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

TL;DR

This paper introduces the axiomatisation of fuzzy bi-G"{o}del modal logic, proves its PSpace-completeness, and explores its paraconsistent variants with two valuations and fuzzy relations, establishing embeddings and frame characterisations.

Contribution

It provides the first axiomatisation and complexity analysis of fuzzy bi-G"{o}del modal logic and extends it to paraconsistent variants with novel semantics.

Findings

Fuzzy bi-G"{o}del modal logic is PSpace-complete.

Paraconsistent variants can be embedded into the fuzzy bi-G"{o}del modal logic.

Characterisation of definable frames for these logics.

Abstract

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