popper-logic-scientific-discovery

probability 153
begin with the segment of the first n elements of α. Next comes the
segment of the elements 2 to n + 1 of α. In general, we take as the xth
element of the new sequence the segment consisting of the elements x
to x + n − 1 of α. The new sequence so obtained may be called the
‘sequence of the overlapping n-segments of α’. This name indicates that
any two consecutive elements (i.e. segments) of the new sequence
overlap in such a way that they have n − 1 elements of the original
sequence α in common.
Now we can obtain, by selection, other n-sequences from a sequence
of overlapping segments; especially sequences of adjoining n-segments.
A sequence of adjoining n-segments contains only such n-segments
as immediately follow each other in α without overlapping. It may
begin, for example, with the n-segments of the elements numbered 1
to n, of the original sequence α, followed by that of the elements n + 1
to 2n, 2n + 1 to 3n, and so on. In general, a sequence of adjoining
segments will begin with the kth element of α and its segments will
contain the elements of α numbered k to n + k − 1, n + k to 2n + k − 1,
2n + k to 3n + k − 1, and so on.
In what follows, sequences of overlapping n-segments of α will be
denoted by ‘α (n)’, and sequences of adjoining n-segments by ‘α n’.
Let us now consider the sequences of overlapping segments α (n) a
little more closely. Every element of such a sequence is an n-segment of
α. As a primary property of an element of α (n), we might consider, for
instance, the ordered n-tuple of zeros and ones of which the segment
consists. Or we could, more simply, regard the number of its ones as the
primary property of the element (disregarding the order of the ones and
zeros). If we denote the number of ones by ‘m’ then, clearly, we have
m � n.
Now from every sequence α (n) we again get an alternative if we select
a particular m (m � n), ascribing the property ‘m’ to each element of
the sequence α (n) which has exactly m ones (and therefore n − m zeros)
and the property ‘m¯ ’ (non-m) to all other elements of α (n). Every
element of α (n) must then have one or the other of these two
properties.
Let us now imagine again that we are given a finite alternative α with
the primary properties ‘1’ and ‘0’. Assume that the frequency of the
ones, αF″ (1), is equal to p, and that the frequency of the zeros, αF″ (0),