popper-logic-scientific-discovery

p(x) = p(x).
Thus the derivability of (f) from (d) and (e) remains unnoticed. Or
in other words, it is not seen that the assumption (a) is completely
redundant if we operate with an axiom system which contains, or
implies, (d) and (e); and that by assuming (a), in addition to (d) and
(e), we prevent ourselves from finding out what kind of relations are implied
by our axioms or postulates. But to find this out is one of the main points of
the axiomatic method.
In consequence, it has also not been noticed that (d) and (e),
although implying (f), i.e. an equation in terms of absolute probability,
do not alone imply (g) and (h), which are the corresponding formulae
in terms of relative probability:
(g)
(h)
p((ab)c, d) = p(a(bc), d)
p(a, (bc)d) = p(a, b(cd)).
In order to derive these formulae (see appendix *v, (41) to (62)),
much more is needed than (d) and (e); a fact which is of considerable
interest from an axiomatic point of view.
I have given this example in order to show that Kolmogorov fails to
carry out his programme. The same holds for all other systems known
to me. In my own systems of postulates for probability, all theorems of
Boolean algebra can be deduced; and Boolean algebra, in its turn, can
of course be interpreted in many ways: as an algebra of sets, or of
predicates, or of statements (or propositions), etc.
Another point of considerable importance is the problem of
a ‘symmetrical’ system. As mentioned above, it is possible to define
relative probability in terms of absolute probability by (d′), as follows:
(d′)
If p(b) ≠ 0 then p(a, b) = p(ab)/p(b).
appendix *iv 333
Now the antecedent ‘If p(b) ≠ 0’ is unavoidable here since division
by 0 is not a defined operation. As a consequence, most formulae of relative
probability can be asserted, in the customary systems, only in conditional
form, analogous to (d′). For example, in most systems, (g) is