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An embedding of classical logic in S4

  • Melvin Fitting (a1)


There are well-known embeddings of intuitionistic logic into S4 and of classical logic into S5. In this paper we give a related embedding of (first order) classical logic directly into (first order) S4, with or without the Barcan formula. If one reads the necessity operator of S4 as ‘provable’, the translation may be roughly stated as: truth may be replaced by provable consistency. A proper statement will be found below. The proof is based ultimately on the notion of complete sequences used in Cohen's technique of forcing [1], and is given in terms of Kripke's model theory [3], [4].



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[1]Cohen, Paul J., Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York, 1966.
[2]Fitting, Melvin, Intuitionistic logic model theory and forcing, Doctoral dissertation, Yeshiva University, 1968, North Holland, Amsterdam, 1969.
[3]Kripke, Saul, Semantical considerations on modal logic, Proceedings of a colloquium on modal and many-valued logics, Helsinki, August 1962, Acta phiiosophica Fennica No. 16, Helsinki (1963), pp. 8394.
[4]Kripke, S., Semantical analysis of modal logic. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963). pp. 6796.
[5]Kripke, Saul, Semantical analysis of intuitionistic logic. I, Formal systems and recursive functions, North Holland, Amsterdam, 1965, pp. 92130.
[6]McKinsey, J. C. C. and Tarski, A., Some theorems about the sentential calculi of Lewis and Heyting, this Journal, vol. 13 (1948), pp. 115.
[7]Prawitz, Dag, An interpretation of intuitionistic predicate logic in modal logic. Contributions to mathematical logic, Proceedings of the logic colloquium, Hanover, 1966.
[8]Schütte, Kurt, Vollständige Systeme modaler und intuitionistischer Logik, Springer-Verlag, Berlin, 1968.
[9]Smullyan, Raymond M., First order logic, Springer-Verlag, Berlin, 1968.

An embedding of classical logic in S4

  • Melvin Fitting (a1)


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