popper-logic-scientific-discovery

APPENDIX *v
Derivations in the Formal Theory
of Probability
In this appendix I propose to give the most important derivations from
the system of postulates which has been explained in appendix *iv. I
am going to show how the laws of the upper and lower bounds, of
idempotence, commutation, association, and distribution are obtained,
as well as a simpler definition of absolute probability. I will also indicate
how Boolean algebra is derivable in the system. A fuller treatment
will be given elsewhere.
As an abbreviation for ‘if . . . then . . .’, I am going to use an arrow
‘...→ . . .’; a double arrow ‘. . . ↔ . . .’, for ‘ . . . if and only
if . . .’; ‘&’ for ‘and’; ‘(Ea) . . .’ for ‘there is an a in S such that . . .’;
and ‘(a) . . .’ for ‘for all a in S, . . .’.
I first re-state postulate 2 and the six operational axioms which will
all be cited in the proofs. (The other postulates will be used implicitly;
even postulate 2 will be cited only once, in the proof of 5.) In reading
the axioms A3 and C, it should be kept in mind that I shall soon
prove—see formula 25—that p(a, a) = 1.
Postulate 2. If a and b are in S, then p(a, b) is a real number.
A1
A2
(Ec)(Ed) p(a, b) ≠ p(c, d),
((c)(p(a, c) = p(b, c)) → p(d, a) = p(d, b),