popper-logic-scientific-discovery

appendix *ix 423
atomic statements (as indicated in appendix in of my Logic of Scientific
Discovery), then we shall have to postulate independence for the atomic,
or relative-atomic, sentences (of course, as far as they are not ‘logically
dependent’, in Kemeny’s sense). On the basis of a probabilistic theory of induction,
it then turns out that we cannot learn if we identify logical and
probabilistic independence in the way here described; but we can
learn very well in the sense of my C-functions; that is to say, we can
corroborate our theories.
Two further points may be mentioned in this connection.
4. The first point is this. On the basis of my axiom systems for
relative probability, 5 P(x, y) can be considered as defined for any value
of x and y, including such values for which P(y) = 0. More especially, in
the logical interpretation of the system, whenever x follows from y,
P(x, y) = 1, even if P(y) = 0. There is thus no reason to doubt that our
definition works for languages containing both singular statements and
universal laws, even if all the latter have zero probability, as is the case,
for example, if we employ Kemeny’s measure function m, by postulating
P(x) = m(x). (In the case of our definitions of E and C, there is no
need whatever to depart from the assignment of equal weight to the
‘models’; cf. Kemeny, op. cit p. 307. On the contrary, any such departure
must be considered as a deviation from a logical interpretation, since it
would violate the equality of logical and probabilistic independence
demanded in 3, above.)
5. The second point is this. Among the derived desiderata, the following
is not satisfied by all definitions of ‘x is confirmed by y’ which
have been proposed by other authors. It might therefore be mentioned
separately as a tenth desideratum: 6
(x) If x is confirmed or corroborated or supported by y so that
C(x, y) > 0, then (a) x¯ is always undermined by y, i.e. C(x¯, y)