popper-logic-scientific-discovery

376
new appendices
favourable possibilities divided by that of all (equal) possibilities. We can
then derive (2), for example, if we identify the favourable possibilities
with the favourable evidence. It is clear that, in this case, p(a,b) = 0; for
the favourable evidence can only be finite, while the possibilities in an
infinite universe must be clearly infinite. (Nothing depends here on
‘infinity’, for any sufficiently large universe will yield, with any desired
degree of approximation, the same result; and we know that our
universe is extremely large, compared with the amount of evidence
available to us.)
This simple consideration is perhaps a little vague, but it can be
considerably strengthened if we try to derive (1), rather than (2), from
the classical definition. We may to this end interpret the universal
statement a as entailing an infinite product of singular statements, each
endowed with a probability which of course must be less than unity. In
the simplest case, a itself may be interpreted as such as infinite product;
that is to say, we may put a = ‘everything has the property A’; or in
symbols, ‘(x)Ax’, which may be read ‘for whatever value of x we may
choose, x has the property A’. 1 In this case, a may be interpreted as the
infinite product a = a 1a 2a 3 . . . where a i = Ak i, and where k i is the name of
the ith individual of our infinite universe of discourse.
We may now introduce the name ‘a n ’ for the product of the first n
singular statements, a 1a 2 ... a n, so that a may be written
1 ‘x’ is here an individual variable ranging over the (infinite) universe of discourse. We
may choose; for example, a = ‘All swans are white’ = ‘for whatever value of x we may
choose, x has the property A’ where ‘A’ is defined as ‘being white or not being a swan’.
We may also express this slightly differently, by assuming that x ranges over the spatiotemporal
regions of the universe, and that ‘A’ is defined by ‘not inhabited by a non-white
swan’. Even laws of more complex form—say of a form like ‘(x)(y)(xRy → xSy)’ may be
written ‘(x)Ax’, since we may define ‘A’ by
Ax ↔ (y)(xRy → xSy).
We may perhaps come to the conclusion that natural laws have another form than the
one here described (cf. appendix *x): that they are logically still stronger than is here
assumed; and that, if forced into a form like ‘(x)Ax’, the predicate A becomes essentially
non-observational (cf. notes *1 and *2 to the ‘Third Note’, reprinted in appendix *ix)
although, of course, deductively testable. But in this case, our considerations remain valid
a fortiori.