popper-logic-scientific-discovery

tion I regard the practicability of Waismann’s proposal as significant.
It is satisfactory to find that a more comprehensive theory can bridge
the gaps—which at first appeared unbridgeable—between the various
attempts to tackle the problem, especially between the subjective
and the objective interpretations. Yet Waismann’s proposal calls for
some slight modification. His concept of a ratio of ranges (cf. note 2
to section 48) not only presupposes that ranges can be compared
with the help of their subclass relations (or their entailment relations);
but it also presupposes, more generally, that even ranges
which only partially overlap (ranges of non-comparable statements)
can be made comparable. This latter assumption, however, which
involves considerable difficulties, is superfluous. It is possible to
show that in the cases concerned (such as cases of randomness) the
comparison of subclasses and that of frequencies must lead to analogous
results. This justifies the procedure of correlating frequencies to
ranges in order to measure the latter. In doing so, we make the
statements in question (non-comparable by the subclass method)
comparable. I will indicate roughly how the procedure described
might be justified.
If between two property classes γ and β the subclass relation
holds, then we have:
γ ⊂ β
(k)[Fsb(k εγ) � Fsb(k εβ)] (cf. section 33)
so that the logical probability or the range of the statement (k εγ) must
be smaller than, or equal to, that of (k εβ). It will be equal only if there
is a reference class α (which may be the universal class) with respect to
which the following rule holds which may be said to have the form of
a ‘law of nature’:
(x) {[x ε (α.β)] → (x εγ)}.
probability 207
If this ‘law of nature’ does not hold, so that we may assume randomness
in this respect, then the inequality holds. But in this case we