popper-logic-scientific-discovery

the same likelihood—i.e.P(e, h) = 0.9930—as would statistical evidence e′,
based on only a hundred tosses and δ = 0.135.* 5 (But it is quite
satisfactory to find that E(h, e) = 0.9946 while E(h, e′) = 0.7606.)
11. It should be noticed that the absolute logical probability of a
universal law h—that is, P(h)—will be in general zero, in an infinite
universe. For this reason, P(e, h)—that is, the likelihood of h—will
become indefinite, in most systems of probability, since in most systems
P(e, h) is defined as P(eh)/P(h) = 0/0. We therefore need a formal
calculus of probability which yields definite values for P(e, h) even if
P(h) = 0, and which will always and unambiguously yield P(e, h) = 1
whenever e follows (or ‘almost follows’) from h. A system answering
these demands was published by me some time ago. 7
12. Our E(h, e) may be adequately interpreted as a measure of the
explanatory power of h with respect to e, even if e is not a report
of genuine and sincere attempts to refute h. But our C(h, e) can be
adequately interpreted as degree of corroboration of h—or of the
rationality of our belief in h, in the light of tests—only if e consists of
reports of the outcome of sincere attempts to refute h, rather than of
attempts to verify h.
As hinted in the preceding sentence, I suggest that, while it is a
mistake to think that probability may be interpreted as a measure of the
rationality of our beliefs (this interpretation is excluded by the paradox
of perfect evidence), degree of corroboration may be so interpreted. 8
As to the calculus of probability, it has a very large number of different
interpretations. 9 Although ‘degree of rational belief’ is not among
* 5 Fisher’s ‘likelihood’ turns out to be in many cases intuitively unsatisfactory. Let x be
‘the next throw with this die is a six’. Then the likelihood of x in the light of the evidence
y will be 1, and thus at its maximal value, if we take y to mean, for example, ‘the next
throw is even’, or ‘the next throw is a number > 4’ or even ‘the next throw is a number
other than 2’. (The values of C(x, y) are satisfactory, it seems: they are, respectively, 3/8;
4/7; and 1/10.)
7 This Journal, 1955, 6; see esp. 56 sq. A simplified form of this axiom system may be
found in my papers ‘Philosophy of Science: A Personal Report’ (p. 191) and ‘The Propensity
Interpretation’, etc., referred to in note 3 above. (In the latter paper, p. 67, note 3,
the last occurrence of ‘ < ’ should be replaced by ‘≠’, and in (B) and (C) a new line
should commence after the second arrows.) *See the new appendices *iv and *v.
8 Cf. this Journal, 1955, 6, 55 (the title of the section).
9 Cf. my note in Mind, 1938, 47, 275 sq.
appendix *ix 433