popper-logic-scientific-discovery

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some structural components of a theory of experience
Now the structure of an alternative can be regular, or it can be more or
less irregular. In what follows we will study this regularity or irregularity
of certain finite alternatives more closely.* 1
55 N-FREEDOM IN FINITE SEQUENCES
Let us take a finite alternative α, for example one consisting of a
thousand ones and zeros regularly arranged as follows:
(α)
1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 . . .
In this alternative we have equal distribution, i.e. the relative frequencies
of the ones and the zeros are equal. If we denote the relative
frequency of the property 1 by ‘F″ (1)’ and that of 0 by ‘F″ (0)’, we can
write:
(1)
αF″ (1) = αF″ (0) = 1 2
We now select from α all terms with the neighbourhood-property of
immediately succeeding a one (within the sequence α). If we denote this
property by ‘β’, we may call the selected sub-sequence ‘α.β’. It will
have the structure:
(α.β)
1 0 1 0 1 0 1 0 1 0 . . .
This sequence is again an alternative with equal distribution. Moreover,
neither the relative frequency of the ones nor that of the zeros has
changed; i.e. we have
(2)
α.βF″ (1) = αF″ (1); α.βF″ (0) = αF″ (0).
In the terminology introduced in section 53, we can say that the primary
properties of the alternative α are insensitive to selection according
to the property β; or, more briefly, that α is insensitive to selection
according to β.
* 1 I suggest that sections 55 to 64, or perhaps only 56 to 64, be skipped at first reading.
It may even be advisable to turn from here, or from the end of section 55, direct to
chapter 10.