popper-logic-scientific-discovery

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new appendices
simplified system first in my paper ‘Philosophy of Science: A Personal
Report’. 5 It is the same system of six axioms which is more fully
presented in the present appendix.
The system is surprisingly simple and intuitive, and its power, which
far surpasses that of any of the customary systems, is merely due to the
fact that I omit from all the formulae except one (axiom C), any condition
like ‘If p(b) ≠ 0 then . . . ’. (In the customary systems, these conditions
either are present, or they ought to be present, in order to avoid
inconsistencies.)
I intend to explain, in the present appendix, first the axiom system,
with proofs of consistency and independence, and afterwards a few
definitions based upon the system, among them that of a Borel field of
probabilities.
First the axiom system itself.
Four undefined concepts appear in our postulates: (i) S, the universe of
discourse, or the system of admissible elements; the elements of S are
denoted by lower case italics, ‘a’, ‘b’, ‘c’, . . . etc.; (ii) a binary numerical
function of these elements, denoted by ‘p(a, b)’, etc.; that is to say, the
probability a given b; (iii) a binary operation on the elements, denoted
by ‘ab’, and called the product (or meet or conjunction) of a and b; (iv)
the complement of the element a, denoted by ‘ā’.
To these four undefined concepts we may add a fifth—one that
can be treated, according to choice, as an undefined or as a defined
concept. It is the ‘absolute probability of a’, denoted by ‘p(a)’.
Each of the undefined concepts is introduced by a Postulate. For an
intuitive understanding of these postulates, it is advisable to keep in
mind that p(a, a) = 1 = p(b, b) for all elements a and b of S, as can of
course be formally proved with the help of the postulates.
Postulate 1. The number of elements in S is at most denumerably
infinite.
Postulate 2. If a and b are in S, then p(a, b) is a real number, and the
following axioms hold:
5 In British Philosophy in the Mid-Century, ed. by C.A. Mace, 1956, p. 191; now ch. 1 of
my Conjectures and Refutations. The six axioms given there are B1, C, B2, A3, A2, and A1
of the present appendix; they are there numbered B1, B2, B3, C1, D1, and E1,
respectively.