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A completeness theorem in modal logic1

Published online by Cambridge University Press:  12 March 2014

Extract

The present paper attempts to state and prove a completeness theorem for the system S5 of [1], supplemented by first-order quantifiers and the sign of equality. We assume that we possess a denumerably infinite list of individual variables a, b, c, …, x, y, z, …, xm, ym, zm, … as well as a denumerably infinite list of n-adic predicate variables Pn, Qn, Rn, …, Pmn, Qmn, Rmn,…; if n=0, an n-adic predicate variable is often called a “propositional variable.” A formula Pn(x1, …,xn) is an n-adic prime formula; often the superscript will be omitted if such an omission does not sacrifice clarity.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1959

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Footnotes

1

My thanks to the referee and to Professor H. B. Curry for their helpful comments on this paper and their careful reading of it. I must express an added debt of gratitude to Curry; without his constant encouragement of my research, publication of these results might have been delayed for years.

References

[1]Lewis, C. I. and Langford, C. H., Symbolic logic, Century Company, 1932.Google Scholar
[2]Rosser, J. B., Logic for mathematicians, McGraw-Hill, 1953.Google Scholar
[3]Carnap, Rudolf, Introduction to semantics, Harvard University Press, 1942.Google Scholar
[4]Beth, E. W., Semantic entailment and formal derivability, Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Ajd Letterkunde, Nieuwe Reeks, Deel 18, no. 13, pp. 309342 (1955).Google Scholar
[5]Quine, W. V., Three grades of modal involvement, Proceedings of the XIth International Congress of Philosophy, Vol. XIV, pp. 6581.Google Scholar
[6]Prior, A. N., Modality and quantification in S5, this Journal, Vol. 21 (1956), pp. 6062.Google Scholar
[7]Kleene, S. C., Introduction to metamathematics, Van Nostrana, 1952.Google Scholar
[8]Curry, H. B., A theory of formal deducibility, Notre Dame Mathematical Lectures, no. 6, 1950.Google Scholar
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