Abstract
AbstractWe introduce a paraconsistent modal logic $$\mathbf {K}\mathsf {G}^2$$
K
G
2
, 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 $$\mathbf {K}\mathsf {G}^2$$
K
G
2
is two-dimensional: we interpret $$\mathbf {K}\mathsf {G}^2$$
K
G
2
on crisp frames with two valuations $$v_1$$
v
1
and $$v_2$$
v
2
, 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 $$\mathbf {K}\mathsf {G}^2$$
K
G
2
is strictly more expressive than the classical modal logic $$\mathbf {K}$$
K
by proving that finitely branching frames are definable and by establishing a faithful embedding of $$\mathbf {K}$$
K
into $$\mathbf {K}\mathsf {G}^2$$
K
G
2
. We also construct a constraint tableau calculus for $$\mathbf {K}\mathsf {G}^2$$
K
G
2
over finitely branching frames, establish its decidability and provide a complexity evaluation.
Publisher
Springer International Publishing
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