Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 63, Issue 3 (1998), 869-890.
Completeness of the Propositions-as-Types Interpretation of Intuitionistic Logic into Illative Combinatory Logic
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, , 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, , 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.
J. Symbolic Logic, Volume 63, Issue 3 (1998), 869-890.
First available in Project Euclid: 6 July 2007
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Dekkers, Wil; Bunder, Martin; Barendregt, Henk. Completeness of the Propositions-as-Types Interpretation of Intuitionistic Logic into Illative Combinatory Logic. J. Symbolic Logic 63 (1998), no. 3, 869--890. https://projecteuclid.org/euclid.jsl/1183745571