  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).
