Correspondence Theory for Modal Fairtlough-Mendler Semantics of Intuitionistic Modal Logic
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We study the correspondence theory of intuitionistic modal logic in modal Fairtlough-Mendler semantics (modal FM semantics) <cit.>, which is the intuitionistic modal version of possibility semantics <cit.>. We identify the fragment of inductive formulas <cit.> in this language and give the algorithm 𝖠𝖫𝖡𝖠 <cit.> in this semantic setting. There are two major features in the paper: one is that in the expanded modal language, the nominal variables, which are interpreted as atoms in perfect Boolean algebras, complete join-prime elements in perfect distributive lattices and complete join-irreducible elements in perfect lattices, are interpreted as the refined regular open closures of singletons in the present setting, similar to the possibility semantics for classical normal modal logic <cit.>; the other feature is that we do not use conominals or diamond, which restricts the fragment of inductive formulas significantly. We prove the soundness of the 𝖠𝖫𝖡𝖠 with respect to modal FM frames and show that the 𝖠𝖫𝖡𝖠 succeeds on inductive formulas, similar to existing settings like <cit.>.
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