popper-logic-scientific-discovery

than upon rational arguments; thus, by a slight change, probability is
then interpreted as the intensity, or the degree, of a belief in so far as it is
rationally justifiable. But at this stage, the reference to the intensity of a
belief, or to its degree, clearly becomes redundant; and ‘degree of
belief’ should therefore be replaced by ‘degree of the rationality of
a belief’. These remarks should not be taken to mean that I am prepared
to accept any form of the subjective interpretation; see point 12, below,
and chapter *ii of my Postscript: After Twenty Years.)
3. In order to save space, I shall explain the problem of the weight
of evidence merely by giving one instance of the paradoxes to which I
referred above. It may be called the ‘paradox of ideal evidence’.
Let z be a certain penny, and let a be the statement ‘the nth (as yet
unobserved) toss of z will yield heads’. Within the subjective theory,
it may be assumed that the absolute (or prior) probability of the
statement a is equal to 1/2, that is to say,
(1)
P(a) = 1/2
Now let e be some statistical evidence; that is to say, a statistical report,
based upon the observation of thousands or perhaps millions of
tosses of z; and let this evidence e be ideally favourable to the hypothesis
that z is strictly symmetrical—that it is a ‘good’ penny, with equidistribution.
(Note that here e is not the full, detailed report about the
results of each of these tosses—this report we might assume to have
been lost—but only a statistical abstract from the full report; for
example, e may be the statement, ‘among a million of observed tosses
of z, heads occurred in 500,000 ± 20 cases’. It will be seen, from
point 8, below, that an evidence e′ with 500,000 ± 1,350 cases would
still be ideal, if my functions C and E are adopted; indeed, from the
point of view of these functions, e is ideal precisely because it entails
e′.) We then have no other option concerning P(a, e) than to assume
that
(2)
P(a, e) = 1/2
appendix *ix 425
This means that the probability of tossing heads remains unchanged,
in the light of the evidence e; for we now have