Fuzzy bi-Gödel modal logic and its paraconsistent relatives

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.