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We present the axiomatisation of the fuzzy bi-G\"{o}del modal logic(formulated in the language containing $\triangle$ and treating thecoimplication as a defined connective) and establish its PSpace-completeness.We also consider its paraconsistent relatives defined on fuzzy frames with twovaluations $e_1$ and $e_2$ standing for the support of truth and falsity,respectively, and equipped with \emph{two fuzzy relations} $R^+$ and $R^-$ usedto determine supports of truth and falsity of modal formulas. We establishembeddings of these paraconsistent logics into the fuzzy bi-G\"{o}del modallogic and use them to prove their PSpace-completeness and obtain thecharacterisation of definable frames.

Fuzzy bi-G\"{o}del modal logic and its paraconsistent relatives