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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) (Fairtlough and Mendler in Inf Comput 137(1):1–33, 1997), which is the intuitionistic modal version of possibility semantics (Holliday in UC Berkeley working paper in logic and the methodology of science, 2022. http://escholarship.org/uc/item/881757qn). We identify the fragment of inductive formulas (Goranko and Vakarelov in Ann Pure Appl Logic 141(1–2):180–217, 2006) in this language and give the algorithm \(\textsf{ALBA}\) (Conradie and Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012) 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 (Zhao in J Logic Comput 31(2):523–572, 2021); 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 \(\textsf{ALBA}\) with respect to modal FM-frames and show that \(\textsf{ALBA}\) succeeds on inductive formulas, similar to existing settings like (Conradie and Palmigiano in Ann Pure Appl Logic 163(3):338–376, 2012; Zhao 2021, in: Cia-battoni, Pimentel, Queiroz (eds) Logic, language, information, and computation, Springer International Publishing, Cham, 2022).

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Acknowledgements

The research of the author is supported by the Taishan Young Scholars Program of the Government of Shandong Province, China (No.tsqn201909151).

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    Zhiguang Zhao

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Presented by Heinrich Wansing; Received October 4, 2022.

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Zhao, Z. Correspondence Theory for Modal Fairtlough–Mendler Semantics of Intuitionistic Modal Logic. Stud Logica 111, 1057–1082 (2023). https://doi.org/10.1007/s11225-023-10064-3

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