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A logical measure function1

Published online by Cambridge University Press:  12 March 2014

John G. Kemeny
John G. Kemeny*
Affiliation:
Princeton University
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Extract

By a logical measure (for a given language) we mean a syntactically defined function which associates some value with each well-formed formula of the language. Various such logical measures have played a fundamental role in the development of Logic and the Philosophy of Science. The purpose of this paper is to define a logical measure which has much wider applications than measures so far studied.

The new measure has two fundamental advantages, which will be referred to throughout the paper. First of all it can be applied to more (and richer) languages than the older measures. Most of the measures now in use are restricted to the first-order functional calculus, and frequently even to a first-order calculus with one-place predicates only. The measure to be defined will be applicable to richer languages as well, e.g., to functional calculi of all finite orders. But even as far as the first-order functional calculus is concerned, the new measure has a great advantage: We do not have to require that the atomic sentences of the calculus be independent. The previous measures depended in their construction on the requirement that “the basic statements of the language express independent facts.”

By an atomic sentence we mean a well-formed formula formed by applying an h-place primitive predicate to h individuals (a well-formed formula no part of which is well-formed); by a permissible conjunction we mean a conjunction of atomic sentences and negations of other atomic sentences; and the requirement of independence is that an atomic sentence (or its negation) is logically implied by a permissible conjunction only if it (its negation) is one of the components of the conjunction.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 18 , Issue 4 , December 1953 , pp. 289 - 308
Copyright
Copyright © Association for Symbolic Logic 1953

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Footnotes

1

The basic ideas of this paper developed out of many discussions with Dr. Paul Oppenheim. The author is indebted for constructive suggestions to Dr. Oppenheim as well as to Professors C. G. Hempel and K. Gödel.

References

2 Models of logical systems, this Journal, vol. 13 (1948), pp. 16–30. This paper by the present author contains a complete definition, and hence will be referred to frequently. References to it will be abbreviated to “Models.”Google Scholar

3 (Added October 31, 1952.) More precisely: a synthetic universally quantified formula.

4 See this Journal, vol. 16, p. 206.

5 This term is used for lack of a better one. The asymptotic method is the method of using functions rather than numbers as values, and ordering the values of the measure according to the asymptotic properties of functions.

6 Models, see footnote 4.

7 More precisely, in the terminology of Models, according to whether the cardinal number of the union of the Ri's is finite.

8 No such convention is necessary for the ranges of higher type, since the Ri's together with the structure of the language determine all the R's (see Models.)

9 ‘W.f.f.’ is an abbreviation for ‘well-formed formula.’

10 For a definition of validity see Models. This concept corresponds to the intuitive idea of a w.f.f. being true under a given interpretation.

11 The circularity in this definition is only apparent. We define the relation greater for functions in terms of the corresponding relation between values of these functions. Since the values are rational numbers, “greater” has a well defined meaning in the latter case.

12 The existence of the limit and the fact that it is a rational number depends on a series expansion of the function. Each term in this expansion is a function of n with a rational coefficient, and each term dominates the succeeding ones asymptotically. The limit is the first (constant) term. (The existence of this expansion will not be proved here.) Strictly speaking it is the constant function identically equal to the limit that is a first approximation of the function m, but it is convenient to think of the constant functions as if they were numbers.

13 See Church, Alonzo, Introduction to mathematical logic Part I, 1944, theorem [30], p. 91Google Scholar.

14 A statement of infinity is a w.f.f. which can be satisfied only in an infinite model. These are the w.f.f.'s which, when used as axioms, are called axioms of infinity.

15 “Axioms of infinity” is here used in the terminology common to type theories (see footnote 14)..In set theories one often uses this term to designate a stronger axiom, which requires not only the existence of an infinite number of sets, but of infinite sets.

16 Functional calculi all of whose variables are individual variables.

17 For the sake of convenience we deviate slightly from Models, in that we assign classes of h-tuples rather than functions of the type there specified.

18 These are the semi-models of Models.

19 See definition 6, p. 21, of Models.

20 The basic idea for this concept comes from number 5ff. of Wittgenstein's, Tractatus Logico-Philosophicus, 1922Google Scholar. We find it fully developed in several papers by Carnap, a good summary of which is found in section 18 of Carnap's, Logical foundations of probability, 1950Google Scholar; and by Helmer and Oppenheim in Probability and degree of confirmation, this Journal, vol. 10 (1945), p. 31Google Scholar. (The latter use the term ‘singular state-description.’)

21 Wittgenstein (op. Cit., 5.15) calls it a “measure of probability”; Carnap (op. cit., chapter V) calls it a “regular measure-function” (other such functions will be discussed later); and Hempel, and Oppenheim, call it a “measure of range” in Studies in the logic of explanation, Philosophy of science, vol. 15 (1948), p. 170CrossRefGoogle Scholar.

22 Which in effect states that every semi-model should be a model.

23 Hempel and Oppenheim (op. cit., p. 169) describe such a function as “a measure of content.” Carnap (op. cit., section 73) develops the concept of “content,” referring to Popper's analysis (see next footnote).

24 Popper, Karl, Logik der Forschung, 1935, p. 13 and pp. 67ffCrossRefGoogle Scholar.

25 According to the best of my knowledge this has not been discussed before in the literature. The basic idea for this measure occurred to me after Dr. Oppenheim and Prof. Hempel had pointed out to me some disadvantages of the usual measures of content (see footnote 23).

26 This will be proved in the following section.

27 By this we can understand any equivalent w.f.f. which is the disjunction of mutually exclusive w.f.f.'s, each being a non-contradictory conjunction of atomic sentences or their negations, with the same number in each conjunction.

28 (Added October 31, 1952.) Two sets of extra-logical constants are independent of each other if the number of possible assignments to the elements of one set does not depend on what assignments have been made to the elements of the other set. This requirement can be fulfilled even in languages where the atomic sentences are not independent, and hence theorem 7 has much wider applicability than if it were restricted to languages with independent atomic sentences.

29 This is identical (for F) with the concept of independence introduced in Degree of factual support (loc. cit.); it represents the case where either w.f.f. gives 0 factual support to the other one.

30 This application was suggested to me by Dr. Oppenheim.

31 It being understood that each individual name is assigned to itself in the new models.

32 This is, in effect, what Carnap does (loc. cit.). Thus we see that wherever Carnap's method applies, it gives results identical with those of the present method.

33 See Helmer and Oppenheim's development (op. cit., pp. 30—32).

34 A proof of this theorem was outlined to the author by K. Gödel some years ago. The first published proof was given by B. A. Trahtenbrot in a Russian journal. This paper was reviewed in this Journal, vol. 15 (1950), p. 229.

35 Op. cit.

36 Op. cit.

37 Ibid., see the appendix.

38 As in the last example, p. 306.

39 We could also work out an example of factual suuport in this language. Let H be Φ(P) · Φ(P′), E be Φ(P). Then we see that H ⊃ E is analytic, hence by the formula of the previous paper (loc.cit.), . Using the calculated values we find that

40 Kemeny, John G., Extension of the methods of inductive logic, Philosophical studies, vol. 3 (1952), pp. 38–42CrossRefGoogle Scholar.