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Coherence and the axioms of confirmation1

  • Abner Shimony (a1)
Abstract

It has been pointed out by Carnap that ‘probability’ is an equivocal term, which is used currently in two senses: (i) the degree to which it is rational to believe a hypothesis h on specified evidence e, and (ii) the relative frequency (in an indefinitely long run) of one property of events or things with respect to another. This paper is concerned only with the first of these two senses, which will be referred to as ‘the concept of confirmation,’ in order to avoid equivocation.

We may distinguish a quantitative and a comparative concept of confirmation. The general form of statements involving the former is

The degree of confirmation of the proposition h, given the proposition

e as evidence, is r (where r is a real number between 0 and 1).

In this paper the notation for a statement of this form is

The general form of statements involving the comparative concept is The proposition h is equally or less confirmed on e than is the proposition h′ on e′,

which may be symbolized as

The following abbreviations will also be used in discussing comparative confirmation:

The former of these two abbreviations means intuitively

The proposition h is less confirmed on evidence e than h′ is on e′. The latter means

The proposition h is confirmed on evidence e to the same degree that h′ is on e′.

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1

From a dissertation in partial fulfillment of the requirements for the degree of Ph. D. in the Department of Philosophy of Yale University, and written in part while the author was a Sterling Fellow in 1951–52. I am grateful to Professors Rudolf Carnap, John Myhill, John Kemeny, Frederic Fitch, and Howard Stein for their very helpful suggestions on the material in this paper.

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2 Logical foundations of probability, Chicago, 1950, pp. 2326.

3 R. Carnap, op. cit., p. 431.

4 Koopman B. O., The axioms and algebra of intuitive probability, Annals of mathematics, Series 2, vol. 41 (1940), pp. 269292.

5 von Kries J., Die Prinzipien der Wahrscheinlichkeitsrechnung. Eine logische Untersuchung, Freiburg i.B., 1886, pp. 29 ff; Keynes J. M., A treatise on probability, London and New York, 1921, ch. III.

6 Theory of probability (second edition), Oxford, 1948, pp. 1725.

7 Zur Axiomatik der Wahrscheinlichkeitsrechnung, Comptes rendus de la Société des Sciences de Varsovie, C1. III, xxv (1932), pp. 14.

8 On confirmation, this Journal, vol. 5 (1940), pp. 133148. Induction et analogie: comparaison de leur fondement, Mind, vol. 50 (1941), pp. 351365.

9 The logical problem of induction, Helsingfors, 1941, pp. 106–7.

10 Op. cit., p. 285.

11 Probability and induction, Oxford, 1949, pp. 125–27.

12 For a comparison of some of these axiomatizations see section 62 of Carnap's Logical foundations of probability. It should be particularly noted that the arguments h, e of the function C in the above set of axioms are propositions rather than sentences. Consequently, the modal operator ‘⥽’ is used rather than the corresponding metalinguistic operator; likewise the modal operator will be used in subsequent discussions, ‘◊p’ means ‘p is possible’, and ‘pq’ means ‘p logically implies q’ but for a detailed discussion of these operators see Fitch F. B., Symbolic logic: an introduction, especially p. 66 and p. 71. The assumption made at the beginning of this paper that all evidential propositions are logically possible makes it superfluous to write explicitly the propositions ‘◊e’ ‘◊(h&c)’ and ’◊(h′&e)’ as antecedents to the above axioms. Finally it should be noted that in subsequent discussions I shall consider a stronger set of axioms in which (2) is replaced by the following (2′) : C(h/e) = 1 if and only if eh.

13 See section 5 of this paper.

14 Hosiasson-Lindonbaum J., Induction et analogie: comparaison de leur fondement, Mind, vol. 50 (1941), p. 353.

15 Kemeny J., Carnap on confirmation, The review of metaphysics, vol. 5 (1941), pp. 152–53.

16 Poincaré , Calcul des probabilités (second edition), Paris, 1912, pp. 37–9; Kneale W., Probability and induction, pp. 125–27.

