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On the proof theory of the intermediate logic MH1

Published online by Cambridge University Press:  12 March 2014

Jonathan P. Seldin
Jonathan P. Seldin*
Affiliation:
Department of Mathematics, Concordia University, Montréal, Québec H4B 1R6, Canada
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Abstract

A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

Keywords

Intermediate logic MHnormalizationGlivenko theoremCraig-Lyndon interpolation theorem.
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 3 , September 1986 , pp. 626 - 647
Copyright
Copyright © Association for Symbolic Logic 1986

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Footnotes

1

A preliminary version of this paper was presented to the meeting of the Association for Symbolic Logic held in Boston, Massachusetts, December 29–30, 1983. This research was supported in part by grants EQ1648 and CEllOof the program Formation de Chercheurs et Action Concertée (F.C.A.C.) of the Québec Ministry of Education.

References

REFERENCES

[1]Curry, Haskell B., Foundations of mathematical logic, McGraw-Hill, New York, 1963; reprint, Dover, New York, 1977.Google Scholar
[2]Curry, Haskell B., A deduction theorem for inferential predicate calculus, Contributions to mathematical logic (Schütte, K., editor), North-Holland, Amsterdam, 1968, pp. 91–108.Google Scholar
[3]Gabbay, Dov M., Semantic proof of Craig's interpolation theorem for intuitionistic logic and extensions. Part II, Logic Colloquium '69 (Gandy, R. O. and Yates, C. M. E., editors), North-Holland, Amsterdam, 1971, pp. 403–410.CrossRefGoogle Scholar
[4]Gabbay, Dov M., Applications of trees to intermediate logics, this Journal, vol. 37 (1972), pp. 135–138.Google Scholar
[5]Prawitz, Dag, Natural deduction, Almqvist & Wiksell, Stockholm, 1965.Google Scholar
[6]Rauszer, Cecylia, Formalizations of intermediate logics. Part 1, Methods in mathematical logic (proceedings of the sixth Latin American symposium, Caracas, 1983), Lecture Notes in Mathematics, vol. 1130, Springer-Verlag, Berlin, 1985, pp. 360–384.Google Scholar
[7]Seldin, Jonathan P., Proof normalizations and generalizations of Glivenko's theorem, Contributed Papers, Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, 1975, pp. I-43-I-44.Google Scholar