Combining First-Order Classical and Intuitionistic Logic

Masanobu Toyooka
(Graduate School of Letters, Hokkaido University)
Katsuhiko Sano
(Faculty of Humanities and Human Science, Hokkaido University)

This paper studies a first-order expansion of a combination C+J of intuitionistic and classical propositional logic, which was studied by Humberstone (1979) and del Cerro and Herzig (1996), from a proof-theoretic viewpoint. While C+J has both classical and intuitionistic implications, our first-order expansion adds classical and intuitionistic universal quantifiers and one existential quantifier to C+J. This paper provides a multi-succedent sequent calculus G(FOC+J) for our combination of the first-order intuitionistic and classical logic. Our sequent calculus G(FOC+J) restricts contexts of the right rules for intuitionistic implication and intuitionistic universal quantifier to particular forms of formulas. The cut-elimination theorem is established to ensure the subformula property. As a corollary, G(FOC+J) is conservative over both first-order intuitionistic and classical logic. Strong completeness of G(FOC+J) is proved via a canonical model argument.

In Andrzej Indrzejczak and Michał Zawidzki: Proceedings of the 10th International Conference on Non-Classical Logics. Theory and Applications (NCL 2022), Łódź, Poland, 14-18 March 2022, Electronic Proceedings in Theoretical Computer Science 358, pp. 25–40.
Published: 14th April 2022.

ArXived at: https://dx.doi.org/10.4204/EPTCS.358.3 bibtex PDF
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