popper-logic-scientific-discovery

420
new appendices
to be slightly amended: in (ii) and in (v), we must replace ± 1 by ± ∞;
and (iii) becomes:
(iii)
(1)
0 � C(x, xy) = C(x, x) = C(x) =−log 2P(x) � +∞.
The other desiderata remain as they were.
Dr. Hamblin suggests 4 that we define degree of confirmation by
C(x, y) = log 2(P(xy)/P(x)P(y))
which for finite systems, but not necessarily for infinite systems, is the
same as
(2)
C(x, y) = log 2(P(y, x)/P(y)),
a formula which has the advantage of remaining determinate even if
P(x) = 0, as may be the case if x is a universal theory. The corresponding
relativized formula would be
(3)
C(x, y, z) = log 2(P(y, xz)/P(y, z)).
The definition (1) does not, however, satisfy my desideratum viii (c),
as Dr. Hamblin observes; and the same holds for (2) and (3). Desiderata
ix (b) and (c) are also not satisfied.
Now my desideratum viii (c) marks, in my opinion, the difference
between a measure of explanatory power and one of confirmation. The
former may be symmetrical in x and y, the latter not. For let y follow
from x (and support x) and let x be unconfirmed by y. In this case it
does not seem satisfactory to say that ax is always as well confirmed by y
as is x. (But there does not seem to be any reason why ax and x should
not have the same explanatory power with respect to y, since y is
completely explained by both.) This is why I feel that viii(c) should not
be dropped.
4 C. L. Hamblin, op. cit., p. 83. A similar suggestion (without, however, specifying 2 as
basis of the logarithm) is made in Dr. I. J. Good’s review of my ‘Degree of Confirmation’;
cf. Mathematical Review, 16, 1955, 376.