions of

truth or of probability in single inductive conclusions, it has become a problem of
showing the superiority of the inductive policy, as such, over rival policies. The change
... in order to justify
in attitude is expressed by Kneale ([1], p. 225f.) as follows:
induction we must show it to be rational without reference to the truth or even to the
of it as a policy to be adopted or
probability of its conclusions, we must conceive
can fail to
understands his situation
rejected and then make clear that no one who
choose this policy.' For a critical examination of this 'pragmatic' or 'practicalist'
approach to the problem of induction see Black [2], pp. 157-90.
2
Peirce, vol. I, p. 28. For Peirce's opinions on induction and their development see
Braithwaite [4], p. 500ff. and Goudge.
4

4
5

II, p. 455f.
Peirce, vol. II, p. 501 f.

Peirce, vol.

Chap,

i,

2.

gives an interesting analysis of this assumption. He appears to have

been of the opinion that the statement 'on a long run of similar trials, every possible
event tends ultimately to recur in a definite ratio of frequency were a kind of synthetical
truth a priori, following from the nature of genera and their species. (Cf. chap, vn, 4,
fn. 4.) Marbe has tried to show that there exist series of statistical observation not
having definite proportions, i.e. limiting-frequencies, for the occurrence of their characteristics. It appears however, as though Marbe had drawn unwarranted conclusions
from his experiments. See von Mises [2], p. 166ff.
7
Reichenbach [4], p. 396. Reichenbach himself does not seem to be aware of the
intimate relationship between his solution of the inductive problem and the ideas of
Peirce. Actually almost everything that is true and essential in the views of Reichenbach
on the justification of induction has already been explicitly stated by Peirce.
8
The proof is roughly as follows: Suppose the number m did not exist. This would
imply that over and over again it happened that the proportion of A 's which are B fell
6

Ellis

[4], p. 49f.)

'

outside the interval p

From this again it follows that there must exist a
e, for some e.
f
value p outside the interval such that the frequency over and over again falls in the
interval p'
But this contradicts (3), which asserts that there exists one
e, for any e.
and only one value p such that the relative-frequency of A's which are B for any e over
and over again falls in the interval/? e. Thus if (4) is false (3) would also be false.

Conversely,

if (3) is true, (4)

must be

true.

224

NOTES
Thi
r,!
|4J, p.

2.

?A

Reichenbach

415.

'

main addition to the

argument of Peirce.

Cf. Rieichenbacfa

Reichenbach's Method of Correction.

The

hi

The

full

description is in Reichenbach [4], 77,

U gh
S Ca
ng ' the C0ntent of Rei ^enbach's Rule of Induction.
-I
'u ?, i |L
Reichenbach

iS

/ ^

See

76 and 80.
posit of the first order consists,
[4],

strictly speaking, in the assumption that the

within certain limits
6 of
p. Reference to the e will be omitted
from our simplified account of the method.
4
That is, falls within some interval
s round a
'mean'. For the sake of simplicity
however, we snail assume that we need only consider exact coincidences of values

limiting-frequency

falls

See Fveichenbach [4],

p. 399ff.
This problem is substantially the same as the well-known
problem of inverse probability which we treated in outline above in chap, vi, 3. The
values/fa/) answer to the
a priori probabilities, the values
/(?/, q k ) answer to the eductive probabilities, and the
calculated values Fi, k to the a
posteriori probabilities.
7
Sometimes there may, for a given /, exist more than one value
/', such that F,FI
It
is
not
how
the correction should be carried out, if this
clear,
equals Fimax.
possibility
happened to be true. Reichenbach does not consider the case. (It might be added that
for sufficiently great values of n, this
possibility does no longer occur.)
8
To say that a posited value is corrected, comes therefore to the
following: A sequence
in which the
limiting frequency of A is the same as the recorded relative frequency of A,
is regarded as less usual than a
sequence in which the limiting-frequency differs (in a
certain assigned way) from the recorded relative
frequency of A in the observed initial
sequence of n members. We may, of course, be mistaken. The sequence may, after all,
be of the unusual kind, and the correction unwarranted. But if this be the
case, we can
be sure that continued use of the method of correction will
finally 'put things right*,
i.e. make us revert to the value first
in accordance with the inductive
posited
principle
on the basis of the recorded relative
frequency of A in the intial segment of n members
of the sequence. That such will
happen follows from the fact that, independently of the
values /(<//), the values F/, t
converge towards I with increasing n. (Cf. above chap, vi,
We can thus be sure that, for a sufficiently large n, the value of Ft will
3, p. 107ft)
equal Frnax. (See Reichenbach [4], p. 401f.)
9
It is not
quite clear from the exposition given in Reichenbach [4], whether the author
has realized the significance of this
point. See, for example, the discussion in Reichenbach [4], p. 41 3f. of the question, whether another
policy than induction (e.g. that of
consulting an oracle about the true values of the proportions) might lead to a quicker
approximation to the truth.
10
On this point it will be useful to remember a passage in the dispute between Leibniz
and James Bernoulli on the epistemological value of the inverse Law of Great Numbers.
Bernoulli assumed that the records of statistical observation
supply us with approximate
values of probabilities, which values
may be corrected by extended observations. This
process of correcting the values he compares to the calculation of new digits of n, i.e.
to a calculation which can
correctly be called an approximation to the truth. (Leibniz
[4], vol. Ill, p. 9 If.) To this Leibniz (ibid., p. 94) acutely observes that the
analogy is
fallacious. In calculating the
digits of n, each new digit is known to take us nearer the
true value, But whether new observations will take us nearer to the true values of the
proportions about which we generalize is uncertain. That they will do so is an assumption
which is essential to the use of induction, but whether it is true or not we cannot know,
not even with 'probability .
6

,-

225

,-'

THE LOGICAL PROBLEM OF INDUCTION

The goodness of inductive policies reconsidered.

3.

The

idea that inductive policies are self-correcting has been severely criticized by
Black ([2], pp. 168-73). According to Black (ibid., p. 170) the term 'self-corrective' is
a misnomer. A modification which experience may lead us to make in our generalizations, can properly be called a correction only if there is some assurance that the modifications will progressively take us nearer to the truth. As we have seen (this chapter, 1)
such an assurance can exist only relative to the (unprovable) assumption that the proportions, about which we generalize, do really exist. Of the necessity of making this
assumption Peirce, as mentioned above (p. 161), was not even aware. Reichenbach
([4], 80) explicitly avowes it and calls it the assumption that the world is 'predictable'.
J. O. Wisdom introduces a related assumption, which he calls the assumption of a
1

'favourable' universe.

(Wisdom

[2], p. 226ff.)

See above chap. I, 1 and 2 on the notion of a generalization.

3
We shall not here substantiate this doubt with further reasons for it. If it is wellgrounded, it puts a serious limitation upon the value of the Peirce-Reichenbach approach
to the problem of induction. For the Peircean idea of induction as a self-correcting
approximation to the truth has no immediate significance, it would seem, for other
types of inductive reasoning than statistical generalization.
*
Cf. Black [2], p. 158 and p. 172.
5
For the notion of a counter-inductive policy see Black [2], p. 171ff.
*
See above chap. I, 2 and chap, vm, 1.
7
See above chap. I, 2, fn. 8.
8
The possibility of a policy for purposes of prediction and generalization about
of
oscillating frequencies shows that material assumptions concerning the constitution
the universe such as those made by Reichenbach and Wisdom (see above fn. 1) are not
needed in order to warrant successful use of induction.
9

Cf.

10

Reichenbach

80.

[4],

Cf. above chap, v,

2, fn. 1.

CHAPTER

ix.

SUMMARY AND CONCLUSIONS

The thesis of the 'impossibility' ofjustifying induction.

1
Whitehead [1], p. 30.
2
Inductive Reasoning ... the glory of Science ... the scandal of Philosophy'. This
often quoted characterization is from the concluding sentence in Broad [6]. See also
Ramsey, p. 197. Ramsey 's remarks on the nature of the problem of Hume seem to us to
1.

'hit
3

on the head.

the nail
Russell
Russell

'

[4], p. 167.

[3], p. 14.

earlier opinions

See also Russell

on induction

views on the topic see Edwards,

2.
1

The

logical nature

For the

idea that

[5], p.

see Smart,

of Hume

Hay

481.

and for a

[1],

For a

criticism of

critical appraisal

some of Russell's
some of his later

of

McLendon and Reichenbach

[14].

s 'scepticism*.

a contradiction or antinomy

is

inherent in the

demand

for a

justification of induction see the acute analysis in Oxenstierna, especially p. 27ff.

2
Although Hume's results as to the impossibility of justifying induction are, in

our

opinion, fundamentally right and expressed with extraordinary clarity and convincingness, it is obvious that he himself did not take the view that they were 'grammatical' in
nature. This is clearly seen from Hume [1], bk. I, pt. rv, 7, where he considers the
consequences of his results for practical life.

