Paraconsistent Gödel modal logic

Abstract

We introduce a paraconsistent modal logic KG 2 , based on Gödel logic with coimplication (bi-Gödel logic) expanded with a De Morgan negation ¬. We use the logic to formalise reasoning with graded, incomplete and inconsistent information. Semantics of KG 2 is two-dimensional: we interpret KG 2 on crisp frames with two valuations v1 and v2, connected via ¬, that assign to each formula two values from the real-valued interval [0, 1]. The first (resp., second) valuation encodes the positive (resp., negative) information the state gives to a statement. We obtain that KG 2 is strictly more expressive than the classical modal logic K by proving that finitely branching frames are definable and by establishing a faithful embedding of K into KG 2 . We also construct a constraint tableau calculus for KG 2 over finitely branching frames, establish its decidability and provide a complexity evaluation.