popper-logic-scientific-discovery

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form: it is unconditional and free of the unusual ‘ + p(c¯, c)’. (79) can be
further generalized:
(80) p(b¯c¯, ad) = p(b, ad) + p(c, ad) − p(bc, ad) 79
(81) p(a b¯c¯, d) = p(ab, d) + p(ac, d) − p(a(bc), d) 80, B2, 40
This is a generalization of (79).
We now proceed to the derivation of the law of distribution. It may
be obtained from (79), (81), and a simple lemma (84) which I propose
to call the ‘distribution lemma’, and which is a generalization of
(30):
(82) p(a(bc), d) = p(a, (bc)d)p(bc, d) = p((aa)(bc),d) B2, 30
(83) p(((aa)b)c, d) = p(a(ab), cd)p(c, d) = p(((ab)a)c, d) B2, 62, 40
(84) p(a(bc), d) = p((ab)(ac), d) 82, 83, 62
This is the ‘distribution lemma’.
(85) p(ab ac, d) = p(ab, d) + p(ac, d) − p((ab)(ac), d) 79 (subst.)
To this formula and (81) we can now apply the ‘distribution lemma’;
and we obtain:
(86) p(a b¯c¯, d) = p(ab ac, d) 81, 85, 84
This is a form of the first law of distribution. It can be applied to the
left side of the following formula
(87) p(b¯b¯a, c) = p(b¯b¯, ac)p(a, c) = p(a, c) B2, 74
We then obtain,
(88) p(ab ab¯, c) = p(a, c). 86, 87, 40
It may be noted that
(89) p(ā¯b, c) = p(ab, c), 68 (subst.)