popper-logic-scientific-discovery

334
new appendices
invalid, and should be replaced by the much weaker conditional
formula (g − ):
(g − )
If p(d) ≠ 0 then p((ab)c, d) = p(a(bc), d)
and an analogous condition should be prefixed to (h).
This point has been overlooked by some authors (for example,
Jeffreys, and von Wright; the latter uses conditions amounting to b ≠ 0,
but this does not ensure p(b) ≠ 0, especially since his system contains
an ‘axiom of continuity’). Their systems, accordingly, are inconsistent
as they stand, although they can sometimes be mended. Other authors
have seen the point. But as a consequence, their systems are (at least as
compared with my system) very weak: it can occur in their systems
that
p(a, b) = r
is a meaningful formula, while at the same time, and for the same
elements
p(b, a) = r
is not meaningful, i.e. not properly defined, and not even definable,
because p(a) = 0.
But a system of this kind is not only weak; it is also for many
interesting purposes inadequate: it cannot, for example, be properly
applied to statements whose absolute probability is zero, although this
application is very important: universal laws, for example, have, we
may here assume (cf. appendices *vii and *viii), zero probability. If we
take two universal theories, s and t, say, such that s is deducible from t,
then we should like to assert that
p(s, t) = 1
But if p(t) = 0, we are prevented from doing so, in the customary
systems of probability. For similar reasons, the expression
p(e, t)