popper-logic-scientific-discovery

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some structural components of a theory of experience
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 decidability
of probability statements.
My hope is that these investigations will help to relieve the present
unsatisfactory situation in which physicists make much use of probabilities
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 frequency
by another objective interpretation—the propensity interpretation—and the replacement
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.