226

BIBLIOGRAPHY
In Keynes, readers

will find

an extensive bibliography of mathematical

and philosophical writings on probability and related subjects such as

induction. and statistics, up to 192L Carnap [1 1] lists most of the relevant
publications on inductive logic and the foundations of probability and
statistical methods up to 1950.
The primary aim of the present Bibliography is to list books and papers
on the epistemological problems of induction and probability. It does not
include works on probability mathematics and statistical
theory, with the
exception of a few works of a historical or synoptic nature and some major
contributions to the axiomatic foundations of probability and to the
doctrine of inverse probability. Nor does it list the rapidly
growing
on the problem of conditionals, a topic related to inductive

literature

theory, nor literature on the causal versus statistical nature of physical

laws, nor on scientific explanation and scientific method in general. But it
aims at including all more important contributions from recent years to

Confirmation-Theory.
The titles of books and longer publications are in italics, the titles of
papers are in quotes. The classification of a work as belonging to the one
or to the other of these two categories is sometimes a matter of taste.
Following continental usage., we have listed names with the prefix de,
del, van, or von under the first letter after the prefix.
The following abbreviations of the names of some of the best known
periodicals in the field of the present work have been used: AJP for the
Australasian Journal of Philosophy, AP for Annee Philosophique, BJPS for
The British Journal for the Philosophy of Science, JP for The Journal of
Philosophy, JSL for The Journal of Symbolic Logic, JUS for The Joufnal

of Unified Science, PAS for Proceedings of the Aristotelian Society, PPR

for Philosophy, and Phenomena logical Research, PR for The Philosophical
for The Review of Metaphysics,
Review, PS for Philosophy of Science,
for Revue de Metaphysique et de Morale and RP for Revue Philoso-

RMM

RM

phique.
'

ACTON, H. B.: The Theory of Concrete Universals. I-II. Mind 45, 1936
and 46, 1937.
AJDUKIEWICZ, K.: *Das Weltbild und die Begriffsapparatur*. Erkenntnis,
4, 1934.

ALDRICH, V. Q: 'Renegade Instances/ PS 3, 1936.

AMBROSE, A,: The Problem of Justifying Inductive Inference/

JP 44 9

1947.

ANCILLON, M.: 'Doutes sur les bases du calcul des probability.' Memoires
de VAcademie Royale, Berlin, 1794-5.
227

BIBLIOGRAPHY
ANDERSON,

O.:

'

'Zur Axiomatik der Wahrscheinlichkeitslehre.

schriftfur Wilhelm Breitzchmayr. Munich, 1951.

ANSCOMBE, F. J.: 'Mr. Kneale on Probability and Induction.'

Fest*

Mind

60,

1951.
F.: Die Theorie der Induction. Leipzig, 1854.
Metaphysik. Leipzig, 1857. [2].
ARISTOTLE: Topics. [1].

APELT, E.

Prior Analytics. [2].

Posterior Analytics.
Rhetoric.

[1],

[3].

[4].

The above works

are here quoted

from The Works of Aristotle transD. W. Ross. Vol. I

lated into English under the editorship of

Oxford, 1928.

VON ASTER, E.- VOGEL, TH.: 'Kritische Bemerkungen zu Hugo

Buch "Das Experiment".' Erkenntnisl, 1931.

Dingier,

'On the Scope of Empirical Knowledge.' Erkemtnis

7, 1938.

BACHELIER, L.: Calcul des probabiiites. Paris, 1912. [1].

Les lots des grands nombres du calcul des probabiiites.

Paris,

AYER, A.

J.:

1937.

BACON,

[2].

F.: Distributio Operis.

Novum Organum.
Valerius Terminus.

[1].

[2].

[3].

The Advancement of Learning.

Cogitata et Visa.

[4].

[5].

from The Works of Francis Bacon, ed.

Heath.
and
London, 1857-8.
by Spedding,
BAIN, A.: Logic. London, 1870.
BAR-HILLEL, Y.: 'A Note on Comparative Inductive Logic.* BJPS 3,

The above works

are quoted

Ellis

1952-3.

[1].

Comments on Popper
BARRETT, W.:

The

[7].

BJPS

5, 1955.

[2].

Present State of the Problem of Induction.' Theoria

6, 1940.

BAYES, TH.: 'An Essay towards solving a Problem in the Doctrine of

Chances.' Philosophical Transactions 53,1763. [1].
*A Demonstration of the Second Rule in the Essay
.'
Philo.

sophical Transactions 54, 1764.

[2].

The papers were communicated by the Rev. R. Price and are

German translation of [1] and [2] by H. E.
partly due to him.
Timerding

in Ostwalds Klassiker, Bd. 169, Leipzig,

of

Facsimiles

[1]

in

of two papers by Bayes.

228

1908.

Ed.

Reprint

by W. E.

BIBLIOGRAPHY
Deming. With comments by E. C. Molina. Washington, D.C., 1940.
E.: Lehrbuch der
Logik als Kunstlehre des Denkens. Berlin,
1832. Quoted from the 2nd edn., Berlin, 1842.

BENEKE, FR.

G.: 'The Logic of Probability.

BERGMANN,
9, 1941.

'

American Journal of Physics

[1].

and Positivism.' PPR 6, 1945-6. [2].

'Some Comments on Carnap 's Logic of Induction. PS 13, 1 946. [3]
BERLIN, L: Induction and Hypothesis.' Symposium. PAS, Suppl Vol.
'Frequencies, Probabilities,

'

16, 1937.
CL.: Introduction a la medicine experimentale. Paris, 1865. [1],
La science experimentale. Paris, 1878. [2].

BERNARD,

BERNOULLI, JAMES: Ars conjectandi. Baal, 1713. [1].

*
Lettre a un Amy surles Parties du Jeu de Paume. (Printed with [ 1 ] .)
'

[2]

Correspondence with Leibniz. (Printed with Leibniz [4],) [3].

German translation of [1] and [2] by R. Haussner in Ostwalds
Klassiker, Bd. 107-8. Leipzig, 1899.
S.:

BERNSTEIN,

'Versuch einer axiomatischen Begriindung der Wahrschein*

Communications of the Mathematical Society

lichkeitsrechnung.
at Charkow, 1917.

BERTRAND, J.: Calcul des probabilites. Paris, 1889.

BIEDERMANN, .: Die Bedeutung der Hypothese. Dresden, 1894.
BLACK, M.: Language and Philosophy. Ithaca, N.Y., 1949. [1].
Problems of Analysis. Ithaca, N.Y., 1954. [2].
S.: 'Concerning a Controversy on the Meaning of 'Probability".'
*

BLOM,

Theoria 21, 1955.

'Demonstration and Inference in the Sciences and

The
Monist, 42, 1932. [1].
Philosophy.'
'
'mile Meyerson's Critique of Positivism. The Monist 42, 1932, [2].
BOBEK, K. J.: Lehrbuch der Wahrscheinlichkeitsrechnung. Stuttgart, 1891.

BLUMBERG, A.

E.:

BOHLMAN,

Article

L.:

'Lebensversicherungs-Mathematik' in Encyklo-

pddie der mathematischen Wissenschaften, Bd.

1901.

BOLZANO,

B.: Wissenschaftslehre.

BOOLE, G.:

An

Investigation of the

Studies in Logic

I,

Teil

Laws of Thought. London,

and Probability. London,

1952.

Probabilite et certitude.

Paris 1943.

Paris 1950.

1854.

[1].

Paris 1941.

[2].

[3].

[4].

et de ses applications.
(editor): Traite du calcul des probabilites
Paris, 1925-39. [5].

229

[1].

[2].

Paris 1914. New edn, Paris 1938.

fi. Le Hasard.
Le Jeu, la Chance et les theories scientifiques modernes.
et la vie.

4b. Leipzig,

Sulzbach, 1837.

BOREL,

Les Probabilities

n D,

vols.

BIBLIOGRAPHY
BOSANQUET, E.: Logic. Oxford, 1911. [1].
The Principle of Individuality andValue. London, 1912. [2].
Implication and Linear Inference. London, 1920. [3].
BRADLEY, F. H.: The Principles of Logic. London, 1883. Quoted from the
2nd edn., London, 1920.
BRAITHWAITE, R. B.: Rev. of Nicod. Mind 34, 1925. [1].
"The Idea of Necessary Connection".' I-II. Mind 36, 1927 and
'

37,1928. [2].
Rev. of Jeffreys [1]. Mind 40, 1931. [3].
Rev. of The Collected Papers of Charles Sanders Pierce/
1934.

Mind

43,

[4].

'Moral Principles and Inductive Policies.' Proceedings of the British

Academy, vol. 36, 1950. [5].
Scientific Explanation. Cambridge, 1953. [6].
BRITTON, K.: Communication, A Philosophical Study ofLanguage. London,
1939.

BROAD, C. D.: 'On the Relation between Induction and Probability.'

