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On the definition of an infinitely-many-valued predicate calculus1

Published online by Cambridge University Press:  12 March 2014

Joseph D. Rutledge
Joseph D. Rutledge*
Affiliation:
Yorktown Heights, New York
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Extract

This paper presents a class of plausible definitions for validity of formulas in the infinitely-many-valued extension of the Łukasiewicz predicate calculus, and shows that all of them are equivalent. This extended system is discussed in some form in [3] and [4]; the questions discussed here are raised rather briefly in the latter.

We first describe the formal framework for the validity definition. The symbols to be used are the following: the connectives + and ‐, which are the strong disjunction B of [2] and negation respectively; the predicate variables Pi for i ∈ I, where I may be taken as the integers; the existential quantifiers E(J), where J⊆I, and I may be thought of as the index set on the individual variables, which however do not appear explicitly in this formulation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 25 , Issue 3 , September 1960 , pp. 212 - 216
Copyright
Copyright © Association for Symbolic Logic 1960

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Footnotes

1

From a thesis in partial fulfillment of the requirements for the degree of Ph.D. in Mathematics at Cornell University. The author wishes to thank Professors E. P. Specker and J. Barkley Rosser for their helpful suggestions. The preparation of this paper was supported in part by the U. S. Navy under contract No. NONR-401(20)-NR 043–167 monitored by the ONR, and in part by IBM.

References

[1]Halmos, Paul, Algebraic logic II — Homogeneous locally finite poly adic Boolean algebras of infinite degree, Fundamenta mathematicae, vol. 43 (1956) pp. 255–325.Google Scholar
[2]Rose, A. and Rosser, J. B., Fragments of the many-valued statement calculi, Transactions of the American Mathematical Society, vol. 87 (1958) pp. 1–53.CrossRefGoogle Scholar
[3]Rosser, J. Barkley, Axiomatization of infinite valued logics, Logique et analyse, (Louvain), n.s. vol. 3 (1960), pp 137–153.Google Scholar
[4]Skolem, Th., Bemerkungen zum Komprehensionsaxiom, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 3 (1957) pp. 1–17.CrossRefGoogle Scholar