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• W4308244853 abstract "In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-G{o}del modal logic $KbiG$. We prove its completeness w.r.t. crisp Kripke models where formulas at each state are evaluated over the standard bi-G{o}del algebra on $[0,1]$. We also consider a paraconsistent expansion of $KbiG$ with a De Morgan negation $neg$ which we dub $KGsquare$. We devise a Hilbert-style calculus for this logic and, as a~con-se-quence of a~conservative translation from $KbiG$ to $KGsquare$, prove its completeness w.r.t. crisp Kripke models with two valuations over $[0,1]$ connected via $neg$. For these two logics, we establish that their decidability and validity are $mathsf{PSPACE}$-complete. We also study the semantical properties of $KbiG$ and $KGsquare$. In particular, we show that Glivenko theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in $mathbf{K}$ (the classical modal logic) and the crisp G{o}del modal logic $KG^c$. We show that, among others, all Sahlqvist formulas and all formulas $phirightarrowchi$ where $phi$ and $chi$ are monotone, define the same classes of frames in $mathbf{K}$ and $KG^c$." @default.
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• W4308244853 date "2023-09-28" @default.
• W4308244853 modified "2023-09-29" @default.
• W4308244853 title "Crisp Bi-Gödel modal logic and its paraconsistent expansion" @default.
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• W4308244853 doi "https://doi.org/10.1093/jigpal/jzad017" @default.
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