Degree-Preserving Gödel Logics with an Involution: Intermediate Logics and (Ideal) Paraconsistency

Coniglio, Marcelo E.; Esteva, Francesc; Gispert, Joan; Godo, Lluı́s

doi:10.1007/978-3-030-71258-7_6

Cited by 3 publications

(3 citation statements)

References 28 publications

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“…φ χ := ∼ i (∼ i χ → ∼ i φ) △φ := 1 (1 φ) Moreover, G inv is paraconsistent in the following sense: (1) (p ∧ ∼ i p) → q is not valid; (2) p, ∼ i p |= Ginv q if φ |= Ginv χ is interpreted as 'v(φ) ≤ v(χ), for each valuation v'. These properties were investigated in [22,20]. Furthermore, G inv is closely connected to G 2 that was first discussed in [24] and independently studied in more detail in [4,8].…”

Section: Introductionmentioning

confidence: 94%

Constraint Tableaux for Two-Dimensional Fuzzy Logics

Bílková

Frittella

Kozhemiachenko

2021

Lecture Notes in Computer Science

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This paper considers two logics. The first one, KGinv, is an expansion of the Gödel modal logic KG with the involutive negation ∼i defined as v(∼iφ, w) = 1 − v(φ, w). The second one, KG bl , is the expansion of KGinv with the bi-lattice connectives and modalities. We explore their semantical properties w.r.t. the standard semantics on [0, 1]-valued Kripke frames and define a unified tableaux calculus that allows for the explicit countermodel construction. For this, we use an alternative semantics with the finite model property. Using the tableaux calculus, we construct a decision algorithm and show that satisfiability and validity in KGinv and KG bl are PSpace-complete.

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Section: Introductionmentioning

confidence: 94%

Constraint Tableaux for Two-Dimensional Fuzzy Logics

Bílková

Frittella

Kozhemiachenko

2021

Lecture Notes in Computer Science

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“…Example 5. 13. Let A be the 14-element IG -chain whose support is A = a ∪ b , where a > b and, for x = a, b, x is made of the elements 1 > x = ∼ ∼ 2 x > x > ∼ 2 x > 2 x > ∼ x > ∼ 2 x > 0 as in Figure 2.…”

Section: Adding a -Operator To Involutive Gödel Algebrasmentioning

confidence: 99%

“…Indeed, within the degree-preserving consequence relations all the truth-values play an equally important role. As an intermediate alternative, it is possible to consider matrix logics in which the designated truth-values are given by (products of) order filters, see for instance [12] and [13] for the case of (products of) Lukasiewicz logics or Gödel's logics (possibly expanded with an involution) respectively.…”

Section: Introductionmentioning

confidence: 99%

On the expressive power of Lukasiewicz's square operator

Coniglio,

Esteva,

Flaminio

et al. 2021

Preprint

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The aim of the paper is to analyze the expressive power of the square operator of Lukasiewicz logic: * x = x x, where is the strong Lukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the Lukasiewicz square if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form n + 1 where n belongs to a class of prime numbers that we fully characterise. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Lukasiewicz's square operator. Finally, we propose an alternative way to account for Lukasiewicz square operator on involutive Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.

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On the expressive power of Łukasiewicz square operator

Coniglio

Esteva

Flaminio

et al. 2021

Journal of Logic and Computation

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The aim of the paper is to analyze the expressive power of the square operator of Łukasiewicz logic: $\ast x=x\odot x$, where $\odot $ is the strong Łukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the Łukasiewicz square operator if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form $n+1$ where $n$ belongs to a class of prime numbers that we fully characterize. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Łukasiewicz square operator. Finally, we propose an alternative way to account for Łukasiewicz square operator on involutive Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.

show abstract

Degree-Preserving Gödel Logics with an Involution: Intermediate Logics and (Ideal) Paraconsistency

Cited by 3 publications

References 28 publications

Constraint Tableaux for Two-Dimensional Fuzzy Logics

Constraint Tableaux for Two-Dimensional Fuzzy Logics

On the expressive power of Lukasiewicz's square operator

On the expressive power of Łukasiewicz square operator

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