popper-logic-scientific-discovery

296
appendices
(b) An analogous method of construction can now be used to show
that sequences exist which possess an ‘absolutely free’ middle frequency
(cf. section 64) even though they have no frequency-limit. We need only
change procedure (a) in such a way that after some given number of
increases in length, we always add to the sequence a finite ‘block’ (or
‘iteration’)—of ones, for example. This block is made so long that some
given frequency p which is different from 1 2 is reached. After attaining
this frequency the whole sequence now written down (it may now have
the length m i) is regarded as the commencing sequence of a period
which is m i − 1-free (with equal distribution), and so on.
(c) Finally, it is possible to construct in an analogous way a model
of a sequence which has more than one ‘absolutely free’ middle frequency.
According to (a), there are sequences which do not have equal
distribution and are ‘absolutely free’. Thus all we have to do is to
combine two such sequences, (A) and (B) (with the frequencies p and
q), in the following way. We write down some commencing sequence
of (A), then search (B) until we find in it this sequence, and rearrange
the period of (B) preceding this point in such a way that it begins with
the sequence written down; we then use this whole rearranged period
of (B) as commencing sequence. Next we search (A) until we find this
new written-down sequence, rearrange (A), and so on. In this way we
obtain a sequence in which again and again terms occur up to which it
is n i-free for the relative frequency p of the sequence (A), but in which
also again and again terms occur up to which the sequence is n i-free for
the frequency q of (B). Since in this case the numbers n i increase without
limit, we obtain a mode of construction for a sequence which has
two distinct ‘middle frequencies’ both of which are ‘absolutely free’.
(For we did determine (A) and (B) in such a way that their frequency
limits are distinct.)
Note. The applicability of the special multiplication theorem to the
classical problem of throwing two dice X and Y at a time (and related
problems) is assured if, for example, we make the hypothetical estimate
that the ‘combination sequence’ (as we may call it)—i.e. the sequence
α that has the throws with X as its odd terms and the throws with Y as
its even terms—is random.