popper-logic-scientific-discovery

appendix *ix 409
here merely following the tradition of Keynes, Jeffreys, Reichenbach,
Kaila, Hosiasson, and others. Moreover, both Bar-Hillel and Kemeny
suggest that this criticism, as far as it applies to Carnap’s theory, is
purely verbal, and that there is no reason why Carnap’s theory should
be given up; and I therefore feel compelled to say now quite clearly that
Carnap’s theory is self-contradictory, and that its contradictoriness is
not a minor matter which can be easily repaired, but is due to mistakes
in its logical foundations.
First, both the assumptions (a) and (b) which, as we have seen,
suffice for the proof that degree of confirmation must not be identified
with probability, are explicitly asserted in Carnap’s theory: (a), that is
to say our formula (1), can be found in Carnap’s book as formula (4)
on p. 464; 7 (b), that is to say (***), or the assumption that our (**) is
self-contradictory, can be found on p. 73 of Carnap’s book where he
writes: ‘If the property Warm and the relation Warmer were designated
by . . . , say, ‘P’ and ‘R’, then ‘Pa.∼Pb.Rba’ would be self-contradictory.’
But this is our (***). Of course, in a way it is quite irrelevant to my
argument that shows the absurdity of the identification of C and p
whether or not (a) and (b) are explicitly admitted in a book; but it so
happens that in Carnap’s book, they are.
Moreover, the contradiction here explained is crucial for Carnap: by
accepting (1), or more precisely, by defining on pp. 463 f. ‘x is confirmed
by y’ with the help of ‘p(x, y) >p(x)’ (in our symbolism),
Carnap shows that the intended meaning of ‘degree of confirmation’
(his ‘explicandum’) is, roughly, the same as the one intended by myself. It is
the intuitive idea of degree of support by empirical evidence. (Kemeny
loc. cit. is mistaken when he suggests the opposite. In fact, ‘a careful
reading’ of my paper—and, I should add, of Carnap’s book—will not
‘show that Popper and Carnap have two different explicanda in mind’, but
it will show that Carnap had inadvertently two different and incompatible
‘explicanda’ in mind with his probability 1, one of them my C, the
other my p; and it will show that I have repeatedly pointed out the
dangers of this confusion—for example in the paper reviewed by
7 See also formula (6) on p. 464. Carnap’s formula (4) on p. 464 is written as an
equivalence, but this does not make any difference. Note that Carnap writes ‘t’ for
tautology; a usage which would allow us to write p(x, t) instead of p(x).