Semantic Incompleteness of del Cerro and Herzig's Hilbert System for a Combination of Classical and Intuitionistic Propositional Logic


  • Masanobu Toyooka Graduate School of Letters, Hokkaido University
  • Katsuhiko Sano Faculty of Humanities and Human Sciences, Hokkaido University



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).


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Author Biography

Katsuhiko Sano, Faculty of Humanities and Human Sciences, Hokkaido University

Katsuhiko Sano is an Associate Professor of Humanities and Human Sciences, Hokkaido University.