Semantic Incompleteness of Hilbert System for a Combination of Classical and Intuitionistic Propositional Logic

07/15/2022

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by   Masanobu Toyooka, et al.

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This paper shows Hilbert system (𝐂+𝐉)^-, given by del Cerro and Herzig (1996) is semantically incomplete. This system is proposed as a proof theory for Kripke semantics for a combination of intuitionistic and classical propositional logic, which is obtained by adding the natural semantic clause of classical implication into intuitionistic Kripke semantics. Although Hilbert system (𝐂+𝐉)^- contains intuitionistic modus ponens as a rule, it does not contain classical modus ponens. This paper gives an argument ensuring that the system (𝐂+𝐉)^- is semantically incomplete because of the absence of classical modus ponens. Our method is based on the logic of paradox, which is a paraconsistent logic proposed by Priest (1979).