Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation

Pub Date : 2024-01-06 DOI:10.1007/s11225-023-10067-0

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Abstract

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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直觉逻辑与古典命题逻辑的结合：根岑化和克雷格插值法

摘要本文从证明论的角度研究了直观命题逻辑和经典命题逻辑的组合系统。基于 Humberstone (J Philos Log 8:171-196, 1979) 以及 del Cerro 和 Herzig (Frontiers of combining systems: FroCoS, Springer, 1996) 的语义处理方法，提出了一个序列微积分（sequent calculus）（\textsf{G}(\textbf{C}+\textbf{J})\）。获得 \(\textsf{G}(\textbf{C}+\textbf{J})\) 的一个近似想法是在前原（Maehara）的直观多成功序列微积分（Nagoya Math J 7:45-64，1954）的基础上增加经典蕴涵规则。然而，在语义处理中，有些公式并不满足heredity，这就导致必须对直观蕴涵的正确规则加以限制，以保持微积分的合理性。微积分（\textsf{G}(\textbf{C}+\textbf{J})）享有切分消除和克雷格插值，本文将对它们进行详细的证明。切分消除使我们能够直接并从语法上证明这种组合的可解性。本文还利用典范模型论证建立了德尔塞罗和赫尔茨格（Frontiers of combining systems: FroCoS, Springer, 1996）提出的希尔伯特系统 \(\mathbf {C+J}\) 的强完备性。

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