Paraconsistent Gödel modal logic on bi-relational frames
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We further develop the paraconsistent Gödel modal logic. In this paper, we consider its version endowed with Kripke semantics on [0,1]-valued frames with two fuzzy relations R^+ and R^- (degrees of trust in assertions and denials) and two valuations v_1 and v_2 (support of truth and support of falsity) linked with a De Morgan negation . We demonstrate that it does not extend Gödel modal logic and that and ◊ are not interdefinable. We also show that several important classes of frames are definable (in particular, crisp, mono-relational, and finitely branching). For over finitely branching frames, we create a sound and complete constraint tableaux calculus and a decision procedure based upon it. Using the decision procedure we show that satisfiability and validity are in PSPACE.
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