popper-logic-scientific-discovery

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some structural components of a theory of experience
predecessors, might be expressed, from the point of view of the subjective
theory, as follows. Information about the property of one predecessor
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 properties
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 specification
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 predecessors’
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 predecessors’,
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, however,
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.