popper-logic-scientific-discovery

construct other sequences, again with equal distribution, which are not
only free from the after effects of one predecessor, i.e. 1-free (like α),
but which are, in addition, free from the after effects of a pair of
predecessors, i.e., 2-free; and after this, we can go on to sequences
which are 3-free, etc. In this way we are led to a general idea which is
fundamental for what follows. It is the idea of freedom from the aftereffects
of all the predecessors up to some number n; or, as we shall say,
of n-freedom. More precisely, we shall call a sequence ‘n-free’ if, and
only if, the relative frequencies of its primary properties are ‘ninsensitive’,
i.e. insensitive to selection according to single predecessors
and according to pairs of predecessors and according to triplets of
predecessors . . . and according to n-tuples of predecessors. 1
An alternative α which is 1-free can be constructed by repeating the
generating period
(A)
1 1 0 0 . . .
any number of times. Similarly we obtain a 2-free alternative with
equal distribution if we take
(B)
1 0 1 1 1 0 0 0 . . .
as its generating period. A 3-free alternative is obtained from the
generating period
(C)
1 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 . . .
and a 4-free alternative is obtained from the generating period
(D)
probability 151
01100011101010010000010111110011...
It will be seen that the intuitive impression of being faced with an
irregular sequence becomes stronger with the growth of the number n
of its n-freedom.
1 As Dr. K. Schiff has pointed out to me, it is possible to simplify this definition. It is
enough to demand insensitivity to selection of any predecessor n-tuple (for a given n).
Insensitivity to selection of n−1-tuples (etc.) can then be proved easily.