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On Some Intuitionistic Modal Logics

Some modal logics based on logics weaker than the classical logic have been studied by Fitch [4], Prior [7], Bull [1], [2], [3], Prawitz [6] etc. Here we treat modal logics based on the intuitionistic propositional logic, which call intuitionistic modal logics (abbreviated as IML’s). Let H be the intuitionistic propositional logic formulated in the Hilbertstyle. The rules of inference of H are modus ponens and the rule of substitution. The IML L0 is obtained from H by adding the following three axioms,