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Marta Bílková, Sabine Frittella, Daniil Kozhemiachenko, Fuzzy bi-Gödel modal logic and its paraconsistent relatives, Journal of Logic and Computation, Volume 35, Issue 2, March 2025, exae011, https://doi.org/10.1093/logcom/exae011
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Abstract
We present an axiomatization of the fuzzy bi-Gödel modal logic |${\textbf{K}\textsf{biG}}^{\textsf{f}}$| formulated in the language containing |$\triangle $| (Baaz Delta operator) and treating |$-\!-\!< $| (co-implication) as the defined connective. We also consider two paraconsistent relatives of |${\textbf{K}\textsf{biG}}^{\textsf{f}}$| — |$\textbf{K}\textsf{G}^{2\pm \textsf{f}}$| and |$\textsf{G}^{2\pm \textsf{f}}_{\blacksquare ,\blacklozenge }$|. These logics are defined on fuzzy frames with two valuations |$e_{1}$| and |$e_{2}$| standing for the support of truth and falsity, respectively, and equipped with two fuzzy relations |$R^{+}$| and |$R^{-}$| used to determine supports of truth and falsity of modal formulas. We construct embeddings of |$\textbf{K}\textsf{G}^{2\pm \textsf{f}}$| and |$\textsf{G}^{2\pm \textsf{f}}_{\blacksquare ,\blacklozenge }$| into |${\textbf{K}\textsf{biG}}^{\textsf{f}}$| and use them to obtain the characterization of |$\textbf{K}\textsf{G}^{2}$|- and |$\textsf{G}^{2}_{\blacksquare ,\blacklozenge }$|-definable frames. Moreover, we study the transfer of |${\textbf{K}\textsf{biG}}^{\textsf{f}}$| formulas into |$\textbf{K}\textsf{G}^{2\pm \textsf{f}}$|, i.e., formulas that are |${\textbf{K}\textsf{biG}}^{\textsf{f}}$|-valid on mono-relational frames |$\mathfrak{F}$| and |$\mathfrak{F}^{\prime}$| iff they are |$\textbf{K}\textsf{G}^{2\pm \textsf{f}}$|-valid on their bi-relational counterparts. Finally, we establish |$\textsf{PSpace}$|-completeness of all considered logics.
