Journal of Symbolic Logic

Systems of Illative Combinatory Logic Complete for First-Order Propositional and Predicate Calculus

Henk Barendregt, Martin Bunder, and Wil Dekkers

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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. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. 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. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of $\lambda$-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).

Article information

Source
J. Symbolic Logic, Volume 58, Issue 3 (1993), 769-788.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744297

Mathematical Reviews number (MathSciNet)
MR1242038

Zentralblatt MATH identifier
0791.03006

JSTOR
links.jstor.org

Citation

Barendregt, Henk; Bunder, Martin; Dekkers, Wil. Systems of Illative Combinatory Logic Complete for First-Order Propositional and Predicate Calculus. J. Symbolic Logic 58 (1993), no. 3, 769--788. https://projecteuclid.org/euclid.jsl/1183744297


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