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# K. Popper - Logic scientific discovery

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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 makes it possible to say of a given alternative A that it is approximately random; and it even allows us to deﬁne a degree of approximation. A more elaborate deﬁnition may be based upon the method (of maximizing my E-function) described under points 8 ﬀ. of my Third Note reprinted in appendix *ix. some structural components of a theory of experience152 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