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# K. Popper - Logic scientific discovery

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(γ) = αF″ (γ) 1 Von Mises’s term is ‘choice’ (‘Auswahl’). 2 The answer to this question is given by the general division theorem (cf. appendix ii). probability 145 In this case we say (following Hausdorﬀ3) that the properties β and γ are ‘mutually independent, within the reference-class α’. The relation of independence is a three-termed relation and is symmetrical in the properties β and γ.4 If two properties β and γ are (mutually) independent within a reference-class α we can also say that the prop- erty γ is, within α, insensitive to the selection of β-elements; or perhaps that the reference-class α is, with respect to this property γ, insensitive to a selection according to the property β. The mutual independence, or insensitiveness, of β and γ within α could also—from the point of view of the subjective theory—be inter- preted as follows: If we are informed that a particular element of the class α has the property β, then this information is irrelevant if β and γ are mutually independent within α; irrelevant namely, to the question whether this element also has the property γ, or not.*1 If, on the other hand, we know that γ occurs more often (or less often) in the subclass α.β (which has been selected from α according to β), then the infor- mation that an element has the property β is relevant to the question whether this element also has the property γ or not.5 3 Hausdorﬀ, Berichte über die Verhandlungen der sächsischen Ges. d. Wissenschaften, Leipzig, mathem.- physik. Klasse 53, 1901, p. 158. 4 It is even triply symmetrical, i.e. for α, β and γ, if we assume β and γ also to be ﬁnite. For the proof of the symmetry assertion cf. appendix ii, (1s) and (1s). *The condition of ﬁnitude for triple symmetry asserted in this note is insuﬃcient. I may have intended to express the condition that β and γ are bounded by the ﬁnite reference class α, or, most likely, that α should be our ﬁnite universe of discourse. (These are suﬃcient conditions.) The insuﬃciency of the condition, as formulated in my note, is shown by the following counter-example. Take a universe of 5 buttons; 4 are round (α); 2 are round and black (αβ); 2 are round and large (αγ); 1 is round, black, and large (αβγ); and 1 is square, black, and large (�βγ). Then we do not have triple symmetry since αF″ (γ) ≠ βF″ (γ). *1 Thus any information about the possession of properties is relevant, or irrelevant, if and only if the properties in question are, respectively, dependent or independent. Rele- vance can thus be deﬁned in terms of dependence, but the reverse is not the case. (Cf. the next footnote, and note *1 to section 55.) 5 Keynes objected to the frequency theory because he believed that it was impossible to deﬁne relevance in its terms; cf. op. cit., pp. 103 ﬀ. *In fact, the subjective theory cannot deﬁne (objective) independence, which is a serious objection as 1 show in my Postscript, chapter *ii, especially sections *40 to *43. some structural components of a theory of experience146 54 FINITE SEQUENCES. ORDINAL SELECTION AND NEIGHBOURHOOD SELECTION Let us suppose that the elements of a ﬁnite reference-class α are numbered (for instance that a number is written on each button in the box), and that they are arranged in a sequence, in accordance with these ordinal numbers. In such a sequence we can distinguish two kinds of selection which have special importance, namely selection according to the ordinal number of an element, or brieﬂy, ordinal selection, and selection according to its neighbourhood. Ordinal selection consists in making a selection, from the sequence α, in accordance with a property β which depends upon the ordinal number of the element (whose selection is to be decided on). For example β may be the property even, so that we select from α all those elements whose ordinal number is even. The elements thus selected form a selected sub-sequence. Should a property γ be independent of an ordinal selection according to β, then we can also say that the ordinal selection is independent with respect to γ; or we can say that the sequence α is, with respect to γ, insensitive to a selection of β-elements. Neighbourhood selection is made possible by the fact that, in ordering the elements in a numbered sequence, certain neighbourhood relations are created. This allows us, for example, to select all those members whose immediate predecessor has the property γ; or, say, those whose ﬁrst and second predecessors, or whose second successor, have the property γ; and so on. Thus if we have a sequence of events—say tosses of a coin—we have to distinguish two kinds of properties: its primary properties such as ‘heads’ or ‘tails’, which belong to each element independently of its position in the sequence; and its secondary properties such as ‘even’ or ‘successor of tails’, etc., which an element acquires by virtue of its position in the sequence. A sequence with two primary properties has been called ‘alterna- tive’. As von Mises has shown, it is possible to develop (if we are careful) the essentials of the theory of probability as a theory of alterna- tives, without sacriﬁcing generality. Denoting the two primary proper- ties of an alternative by the ﬁgures ‘1’ and ‘0’, every alternative can be represented as a sequence of ones and zeros. probability 147 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: 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 . . .(α) In this alternative we have equal distribution, i.e. the relative frequen- cies 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: αF″ (1) = αF″ (0) = 1 2(1) 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: 1 0 1 0 1 0 1 0 1 0 . . .(α.β) 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 α.βF″ (1) = αF″ (1); α.βF″ (0) = αF″ (0).(2) In the terminology introduced in section 53, we can say that the pri- mary 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. some structural components of a theory of experience148 Since every element of α has either the property β (that of being the successor of a one) or that of being the successor of a zero, we can denote the latter property by ‘β - ’. If we now select the members having the property β - we obtain the alternative: 0 1 0 1 0 1 0 1 0 . . .(α.β - ) This sequence shows a very slight deviation from equal distribution in so far as it begins and ends with zero (since α itself ends with ‘0, 0’ on account of its equal distribution). If α contains 2000 elements, then α.β - will contain 500 zeros, and only 499 ones. Such deviations from equal distribution (or from other distributions) arise only on account of the ﬁrst or last elements: they can be made as small as we please by making the sequence suﬃciently long. For this reason they will be neglected in what follows; especially since our investigations are to be extended to inﬁnite sequences, where these deviations vanish. Accord- ingly, we shall say that the alternative α.β - has equal distribution, and that the alternative α is insensitive to the selection of elements having the property β - . As