17 On some aspects of the theory of probability, Philosophical magazine, 6th series, vol. 38 (1919), pp. 715731, especially, p. 722.

18 Op. cit., pp. 135–38.

19 Op. cit., pp. 275–76.

20 Probability, frequency, and reasonable expectation, American journal of physics, vol. 14 (1946), pp. 113.

21 Ramsey F. P., Truth and probability, The foundations of mathematics and other logical essays, London and New York, 1931; DeFinetti B., Sul significato soggetivo della probabilità, Fundamenta mathematicae, vol. 17 (1931), pp. 298329, and La prévision: ses lois logiques, ses sources subjectives, Annales de l'Institut Henri Poincaré, vol. 7 (1937), pp. 168.

22 R. Carnap, op. cit., section 51.

23 La prévision: ses lois logiques, ses sources subjectives, Annales de l'Institut Henri Poincaré, vol. 7 (1937), p. 7.

24 Op. cit., pp. 182–83.

25 R. Carnap, op. cit., pp. 83–84.

26 Gödel K., Russell's mathematical logic, The philosophy of Bertrand Russell (edited by Schilpp P. A.), Menasha, Wisconsin, 1946, pp. 150–52.

27 Op. cit., p. 150.

28 K. Gödel, op. cit., pp. 150–52; also his What is Cantor's continuum hypothesis, American mathematical monthly, vol. 54 (1947), pp. 520–22.

29 The difficulty of extending the notion of coherence so as to apply to infinite sets will be pointed out in § 5.

30 If this condition is satisfied we shall often use the expression ‘(h, e) and (h′, e′) are comparable’.

31 However, the assumption has been made that ‘C(h/e)’ and ‘SC(h/e, h′/e′)’ axe well-defined only if e and e′ are logically possible propositions.

32 Fair beis and inductive probabilities, (unpublished manuscript).

33 Kemeny also proved the following : a sufficient condition for C to be such that every finite set of quantitative beliefs regulated by C is coherent in the usual sense of Ramsey and DeFinetti is that C should satisfy axioms (1), (2), (3), and (4).

34 Kolmogoroff A., The foundations of probability, New York, 1950, p. 15.

35 Cf. footnote 19.

36pq’ means ‘p is logically or strictly equivalent to q.’ See F. B. Fitch, op. cit., pp. 77–78.

37 Op. cit., footnote 4.

38 Cf. the discussion of weight measurements in Cohen M. and Nagel E., An introduction to logic and scientific method, New York, 1934, pp. 297–8.

39 B. O. Koopman, op. cit., pp. 290–91, especially theorems 20, 24, 25.

40 The bases of probability, Bulletin of the American Mathematical Society, vol. 46 (1940), p. 769.

41 For a discussion of this principle, see J. M. Keynes, op. cit., pp. 41–64. In my dissertation, A theory of confirmation (submitted to the Graduate School of Yale University in 1953), I attempted to establish the analyticity of a strict formulation of the principle.

42 Op. cit., pp. 29 ff.

43 Op. cit., pp. 34–40.

44 Sur l'extension de l'ordre partiel, Fundamenta mathematicae, vol. 16 (1930), pp. 386–89. This theorem was independently proved by Professor E. Begle of the Department of Mathematics of Yale University.

45 Prolegomena to any future metaphysics, translated by Carus Paul, Chicago, 1912, section 36.

46 A system of logic, BK. III, ch. III.

47 Op. cit., chapters xx–xxii.

48 Op. cit., p. 113.

1 From a dissertation in partial fulfillment of the requirements for the degree of Ph. D. in the Department of Philosophy of Yale University, and written in part while the author was a Sterling Fellow in 1951–52. I am grateful to Professors Rudolf Carnap, John Myhill, John Kemeny, Frederic Fitch, and Howard Stein for their very helpful suggestions on the material in this paper.

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