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Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic

Published online by Cambridge University Press:  12 March 2014

Wil Dekkers
Affiliation:
Faculty of Mathematics and Computer Science, Catholic University, Nijmegen, The Netherlands E-mail: wil@cs.kun.nl
Martin Bunder
Affiliation:
Faculty of Informatics, Department of Mathematics, University of Wollonoong, Nsw Australia E-mail: martin_bunder@uow.edu.au
Henk Barendregt
Affiliation:
Faculty of Mathematics and Computer Science, Catholic University, Nijmegen, The Netherlands E-mail: henk@cs.kun.nl

Abstract

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1]Barendregt, H., Lambda calculi with types, Handbook of logic in computer science (Abramski, S., Gabbay, D. M., and Maibaum, T. S. E., editors), vol. II, Oxford University Press, 1992.Google Scholar
[2]Barendregt, H., Bunder, M., and Dekkers, W., Systems of illative combinatory logic complete for first-order propositional and predicate calculus, this Journal, vol. 58 (1993), pp. 769788.Google Scholar
[3]Dekkers, W., Bunder, M., and Barendregt, H., Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus, Archive for Mathematical Logic (1998), to appear.Google Scholar