Algebraic Semantics for Quasi-Nelson Logic

Liang, Fei; Nascimento, Thiago

doi:10.1007/978-3-662-59533-6_27

Cited by 9 publications

(8 citation statements)

References 8 publications

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“…The papers Spinks (2019, 2020) and Liang and Nascimento (2019) concern the logic obtained by extending F L ew (including a 0 constant) with the addition of the Nelson axiom. We dubbed this logic quasi-Nelson logic (QN ), and the corresponding algebras QNAs or QNRLs.…”

Section: Quasi-nelson Logic Algebras and Residuated Latticesmentioning

confidence: 99%

“…We now turn our attention to the {→, ∼}-fragment of the quasi-Nelson algebraic language, which has not been considered in any earlier paper. We dub it the 'algebraisable fragment' because the negation and the weak implication are the two connectives that witness the algebraisability (in the sense of Blok and Pigozzi 1989) of quasi-Nelson logic as presented in Liang and Nascimento (2019). 6 We begin with a quasi-equational presentation of the abstract class of algebras corresponding to this fragment.…”

Section: The Algebraisable Fragmentmentioning

confidence: 99%

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Quasi-Nelson algebras and fragments

Rivieccio

Jansana

2021

Math. Struct. Comp. Sci.

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The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒ ew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

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Section: Quasi-nelson Logic Algebras and Residuated Latticesmentioning

confidence: 99%

Section: The Algebraisable Fragmentmentioning

confidence: 99%

Quasi-Nelson algebras and fragments

Rivieccio

Jansana

2021

Math. Struct. Comp. Sci.

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“…The recent series of papers [16,7,17,15] introduced and developed the basic theory of a non-involutive generalization of Nelson constructive logic with strong negation [9]. The new logic was dubbed quasi-Nelson logic [7], and the corresponding algebraic models quasi-Nelson algebras.…”

Section: Introductionmentioning

confidence: 99%

Quasi-N4-lattices

Rivieccio

2022

Soft Comput

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Within the Nelson family, two mutually incomparable generalizations of Nelson constructive logic with strong negation have been proposed so far. The first and more well-known, Nelson paraconsistent logic, results from dropping the explosion axiom of Nelson logic; a more recent series of papers considers the logic (dubbed quasi-Nelson logic) obtained by rejecting the double negation law, which is thus also weaker than intuitionistic logic. The algebraic counterparts of these logical calculi are the varieties of N4-lattices and quasi-Nelson algebras. In the present paper we propose the class of quasi-N4-lattices as a common generalization of both. We show that a number of key results, including the twist-structure representation of N4-lattices and quasi-Nelson algebras, can be uniformly established in this more general setting; our new representation employs twist-structures defined over Brouwerian algebras enriched with a nucleus operator. We further show that quasi-N4-lattices form a variety that is arithmetical, possesses a ternary as well as a quaternary deductive term, and enjoys EDPC and the strong congruence extension property.

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“…Nelson algebras and N4-lattices) have been extensively investigated [Sendlewski 1990, Odintsov 2003, Odintsov 2004, Spinks and Veroff 2018. A recent series of papers introduced and investigated the class of quasi-Nelson algebras (a subvariety of commutative integral bounded residuated lattices) as a non-involutive generalization of Nelson algebras [Rivieccio and Spinks 2021]; the corresponding logic was axiomatized in [Liang and Nascimento 2019]. A similar abstraction was applied to the class of N4-lattices in [Rivieccio 2022], introducing the class of quasi-N4-lattices (i.e.…”

Section: Introductionmentioning

confidence: 99%

Quasi-N4-lattices and their logic

Neto¹,

Nascimento

Rivieccio³

2022

Anais Do III Workshop Brasileiro De Lógica (WBL 2022)

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The variety of quasi-N4-lattices (QN4) was recently introduced as a non-involutive generalization of N4-lattices (algebraic models of Nelson's paraconsistent logic). While research on these algebras is still at a preliminary stage, we know that QN4 is an arithmetical variety which possesses a ternary as well as a quaternary deductive term, enjoys equationally definable principal congruences and the strong congruence extension property. We furthermore have recently introduced an algebraizable logic having QN4 as its equivalent semantics. In this contribution we report on the results obtained so far on this class of algebras and on its logical counterpart.

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Algebraic Semantics for Quasi-Nelson Logic

Cited by 9 publications

References 8 publications

Quasi-Nelson algebras and fragments

Quasi-Nelson algebras and fragments

Quasi-N4-lattices

Quasi-N4-lattices and their logic

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