popper-logic-scientific-discovery

This is the generalization of the unconditional form of C and of
formula (64).
(71) p(aa, b) + p(āa, b) = p(a, b) + p(b¯, b) 70
(72) p(āa, b) = p(aā, b) = p(b¯, b) 40, 71, 30
(73) p(āa, b) + p(āa, b) = p(aā, b) + p(aā, b) = 1 + p(b¯, b) 64
(74) p(āa, b) = 1 = p(aā, b) 72, 73
This establishes the fact that the elements aā satisfy the condition of
Postulate AP. We obtain, accordingly,
(75) p(a) = p(a, aā) = p(a, āa) = p(a, bb¯) = p(a, b¯b); 25, 74, AP
that is, a definition of absolute probability in a more workable form.
We next derive the general law of addition.
appendix *v 361
(76) p(ab¯, c) = p(a, c) − p(ab, c) + p(c¯, c) 70, 40
(77) p(āb¯, c) = p(ā, c) − p(āb, c) + p(c¯, c) 76
(78) p(āb¯, c) = 1 − p(a, c) − p(b, c) + p(ab, c) + p(c¯, c) 77, 76, 64, 40
(79) p(āb¯, c) = p(a, c) + p(b, c) − p(ab, c) 78, 64
This is a form of the general law of addition, as will be easily seen if
it is remembered that ‘āb¯’ means the same in our system as ‘a + b’
in the Boolean sense. It is worth mentioning that (79) has the usual
(28′) p(ab, ab) = 1 = p(b, ab) 25, 32, 17
(29′) p(ba, b) = p(b, ab)p(a, b) = p(a, b). B2, 28′
To this we may add the law of idempotence for the second argument
(30′) p(ab, b) = p(a, bb) = p(a, b). B2, 25, 29′, 40
Moreover, from (28) we obtain by substitution
(31′) p(a, aā) = 1 28
and likewise from (28′)
(32′) p(ā, aā) = 1 28′
This yields, by C,
(33′) p(a, b¯b) = 1 31′, 32′, C
We therefore have
(34′) (Eb)(a) p(a, b) = 1 33′
(35′) (Ea) p(ā, a) = 1 34′
See also (27). Formulae (31′) to (35′) do not belong to the theorems of the usual systems.