Intuitionistic logic
Modal logic

Any formula of the intuitionistic propositional logic (IPC) may be translated into the language of the normal modal logic S4 as follows:

and it has been demonstrated that the translated formula is valid in the propositional modal logic S4 if and only if the original formula is valid in IPC. The above set of formulae are called the Gödel–McKinsey–Tarski translation.There is also an intuitionistic version of modal logic S4 called Constructive Modal Logic CS4.
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