popper-logic-scientific-discovery

384
new appendices
the required a always exists; and there always is also another law, a′,
yielding the same first n − 1 results but predicting, for the nth toss, the
opposite result. It would be paradoxical, therefore, to accept Jeffreys’s
case (3), since for a sufficiently large n we would always obtain p(b n,
b n − 1 ) close to 1, and also (from another law, a′) p(b¯ n, b n − 1 ) close to 1.
Accordingly, Jeffreys’s argument, which is mathematically inescapable,
can be used to prove his case (2), which happens to coincide with my
own formula (2), as stated at the beginning of this appendix. 10
We may sum up our criticism of ( + ) as follows. Some people believe
that, for purely logical reasons, the probability that the next thing we
meet will be red increases in general with the number of red things
seen in the past. But this is a belief in magic—in the magic of human
language. For ‘red’ is merely a predicate; and there will always be
predicates A and B which both apply to all the things so far observed,
but lead to incompatible probabilistic predictions with respect to the
next thing. These predicates may not occur in ordinary languages, but
they can always be constructed. (Strangely enough, the magical belief
here criticized is to be found among those who construct artificial
model languages, rather than among the analysts of ordinary language.)
By thus criticizing ( + ) I am defending, of course, the principle
of the (absolute logical) independence of the various a n from any combination
a ia j . . . ; that is to say, my criticism amounts to a defence of (4) and (1).
There are further proofs of (1). One of them which is fundamentally
due to an idea of Jeffreys and Wrinch 11 will be discussed more fully in
appendix *viii. Its main idea may be put (with slight adjustments) as
follows.
Let e be an explicandum, or more precisely, a set of singular facts or data
which we wish to explain with the help of a universal law. There will
be, in general, an infinite number of possible explanations—even an
infinite number of explanations (mutually exclusive, given the data e)
such that the sum of their probabilities (given e) cannot exceed unity.
But this means that the probability of almost all of them must be
zero—unless, indeed, we can order the possible laws in an infinite
10 Jeffreys himself draws the opposite conclusion: he adopts as valid the possibility stated
in case (3).
11 Philos. Magazine 42, 1921, pp. 369 ff.