The class of Kleene algebras satisfying an interpolation property and Nelson algebras

Cignoli, Roberto

doi:10.1007/bf01230621

Cited by 72 publications

(91 citation statements)

References 15 publications

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“…Due to this reason we have to find a rather complicated formula for the weak implication. However, the resulting duality theory for N4 ⊥ -lattices agrees very well with the duality theory developed by R. Cignioli [1] and A. Sendlewski [18].…”

Section: Introductionsupporting

confidence: 66%

“…Both Sendlewski [18] and Cignioli [1] used the technique of Priestley to obtain the representation of N3-lattices as Heyting algebras with distinguished filter [18] and to prove that the congruence lattice of N3-lattice is isomorphic to the congruence lattice of the underlying Heyting algebra. In [10], the generalizations of both results to N4-lattices was obtained via a direct algebraic proof, but this generalization and finding out a simple proof would be impossible without results on N3-lattices obtained with the help of Priestley duality.…”

Section: Introductionmentioning

confidence: 99%

“…For N3-lattices, i.e., for models of explosive Nelson's logic N3 [8], the elegant representation theory of Priestley type was developed independently by R. Cignoli [1] and A. Sendlewski [17,16,18] (they used the term Nelson algebras). Both Cignoli and Sendlewski essentially used the original algebraic representation of N3-lattices obtained by A. Monteiro [7] and based on the so called interpolation property (see Section 3).…”

Section: Introductionmentioning

confidence: 99%

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Priestley Duality for Paraconsistent Nelson’s Logic

Odintsov

2010

Stud Logica

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The variety of N4 ⊥ -lattices provides an algebraic semantics for the logic N4 ⊥ , a version of Nelson's logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4 ⊥ -lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.

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Section: Introductionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Priestley Duality for Paraconsistent Nelson’s Logic

Odintsov

2010

Stud Logica

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“…The semantics is given by Nelson algebras (Cignoli, 1986), that is Kleene algebras where a further implication a → N b = a → G (¬a b) always exists for any a, b and it satisfies (a ∧ b)…”

Section: Nelson Logicmentioning

confidence: 99%

A modal theorem-preserving translation of a class of three-valued logics of incomplete information

Ciucci

Dubois

2013

Journal of Applied Non-Classical Logics

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There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, Lukasiewicz and Nelson logics. We show that Priest logic of paradox, closely connected to Kleene's, can also be translated into our modal setting, just exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management.

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“…• BZMV dM algebras, a pasting of MV algebras and BZ lattices [Cattaneo et al(1999)Cattaneo, Giuntini, an On Table 4, we can see that among the 14 implications we also have the Kleene, the Gaines-Rescher, the Nelson [Vakarelov(1977), Cignoli(1986) Remark 3. In his study on invariant fuzzy implications [Drewniak(2006)], Drewniak points out 18 implications which are minimal "invariant with respect to the family of bijections".…”

Section: Remark 1 Other Definitions Of Conjunctionmentioning

confidence: 99%

A map of dependencies among three-valued logics

Ciucci

Dubois

2013

Information Sciences

100

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Abstract. Three-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value.

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The class of Kleene algebras satisfying an interpolation property and Nelson algebras

Cited by 72 publications

References 15 publications

Priestley Duality for Paraconsistent Nelson’s Logic

Priestley Duality for Paraconsistent Nelson’s Logic

A modal theorem-preserving translation of a class of three-valued logics of incomplete information

A map of dependencies among three-valued logics

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