popper-logic-scientific-discovery

favour, more probable than its negation. But this is only possible if (1)
is false, that is to say, if we have p(a) >0.
A more direct disproof of ( + ) and a proof of (2) can be obtained
from an argument which Jeffreys gives in his Theory of Probability, § 1.6. 8
Jeffreys discusses a formula which he numbers (3) and which in our
symbolism amounts to the assertion that, provided p(b i, a) = 1 for every
i � n, so that p(ab n ) = p(a), the following formula must hold:
(10)
p(a, b n ) = p(a)/p(b n ) = p(a)/p(b 1)p(b 2, b 1 ) ... p(b n, b n − 1 )
appendix *vii 383
Discussing this formula, Jeffreys says (I am still using my symbols in
place of his): ‘Thus, with a sufficient number of verifications, one of
three things must happen: (1) The probability of a on the information
available exceeds 1. (2) it is always 0. (3) p(b n, b n − 1 ) will tend to 1.’ To
this he adds that case (1) is impossible (trivially so), so that only (2)
and (3) remain. Now I say that the assumption that case (3) holds
universally, for some obscure logical reasons (and it would have to
hold universally, and indeed a priori, if it were to be used in induction),
can be easily refuted. For the only condition needed for deriving (10),
apart from 0 < p(b i) < 1, is that three exists some statement a such that
p(b n , a) = 1. But this condition can always be satisfied, for any sequence
of statements b i. For assume that the b i are reports on penny tosses; then
it is always possible to construct a universal law a which entails the
reports of all the n − 1 observed penny tosses, and which allows us to
predict all further penny tosses (though probably incorrectly). 9 Thus
8 I translate Jeffreys’s symbols into mine, omitting his H since nothing in the argument
prevents us from taking it to be either tautological or at least irrelevant; in any case, my
argument can easily be restated without omitting Jeffreys’s H.
9 Note that there is nothing in the conditions under which (10) is derived which would
demand the b i to be of the form ‘B(k i)’, with a common predicate ‘B’, and therefore
nothing to prevent our assuming that b i = ‘k i is heads’ and b j = ‘k j is tails’. Nevertheless, we
can construct a predicate ‘B’ so that every b i has the form ‘B(k i)’: we may define B as
‘having the property heads, or tails, respectively, if and only if the corresponding element
of the sequence determined by the mathematical law a is 0, or is 1, respectively’. (It
may be noted that a predicate like this can be defined only with respect to a universe of
individuals which are ordered, or which may be ordered; but this is of course the only case
that is of interest if we have in mind applications to problems of science. Cf. my Preface,
1958, and note 2 to section *49 of my Postscript.)