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Coherence and the axioms of confirmation1

Published online by Cambridge University Press:  12 March 2014

Abner Shimony
Abner Shimony*
Affiliation:
Neptune, New Jersey
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Extract

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′.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 20 , Issue 1 , March 1955 , pp. 1 - 28
Copyright
Copyright © Association for Symbolic Logic 1955

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Footnotes

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.

References

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

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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 ‘p ⥽q’ 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. 71Google Scholar. 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 e ⥽ h.

13 See section 5 of this paper.

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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).

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35 Cf. footnote 19.

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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. 297–8Google Scholar.

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

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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. 386–89CrossRefGoogle Scholar. This theorem was independently proved by Professor E. Begle of the Department of Mathematics of Yale University.

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47 Op. cit., chapters xx–xxii.

48 Op. cit., p. 113.