Free and Projective Bimodal Symmetric Gödel Algebras

Grigolia, Revaz; Kiseliova, Tatiana; Odisharia, Vladimer

doi:10.1007/s11225-015-9630-3

Cited by 7 publications

(6 citation statements)

References 16 publications

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“…Let us now present the Hilbert-style calculus for biG. Since biG is an extension of the biintuitionistic logic HB [22], we first begin with a Hilbert-style calculus for it.…”

Section: Logical Preliminaries 21 Bi-gödel Logicmentioning

confidence: 99%

“…bi-Heyting algebras and bi-intuitionistic frames [34,35]. The bi-Gödel logic (or symmetric Gödel logic in the terminology of [22]) biG is axiomatised by adding two prelinearity axioms which make it complete w.r.t. linearly ordered bi-intuitionistic frames (bi-Gödel frames) and linearly ordered bi-Heyting algebras (bi-Gödel algebras):…”

Section: Definition 23 (Hhb)mentioning

confidence: 99%

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Qualitative reasoning in a two-layered framework

Bílková¹,

Frittella²,

Kozhemiachenko³

et al. 2022

Preprint

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The reasoning with qualitative uncertainty measures involves comparative statements about events in terms of their likeliness without necessarily assigning an exact numerical value to these events.The paper is divided into two parts. In the first part, we formalise reasoning with the qualitative counterparts of capacities, belief functions, and probabilities, within the framework of two-layered logics. Namely, we provide two-layered logics built over the classical propositional logic using a unary belief modality B that connects the inner layer to the outer one where the reasoning is formalised by means of Gödel logic. We design their Hilbert-style axiomatisations and prove their completeness. In the second part, we discuss the paraconsistent generalisations of the logics for qualitative uncertainty that take into account the case of the available information being contradictory or inconclusive.

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“…Let us now present the Hilbert-style calculus for biG. Since biG is an extension of the biintuitionistic logic HB [22], we first begin with a Hilbert-style calculus for it.…”

Section: Logical Preliminaries 21 Bi-gödel Logicmentioning

confidence: 99%

Section: Definition 23 (Hhb)mentioning

confidence: 99%

Qualitative reasoning in a two-layered framework

Bílková¹,

Frittella²,

Kozhemiachenko³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Furthermore, adding coimplication or, equivalently, Baaz' Delta operator △ (cf. [2] for details), results in bi-Gödel ('symmetric Gödel' in the terminology of [20]) logic that can additionally express strict order.…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, the only work on bi-Gödel (symmetric Gödel) modal logic is[20]. There, the authors propose an expansion of biG with and ♦ equipped with proof-theoretic interpretation and provide its algebraic semantics.…”

mentioning

confidence: 99%

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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“…To the best of our knowledge, the only work on bi-Gödel (symmetric Gödel) modal logic is[17]. There, the authors propose an expansion of biG with and ♦ equipped with proof-theoretic interpretation and provide its algebraic semantics.…”

mentioning

confidence: 99%

Paraconsistent Gödel modal logic

Bílková,

Frittella,

Kozhemiachenko

2022

Preprint

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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.

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Free and Projective Bimodal Symmetric Gödel Algebras

Cited by 7 publications

References 16 publications

Qualitative reasoning in a two-layered framework

Qualitative reasoning in a two-layered framework

Paraconsistent Gödel Modal Logic

Paraconsistent Gödel modal logic

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