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Undefinability of propositional quantifiers in the modal system S4. (English) Zbl 0831.03008
Summary: We show that (contrary to the parallel case of intuitionistic logic) there does not exist a translation from \(\text{S}4^2\) (the propositional modal system S4 enriched with propositional quantifiers) into S4 that preserves provability and reduces to identity for Boolean connectives and \(\square\).

MSC:
03B45 Modal logic (including the logic of norms)
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
18C10 Theories (e.g., algebraic theories), structure, and semantics
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