popper-logic-scientific-discovery

ook (sections 80, 81, and 83) by a discussion of certain ideas of
Reichenbach’s, Keynes’s and Kaila’s. One result of this discussion is
that in an infinite universe (it may be infinite with respect to the number of
distinguishable things, or of spatio-temporal regions), the probability of
any (non-tautological) universal law will be zero.
(Another result was that we must not uncritically assume that scientists
ever aim at a high degree of probability for their theories. They
have to choose between high probability and high informative content,
since for logical reasons they cannot have both; and faced with this choice, they
have so far always chosen high informative content in preference to
high probability—provided that the theory has stood up well to its
tests.)
By ‘probability’, I mean here either the absolute logical probability of
the universal law, or its probability relative to some evidence; that is to say,
relative to a singular statement, or to a finite conjunction of singular
statements. Thus if a is our law, and b any empirical evidence, I assert
that
(1)
and also that
(2)
p(a) = 0
p(a, b) = 0
appendix *vii 375
These formulae will be discussed in the present appendix.
The two formulae, (1) and (2), are equivalent. For as Jeffreys and
Keynes observed, if the ‘prior’ probability (the absolute logical probability)
of a statement a is zero, then so must be its probability relative
to any finite evidence b, since we may assume that for any finite
evidence b, we have p(b) ≠ 0. For p(a) = 0 entails p(ab) = 0, and since
p(a, b) = p(ab)/p(b), we obtain (2) from (1). On the other hand, we
may obtain (1) from (2); for if (2) holds for any evidential b, however
weak or ‘almost tautological’, we may assume that it also holds for the
zero-evidence, that is to say, for the tautology t = bb¯; and p(a) may be
defined as equal to p(a, t).
There are many arguments in support of (1) and (2). First, we may
consider the classical definition of probability as the number of the