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Axiomatization of the infinite-valued predicate calculus1

Published online by Cambridge University Press:  12 March 2014

Louise Schmir Hay
Louise Schmir Hay*
Affiliation:
Mount Holyoke College
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Extract

The infinite-valued statement calculus to which this paper refers is that of Łukasiewicz [10], whose axiomatization was proved complete in [5]. In [9], Rutledge extended this system to include predicates and quantifiers2 and presented a deductively complete set of axioms for the monadic predicate calculus. This paper represents an attempt to axiomatize the full predicate calculus; for the proposed axiomatization, a property akin to but weaker than completeness is proved. An attempt to prove full completeness along similar lines failed; it has since been shown [11] that the set of valid formulas of the infinite-valued predicate calculus is not recursively enumerable. The method of this paper was suggested by Professor J. Barkley Rosser.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 28 , Issue 1 , March 1963 , pp. 77 - 86
Copyright
Copyright © Association for Symbolic Logic 1964

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Footnotes

1

From a thesis in partial fulfillment of the requirements for the degree of M. A. in the Department of Mathematics at Cornell University, research sponsored by the Office of Naval Research under Contract No. NONT 401(20)–NR 043–167. The author would like to thank Professor J. B. Rosser for his kind assistance and valuable suggestions regarding the subject of this paper.

References

[1]Belluce, L. P., Some remarks on the completeness of infinite-valued predicate logic, abstract, American Mathematical Society, notices vol. 7 (1960), p. 633.Google Scholar
[2]Chang, C. C., Algebraic analysis of many-valued logics, Transactions of the American Mathematical Society, vol. 88 (1958), pp. 467–490.CrossRefGoogle Scholar
[3]Church, A., Introduction to mathematical logic, Princeton University Press, 1956.Google Scholar
[4]Rose, A., The degree of completeness of the Revalued Lukasiewicz propositional calculus, Journal of the London Mathematical Society, vol. 28 (1953), pp. 176–184.CrossRefGoogle Scholar
[5]Rose, A. and Rosser, J. B., Fragments of many-valued statement calculi, Transactions of the American Mathematical Society, vol. 87 (1958), pp. 1–53.CrossRefGoogle Scholar
[6]Rosser, J. B., Axiomatization of infinite-valued logics. Logique et analyse, vol. 3 (1960), pp. 137–153.Google Scholar
[7]Rosser, J. B., Deux esquisses de logique, Collection de logique mathématique, Série A, Gauthier-Villars, Paris, 1955.Google Scholar
[8]Rosser, J. B. and Turquette, A. R., Many-valued logics, North-Holland Publishing Co., Amsterdam, 1952.Google Scholar
[9]Rutledge, J. D., A preliminary investigation of the infinitely many-valued predicate calculus, Ph.D. Thesis, Cornell University, 1959.Google Scholar
[10]Tarski, A., Logic, semantics, metamathemattcs, Oxford, 1956.Google Scholar
[11]Scarpellini, B., Die Nichtaxiomatisierbarkeit der unendlichwertigen Logik von Lukasiewicz-Tarski, forthcoming in this Journal.Google Scholar