Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-10T09:36:59.595Z Has data issue: false hasContentIssue false
  • English
  • Français

A completeness theorem in modal logic1

Published online by Cambridge University Press:  12 March 2014

Saul A. Kripke
Get access Rights & Permissions [Opens in a new window]

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
Information
The Journal of Symbolic Logic , Volume 24 , Issue 1 , March 1959 , pp. 1 - 14
Copyright
Copyright © Association for Symbolic Logic 1959

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 309–342 (1955).Google Scholar
[5]Quine, W. V., Three grades of modal involvement, Proceedings of the XIth International Congress of Philosophy, Vol. XIV, pp. 65–81.Google Scholar
[6]Prior, A. N., Modality and quantification in S5, this Journal, Vol. 21 (1956), pp. 60–62.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