popper-logic-scientific-discovery

360
new appendices
(Applying B2 twice to each side of (62) only leads to a conditional
form with ‘p(bc, d)≠0 → ’ as antecedent.)
I now turn to a generalization of the axiom of complementation, C. I
shall be a little more concise in my derivations from now on.
(63) p(b¯, b) ≠ 0 ↔ (c)p(c, b) = 1 7, 25
(64) p(a, b) + p(ā, b) = 1 + p(b¯, b) C, 25, 63
This is an unconditional form of the principle of complementation,
C, which I am now going to generalize.
In view of the fact that (64) is unconditional, and that ‘a’ does not
occur on the right-hand side, we can substitute ‘c’ for ‘a’ and assert
(65) p(a, b) + p(ā, b) = p(c, b) + p(c¯, b) 64
(66) p(a, bd) + p(ā, bd) = p(c, bd) + p(c¯, bd) 65
By multiplying with p(b, d) we get:
(67) p(ab, d) + p(āb, d) = p(cb, d) + p(c¯b, d) B2, 66
This is a generalization of (65). By substitution, we get:
(68) p(ab, c) + p(āb, c) = p(cb, c) + p(c¯b, c). 67
In view of
(69) p(c¯b, c) = p(c¯, c), 7, B1, 25, 63
we may also write (68) more briefly, and in analogy to (64),
(70) p(ab, c) + p(āb, c) = p(b, c) + p(c¯, c). 68, 69 1
1 In the derivation of (70) we also need the following formula
p(cb, c) = p(b, c),
which may be called ‘(29′)’. Its derivation, in the presence of (40) and (32) is analogous
to the steps (28) and (29):