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).

翻译：本文更新版已发表于《澳大利亚逻辑学刊》，可通过以下链接获取：https://ojs.victoria.ac.nz/ajl/article/view/7696。本文证明，由del Cerro与Herzig（1996）提出的Hilbert系统$(\mathbf{C+J})^{-}$在语义上是不完全的。该系统作为用于直觉主义与经典命题逻辑组合的Kripke语义的证明论而提出，其通过将经典蕴涵的自然语义从句引入直觉主义Kripke语义中构建。尽管Hilbert系统$(\mathbf{C+J})^{-}$包含直觉主义分离规则作为推理规则，但不包含经典分离规则。本文论证了由于缺少经典分离规则，系统$(\mathbf{C+J})^{-}$在语义上是不完全的。我们的方法基于悖论逻辑——由Priest（1979）提出的一种次协调逻辑。