popper-logic-scientific-discovery

366
new appendices
which is nevertheless invalid in the usual systems—provided they are
consistent. For in order to make it valid, we must make allowance for 4
even though we have
p(a, aā) + p(ā, aā) = 2,
p(a + ā, aā) = 1.
That is to say, such formulae as p(a + ā, b) = p(a, b) + p(ā, b) must not be
unconditionally asserted in the system. (Cf. our axiom C; see also footnote
1, above.)
The converse of (+), that is to say,
p(a, b) = 1 → a � b
must not be demonstrable, of course, as our second and third examples
proving consistency show. (Cf. also the formula (E) in the present and
in the preceding appendices.) But there are other valid equivalences in
our system such as
(‡)
a � b ↔ p(a, āb) ≠ 0
a � b ↔ p(a, āb) = 1
None of these can hold in the usual systems in which p(a, b) is
undefined unless p(b) ≠ 0. It seems to be quite clear, therefore, that the
usual systems of probability theory are wrongly described as generalizations
of logic: they are formally inadequate for this purpose, since
they do not even entail Boolean algebra.
The formal character of our system makes it possible to interpret it,
for example, as a many-valued propositional logic (with as many
values as we choose—either discrete, or dense, or continuous), or as a
system of modal logic. There are in fact many ways of doing this; for
example, we may define ‘a necessarily implies b’ by ‘p(b, ab¯) ≠ 0’, as just
indicated, or ‘a is logically necessary’ by ‘p(a, ā) = 1’. Even the problem
4 See formulae 31′ ff. in footnote 1, above