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

This paper shows that the Hilbert system H(C + J)-, given by del Cerro and Herzig (1996), is semantically incomplete. This system is proposed as a proof theory for a combination of classical and intuitionistic propositional logic, which is called C + J. The Kripke semantics for this combination is obtained by adding the semantic clause of classical implication directly to the Kripke semantics for intuitionistic propositional logic. The Hilbert system H(C + J)- does not contain the classical modus ponens, although it contains the intuitionistic modus ponens as a rule, which makes this system semantically incomplete. This paper gives an argument that the system H(C + J)- is semantically incomplete because of the absence of the classical modus ponens. Our way to prove this is based on the logic of paradox, which is a paraconsistent logic proposed by Priest (1979). [ABSTRACT FROM AUTHOR]