Paraconsistent Gödel Modal Logic

Abstract

We 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.

Two-layered logics for paraconsistent probabilities

“…-we use the term '±-probability' to stand for µ from Definition 2; -we call µ 4 from Defintion 3 a '4-probability' or a 'four-valued probability'. Recall that ±-probabilities are referred to as 'non-standard' in [25] and [7]. As this term is too broad (four-valued probabilities are not 'standard' either), we use a different designation.…”

“…[8,5] for details) with two valuations -v 1 (support of truth) and v 2 (support of falsity) -on [0, 1]. This was done in [7] -the resulting logic Pr △ was proven to be complete w.r.t. BD models with ±-probabilities.…”

“…Likewise, Pr L 2 △ proofs are also reducible to L 2 proofs (cf. [7,Theorem 4.24]) from substitution instances of axioms Prφ → Prχ (for φ |= BD χ), ¬Prφ ↔ Pr¬φ, and Pr(φ ∨ χ) ↔ (Prφ ⊖ Pr(φ ∧ χ)) ⊕ Prχ. Thus, it is clear that the validity and satisfiability of 4Pr L△ and Pr L 2 △ are coNP-hard and NP-hard, respectively.…”

“…Second, employing a two-layered formalism (cf. [16,15], [2], and [7,6] for examples). There, the logic is split into two levels: the inner layer describes events, and the outer layer describes the reasoning with the measure defined on events.…”

“…Plan of the paper Our paper continues the project proposed in [8] and continued in [7] and [6]. Here, we set to provide a logic that formalises the reasoning with four-valued probabilities as presented in [25].…”