popper-logic-scientific-discovery

(43) p(a, (bc)((ab)c)) = 1 42 (subst.)
(44) p(a(bc), (ab)c) = p(bc, (ab)c) 43, B2
(45) p(bc, (ab)c) = p(b, c((ab)c))p(c, (ab)c) B2
(46) p(b, c((ab)c)) = 1 42 (subst.)
(47) p(c, (ab)c) = 1 25, 32, 17
(48) p(a(bc), (ab)c) = 1 44 to 47
This is a preliminary form of the law of association. (62) follows
from it by A2 + (and B2), but I avoid where possible using A2 or A2 + .
(49) p(a(b(cd)), d) = p(cd, b(ad))p(b, ad)p(a, d) 40, B2
(50) p(a(bc), d) = p(c, b(ad))p(b, ad)p(a, d) 40, B2
(51) p(a(bc), d) � p(a(b(cd)), d) 49, 50 B1
This is a kind of weak generalization of the first monotony law, B1.
(52) p(a(b(cd)), (ab)(cd)) = 1 48 (subst.)
(53) p((a(b(cd))(ab), cd) = p(ab, cd) 52, B2
(54) p(a(b(cd)), cd) � p(ab, cd) 53, B1
(55) p((a(b(cd)))c, d) � p((ab)c, d) 54, B2
(56) p(a(b(cd)), d) � p((ab)c, d) 55, B1
(57) p(a(bc), d) � p((ab)c, d) 51, 56
This is one half of the law of association.
(58) p((bc)a, d) � p((ab)c, d) 57, 40
(59) p((ab)c, d) � p(b(ca), d) 58 (subst.), 40
(60) p((bc)a, d) � p(b(ca), d) 58, 59
(61) p((ab)c, d) � p(a(bc), d) 60 (subst.)
This the second half of the law of association.
appendix *v 359
(62) p((ab)c, d) = p(a(bc), d) 57, 61
This is the complete form of the law of association, for the
first argument (see also formula (g) at the beginning of appendix *iv).
The law for the second argument can be obtained by applying A2.