MI. Mind 27, 1918 and 29, 1920. [1].
Rev. ofKeynes. Mind 31, 1922. [2].
Rev.of Johnson [1], vol.11. Mind 31, 1922. [3].
'Mr. Johnson on the Logical Foundations of Science.
1924.

'

I-II.

Mind 33,

[4].

The Mind and its Place in Nature. Cambridge, 1925. [5].

The Philosophy of Francis Bacon. Cambridge, 1926. [6],
'The Principles of Problematic Induction/ PAS 28, 1927-8. [7].
d 39, 1930. [8].
'The Principles of Demonstrative Induction.' I-II.
Examination of McTaggart's Philosophy. Cambridge, 1933-8. [9].
*
'Mechanical and Teleological Causation. Symposium. PAS, Suppl.

Mm

vol. 14, 1935.

[10].

Rev. ofvonMises
'Hr.

[2].

Von Wright on the

Mind 46,

1937.

[11].

Logic of Induction.'

MIL Mind

53, 1944.

[12].

Rev. of Kneale

[1].

Mind 59,

1950.

[13].
1

BRODBECK, M.: 'The New Rationalism: Dewey's Theory of Induction.

JP46, 1949. [1].
'An Analytic Principle of Induction?' JP 49, 1952. [2].
BRUNS, H.: Wahrscheinlichkeitsrechnung und Kollektivmasslehre. Leipzig,
1906.

BRYANT, S.: 'On the Failure of the Attempt

from the Mathematical Theory of
Magazine, Suppl.

vol. 17, 1884.

230

to

deduce Inductive Principles

Probabilities.'

Philosophical

BIBLIOGRAPHY
BUCHDAHL,

G.: 'Induction

and

Scientific

Method/ Mind

60, 1951.

BURES, CH. E.: The Concept of Probability/ PS 5, 1938.

BURKS, A. W.: 'Peirce's Theory of Abduction. PS 13, 1946.
The Logic of Causal Propositions. Mind 6, 1951. [2].
'Reichenbach's Theory of Probability and Induction.
'

[1].

'

'

RM

4, 1951.

[3].

'Justification in Science.'

Symposium. American Philosophical Asso-

ciation, Eastern Division, vol. 2, Philadelphia, 1953.

'Presupposition Theory of Induction.'

'On the Presuppositions of Induction.'

BUTLER,

J.:

PS 20,

[4].

1953,

RMS,

[5].

1955.

[6].

The Analogy of Religion. London, 1736.

CANTOR, G.:

'Historische Notizen liber die Wahrscheinlichkeitsrechnung.'

Reprinted in Gesammelte Abhandlungen mathematischen undphilosophischen Inhalts. Berlin, 1932.
CANTOR, M.: Das Gesetz im Zufall Berlin, 1877. [1].
Vorlesungen uber Geschichte der Mathematik. Leipzig, 1880-1908. [2].
Politische Arithmetik.

Leipzig, 1898.

[3].

CARLSSON, G.: 'Sampling, Probability and Causal Inference.' Theoria 18,

1952.

CARMICHAEL, R. D.: The Logic of Discovery. London, 1930.

'
PS 3, 1936 and 4, 1937.
CARNAP,R.: Testability and Meaning.
'On Inductive Logic/ PS 12, 1945. [2].

ML

The Two

Concepts of Probability.'

PPR 5,

1944-5.

[1].

[3].

Reply to Kaufmann [3]. PPR 6, 1945-6. [4].

'Remarks on Induction and Truth.' PPR 6, 1945-6. [5].
'Theory and Prediction in Science.' Science 104, 1946. [6].
'Probability as a Guide in Life. JP 44, 1947. [7],
'On the Application of Inductive Logic. PPR 8, 1947-8. [8].
'

'

Reply to

Goodman

[2].

PPR 8,

Truth and Confirmation.

Analysis,

New

'

York," 1949.

1947-8.

[9].

In Feigl-Sellars, Readings in Philosophical

[10].

Logical Foundations of Probability. Chicago, 1950.

[11].

The Nature and Application of Inductive Logic. Six lectures from

Carnapfll]. Chicago, 1951. [12].
The Problem of Relations in Inductive Logic.' Philosophical Studies
2, 1951.

[13].

The Continuum of Inductive Methods. Chicago, 1952. [14].

'On the Comparative Concept of Confirmation. BJPS 3, 1952-3. [15].
'Inductive Logic and Science. Proceedings of the American Academy
of Arts and Sciences 80, 1953. [16].
'

'

231

BIBLIOGRAPHY
'What is Probability?' Scientific American 189, 1953. [17].
'On the Comparative Concept of Confirmation.' BJPS 5, 1955. [18].
For reviews of Carnap's opinions see Bergmann [3], Hay [2] and von
Wright

[10].

CASSIRER, E.: Das Erkenntnisproblem in der Philosophic und Wissenschaft

derneuerenZeit. Berlin, 1906-20.
CASSIRER, W. H.: A Commentary on Kant's Critique of Judgment. London,
1938.

CAVAILLES,

'Du

J.:

collectif

au pan.
'

recentes sur les probabilites.

CHARPENTIER, T. V.: 'La logique

RP6,

propos de quelques theories

RMM 47, 1940.

du hasard d'aprs M. John Venn.'

1878.

CHATALIAN, G.:

'Probability: inductive versus deductive.'

Studies 3, 1952. [1].

'Induction and the Problem of the External World.

'

Philosophical

JP 49,

952.

[2]

'On the Concept of a Random Sequence. Bulletin of the

American Mathematical Society 46, 1940.
CHURCH, R. W.: Hume's Theory of the Understanding. London, 1935.
CHURCHMAN, C. W.: 'Probability Theory. PS 12, 1945. [1].
Theory of Experimental Inference. New York, 1948. [2].

CHURCH,

A,:

'

Pragmatics, Induction.' PS 15, 1948. [3].

CONDORCET, J. A. N. C.: Essaisur V application de P analyse a la probability
des decisions r endues a la pluralite des voix. Paris, 1785.
'Statistics,

COPELAND, A. H.: 'Probability and Prediction.' Erkenntnis 6 1936 [1].

'Consistency of the Conditions Determining Kollektivs.' Transactions of American Mathematical Society 42, 1937. [2].
9

CORNELIUS, H.: Einleitung in die Philosophie. Leipzig, 1903. [1].

'Zur Kritik der wissenschaftlichen Grundbegriffe. Erkenntnis
'

1931.

2,

[2].

COURNOT, A.

A.: Exposition de la theorie des chances et des probabilites.

Paris, 1843.

[I].

fondements de nos connaissances. Paris, 1851. [2].

COUTURAT, L.: La logique de Leibniz d'apres des documents intfdits. Paris,
Essai sur

1901.

les

[1].

Opuscles et fragments intdits de Leibniz. Paris, 1903. [2].

'
*La logique algorithmique et le calcul des probability.
1917.

RMM 24,

[3].

Cox, R. T.: 'Probability, Frequency, and Reasonable Expectation.'

American Journal of Physics 14, 1946.
CRAIG, J.: Theologiae Christianae Principia Mathematica. London, 1699.
232

BIBLIOGRAPHY
CRAMER, H.: Random
1937.

Variables

and Probability

Distributions.

Cambridge

'

[!].

Mathematical Methods of Statistics. Princeton, 1946.

I. P.: The Justification of the Habit of Induction.

CREED,

CRAWSHAY, R.: 'Equivocal Confirmation.'

CZUBER, E.iZum Gesetz der grossenZahlen.

31, 1940.

ihrer

Anwend-

[2].

Leipzig, 1903.

Wahrscheinlichkeitsrechnung.
edn., reprinted Leipzig, 1932-8.

Quoted from the 4th

[3],

Die philosophischen Grundlagen

Leipzig, 1923.

JP

Analysis 11, 1950-1.

Prague, 1889. [1].

Die Entwicklung der Wahrscheinlichkeitstheorie und

ungen. Leipzig, 1899.

[2].
'

der

Wahrscheinlichkeitsrechnung.

[4].

VAN DANTZIG,

D.: 'Sur Panalyse logique des relations entre le calcul

des probability et ses applications.' Actualites scientifique et industrielleslU6. Paris, 1951.
DARWIN, CH. G.: 'Logic and Probability in Physics.' PS 6, 1939.

DINGLER, H.: Die Grundlagen der Physik. Berlin, 1919. [1].

Physik und Hypothese. Versuch einer induktiven Wissenschaftslehre.
Berlin, 1921.

[2].

Der Zusammenbruch der Wissenschaft. Munich, 1926. [3].

Das System. Munich, 1930. [4],
*t)ber den Aufbau der experimentellen Physik. Erkenntnis2,
'

1931.

[5].

Die Methode der Physik. Berlin, 1938. [6].

'Was ist Konventionalismus?* Actes du XI*me Congres International de
Philosophic, vol. 5. Amsterdam, 1953. [7].
DONKIN, W. F.: 'On Certain Questions relating to the Theory of Proba'

bilities.

