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On the strong semantical completeness of the intuitionistic predicate calculus

  • Richmond H. Thomason (a1)

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

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[1] E. Beth , Formal methods, Reidel, Dordrecht, 1962.

[3] F. Fitch , Symbolic logic, Ronald, New York, 1952.

[4] C. Gentzen , Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934), pp. 176210 and 405–431.

[8] S. Kripke , Semantical analysis of intuitionistic logic. I, Formal systems and recursive functions, ed. J. Crossley and M. Dummett , Amsterdam, 1965, pp. 92130.

[9] H. Leblanc and R. Thomason , On the demarcation line between intuitionist logic and classical logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 257262.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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