popper-logic-scientific-discovery

interpretation according to which a and b are statements (or ‘propositions’,
if you like). He says, rightly, ‘what the members of this set
represent is of no importance’; but this remark is not sufficient to
establish the formal character of the theory at which he aims; for in
some interpretations, a and b have no members, nor anything that might
correspond to members.
All this has grave consequences in connection with the actual
construction of the axiom system itself.
Those who interpret the elements a and b as statements or propositions
very naturally assume that the calculus of statement-composition
(the propositional calculus) holds for these elements. Similarly,
Kolmogorov assumes that the operations of addition, multiplication,
and complementation of sets hold for his elements, since they are
interpreted as sets.
More concretely, it is always presupposed (often only tacitly), that
such algebraic laws as the law of association
(a)
or the law of commutation
(b)
or the law of indempotence
(c)
(ab)c = a(bc)
ab = ba
a = aa
hold for the elements of the system; that is to say, for the arguments of
the function p( ..., ... ).
Having made this assumption either tacitly or explicitly, a number of
axioms or postulates are laid down for relative probability,
p(a, b)
that is to say for the probability of a, given the information b; or else for
absolute probability,
p(a)
appendix *iv 331