Abstract
In this paper I intend to put forward some arguments in favour of what I am going to call the propensity interpretation of probability.
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Notes
See my Logic of Scientific Discovery (1934, 1959), section 48, and appendix* ii
The most characteristic laws of the calculus of probability are (1) the addition theorems, pertaining to the probability of a v b (that is, of a-or-b); (2) the multiplication theorems, pertaining to the probability of ab (that is, of a-and-b); and (3) the complementation theorems, pertaining to the probability of a (that is, of non-a). They may be written (1) p(a v b, c) = p(a, c) + p(b, c) — p(ab, c) (2) p(ab, c) = p(a, bc) p(b, c) (3) p(d, c) = 1 — p(a, c), provided p(c, c) + 1. The form of (3) here given is somehwat unusual: it is characteristic of a probability theory in which (4) p(a, cc) = 1 is a theorem. The first axiom system for a theory of this kind was presented, as far as I know, in BJPS,1955, 6, 56. See also my Logic of Scientific Discovery,appendix* iv, and the appendix to the present paper.
W. C. Kneale said in this discussion: “More recently the difficulties of the frequency interpretation, i.e. the muddles, if not the plain contradictions, which can be found in von Mises, have become well known, and I suppose that these are the considerations which have led Professor Popper to abandon that interpretation of probability.” See Observation and Interpretation,edited by S. Körner, 1957, p. 80. I am not aware of any ‘muddles’ or ‘contradictions’ in the frequency theory which have become well known more recently; on the contrary, I believe that I have discussed all objections of any importance in my Logic of Scientific Discovery when it was first published in 1934, and I do not think that Kneale’s criticism of the frequency theory in his Probability and Induction,1949, presents a correct picture of the situation prevailing at any time since 1934. One objection of Kneale’s (see especially p. 156 of his book) was not discussed in my book—that, in the frequency theory, a probability equal to one does not mean that the event in question will occur without exception (or ‘with certainty’). But this objection is invalid; it can be shown that every adequate theory of probability (if applicable to infinite sets) must lead to the same result.
For the remaining sections 3 to 5, see also my paper ‘The Propensity Interpretation of the Calculus of Probability and The Quantum Theory’, in Observation and Interpretation,edited by S. Körner, 1957.
A criticism of the subjective theory of probability will be found in my notes in BJPS,quoted above, and in my paper ‘Probability Magic, or Logic out of Ignorance’, Dialectica,1957, 354-374.
See my ‘Note on Berkeley as a Precursor of Mach’, BJPS,1953, 4, 21 (4).
See my paper ‘Philosophy of Science: A Personal Report’, in British Philosophy in the Mid-Century,edited by C. A. Mace, 1956; the axiom system can be found on p. 191.
As compared with the system of op. cit.,p. 332, the present system combines, in B, A2 with BI and B2. C is the Cs of p. 334.
The following abbreviations are used: ‘(x)’ for ‘for all elements x in S’; ‘(Ex)’ for ‘there is at least one element x in S such that’; ‘... –...: for ‘if... then...’; for ‘if and only if’; ’ for ‘and’.
This is due to the fact that Cd is logically stronger than C since it allows us to replace A by a logically weaker conditional formula; for in the presence of Cd, we may add to A the proviso, ‘provided (Ee) (Ef) p (e, f) $ 0’ (or in words, ‘provided not all probabilities are equal to zero’). The strength of Cd is due to the fact that, with the arrow from right to left only, Cd would be the same as C, while the arrow from left to right allows us, in addition, to deduce that not all probabilities are zero.
It may also be mentioned here that the condition of B, as formulated in the text, may be replaced by the (stronger) condition, ‘(e) p(bc, e) = p(d, e)’. (This replacement corresponds to the transition from formula A2+, on p. 335 of op. cit., to A2 on p. 332.)
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Popper, K.R. (1978). The Propensity Interpretation of Probability. In: Tuomela, R. (eds) Dispositions. Synthese Library, vol 113. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1282-8_15
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