Philosophical Magazine, 1851.

DOTTERER, R. H.: 'Ignorance and Equal Probability.' PS 8, 1941.

DROBISCH, M. W.: Neue Darstellung der Logik. Leipzig, 1836. Quoted
from the 2nd revised edn. Leipzig, 1851.
DUBS, H. H.: Rational Induction. Chicago, 1930. [1].
?

The Principle of Insufficient Reason.' PS 9, 1942. [2],

DUCASSE, C. J.: 'Whewell's Philosophy of Scientific Discovery.'
PjR60, 1951. [1].
'

'Deductive Probability Arguments. Philosophical Studies

EDGEWORTH, F. Y.: 'Philosophy of Chance.' Mind 9 1884.

EDWARDS, P.: 'Russell's Doubts about Induction.' Mind 58,

3, 1952. [2].

233

I-IL

1949.

BIBLIOGRAPHY
Works of Francis Bacon, ed. by
and Heath. London, 1857. [1].
'Preface' to Novum Organum. Ibid. [2],
'On the Foundations of the Theory of Probabilities. Transactions of
the Cambridge Philosophical Society 8, 1844. (Paper read 1842.) [3].
'Remarks on the Fundamental Principle of the Theory of Probabilities.' Transactions of the Cambridge Philosophical Society 9, 1856.

ELLIS, R. L.: 'General Preface' to The

Ellis

Spedding,

'

(Paper read 1854.)

[4].

are here quoted from The Mathematical and otherWritings of Robert Leslie Ellis. Ed. by Walton. Cambridge, 1863.
ERDMANN, B.:LogikI. LogischeElementarlehre. Halle, 1892. Quoted from
the 3rd edn. Berlin, 1923.

and

[3]

[4]

EWING, A. C: 'A Defense of

'

Causality.

PAS 33,

1932-3.

FALES, W.: 'Causes and Effects.' PS 20, 1953.

FEIGL,H.: 'WahrscheinUchkeit und Erfahrung. Erkenntnisl, 1930-1. [1].
The Logical Character of the Principle of Induction.' PS 1, 1934.
'

[2].

Principiis non
ed. by M. Black.

'De

.
?'
In Philosophical AnalyDisputandum
Ithaca, N.Y., 1950. [3].
FEEBLEMAN, J.: 'Pragmatism and Inverse Probability.' PPR 5, 1944-5.
FELLER, W.: 'tJber die Existenz von sogenannten Kollektiven.' Funda-

sis,

est

menta Mathematicae 32, 1939.

FEN, S.: 'Has James answered Hume?' JP 49, 1952.
FICK, A.: Philosophischer Versuch iiber die WahrscheinUchkeit en.

Wurz-

burg, 1883.
DE FINETTI, BR.: 'Sul significato soggettivo della probability.'

Funda-

menta Mathematicae 17, 1931. [1].

*La Logiques de la probability Actes du Congres International de
'

Philosophie Scientifique. Paris, 1936.

'La pr6vision: ses

lois logiques, ses

H.Poincarel, 1937.

[2].

sources subjectives.' Ann. Inst.

[3].

FISHER, R. A.: 'Theory of Statistical Estimation.'

Cambridge

Philosophical Society 22, 1923-5.

Proceedings of the

[1].

Methods for Research Workers. London, 1925. [2].

'Inverse Probability.'
Proceedings of the Cambridge Philosophical

Statistical

Society 26, 1930.

[3].

'Inverse Probability

and the Use of Likelihood.

'

Proceedings of the
1932.
28,
[4].
Society
Cambridge Philosophical
The Concepts of Inverse Probability and Fiducial Probability Re234

BIBLIOGRAPHY

Unknown Parameters/ Proceedings of the Royal Society

Section A, 139, 1933. [5].
The Logic of Inductive Inference.' Journal of the Royal Statistical

ferring to

Society 98, 1935.

[6].

The Design of Experiments. London, 1935. [7].

FOWLER, TH.: Inductive Logic.. Oxford, 1869. Quoted from the 6th edn
Oxford, 1892.
FRECHET, M.: 'The Diverse Definitions of Probability. JUS (Erkenntnis)
'

8, 1939-40.

FREUDENTHAL, H.:

'Is

MindS,

bility?'

there a Specific
1941.

Problem of Application for Proba-

J. E.: 'On the Confirmation of Scientific Theories.' PS

17, 1950.
FRIES, J. FR.: Neue oder anthropologische Kritik der Vernunft. Heidelberg,
1807. [1].

FREUND,

J. FR.:
System der Logik. Heidelberg, 1811. [2].
Versuch einer Kritik der Prinzipien der
Wahrscheinlichkeitsrechnung.
Braunschweig, 1842. [3].

FRIES,

GALILEO, G.:

D is com e dimostrazioni intorno

Quoted from German

25.

a due nuove scienze. 1638.

translation in Ostwalds Klassiker, Bd. 24 and

Leipzig, 1890-1.

GEIRINGER, H.: *t)ber die Wahrscheinlichkeit von Hypothesen.'

(Erkenntnis) 8, 1939-40.
GIBSON, Q.: 'Argument from Chances.' AJP 31, 1953.
GOODMAN, N.: 'A Query on Confirmation.' JP 43, 1946.
'

'On

JUS

[1].

PPR

Infirmities of Confirmation-Theory.
8, 1947-8 [2].
The Logical Simplicity of Predicates.' JSZ,14, 1949. [3].

'New Notes on

Simplicity.
Fact, Fiction and Forecast.

'

JSL

17, 1952.

[4].

London, 1955. [5].

'On Von Mises's Theory of Probability.

'

Mind 49, 1940.

GOODSTEIN, R. L.:
Induction.
PS
Treatment
of
1940.
A.:
Teirce's
TH.
7,
GOUDGE,
9 1955.
GREENBERG, L.: 'Necessity in Hume's Causal Theory.'
'

RMS

GRELLING, K.: 'Die philosophischen Grundlagen der Wahrscheinlichkeitsrechnung. Abhandlungen der Fries schen Schule y Neue Folge 3.
9

'

Gdttingen, 1912.

GROSS, M. W.: 'Whitehead's Answer to Hume.' JP 38, 1941.

HAILPERIN, TH.: 'Foundations of Probability in Mathematical Logic/

PS 4,

1937.

HALMOS,]?. R.:

Monthly

The Foundations of Probability.' American Mathematical

51, 1944.
235

BIBLIOGRAPHY
VON HARTMANN, E.: Philosophie des Unbewussten. Berlin, 1869.
HARTSHORNE, CH.: 'Causal Necessities: An Alternative to Hume.'

PR

63, 1954.
HAWKINS, D.: 'Existential

and Epistemic Probability.' PS 10, 1943.

HAY, W. H.: 'Bertrand Russell on the Justification of Induction.' PS
1950.

17,

[1].

'Professor

Carnap and Probability.' PS

19, 1952.

[2].

Rev. of von Wright [11]. JP 50, 1953. [3].

OPPENHEIM, P.: 'A Syntactical Definition of Probability
HELMER, O.
and of Degree of Confirmation. JSL 10, 1945.
'

Handbuch der physiologischen Optik. Leipzig,

VON HELMHOLTZ,
1856-66. Quoted from the 3rd edn., Leipzig 1909-1 1. [1].
Die Tatsachen in der Wahrnehmung. Berlin, 1879. [2].
HEMPEL, C. G.: Beitrage zur logischen Analyse des WahrscheinlichkeitsH.:

Jena, 1934.

begriffs.

[1].

'Uber den Gehalt von Wahrscheinlichkeitsaussagen.'

5, 1935.

Erkenntnis

[2].

'LaproblemedelavSriteV Theoria 3, 1937 [3].

'On the Logical Form of Probability-Statements/ Erkenntnis
.

1937-8.

'A Purely Syntactical Definition of Confirmation.' JSL

ML

'Studies in the Logic of Confirmation.'

'A Note on the Paradoxes of Confirmation.'

OPPENHEIM,

PS

7,

[4].

12, 1945.

P.:

[5].

1945.

[6].

1946.

[7].

'A Definition of "Degree of Confirmation".'

[1].

OPPENHEIM,
1948.

8, 1943.

Mind 54,
Mind 55,

P.:

'Studies in the Logic of Explanation.

'

PS

15,

[2].

J. F. W.: A
Preliminary Discourse on the Study of Natural
Philosophy. London, 1830.
HERTZ, P.: 'Kritische Bemerkungen zu Reichenbachs Behandlung des

HERSCHEL,

Humeschen Problems.

'

Erkenntnis 6, 1936.
GR.: Inductive Logic. Edinburgh, 1896.
HINTON, J. M.: 'Quasi-Inductive Scepticism.' Mind 60, 1951.
HOBART, R. E.: 'Hume without Scepticism.' HI. Mind 39, 1930.
HOBBES, TH.: The Elements of Law. Ed. by Tonnies. London, 1889.

