popper-logic-scientific-discovery

358
new appendices
(26) (Eb) (Ea) p(b, a) ≠ k A1, 1
(27) (Ea) p(ā, a) = 0 7, 26
We have now established all the laws of the upper and lower bounds:
(14) and (17), summed up in (18), show that probabilities are
bounded by 0 and 1. (25) and (27) show that these bounds are actually
reached. We now turn to the derivation of the various laws usually
taken either from Boolean algebra or from the propositional calculus.
First we derive the law of idempotence.
(28) 1 = p(ab, ab) � p(a, ab) = 1 25, B1, 17
(29) p(aa, b) = p(a, ab)p(a, b) B2
(30) p(aa, b) = p(a, b) 28, 29
This is the law of idempotence, sometimes also called the ‘law of
tautology’. We now turn to the derivation of the law of commutation.
(31) p(a, bc) � 1 17
(32) p(ab, c) � p(b, c) B2, 31, 14
This is the second law of monotony, analogous to B1.
(33) p(a(bc), a(bc)) = 1 25
(34) p(bc, a(bc)) = 1 33, 32, 17
(35) p(b, a(bc)) = 1 34, B1, 17
(36) p(ba, bc) = p(a, bc) 35, B2
(37) p((ba)b, c) = p(ab, c) 36, B2
(38) p(ba, c) � p(ab, c) 37, B1
(39) p(ab, c) � p(ba, c) 38 (subst.)
(40) p(ab, c) = p(ba, c) 38, 39
This is the law of commutation for the first argument. (In order to
extend it to the second argument, we should have to use A2.) It has
been derived from (25), merely by using the two laws of monotony
(B1 and 32) and B2. We now turn to the derivation of the law of
association.
(41) p(ab, d((ab)c)) = 1 35 (subst.)
(42) p(a, d((ab)c)) = 1 = p(b, d((ab)c)) 41, B1, 17, 32