Qualitative reasoning in a two-layered framework

Bílková, Marta; Frittella, Sabine; Kozhemiachenko, Daniil; Majer, Ondrej

doi:10.1016/j.ijar.2022.12.011

Cited by 9 publications

(11 citation statements)

References 40 publications

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“…Moreover, they provide an easy way of the countermodel extraction from complete open branches. Furthermore, while the propositional connectives of G 2 allow for the construction of an analytic proof system, e.g., a display calculus extending that of I 4 C 4 7 [38], the modal ones are not dual to one another w.r.t. ¬ nor the Gödel negation ∼.…”

Section: Tableaux Calculusmentioning

confidence: 99%

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Paraconsistent Gödel Modal Logic

Bílková

Frittella

Kozhemiachenko

2022

Automated Reasoning

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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.

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Section: Tableaux Calculusmentioning

confidence: 99%

“…Finally, just as intuitionistic logic can be expanded with a De Morgan negation, so can be Gödel logic as well (cf. [35] for expansions of intuitionistic propositional logic and [3,5] for expansions of G). The negated formula ¬φ is, as expected, interpreted as 'φ is false'.…”

Section: Fuzzy Modal Logicsmentioning

confidence: 99%

Paraconsistent Gödel Modal Logic

Bílková

Frittella

Kozhemiachenko

2022

Automated Reasoning

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Two-layered logics for paraconsistent probabilities

Bílková¹,

Frittella²,

Kozhemiachenko³

et al. 2023

Preprint

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We discuss two two-layered logics formalising reasoning with paraconsistent probabilities that combine the Lukasiewicz [0, 1]-valued logic with Baaz △ operator and the Belnap-Dunn logic. The first logic Pr L 2 △ (introduced in [7]) formalises a 'two-valued' approach where each event φ has independent positive and negative measures that stand for, respectively, the likelihoods of φ and ¬φ. The second logic 4Pr L △ that we introduce here corresponds to 'four-valued' probabilities. There, φ is equipped with four measures standing for pure belief, pure disbelief, conflict and uncertainty of an agent in φ. We construct faithful embeddings of 4Pr L △ and Pr L 2 △ into one another and axiomatise 4Pr L △ using a Hilber-style calculus. We also establish the decidability of both logics and provide complexity evaluations for them using an expansion of the constraint tableaux calculus for L.

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“…

This paper is an extended version of Bílková et al (2023b). We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Łukasiewicz [0, 1]-valued logic with Baaz △ operator and the Belnap-Dunn logic.

…”

mentioning

confidence: 99%