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Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation

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This paper studies a combined system of intuitionistic and classical propositional logic from proof-theoretic viewpoints. Based on the semantic treatment of Humberstone (J Philos Log 8:171–196, 1979) and del Cerro and Herzig (Frontiers of combining systems: FroCoS, Springer, 1996), a sequent calculus \(\textsf{G}(\textbf{C}+\textbf{J})\) is proposed. An approximate idea of obtaining \(\textsf{G}(\textbf{C}+\textbf{J})\) is adding rules for classical implication on top of the intuitionistic multi-succedent sequent calculus by Maehara (Nagoya Math J 7:45–64, 1954). However, in the semantic treatment, some formulas do not satisfy heredity, which leads to the necessity of a restriction on the right rule for intuitionistic implication to keep the soundness of the calculus. The calculus \(\textsf{G}(\textbf{C}+\textbf{J})\) enjoys cut elimination and Craig interpolation, whose detailed proofs are described in this paper. Cut elimination enables us to show the decidability of this combination both directly and syntactically. This paper also employs a canonical model argument to establish the strong completeness of Hilbert system \(\mathbf {C+J}\) proposed by del Cerro and Herzig (Frontiers of combining systems: FroCoS, Springer, 1996).

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Acknowledgements

We would like to thank Rineke Verbrugge and R. Ramanujam for their comments at ICLA2021: 9th Indian Conference on Logic and its Applications. In addition, two anonymous referees of this paper for giving us very helpful comments. The work of the first author is partially supported by Grant-in-Aid for JSPS Fellows Grant Number JP22J20341. The work of the second author was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number JP 22H00597 and (C) Grant Number JP 19K12113.

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  1. Graduate School of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Masanobu Toyooka

  2. Faculty of Humanities and Human Sciences, Hokkaido University, Sapporo, Japan

    Katsuhiko Sano

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Toyooka, M., Sano, K. Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation. Stud Logica 112, 1091–1121 (2024). https://doi.org/10.1007/s11225-023-10067-0

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