HIBBEN,

J.

HOFSTADTER, A.:

'Universality, Explanation,

and

Scientific

Law.' JP 50,

1953.

'A General Scheme of the Present Slate of Natural PhilosThe Posthumous Works of Dr. Robert Hooke. London 1705.
ophy'
HOPF, E.: 'Remarks on Causality and Probability/ Journal of Mathe-

HOOKE,

R.:

in

matics and Physics 14, 1935.

236

BIBLIOGRAPHY
HOSIASSON (-LINDENBAUM), J.: 'Why do we Prefer Probabilities Relative
to Many Data?* Mind 40, 193 1.
[1].
'La thorie des probability est-elle une
logique gen&ralisee?'
Actes du Congres International de
Philosophic Scientifique. Paris,
1936.

[2],

*On Confirmation.' JSL 5, 1940. [3].

Induction et analogic: comparaison de
1941.

leur fondement.

'

Mind 50,

[4].

D.: A Treatise on Human Nature. London 1739.

[1].
An Abstract of A Treatise on Human Nature. A Pamphlet hitherto
unknown by David Hume. Reprinted with an Introduction byJ. M.

HUME,

Keynes and

An

P. Sraffa.

Cambridge, 1938.

[2].

Human

Understanding. London, 1748. [3].

HUTTEN, E. H.: 'Induction as a Semantic Problem.' Analysis 10, 1949-50.
HUYGHENS, CHR.: Traite de la lumiere. Leiden, 1690. Quoted from Oeuvres

Enquiry Concerning

completes, vol. 19.

The Hague,

1937.

JEFFREYS, ^Scientific Inference. Cambridge 1931.

The Problem

of Inference.'

Mind 45,

Theory of Probability. Oxford, 1939.

1936.

[1],

2nd rev. edn. 1956.

[2].

[3].

'Bertrand Russell on Probability.' Mind 59, 1950. [4].

'The Present Position in Probability Theory. ' BJPS 5, 1955.
JEVONS, W. ST.: Elementary Lessons in Logic. London, 1870.
from the 6th edn. London, 1877. [1].

[5].

Quoted

The Principles of Science. London, 1874. Quoted from the

2nd edn. London, 1877. [2].
Pure Logic and Other Minor Works. Ed. by Adamson and
Jevons. London, 1890. [3].
JOHNSON, W. E.: Logic. Cambridge, 1921-4. [1].
'Probability: The Deductive and Inductive Problems.' Mind 41,
1932.

[2].

JONES, H.: 'Causality and Perception.' JP 47, 1950.

JOSEPH, H. W.: An Introduction to Logic. Oxford, 1906. Quoted from
the 2nd edn. Oxford, 1916.
JOURDAIN, PH. E. B.: 'Causality, Induction and Probability.' Mind 28,
1919.

KAHLE, L. M.: Elementa Logicae Probabilium. Halle, 1735.

KAILA, E.: Der Satz von Ausgleich des Zufalls und das Kausalprinzip.
Annales Universitatis Fennicae Aboensis, Series B, Torn II, Nr. 2,
1925.

[1].

237

BIBLIOGRAPHY
Die Prinzipien der Wahrscheinlichkeitslogik. Annales Universitatis
Fennicae Aboensis, Series B, Tom IV, Nr. 1, 1926. [2].
Der logistische Neupositivismus. Annales Universitatis Fennicae
Aboensis, Series B,

Das System

der

2. Helsinki, 1936.

Tom XIII,

1930.

[3].

Wirklichkeitsbegriffe.

Acta Philosophica Fennica

[4].

Preface to the Finnish translation of Hume

Inhimillinen

Helsinki, 1938. [5].

(Human Knowledge.) Helsinki, 1939. [6].
Gesetz der Serie, eine Lehre von den Wiederholungen
[3].

tieto.

KAMMERER, P.: Das

im Lebens- und im Weltgeschehen. Stuttgart, 1919.
KANT, I Reflexionen Kants zur Kritik der reinen Vernunft. Ed by Erdmann
.

.:

Leipzig, 1884. [1].

Kritik der reinen Vernunft.
edn., Riga, 1787. [2].

Riga,

1781.

Quoted from the 2nd

Prolegomena zu einer jeden kunftigen Metaphysik. Riga, 1783. [3].

Kritik der Vrtheilskraft. Berlin, 1790. [4].
zu einer allgemeinen Geschichte in Weltbtirgerlicher
'Id6e
Absicht.' Berlinische Monatsschrift, 1784. Quoted from Kant's

Gesammelte Schriften herausgegeben von der Koniglich Preussischen

Akademie der Wissenschaften, vol. VII. Berlin, 1912. [5].

On Philosophy in General Ed. by Kabir. Calcutta, 1935. [6].

9
KASTIL, A.: Jakob Friedrich Fries Lehre von der unmittelbaren Erkenntnis.
Gottingen, 1912.

KAUFMANN,
1941-2.

F.:

The

Logical Rules of Scientific Procedure.'

PPR

2,

[1].

'Scientific Procedure and Probability.' PPR 6, 1945-6. [2].

'On the Nature of Inductive Inference.' PPR 6, 1945-6. [3].
KELLY, TH. R.: Explanation and Reality in the Philosophy of Smile Meyer-

son.

Princeton, 1937.

KEMBLE, E. C: The Probability Concept.' PS 8, 1941. [1].

'Is the
Frequency Theory of Probability Adequate for
Purposes?' American Journal of Physics 10, 1942. [2].

KEMENY,

J,

G.: 'Extensions of the

all Scientific

Methads of Inductive Logic.'

Philoso-

Studies 2, 195L [1].

phical
6
Logical Measure Function.' JSL 18, 1953. [2].
5
'A Treatise on Induction and Probability. PR 62, 1953. [3].
The Use of Simplicity in Induction.' PR 62, 1953. [4].
OPPENHEIM, P.: 'Degree of Factual Support.' PS 19, 1952. [5].
'Fair Bets and Inductive Probabilities.' JSL 20, 1955. [6].

KERBY-MILLER, S.: 'Causality.' In Philosophical Essays for Alfred

North Whitehead. London, 1936.
238

BIBLIOGRAPHY
KEYNES, J. M.: A Treatise on Probability. London, 1921.
KIRCHHOFF, G.: 'Untersuchungen iiber das Sonnenspektrum. und die
Spektren der chemischen Elemente. Abhandlungen der Koniglichen
*

Akademie der Wissenschaften zu Berlin, 1861.

KNEALE, W.: Probability and Induction. Oxford, 1949. [1].
'Natural Laws and Contrary-to-fact Conditionals.' Analysis
1950.

10,

[2].

Kneale's opinions are reviewed in Anscombe, Broad

and Will [4].

KOLMOGOROV, A.

N.:

Grundbegriffe

der

[13],

Nagel

[8],

Wahrscheinlichkeitsrechnung.

Ergebnisse der Mathematik und ihrer Grenzgebiete, Berlin, 1933.

English edn. Foundations of the Theory of Probability. New York,
1950.
5

B. O.: 'The Axioms and Algebra of Intuitive Probability.

Annals of Mathematics 41, 1940. [I].
'Intuitive Probabilities and Sequences.'
Annals of Mathematics 42,

KOOPMAN,

1942.

KORNER,

[2],

S.:

'On Laws of Nature.' Mind

62, 1953.

'The Development of the Main Problem in the Philoof

Francis
Bacon.* Studia Philosophica 1, Lwow, 1935.
sophy
KRAMPF, W.: 'Studien zur Philosophic und Methodologie des Kausalprinzips.* Kant-Studien 41, 1936.

KOTARBINSKI,

VON

T.:

KRIES, J.: Die Prinzipien der Wahrscheinlichkeitsrechnung. Freiburg

B. 1886. Quoted from the 2nd edn., Tubingen, 1927.

i.

LACHELIER,

J.:

DuFondementde I* Induction. In Oeuvres de

Paris, 1935.
LACROIX, S. F.: Traiti llementaire

LALANDE, A.: Les

du cakul des

theories de ^induction et de

Jules Lachelier.

probabilites.

Paris, 1816.

V experimentation.

Paris,

1929.

LAMBERT, J. H.: Neues Organon. Leipzig, 1764.

LANGE., FR. A.: Logische Studien. Iserlohn, 1877.
*
LAPLACE, P. S.: *M6moire sur la probability des causes par les evenements.
Memoires presentees a V Academic des Sciences, 1774. [1].

*M6moire sur les probability.' Memoires presentees a PAcademie

des Sciences, 1780. [2].
*M6moire sur les approximations des formules qui sont fonctions
de tres grands nombres.' Memoires deVInstitut, 1810. [3].
*M6moire sur les integrates d6finies et leur application aux probabiliMemoires de Vlnstitut, 1810. [4].

t6s.

