popper-logic-scientific-discovery

(90) p(a, c) = p(b, c) → p(ā, c) = p(b¯, c) 64
Consequently, we have
(91) p(ā b¯ c¯, d) = p(ā b¯ c¯, d) 62, 89, 40
(92) p(ā b¯ c¯, d) = p(ā b¯ c¯, d) 90, 91
This is the law of association for the Boolean sum. By substituting in
(40) the complements of a and b, we find
(93) p(ā b¯, c) = p(b¯ ā, c) 40, 90
This is the law of commutation for the Boolean sum. In the same way
we get
(94) p(ā ā, b) = p(a, b) 30, 89, 90
This is the law of idempotence for the Boolean sum. From (87) we
obtain
(95) p(a, b) = p(a, bcc¯), 87, 40, A2
(96) p(a, b)p(b) = p(ab) 95, B2, 75
This may also be written
(97) p(b) ≠ 0 → p(a, b) = p(ab)/p(b) 96
This formula shows that our generalized concept of relative probability
coincides, for p(b) ≠ 0, with the usual concept, and that our calculus is
a generalization of the usual calculus. That the generalization is a genuine
one can be seen from the examples, given in the preceding appendix
*iv, showing the consistency of our system with the following
formula (E):
(E)
(Ea)(Eb)(Ec) p(a, b) = 1 & p(a, bc) = 0
appendix *v 363
—a formula which is invalid in many finite interpretations of our S but
valid in its normal infinite interpretations.