Non axiomatizability of Modal Lukasiewicz Logic
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In this work we study the decidability of the global modal logic arising from Kripke frames evaluated on certain residuated lattices (including all BL algebras), known in the literature as crisp modal many-valued logics. We exhibit a large family of these modal logics that are undecidable, in opposition to classical modal logic and to the propositional logics defined over the same classes of algebras. These include the global modal logics arising from the standard Lukasiewicz and Product algebras. Furthermore, it is shown that global modal Lukasiewicz and Product logics are not recursively axiomatizable. We conclude the paper by solving negatively the open question of whether a global modal logic coincides with the local modal logic closed under the unrestricted necessitation rule.
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