Theorie analytique des probabilites. Paris, 1812.

239

[5].

BIBLIOGRAPHY
Essai philosophique sur
Introduction to Laplace

Paris

les probabilities.

1814.

(Printed as

from 2nd edn. onwards.) [6],

The above works are quoted from Oeuvres completes, Paris, 1891-8.
German translation of [6] by R. von Mises, Leipzig, 1932. English
translation of [6] by Truscott and Emory, New York, 1902.
LEHMAN; R. S.: 'On Confirmation and Rational Betting.' JSL 20, 1955.
LEIBNIZ, G. W.: Marii Nizolii de Veris Principiis et Vera Rat fone Philoso[5]

phandi con tra Pseudophilosophos. 1 670 [ 1 ]

Monadologie. [2].
De modo' distinguendi phaenomena realia ab imaginariis. [3].
These works are quoted from Die philosophischen Schriften von
Gottfried Wilhelm Leibniz. Ed. by Gerhardt. Berlin, 1875-90. See
also under Couturat.
Correspondence between Leibniz and James Bernoulli. Printed in
Ed. by Gerhardt. Berlin,
Leibnizens mathematische Schriften.
.

1849-63.

[4].

LENZEN, V. F.: *Experience and Convention.* Erkenntnis 7, 1938.

LE ROY, E.: 'Science et philosophic/
7, 1899 and 8, 1900.

RMM
RMM

[I].

'Sur la logique de 1'invention.'

13, 1905. [2].
a
Laws
H,:
LEVY,
Methodological and Historical Survey.'
'Probability
Science and Society 1, 1937.

LEWIS, C.

LEWY,

I.:

C.:

An

Analysis of Knowledge

'On the

and Valuation. La

Salle,

111.,

1946.

"Justification'* of Induction.'

(Discussion in Analysis
C. H. Whiteley, and H.

7,

1940 by

M.

Analysis 6, 1939.
A. Cunningham, L. D. Sass,

W. Chapman.)

PREVOST, P.: 'Sur 1'art d'estimer la probability des

LHUILIER, S. A.
causes par les effets.' Memoires de VAcademie Royale, Berlin, 1799.
VON LEIBIG, J.: Induktion und Deduktion. Munich, 1865.
VON LIECHTENSTEIN, CHR. R.: 'Die Wahrscheinlichkeit als Realkategorie.
Actes du XP me Congres International de Philosophie, vol. 6, Amster'

dam, 1953.
LINDENBAUM, J.: See Hosiasson-Lindenbaum,

J.

LINDSAY, R. B.: 'The Meaning of Simplicity in Physics.' PS 4, 1937.

LOCKE, J.: An Essay Concerning Human Understanding. London, 1690.
LONG, P.: 'Natural Laws and so-called Accidental General Statements.'
Analysis 13, 1952-3.

Los,

J.:
Todstawy analizy metodologicznej kanonow Milla.' ('Foundations of the Methodological Analysis of Mill's Canons.') Annales
Universitatis Mariae Curie-Sklodowska, F 2, 1947.

LOTZE, H.: System der Philosophie. I. Logik. Leipzig, 1874. [1].

GrundzugederLogikundEncydopadieder Philosophie. Leipzig, 1 883. [2]
240

BIBLIOGRAPHY
LUKASIEWICZ,
rechnung.

L:

Die

Krakau

MACDONALD, M.:

logischen
1913.

Grundlagen der

'Induction and

Hypothesis.'

Suppl. vol. 16, 1937.

MACH, E.: Die Mechanik in ihrer Entwickelung.

Wahrscheinlichkeits-

Leipzig, 1883.

Die Principien der Warmelehre. Leipzig, 1896.

PAS,

Symposium.
[1].

[2],

Erkenntnis undlrrtum. Leipzig, 1905. [3].

S.: Versuch einer neuen
Logik. Nebst angehangten Briefen des
Philoletes an Anesidenms. Berlin, 1794.
MAKER, P. TH.: 'A Proof that Pure Induction approaches Certainty as

MAIMON,

its

Limit.'

Mind 42,

1933.

MALEBRANCHE, N.: De la recherche de la write. Paris, 1675.

MALLY, E.: Wdhrscheinlichkeit und Gesetz. Berlin, 1938.
MARBE, K.: Die Gleichforrnigkeit in der Welt. Munich, 1916.
MARGENAU, H.: 'Probability, Many-Valued Logics, and Physics.' PS
1939.

6,

[1].

'Probability and Physics.' JUS (Erkenntnis) 8, 1939-40. [2].

'On the Frequency Theory of Probability.' PPR 6, 1945-6. [3].
MARTIN, N. M.: 'The Explicandum of the Classical Concept of Probability.'

PS

18, 1951.

Hume's Skepsis und die Wahrscheinlichkeitsrechnung. Wien, 1884.

MAXWELL, C. J.: Matter and Motion. London, 1876.
MAZURKIEWICZ, S.: 'Zur Axiomatik der Wahrscheinlichkeitsrechnung.
Comptes rendus des seances de la Societe des Sciences et des Lett res

MASARYK, TH.

G.: Dav.

'

deVarsovie, Cl. Ill 25, 1932. [1].

'tJber die Grundlagen der Wahrscheinlichkeitsrechnung.' Monatsheftefur Mathematik undPhysik 41, 1934. [2].
J.: 'Has Russell Answered Hume?' JP 49, 1952.
MELHLBERG, H.: 'Essai sur la th6orie causale du temps.' I-II.
Philosophica 1, Lwow, 1935 and 2, Lwow, 1937.

McLENDON, H.

Studia

A.: Uber Moglichkeit und Wahrscheinlichkeit. Leipzig, 1915. [1]

Rev. of von Kries. Gdttingische gelehrte Anzeigen 2, 1890. [2].
MENDELSSOHN, M.: t)ber die Wahrscheinlichkeit.' In Philosophische

MEINONG,

Schriften, vol. 2, Berlin, 1771.

E.: Identite et Re'alite. Paris, 1908. [1].
De ['explication dans les sciences. Paris 1921.

MEYERSON,

2nd edn., Paris, 1927

MILHAUD, G.: 'La science

rationelle.'

241

RMM 4,

1896.

Quoted from

the

BIBLIOGRAPHY
J. STUART: A System of Logic.
London, 3843. Quoted from the
8th edn., London, 1872.
MILLER. D. S.: 'Professor Donald Williams versus Hume.' JP 44, 1947.
'
VON MISES, R.: 'Grundlagen der Wahrscheinlichkeitsrechnung. Mathe-

MILL,

matische Zeitschrift 5 9 1919.

[1].

Quoted
Wahrscheinlichkeit, Statistik und Wahrheit. Wien, 1928.
from the 2nd revised edn., Wien, 1936. English translation, New
York, 1939. [2].
'Uber kausale und

statistische

Erkenntnis

[3].

1,

1930-1.

Gesetzmassigkeit in der Physik.*

Comments on Williams [1] and [2]. PPR 6, 1945-6. [4], [5].

DE MOIVRE, A.: The Doctrine of Chances. London, 1718. Quoted from
the 3rd edn., London, 1756.
MOLINA, E. C: The Theory of Probability; some Comments on Laplace's
Theorie analytique.' Bulletin of the American Mathematical Society
36, 1930.

[1].

'Bayes's Theorem; and Expository Presentation.' Annals of Mathematical Statistics 2, 1 93 1 [2]

'
MOORE, A.: 'The Principle of Induction, JP 49, 1952.
.

DE MORGAN, A.: An Essay on Probabilities. London, 1838. [1].

Formal Logic: or, The Calculus of Inference, Necessary and Probable.
London, 1847. [2].
'

'A Frequency Theory of Probability. JP 30, 1933. [1].

Rev. of Reichenbach [4]. Mind 4$, 1936. [2].
The Meaning of Probability.' Journal of American Statistical

NAGEL,

E.:

Association 31, 1936. [3].

[4].
Trobability and the Theory of Knowledge.' PS 6, 1939.
Principles of the Theory of Probability. International Encyclopaedia

of Unified Science /, 6. Chicago, 1939. [5].

Trobability and Non-Demonstrative Inference.' PPR 5, 1944-5. [6].
'Is
the Laplacean Theory of Probability Tenable?' PPR 6,
1945-6.

[7].

Rev. of Williams [4]. JP 44, 1947. [8].

Rev. ofKneale[l]. JP 47, 1950. [9].
Rev. of Reichenbach [4], the English edn. JP41, 1950. [10].
NATORP, P. Geschichte des Erkenntnisproblems im Altertum. Berlin, 1884.
NAVILLE, E.: LaLogique de Phypothese. Paris, 1880.
DEL NEGRO, W.: 'Die Begriindung der Wahrscheinlichkeit und das

Anwendungsproblem des Apriorischen.'

phische Forschung

3, 1948-9.

242

Zeitschrift

fur philoso-

BIBLIOGRAPHY
NELSON, E.

