popper-logic-scientific-discovery

appendix *vii 387
my Postscript.) Here too, Newton’s theory is better testable, because its
content is greater. 13
Now our proof of (1) shows that these differences in content and in
testability cannot be expressed immediately in terms of the absolute
logical probability of the theories a 1 and a 2, since p(a 1) = p(a 2) = 0. And
if we define a measure of content, C(a), by C(a) = 1 − p(a), as suggested
in the book, then we obtain, again, C(a 1) = C(a 2), so that the differences
in content which interest us here remain unexpressed by these measures.
(Similarly, the difference between a self-contradictory statement
aā and a universal theory a remains unexpressed since p(aā) = p(a) = 0,
and C(aā) = C(a) = 1. 14 )
13 Whatever C. G. Hempel may mean by ‘confirming evidence’ of a theory, he clearly
cannot mean the result of tests which corroborate the theory. For in his papers on the
subject (Journal of Symbolic Logic 8, 1943, pp. 122 ff., and especially Mind 54, 1945, pp. 1 ff.
and 97 ff.; 55, 1946, pp. 79 ff.), he states (Mind 54, pp. 102 ff.) among his conditions
for adequacy the following condition (8.3): if e is confirming evidence of several
hypotheses, say h 1 and h 2, then h 1 and h 2 and e must form a consistent set of statements.
But the most typical and interesting cases tell against this. Let h 1 and h 2 be Einstein’s
and Newton’s theories of gravitation. They lead to incompatible results for strong gravitational
fields and fast moving bodies, and therefore contradict each other. And yet, all
the known evidence supporting Newton’s theory is also evidence supporting Einstein’s,
and corroborates both. The situation is very similar for Newton’s and Kepler’s theories,
or Newton’s and Galileo’s. (Also, any unsuccessful attempt to find a red or yellow swan
corroborates both the following two theories which contradict each other in the presence
of the statement ‘there exists at least one swan’: (i) ‘All swans are white’ and (ii) ‘All
swans are black’.)
Quite generally, let there be a hypothesis h, corroborated by the result e of severe tests,
and let h 1, and h 2 be two incompatible theories each of which entails h. (h 1 may be ah, and
h 2 may be āh.) Then any test of h is one of both h 1 and h 2, since any successful refutation of
h would refute both h 1 and h 2; and if e is the report of unsuccessful attempts to refute h,
then e will corroborate both h 1 and h 2. (But we shall, of course, look for crucial tests
between h 1 and h 2.) With ‘verifications’ and ‘instantiations’, it is, of course, otherwise.
But these need not have anything to do with tests.
Yet quite apart from this criticism, it should be noted that in Hempel’s model language
identity cannot be expressed; see his paper in The Journal of Symbolic Logic 8, 1943, the last
paragraph on p. 143, especially line 5 from the end of the paper, and p. 21 of my Preface,
1958. For a simple (‘semantical’) definition of instantiation, see the last footnote of my note in
Mind 64, 1955, p. 391.
14 That a self-contradictory statement may have the same probability as a consistent
synthetic statement is unavoidable in any probability theory if applied to some infinite universe
of discourse: this is a simple consequence of the multiplication law which demands that