K. Popper - Logic scientific discovery
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K. Popper - Logic scientific discovery

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a consequence, α, or rather the relative frequency of the

primary properties of α, is insensitive to both, a selection according to
β and according to β

-
; and we may therefore say that α is insensitive to

every selection according to the property of the immediate predecessor.
Clearly, this insensitivity is due to certain aspects of the structure of

the alternative α; aspects which may distinguish it from other alterna-
tives. For example, the alternatives α.β and α.β

-
 are not insensitive to

selection according to the property of a predecessor.
We can now investigate the alternative α in order to see whether it is

insensitive to other selections, especially to selection according to the
property of a pair of predecessors. We can, for example, select from α all
those elements which are successors of a pair 1,1. And we see at once
that α is not insensitive to the selection of the successor of any of the
four possible pairs 1,1; 1,0; 0,1; 0,0. In none of these cases have the
resulting sub-sequences equal distribution; on the contrary, they all
consist of uninterrupted blocks (or ‘iterations’), i.e. of nothing but ones, or
of nothing but zeros.

The fact that α is insensitive to selection according to single pre-
decessors, but not insensitive to selection according to pairs of

probability 149

predecessors, might be expressed, from the point of view of the sub-
jective theory, as follows. Information about the property of one pre-
decessor of any element in α is irrelevant to the question of the
property of this element. On the other hand, information about the
properties of its pair of predecessors is of the highest relevance; for given
the law according to which α is constructed, it enables us to predict the
property of the element in question: the information about the proper-
ties of its pair of predecessors furnishes us, so to speak, with the initial
conditions needed for deducing the prediction. (The law according to
which α is constructed requires a pair of properties as initial conditions;
thus it is ‘two-dimensional’ with respect to these properties. The speci-
fication of one property is ‘irrelevant’ only in being composite in an
insufficient degree to serve as an initial condition. Cf. section 38.*1)

Remembering how closely the idea of causality—of cause and effect—is
related to the deduction of predictions, I shall now make use of the
following terms. The assertion previously made about the alternative α,
‘α is insensitive to selection according to a single predecessor’, I shall
now express by saying, ‘α is free from any after-effect of single predeces-
sors’ or briefly, ‘α is 1-free’. And instead of saying as before, that α is
(or is not) ‘insensitive to selection according to pairs of predecessors’, I
shall now say: ‘α is (not) free from the after-effects of pairs of predeces-
sors’, or briefly, ‘α is (not) 2-free.’*2

Using the 1-free alternative α as our prototype we can now easily

*1 This is another indication of the fact that the terms ‘relevant’ and ‘irrelevant’, figuring
so largely in the subjective theory, are grossly misleading. For if p is irrelevant, and
likewise q, it is a little surprising to learn that p.q may be of the highest relevance. See also
appendix *ix, especially points 5 and 6 of the first note.
*2 The general idea of distinguishing neighbourhoods according to their size, and of
operating with well-defined neighbourhood-selections was introduced by me. But the
term ‘free from after-effect’ (‘nachwirkungsfrei’) is due to Reichenbach. Reichenbach, how-
ever, used it at the time only in the absolute sense of ‘insensitive to selection according to
any preceding group of elements’. The idea of introducing a recursively definable concept of
1-freedom, 2-freedom, . . . and n-freedom, and of thus utilizing the recursive method for
analysing neighbourhood selections and especially for constructing random sequences is mine.
(I have used the same recursive method also for defining the mutual independence of n
events.) This method is quite different from Reichenbach’s, See also footnote 4 to section
58, and especially footnote 2 to section 60, below. Added 1968: I have now found that
the term was used long before Reichenbach by Smoluchowski.

some structural components of a theory of experience150

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 after-
effects 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 ‘n-
insensitive’, 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

1 1 0 0 . . .(A)

any number of times. Similarly we obtain a 2-free alternative with
equal distribution if we take

1 0 1 1 1 0 0 0 . . .(B)

as its generating period. A 3-free alternative is obtained from the
generating period

1 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 . . .(C)

and a 4-free alternative is obtained from the generating period

0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 . . .(D)

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.

probability 151

The generating period of an n-free alternative with equal distribu-
tion must contain at least 2n + 1 elements. The periods given as examples
can, of course, begin at different places; (C) for example can begin
with its fourth element, so that we obtain, in place of (C)

1 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1 . . .(C′)

There are other transformations which leave the n-freedom of a
sequence unchanged. A method of constructing generating periods
of n-free sequences for every number n will be described
elsewhere.*3

If to the generating period of an n-free alternative we add the first n
elements of the next period, then we obtain a sequence of the length
2n + 1 + n. This has, among others, the following property: every
arrangement of n + 1 zeros and ones, i.e. every possible n + 1-tuple,
occurs in it at least once.*4

56 SEQUENCES OF SEGMENTS. THE FIRST
FORM OF THE BINOMIAL FORMULA

Given a finite sequence α, we call a sub-sequence of α consisting of n
consecutive elements a ‘segment of α of length n’; or, more briefly, an
‘n-segment of α’. If, in addition to the sequence α, we are given some
definite number n, then we can arrange the n-segments of α in a
sequence—the sequence of n-segments of α. Given a sequence α, we may
construct a new sequence, of n-segments of α, in such a way that we

*3 Cf. note *1 to appendix iv. The result is a sequence of the length 2n + n − 1 such that by
omitting its last n − 1 elements, we obtain a generating period for an m-free alternative,
with m = n − 1.
*4 The following definition, applicable to any given long but finite alternative A, with
equidistribution, seems appropriate. Let N be the length of A, and let n be the greatest
integer such that 2n + 1 � N. Then A is said to be perfectly random if and only if the relative
number of occurrences of any given pair, triplet, . . . , m-tuplet (up to m = n) deviates
from that of any other pair, triplet, . . . , m-tuplet, by not more than, say, m/N½

respectively. This characterization