popper-logic-scientific-discovery

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new appendices
that is to say, for the probability of a (given no information, or only
tautological information).
But this procedure is apt to veil the surprising and highly important
fact that some of the adopted axioms or postulates for relative probability,
p(a, b), alone guarantee that all the laws of Boolean algebra hold for the
elements. For example, a form of the law of association is entailed by the
following two formulae (cf. the preceding appendix *iii),
(d)
(e)
p(ab) = p(a, b)p(b)
p(ab, c) = p(a, bc)p(b, c)
of which the first, (d), also gives rise to a kind of definition of relative
probability in terms of absolute probability,
(d′)
If p(b) ≠ 0 then p(a, b) = p(ab)/p(b),
while the second, the corresponding formula for relative probabilities,
is well known as the ‘general law of multiplication’.
These two formulae, (d) and (e), entail, without any further
assumption (except substitutivity of equal probabilities) the following
form of the law of association:
(f)
p((ab)c) = p(a(bc)).
But this interesting fact 3 remains unnoticed if (f) is introduced by
way of assuming the algebraic identity (a)—the law of association—
before even starting to develop the calculus of probability; for from
(a)
(ab)c = a(bc)
we may obtain (f) merely by substitution into the identity
3 The derivation is as follows:
(1) p((ab)c) = p(ab, c)p(c) d
(2) p((ab)c) = p(a, bc)p(b, c)p(c) 1, e
(3) p(a(bc)) = p(a, bc)p(bc) d
(4) p(a(bc)) = p(a, bc)p(b, c)p(c) 3, d
(5) p((ab)c) = p(a(bc)) 2, 4