J.:

'Professor Reichenbach

The

on Induction.' JP38, 1936.

[I].

External World and Induction.' PS 9 9 1942.

[2].
NELSON, L.: 1st metaphysikfreie Naturwissenschaft moglich?' Abhandlungen der Fries' schen Schule, Neue Folge 2, Gottingen, 1907. [1].
'Ober das sogenannte Erkenntnisproblem.' Abhandlungen der Fries"
schen Schule, Neue Folge 3, Gottingen, 1912. [2].
NEWTON, L: Philosophiae Naturalis Prindpia Mathematica. London, 1687.
[1].

London, 1704. Quoted from the 4th edn., London,

Opticks.
1730. [2]. The relevant sections are included in Newton '$
Philosophy

of Nature, ed. by H. S. Tfaayer, New York, 1953.

NipOD, J.: Le probleme logique de ^induction. Paris 1924. Quoted from
the English translation in Nicod, Foundations of Geometry and
Induction, London, 1930.
NISBET, R. H.: The Foundations of Probability. Mind 35, 1926.
NITSCHE, A.: 'Die Dimensionen der Wahrscheinlichkeit und die Evidenz
'

der

Ungewissheit.'
Vierteljahrschrift fur wissenschaftliche
sophie 16, 1892.
NYMAN, A.: 'Induction et intuition.' Theoria 19, 1953.

OLIVER, V. D.: 'A Re-Examination of the Problem of Induction.

'

Philo-

JP

49,

1952.
A.: 'Die Unerweisbarkeit des Kausalgesetzes.
der Naturphilosophie 14, 1921.

OLZELT-NEWIN,

Annakn

OPPENHEIM, P.: See Helmer, O., Hempel, C. G. and Kenaeny, J. G.

OXENSTIERNA, G.: 'Nagra problem i laran om deduktion och induktion.*
('Some Problems in the Theory of Deduction and Induction.')
Festskrift tillagnad Axel Hagerstrom. Uppsala, 1928.
PASSMORE, V. J. A.: 'Prediction and Scientific Law.' AJP 24, 1946.
PEARS, D.: 'Hypothetical Analysis 10, 1949-50.
PEARSON, K,: The Grammar of Science. London, 1892. [1].
'On the Influence of Past Experience on Future Expectation.'
'

[2].
Philosophical Magazine, 1907
PEIRCE CBL S.: Collected Papers of Charles Sanders Peirce. Ed. by Ch.
Hartshorne and P. Weiss. Cambridge, Mass. 1931-5. Peirce's
papers and remarks on induction and probability are scattered among
'.

his writings.

PiCARD, 3. :Essaisur la logique deVinvention dans les sciences. Paris, 1928. [1],
Les conditions positives de ^invention dans les sciences. Paris,
1928.

[2].

243

BIBLIOGRAPHY
'M6thode Inductive

et

Raisonement

RP

Inductif.'

POINCARE, H.: La science et Vhypothese. Paris, 1902.

La valeur de la science. Paris, 1904. [2],

114, 1932.

[3].

[1],

Science et methode. Paris, 1908. [3].

POIRIER, R.: Remarques sur la probabilite des inductions. Paris, 1931.
POISSON, S. D.: Recherches sur la probabilite des jugements. Paris, 1837.
POLYA, G.: 'Heuristic Reasoning and the Theory of Probability.' American

Mathematical Monthly *&, 1941. [1].

'On Patterns of Plausible Inference.' In Courant Anniversary
Volume. New York, 1948. [2].
Mathematics and Plausible Reasoning. Princeton, 1953. [3].
*

POPPER, K. R.:
"Induktionslogik" und "Hypothesenwahrscheinlichkeit'V Erkenntnis 5, 1935. [1].

Logik der Forschung. Wien, 1935. [2].

'A set of Independent Axioms for Probability. Mind 47, 1938.
'A Note on Degree of Confirmation.' BJPS 4, 1954. [4].
'Content and Degree of Confirmation.' A Reply to Bar-Hillel
'

BJPS

5,

1955.

[2].

[5].

P.: See Lhuilier, S.

R.:
See Bayes, Th.
PRICE,

PREVOST,

A.

QUETELET, A.: Instructions populates sur

1828.

[3].

le

calcul des probability. Brussels,

[1].

Lettres sur la theorie des probabilites appliquee aux sciences morales

et politiques, Brussels, 1846. [2].
Du systeme soclale et des lois qui le regnissent. Paris, 1848. [3].

Tables de mortalite et leur developpement.

Brussels, 1872. [4].

RADAKOVIC, TH.: 'Die Axiome der Elementargeometrie und des Aussagenkalkiils.' Monatshefte fur Mathematik und Physik 36, 1929.
RAMSEY, F. P.: The Foundations of Mathematics and other Logical Essays.
London, 1931.
RANKIN, K. W.: 'Linguistic Analysis and the Justification of Induction.'
The Philosophical Quarterly 5, 1955.
REACH, K.: The Foundations of our Knowledge.* Synthese 5, 1946.
REICHENBACH, H.: *Kausalitat und Wahrscheinlichkeit.' Erkenntnis
1930-1.
*

1,

[1].

Axiomatik der Wahrscheinlichkeitsrechnung.

schrift 34,

1932.

'

Mathematische Zeit-

[2].
'

'Die logischen Grundlagen des Wahrscheinlichkeitsbegriffs. Erkenntnis 3,

1932-3.

English translation

Philosophical Analysis,

New

in

York, 1949.
244

Feigl-Sellars,
[3].

Readings

in

BIBLIOGRAPHY
Wahrscheinlichkeitslehre.

English translation with

1935.

Leiden,

additions. Berkeley, 1949.

[4],

'Wahrscheinlichkeitslogik.' Erkenntnis 5, 1935.

[5].

'Bemerkungen zu Carl Hempels Versuch einer finitistiscfaen Deutung des Wahrschemlichkeitsbegriffs.' Erkenntnis 5, 1935-6. [6].
*0ber Induktion und Wahrscheinlichkeit. Erkenntnis 5, 1935. [7].
Induction and Probability.' PS 3, 1936. [8].
Experience and Prediction. Chicago, 1938. [9].
'On Probability and Induction .' PS 5, 1938. [10].
'Ober die semantische und die Objekt-Auffassung von Wahrschein'

lichkeitsausdrucken.'

JUS (Erkenntnis) 8,

1939-40. [11].
'

'Bemerkungen zur Hypothesenwahrschemlichkeit.

8,1939-40.

JUS (Erkenntnis)

[12].

'On the Justification of Induction.' J? 37, 1940. [13].

*A Conversation between Bertrand Russell and David Hume.
45, 1948.

'

JP

[14].

For reviews of Reichenbach's opinions see Burke [3], Hertz, Nagel

[2] and [10] and Nelson [1].
REINACH, A.: 'Kants Auffassung des Humeschen Problems.' Zeitschrift
fur Philosophie und Philosophische Kritik 141, 1911.
RENOUVIER, CH.: 'La methode ph6nom6niste.' AP 1, 1890.
RlCHTER, P. David Humes Kausalitdtstheorie und ihre Be deutung fur die
9

der Theorie der Induktion. Halle, 1893.

RITCHIE, A. D.: Induction and Probability.' Mind 35, 1926.
ROBSON, S. W,: 'Whitehead's Answer to Hume.' JP 38, 1941.

Begmndung

RUSSELL, B.: The Principles of Mathematics. London, 1903.

The Problems of Philosophy. London, 1912. [2].

[1].

An

Outline of Philosophy. London, 1927. [3].

The Analysis of Matter. London, 1927. [4],
Rev. of Ramsey. Mind 40, 1931. [5].

The Limits of Empiricism.' PAS 36, 1935-6. [6].

Human Knowledge, its Scope and Limits. London, 1948.

For reviews of Russell's opinions

McLendon and H. R. Smart.
RYLE, G.: Induction and Hypothesis.

see Edwards,
*

Hay

[7].

[1], Jeffreys [4],

Symposium. PAS, Suppl.

vol. 16,

1937.

RYNIN, D.: 'Probability and Meaning/ JP 44, 1947.

ScBLiCK,M.:Allgemeinerkenntnislehre.

2nd

edn. Berlin, 1925.

[1].

245

Berlin, 1918.

Quoted from the

BIBLIOGRAPHY
'Die

Stellung

der

'Gesetz

in

Kausalitat

Naturwissenschaften 19, 1931.

der

gegenwartigen

Physik.'

[2].

und Wahrscheinlichkeit.

'

Actes du Congres International de

Philosophie Scientifique, Paris, 1936. [3].

[2] and [3] are reprinted in Gesammelte Aufsatze, Wien, 1938.
SCHUPPE, W.: Erkenntnistheoretische Logik. Bonn, 1878.

SERVIEN, P.: Hazard et probabilites. Paris, 1949.

SHIMONY, A.: 'Coherence and the Axioms of Confirmation.

