popper-logic-scientific-discovery

probability 137
probability statement as a statement about the relative frequency with
which an event of a certain kind occurs within a sequence of occurrences. 5
According to this interpretation, the statement ‘The probability of
the next throw with this die being a five equals 1/6’ is not really an
assertion about the next throw; rather, it is an assertion about a whole
class of throws of which the next throw is merely an element. The statement
in question says no more than that the relative frequency of fives,
within this class of throws, equals 1/6.
According to this view, numerical probability statements are only
admissible if we can give a frequency interpretation of them. Those probability
statements for which a frequency interpretation cannot be
given, and especially the non-numerical probability statements, are
usually shunned by the frequency theorists.
In the following pages I shall attempt to construct anew the theory
of probability as a (modified) frequency theory. Thus I declare my faith in
an objective interpretation; chiefly because I believe that only an objective
theory can explain the application of the probability calculus within
empirical science. Admittedly, the subjective theory is able to give a
consistent solution to the problem of how to decide probability statements;
and it is, in general, faced by fewer logical difficulties than is the
objective theory. But its solution is that probability statements are nonempirical;
that they are tautologies. And this solution turns out to be
utterly unacceptable when we remember the use which physics makes
of the theory of probability. (I reject that variant of the subjective
theory which holds that objective frequency statements should be
derived from subjective assumptions—perhaps using Bernoulli’s theorem
as a ‘bridge’: 6 I regard this programme for logical reasons as
unrealizable.)
5 For the older frequency theory cf. the critique of Keynes, op. cit., pp. 95 ff., where special
reference is made to Venn’s The Logic of Chance. For Whitehead’s view cf. section 80 (note
2). Chief representatives of the new frequency theory are: R. von Mises (cf. note 1 to
section 50), Dörge, Kamke, Reichenbach and Tornier. *A new objective interpretation,
very closely related to the frequency theory, but differing from it even in its mathematical
formalism, is the propensity interpretation, introduced in sections *53 ff. of my Postscript.
6 Keynes’s greatest error; cf. section 62, below, especially note 3. *I have not changed my
view on this point even though I now believe that Bernoulli’s theorem may serve as a
‘bridge’ within an objective theory—as a bridge from propensities to statistics. See also
appendix *ix and sections *55 to *57 of my Postscript.