Abstract
A system S 1 can be embedded in a system S 2 by a translation, i.e. one-one mapping, t from the language of S 1 into the language of S 2, when for every formula A of the language of S 1, we have that A is provable in S 1 iff t(A) is provable in S 2. In this sense Heyting’s propositional logic can be embedded in the modal propositional logic S4 by various modal translations, i.e. translations that prefix the necessity operator to certain subformulae of a nonmodal formula. Likewise, classical propositional logic can be embedded by modal translations in S5. Of course, these results are interesting because a modal translation is not any one-one mapping, but a mapping that preserves structure.
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