'

JSL 20,

1955.

SIGWART, CHR.: Logik. Tubingen, 1873-8. Quoted from the 3rd edn.
Tubingen, 1904.
SIMON, H. A,: 'On the Definition of the Causal Relation.' JP 49, 1952.
SMART, H. R.: The Problem of Induction.' JP 25, 1928.,
SMART, J. J. C: 'Excogitation and Induction.' AJP 28, 19*50.
SPEDDING, J.: Preface to the Parasceve ad Historian Naturalem et Expertmentalem. In the Works of Francis Bacon, ed. by Spedding, Ellis and
Heath. London, 1857-8.
SPILSBURY, R. J.: 'A Note on Induction.' Mind 58, 1949.
STERZINGER, O.: Zur Logik und Naturphilosophie der WahrscheinlichkeitsLeipzig, 1911.

lehre.

'

'Epicurean Induction. Mind 34, 1925.

R.:
WhewelV s Philosophy of Induction. Lancaster, U.S., 1931.
M.
STOLL,
STRAUSS, M.: 1st die Limes-Theorie der Wahrscheinlichkeit eine sinnvolle

STOCKS,

J. L.:

Idealisation?' Synthese 5, 1946.

J.: *On the Foundations of the

STRUIK, D.

Theory of

'

Probabilities.

PS

1, 1934.
1
STUMPF, K.: 'Ober den Begriff der mathematischen Wahrscheinlichkeit.
Sitzungsberichte der philosophisch-philologischen und der historischen

Classe der K. b. Akademie der Wissenschaften zu Miinchen, 1892.

TARSKI, A.: 'Wahrscheinlichkeitslehre und mehrwertige Logik.' Erkenntnis 5, 1935.

TAYLOR, D.: 'A Study in Probability.' AJP 13, 1935.

TIMERDING, H. E.: Die Analyse desZufalls. Braunschweig, 1915,
TINTNER, G.: 'Foundations of Probability and Statistical Inference.'
The Journal of the Royal Statistical Society, A 112, 1949.
TISSOT,

J.:

Essai de logique objective. Paris, 1868.

TODHUNTER, L: A History of the Mathematical Theory of Probability from

the Time of Pascal to that of Laplace. Cambridge, 1865.
TORNIER,

E.: 'Eine

neue Grundlegung der Wahrscheinlichkeitsrechnung.'

Zeitschrift fur Physik 63, 1930.

[1],

246

BIBLIOGRAPHY
*Gmndlagen der Wahrscheinlichkeitsrechnung.
60,1933.

TREMBLEY,

J.:

'

Acta Mathematica

[2].

*De probabilitate causarum ab

efiectibus

orhmda.' Com-

mentationes Sodetatis Regiae Scientiamm Gottingiensis 13, 1795-8.

TRENDELENBURG, A.: Logische Untersuchungen. Berlin, 1840. Quoted
from the 3rd edn., Leipzig, 1870.
TWARDOWSKI, K.: *t)ber sogenannte relative Wahrheiten.' Archiv fur
systematische Philosophic

USHENKO, A.

P.:

8, 1902.

'The Principle of Causality.' JP 49, 1952.

VENN, J.: The Logic of Chance. London, 1866. [1].

The Principles of Empirical or Inductive Logic. London,
Quoted from the 2nd edn., London, 1907. [2].
VIETORIS, L.: *t)ber den Begriff der Wahrscheinlichkeit.
fur Mathematik 52, 1948.

1889.

'

Monatshefte

VOGEL, TH.: See von Aster.

FR.: 'Logische Analyse des Wahrscheinlichkeitsbegriffs.*
Erkenntnis 1, 1930-1.
des Kollektivbegriffes der WahrWALD, A.: 'Die Widerspruchsfreiheit
'
scheinlichkeitsrechnung. Ergebnisse eines mathematischen Kolloquiums

WAISMANN,

8, 1937.

WALKER, E. R.: 'Verification and Probability.' JP 44, 1947.

WALKER, H. ML: Studies in the History of Statistical Method.

Baltimore,

1929.
'

H.: 'Notes on the Justification of Induction. JP 44, 1947. [1].

'On Scepticism about Induction.' PS 17, 1950. [2].
WEINBERG, H.: Das Geltungsproblem bei Hugo Dingier. Metzingen, 1934.
WEINBERG, J. R..: An Examination ofLogical Positivism. London, 1936. [1],

WANG,

'Our Knowledge of Other Minds.' PR 55, 1946. [2].

Idea of Causal Efficacy. JP 47, 1950. [3].
WEISMANN, A.: 'Relations of Causality in the Course of Nature.'

The

'

PPR

15, 1954.
WEYL, H.: Philosophy of Mathematics and Natural Science. Princeton,
1949.
WHEWELL, W.: The Philosophy of the Inductive Sciences. London, 1 840. [1].

History of Scientific Ideas. London, 1858.

[2].

Novum Organum Renovatum. London, 1858. [3].

On the Philosophy of Discovery. London, 1860. [4].
247

BIBLIOGRAPHY
For examinations of Whewell's opinions see Ducasse [1] and Stoll.
WHITE, M.: 'Probability and Confirmation.' JP 36, 1939.
WHITEHEAD, A. N.: Science and the Modern World. Cambridge, 1927. [1],
Symbolism, its Meaning and Effect. New York, 1927. [2].
Process and Reality. Cambridge, 1928. [3].
For an examination of Whitehead's opinions on induction see
Kerby-Miller and Robson.
WmrrAKER, E. T.: 'On some Disputed Questions of Probability.' Transacof the Faculty of Actuaries, Scotland 8, 1920.
L.: 'Is there a Problem of Induction?' JP 39, 1942. [1].
'Will the Future be Like the Past?' Mind 56, 1947. [2].
'
'Donald Williams's Theory of Induction. PR 57, 1948. [3].
'Scepticism and the Future.' PS 17, 1950. [4].
'Generalization and Evidence.' In Philosophical Analysis, A Collec-

tions

WILL, FR.

tion of Essays, ed. by M. Black. Ithaca, N.Y., 1950.

'Kneale's Theories of Probability and Induction.'

[5].

PR

WILLIAMS, D.:

PPR 5,

1944-5.

[1].

'The Challenging Situation

PPR

63, 1954. [6].

'On the Derivation of Probabilities from Frequencies.'

5, 1944-5.

in

the

Philosophy

of Probability.'

[2].

'The Problem of Probability.' PPR 6, 1945-6. [3].

The Ground of Induction. Cambridge, Mass., 1947, [4].
'Induction and the Future.' Mind 57, 1948. [5]
For reviews of Williams's opinions see Miller, von Mises [4] and
and Nagel [8].
WISDOM, J.: 'A Note on Probability.' In Philosophical Analysis,
Collection of Essays, ed.

WISDOM,

J.

O.:

'Criteria

by M. Black.

for

Relationship/ Mind 54,

Foundations of Inference

Ithaca, N.Y., 1950.

Causal Determination and Functional

1945.
in

[5]

[1].

Natural Science, London, 1952.

[2].

L.: Tractatus logico-philosophicus.

London, 1922.
VON WRIGHT, G. H.: 'Der Wahrscheinlichkeitsbegriff in der modernen

WITTGENSTEIN,

Erkenntnisphilosophie.' Theoria 4, 1938.

'On Probability.' Mind 49, 1940. [2].

The Logical Problem of Induction.

Helsinki 1941.

[1].

Ada

Philosophica Fennica 3.

[3].

'Induktionsproblemet och kunskapens granser.' ('The Problem of

Induction and the Limits of Knowledge'.) FinskTidskriftl9, 1941. [4].

'Nagra anmarkningar om nodvandiga och tillrackliga betingelser.

('Some Remarks on Necessary and Sufficient Conditions.') Ajatus
'

1H, 1942.

[5].

248

BIBLIOGRAPHY
Tilastollisen todennakoisyysteorian vaiheita'.
('On the History of
the Frequency
Theory of Probability'). With English Summary.
AjatusU, 1943. [6].

Uber Wahrscheinlichkeit,

eine logische und

philosophische Untersuchung. ActaSocietatisScientiarumFennicaeAIIIll.Rdsinki, 1945. [7].
'On Confirmation.' Actes du X^me
Congres International de Philosophic, vol. 1, fasc. 2, Amsterdam, 1948. [8].

'Some

Principles of Eliminative Induction.' AjatuslS, 1949. [9].

'Carnap's Theory of Probability.' PR 60, 1951. [10].
A Treatise on Induction and Probability. London, 1951. [11].
For reviews see Broad [12], Hay [3] and

Kemeny

ZAWIRSKI,

Z.:

[3].

'Uber das Verhaltnis der mehrwertigen

Logik zur Wahr-

scheinlichkeitsrechnung.' Studia Philosophica 1,

ZILSEL, E.: Das Anwendungsproblem. Leipzig, 1916.

249

Lwow,

1935.

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