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Syntactical and semantical properties of generalized quantifiers

Published online by Cambridge University Press:  12 March 2014

Mitsuru Yasuhara
Mitsuru Yasuhara*
Affiliation:
Université de montréal, New York University, University Heights
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In the first-order language, quantifiers (∀x) and (∃#) are understood “to say” that “for all elements” and “there is at least one element such that …”, respectively. We are interested in changing the interpretation to “for all elements with fewer than κ exceptions” and “there are at least κ elements such that”, respectively, where κ is a cardinal. We call this the κ-interpretation of the quantifiers.1 The first question which presents itself is “What is the relationship between the κ-interpretation and the λ-interpretation?” For instance, is a formula valid under one interpretation also valid in all other interpretations? In the second section, it will be shown that as far as infinite interpretations, i.e. κ-interpretations for infinite cardinals κ, are concerned, the validity of a formula is preserved. Actually, a more general result is obtained there by model theoretic methods.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 4 , December 1966 , pp. 617 - 632
Copyright
Copyright © Association for Symbolic Logic 1997

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References

The present paper is based on the author's Ph.D. thesis (The University of California, Berkeley, 1963). The author is most grateful to his thesis adviser Professor Craig for the initiation to this field and the subsequent warm encouragement. It was written while the author was a post-doctorate fellow of N.R.C. of Canada.

1 The ordinary interpretation is the 1-interpretation in our sense.

2 In a relational system of power less than κ, a formula beginning with is never κ-satisfiable and a formula beginning with is always κ-valid according to the phrases (b) and (c). Hence such a degenerate case is excluded from the consideration of κ-satisfiability.

3 More rigorously, we should extend the language L by introducing new individual symbols corresponding to the elements of the universe under consideration and should understand A[X] to be the result of substitution of these constants to free variables. But we use a sloppy notation, its being convenient.

4 Cf. Robinson, A., Complete Theories, Amsterdam, 1956, p. 6.Google Scholar

5 The cofinality cf(κ) of κ is the least cardinal λ such that there is a sequence of ordinals cofinal in κ.

6 From the set [CE], one sees that the exception for the case κ = No in Theorems 6 and 8 is essentially due to the fact that there are denumerably many symbols in L. One may ask if results similar to Theorems 4, 5, 6 and 8 hold for languages having more than denumerably many symbols. The author does not find any reason to the contrary, but he has not and does not intend to look into this matter.

7 Refer: Fuhrken, G., Skolem-type normal forms for first order languages with a generalized quantifier, Fundamenta Mathematicae LIV, (1964)Google Scholar Theorem 3.4.

8 Our new paper, An axiomatic system for the first order languages with an equicardinality quantifier, (this Journal, Vol. 31 (1966), pp. 633–640.) may not be completely uninteresting in this connection. There, the structure of the V+κ is studied to some extent.

9 The recursive enumerability of VI follows from a result of Vaught that the set of valid formulae of the usual first-order language strengthened by introducing and under the N1-interpretation is recursively enumerable and the Corollary to Theorem 6 of the last section. Our deduction system is, however, more explicit and straightforward for our purposes and furnishes a workable tool for the proofs of the cut-elimination theorem and of the non-recursiveness. For the result of Vaught, refer to: Vaught, R., The completeness of logic with the added quantifier “there are uncountably many”, Fundamenta Mathematicae, LIV (1964), pp. 303CrossRefGoogle Scholar, 304.

10 Refer: Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, Band 39, (1934)Google Scholar.

11 Refer: Surányi, J., Reduktionstheorie des Entscheidungsproblems, Budapest, 1959.Google Scholar Satz XIV.

12 For instance, and are in π(1, 2, 3) B.

13 Since the universal quantifier is not available in L, this theorem is formulated in the open formula form.

14 For instance, centering our attention on V5, Vi are subsets of V5 for all i > 5 exept i = 6, 7, 8; 11, 12; 16. For these i, V5 and Vi are incomparable. Also V5 is a subset of V1, V2, and V3, but is incomparable with V4.