K. Popper - Logic scientific discovery
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K. Popper - Logic scientific discovery

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What I attempted was merely this:

Some great scientists and philosophers have made assertions about
simplicity and its value for science. I suggested that some of these

1 Schlick, ibid., p. 148.

simplicity 131

assertions can be better understood if we assume that when speaking
about simplicity they sometimes had testability in mind. This eluci-
dates even some of Poincaré’s examples, though it clashes with his views.

Today I should stress two further points: (1) We can compare theor-
ies with respect to testability only if at least some of the problems they are
supposed to solve coincide. (2) Ad hoc hypotheses cannot be compared
in this way.

some structural components of a theory of experience132

8
PROBABILITY

In this chapter I shall only deal with the probability of events and the
problems it raises. They arise in connection with the theory of games
of chance, and with the probabilistic laws of physics. I shall leave the
problems of what may be called the probability of hypotheses—such ques-
tions as whether a frequently tested hypothesis is more probable than
one which has been little tested—to be discussed in sections 79 to 85
under the title of ‘Corroboration’.

Ideas involving the theory of probability play a decisive part in mod-
ern physics. Yet we still lack a satisfactory, consistent definition of
probability; or, what amounts to much the same, we still lack a satisfac-
tory axiomatic system for the calculus of probability. The relations
between probability and experience are also still in need of clarifica-
tion. In investigating this problem we shall discover what will at first
seem an almost insuperable objection to my methodological views. For
although probability statements play such a vitally important rôle in
empirical science, they turn out to be in principle impervious to strict
falsification. Yet this very stumbling block will become a touchstone
upon which to test my theory, in order to find out what it is worth.

Thus we are confronted with two tasks. The first is to provide new founda-
tions for the calculus of probability. This I shall try to do by developing the
theory of probability as a frequency theory, along the lines followed by

Richard von Mises, but without the use of what he calls the ‘axiom of
convergence’ (or ‘limit axiom’), and with a somewhat weakened
‘axiom of randomness’. The second task is to elucidate the relations between
probability and experience. This means solving what I call the problem of decid-
ability of probability statements.

My hope is that these investigations will help to relieve the present
unsatisfactory situation in which physicists make much use of prob-
abilities without being able to say, consistently, what they mean by
‘probability’.*1

47 THE PROBLEM OF INTERPRETING
PROBABILITY STATEMENTS

I shall begin by distinguishing two kinds of probability statements:
those which state a probability in terms of numbers—which I will call
numerical probability statements—and those which do not.

Thus the statement, ‘The probability of throwing eleven with two
(true) dice is 1/18’, would be an example of a numerical probability
statement. Non-numerical probability statements can be of various
kinds. ‘It is very probable that we shall obtain a homogeneous mixture

*1 Within the theory of probability, I have made since 1934 three kinds of changes.
(1) The introduction of a formal (axiomatic) calculus of probabilities which can be

interpreted in many ways—for example, in the sense of the logical and of the frequency
interpretations discussed in this book, and also of the propensity interpretation discussed
in my Postscript.

(2) A simplification of the frequency theory of probability through carrying out,
more fully and more directly than in 1934, that programme for reconstructing the
frequency theory which underlies the present chapter.

(3) The replacement of the objective interpretation of probability in terms of fre-
quency by another objective interpretation—the propensity interpretation—and the replace-
ment of the calculus of frequencies by the neo-classical (or measure-theoretical)
formalism.

The first two of these changes date back to 1938 and are indicated in the book itself
(i.e. in this volume): the first by some new appendices, *ii to *v, and the second—the one
which affects the argument of the present chapter—by a number of new footnotes to this
chapter, and by the new appendix *vi. The main change is described here in footnote
*1 to section 57.

The third change (which I first introduced, tentatively, in 1953) is explained and
developed in the Postscript, where it is also applied to the problems of quantum theory.

some structural components of a theory of experience134

by mixing water and alcohol’, illustrates one kind of statement which,
suitably interpreted, might perhaps be transformed into a numerical
probability statement. (For example, ‘The probability of obtaining . . .
is very near to 1’.) A very different kind of non-numerical probability
statement would be, for instance, ‘The discovery of a physical effect
which contradicts the quantum theory is highly improbable’; a state-
ment which, I believe, cannot be transformed into a numerical prob-
ability statement, or put on a par with one, without distorting its
meaning. I shall deal first with numerical probability statements; non-
numerical ones, which I think less important, will be considered
afterwards.

In connection with every numerical probability statement, the ques-
tion arises: ‘How are we to interpret a statement of this kind and, in
particular, the numerical assertion it makes?’

48 SUBJECTIVE AND OBJECTIVE INTERPRETATIONS

The classical (Laplacean) theory of probability defines the numerical
value of a probability as the quotient obtained by dividing the number
of favourable cases by the number of equally possible cases. We might
disregard the logical objections which have been raised against this
definition,1 such as that ‘equally possible’ is only another expression
for ‘equally probable’. But even then we could hardly accept this defin-
ition as providing an unambiguously applicable interpretation. For
there are latent in it several different interpretations which I will
classify as subjective and objective.

A subjective interpretation of probability theory is suggested by the fre-
quent use of expressions with a psychological flavour, like ‘mathemat-
ical expectation’ or, say, ‘normal law of error’, etc.; in its original form it is
psychologistic. It treats the degree of probability as a measure of the feel-
ings of certainty or uncertainty, of belief or doubt, which may be

1 Cf. for example von Mises, Wahrscheinlichkeit, Statistik und Wahrheit, 1928, pp. 62 ff.; 2nd
edition, 1936, pp. 84 ff.; English translation by J. Neyman, D. Sholl, and E. Rabinowitsch,
Probability, Statistics and Truth, 1939, pp. 98 ff. *Although the classical definition is often
called ‘Laplacean’ (also in this book), it is at least as old as De Moivre’s Doctrine of Chances,
1718. For an early objection against the phrase ‘equally possible’, see C. S. Peirce, Collected
Papers 2, 1932 (first published 1878), p. 417, para. 2, 673.

probability 135

aroused in us by certain assertions or conjectures. In connection with
some non-numerical statements, the word ‘probable’ may be quite
satisfactorily translated in this way; but an interpretation along these
lines does not seem to me very satisfactory for numerical probability
statements.

A newer variant of the subjective interpretation,*1 however, deserves
more serious consideration here. This interprets probability statements
not psychologically but logically, as assertions about what may be called
the ‘logical proximity’2 of statements. Statements, as we all know, can
stand in various logical relations to one another, like derivability,
incompatibility, or mutual independence; and the