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

The updated version of this paper has already been published in The Australasian Journal of Logic. You can access to the paper from the following link: https://ojs.victoria.ac.nz/ajl/article/view/7696. This paper shows Hilbert system $(\mathbf{C+J})^{-}$, 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 $(\mathbf{C+J})^{-}$ contains intuitionistic modus ponens as a rule, it does not contain classical modus ponens. This paper gives an argument ensuring that the system $(\mathbf{C+J})^{-}$ 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).