popper-logic-scientific-discovery

Then we obtain:
(i) If a and b are in S, then a + b is in S
(Postulate 3, D2, D1, 90, 100)
(ii) If a is in S then ā is in S (Postulate 4)
(iii) a + b = b + a 93, D2
(iv) (a + b) + c = a + (b + c) 92, D2
(v) a + a = a 94, D2
(vi) ab + ab¯ = a 88, D2
(vii) (Ea)(Eb) a ≠ b 27, 74, 90, D1
But the system (A) to (D2) and (i) to (vi) is a well-known axiom
system for Boolean algebra, due to Huntington; 2 and it is known that
all valid formulae of Boolean algebra are derivable from it.
Thus S is a Boolean algebra. And since a Boolean algebra may be
interpreted as a logic of derivation, we may assert that in its logical
interpretation, the probability calculus is a genuine generalization of the logic of
derivation.
More particularly, we may interpret
a � b
which is definable by ‘ab = b’, to mean, in logical interpretation, ‘a
follows from b’ (or ‘b entails a’). It can be easily proved that
(+)
a � b → p(a, b) = 1
appendix *v 365
This is an important formula 3 which is asserted by many authors, but
2 Cf. E. V. Huntington, Transactions Am. Math. Soc. 35, 1933, pp. 274–304. The system (i) to
(vi) is Huntington’s ‘fourth set’, and is described on p. 280. On the same page may be
found (A) to (D), and (D2). Formula (v) is redundant, as Huntington showed on pp. 557 f.
of the same volume. (vii) is also assumed by him.
3 It is asserted, for example, by H. Jeffreys, Theory of Probability, § 1.2 ‘Convention 3’. But if
it is accepted, his Theorem 4 becomes at once contradictory, since it is asserted without a
condition such as our ‘p(b) ≠ 0’. Jeffreys improved, in this respect, the formulation of
Theorem 2 in his second edition, 1948: but as shown by Theorem 4 (and many others)
his system is still inconsistent (even though he recognized, in the second edition, p. 35,
that two contradictory propositions entail any proposition; cf. note *2 to section 23, and
my answer to Jeffreys in Mind 52, 1943, pp. 47 ff.).