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A logic stronger than intuitionism1

Published online by Cambridge University Press:  12 March 2014

Sabine Görnemann
Sabine Görnemann*
Affiliation:
Philosophisches Seminar Der Universität, 23 Kiel Olshausenstr, 40-46, Germany
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Extract

S. A. Kripke has given [6] a very simple notion of model for intuitionistic predicate logic. Kripke's models consist of a quasi-ordering (C, ≤) and a function ψ which assigns to every c ∈ C a model of classical logic such that, if c ≤ c′, ψ(c′) is greater or equal to ψ(c). Grzegorczyk [3] described a class of models which is still simpler: he takes, for every ψ(c), the same universe. Grzegorczyk's semantics is not adequate for intuitionistic logic, since the formula

where х is not free in α. holds in his models but is not intuitionistically provable. It is a conjecture of D. Klemke that intuitionistic predicate calculus, strengthened by the axiom scheme (D), is correct and complete with respect to Grzegorczyk's semantics. This has been proved independently by D. Klemke [5] by a Henkinlike method and me; another proof has been given by D. Gabbay [1]. Our proof uses lattice-theoretical methods.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 249 - 261
Copyright
Copyright © Association for Symbolic Logic 1971

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Footnotes

1

Part of the author's thesis submitted at the Technical University of Hannover.

References

[1]Gabbay, D. M., Montague type semantics for nonclassical logics. I, Scientific report no. 4, The Hebrew University of Jerusalem, 10, 1969.Google Scholar
[2]Görnemann, S., Vergleich von Kripkes Modellbegriff der intuitionistischen Logik mit dem von Rasiowa und Sikorski, mimeographed, Hannover, 1967.Google Scholar
[3]Grzegorczyk, A., A philosophically plausible formal interpretation of intuitionistic logic, Indagationes Mathematicae, vol. 26 (1964), pp. 596–601.CrossRefGoogle Scholar
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[5]Klemke, D., Ein vollständiger Kalkül für die Folgerungsbeziehung der Grzegorczyk-Semantik, Dissertation, Freiburg, 1969.Google Scholar
[6]Kripke, S. A., Semantical analysis of intuitionistic logic. I, Proceedings of the Eighth Logic Colloquium, Oxford, 07 1963, pp. 92–130.Google Scholar
[7]Rasiowa, H. and Sikorski, R., The Mathematics of Metamathematics, Warszawa 1963.Google Scholar
[8]Sikorski, R., Boolean algebras, 2nd edition, Berlin, Heidelberg, New York, 1964.Google Scholar