popper-logic-scientific-discovery

380
new appendices
The same point may be put a little differently thus: in the presence of
some hypothesis, h say, we may of course have
(6)
p(a j, a ih) >p(a j, h)
For h may inform us of the existence of a kind of after-effect.
Consequently, we should then have
(7)
p(a ia j, h) >p(a i, h)p(a j, h),
since (7) is equivalent to (6). But in the absence of h, or if h is tautologous
or, in other words, if we are concerned with absolute logical
probabilities, (7) must be replaced by
(8)
p(a ia j) = p(a i)p(a j)
which means that a i and a j are independent, and which is equivalent to
(9)
p(a j, a i) = p(a j).
But the assumption of mutual independence leads, together with
p(a i) < 1, as before to p(a) = 0; that is to say, to (1).
Thus (8), that is, the assumption of the mutual independence of the
singular statements a i leads to (1); and mainly for this reason, some
authors have, directly or indirectly, rejected (8). The argument has
been, invariably, that (8) must be false because if it were true, we could
not learn from experience: empirical knowledge would be impossible. But
this is incorrect: we may learn from experience even though p(a) =
p(a, b) = 0; for example, C(a, b)—that is to say, the degree of corroboration
of a by the tests b—may none the less increase with new tests. (Cf.
appendix *ix). Thus this ‘transcendental’ argument fails to hit its target;
at any rate, it does not hit my theory. 3
3 An argument which appeals to the fact that we possess knowledge or that we can learn
from experience, and which concludes from this fact that knowledge or learning from
experience must be possible, and further, that every theory which entails the impossibility
of knowledge, or of learning from experience, must be false, may be called a ‘transcendental
argument’. (This is an allusion to Kant.) I believe that a transcendental