Abstract We introduce a paraconsistent modal logic$$\mathbf {K}\mathsf {G}^2$$KG2, based on Gödel logic with coimplication (bi-Gödel logic) expanded with a De Morgan negation$$\lnot $$¬. We use the logic to formalise reasoning with graded, incomplete and inconsistent information. Semantics of$$\mathbf {K}\mathsf {G}^2$$KG2is two-dimensional: we interpret$$\mathbf {K}\mathsf {G}^2$$KG2on crisp frames with two valuations$$v_1$$v1and$$v_2$$v2, connected via$$\lnot $$¬, 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$$\mathbf {K}\mathsf {G}^2$$KG2is strictly more expressive than the classical modal logic$$\mathbf {K}$$Kby proving that finitely branching frames are definable and by establishing a faithful embedding of$$\mathbf {K}$$Kinto$$\mathbf {K}\mathsf {G}^2$$KG2. We also construct a constraint tableau calculus for$$\mathbf {K}\mathsf {G}^2$$KG2over finitely branching frames, establish its decidability and provide a complexity evaluation.
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