popper-logic-scientific-discovery

whether a necessary statement is necessarily necessary finds a natural
place in probability theory: it is closely connected with the relation
between primary and secondary probability statements which plays an
important part in probability theory (as shown in appendix *ix, point
*13 of the Third Note). Roughly, writing ‘� x’ for ‘x is necessary (or
demonstrable)’ and ‘h’ for ‘p(a, ā) = 1’, we may show that
and therefore we find that
� a ↔ � h,
� a → � ‘p(h, h¯) = 1’,
which may be taken to mean that � a entails that a is necessarily
necessary; and since this means something like
� a → � ‘p(‘p(a, ā) = 1’, ‘p(a, ā) = 1’) = 1’,
we obtain (secondary) probability statements about (primary)
probability statements.
But there are of course other possible ways of interpreting the
relation between a primary and a secondary probability statement.
(Some interpretations would prevent us from treating them as
belonging to the same linguistic level, or even to the same language.)
Addendum, 1964
I have found since that the following system of three axioms, A, BD,
and CD, is equivalent to the six axioms on pp. 337 and 356–7.
A (Ea) (Eb)p(a, a) ≠ p(a, b)
BD ((d)p(ab, d) = p(c, d)) ↔ (e) (f) (p(a, b) � p(c, b) & p(a, e) �
� p(c, e) � p(b, c) & ((p(b, e) � p(f, e) & p(b, f) � p(f, f) �
� p(e, f)) → p(a, f)p(b, e) = p(c, e)))
CD p(ā, b) = p(b, b) − p(a, b) ↔ (Ec)p(b, b) ≠ p(c, b)
appendix *v 367
I also have found since an example not satisfying A2 but satisfying all