popper-logic-scientific-discovery

probability 177
conversely: if it has no frequency limit, then it has more than one 2
middle frequency.
The idea of a middle frequency will be found very suitable for our
purpose. Just as previously it was our estimate—perhaps a hypothetical
estimate—that p was the frequency limit of a sequence α, so we now
work with the estimate that p is a middle frequency of α. And provided
we take certain necessary precautions, 3 we can make calculations with
the help of these estimated middle frequencies, in a way analogous to
that in which we calculate with frequency limits. Moreover the concept
of middle frequency is applicable to all possible infinite reference
sequences, without any restriction.
If we now try to interpret our symbol αF′ (β) as a middle frequency,
rather than a frequency limit, and if we accordingly alter the definition
of objective probability (section 59), most of our formulae will still be
derivable. One difficulty arises however: middle frequencies are not
unique. If we estimate or conjecture that a middle frequency is αF′
(β) = p, then this does not exclude the possibility that there are values
of αF′ (β) other than p. If we postulate that this shall not be so, we
thereby introduce, by implication, the axiom of convergence. If on the
other hand we define objective probability without such a postulate of
uniqueness, 4 then we obtain (in the first instance, at least) a concept of
probability which is ambiguous; for under certain circumstances a sequence
may possess at the same time several middle frequencies which are
‘absolutely free’ (cf. section c of appendix iv). But this is hardly
acceptable, since we are accustomed to work with unambiguous or unique
2 It can easily be shown that if more than one middle frequency exists in a reference
sequence then the values of these middle frequencies form a continuum.
3 The concept of ‘independent selection’ must be interpreted more strictly than hitherto,
since otherwise the validity of the special multiplication theorem cannot be proved.
For details see my work mentioned in note 3 to section 51. (*This is now superseded by
appendix *vi.)
4 We can do this because it must be possible to apply the theory for finite classes (with
the exception of the theorem of uniqueness) immediately to middle frequencies. If a
sequence α has a middle frequency p, then it must contain—whatever the term with
which the counting starts—segments of any finite magnitude, the frequency of which
deviates from p as little as we choose. The calculation can be carried out for these. That p
is free from after-effect will then mean that this middle frequency of α is also a middle
frequency of any predecessor selection of α.