popper-logic-scientific-discovery

342
new appendices
obtain p(a, c) = 1 = p(ā, c) so that, in this limiting case, the apparently
‘obvious’ formula breaks down.
This postulate, or the axiom C, has the character of a definition of
p(ā, b), in terms of p(a, b) and p(a, a), as can be easily seen if we write C
as follows. (Note that (ii) follows from (i).)
(i) p(ā, b) = p(a, a) − p(a, b), provided there is a c such that p(c, b) ≠
p(a, a)
(ii) p(ā, b) = p(a, a), provided there is no such c.
A formula CD analogous to an improved version of BD will be found
in an Addendum on p. 367.
The system consisting of A1, BD, and CD is, I think, slightly
preferable to A1, B, and C + , in spite of the complexity of BD.
Postulate AP, ultimately, can be replaced by the simple definition
(.)
p(a) = p(a, āa)
which, however, uses complementation and the product, and accordingly
presupposes both Postulates 3 and 4. Formula (.) will be derived
below in appendix *v as formula 75.
Our axiom system can be proved to be consistent: we may construct
systems of elements S (with an infinite number of different elements:
for a finite S, the proof is trivial) and a function p(a, b) such that all the
axioms are demonstrably satisfied. Our system of axioms may also be
proved to be independent. Owing to the weakness of the axioms, these
proofs are quite easy.
A trivial first consistency proof for a finite S is obtained by assuming that
S = {1, 0}; that is to say, that S consists of the two elements, 1 and
0; product or meet and complement are taken to be equal to
arithmetical product and complement (with respect to 1). We define
p(0, 1) = 0, and in all other cases put p(a, b) = 1. Then all the axioms
are satisfied.
Two further finite interpretations of S will be given before proceeding
to a denumerably infinite interpretation. Both of these satisfy not
only our axiom system but also, for example, the following existential
assertion (E).