Fragments of Quasi-Nelson: The Algebraizable Core

This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructu...

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Detalles Bibliográficos
Autor: Rivieccio, Umberto
Tipo de recurso: artículo
Fecha de publicación:2022
País:España
Institución:Universidad Nacional de Educación a Distancia
Repositorio:e-spacio. Repositorio Institucional de la UNED
Idioma:inglés
OAI Identifier:oai:e-spacio.uned.es:20.500.14468/30670
Acceso en línea:https://hdl.handle.net/20.500.14468/30670
Access Level:acceso abierto
Palabra clave:72 Filosofía
11 Lógica
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spelling Fragments of Quasi-Nelson: The Algebraizable CoreRivieccio, Umberto72 Filosofía11 LógicaThis is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. -algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.Oxford University PressCNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brasile-Spacio UNED20252025-10-2920222022-10-0120222022-10-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14468/30670reponame:e-spacio. Repositorio Institucional de la UNEDinstname:Universidad Nacional de Educación a DistanciaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-nd/4.0/deed.esoai:e-spacio.uned.es:20.500.14468/306702026-06-06T12:38:31Z
dc.title.none.fl_str_mv Fragments of Quasi-Nelson: The Algebraizable Core
title Fragments of Quasi-Nelson: The Algebraizable Core
spellingShingle Fragments of Quasi-Nelson: The Algebraizable Core
Rivieccio, Umberto
72 Filosofía
11 Lógica
title_short Fragments of Quasi-Nelson: The Algebraizable Core
title_full Fragments of Quasi-Nelson: The Algebraizable Core
title_fullStr Fragments of Quasi-Nelson: The Algebraizable Core
title_full_unstemmed Fragments of Quasi-Nelson: The Algebraizable Core
title_sort Fragments of Quasi-Nelson: The Algebraizable Core
dc.creator.none.fl_str_mv Rivieccio, Umberto
author Rivieccio, Umberto
author_facet Rivieccio, Umberto
author_role author
dc.contributor.none.fl_str_mv CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brasil
e-Spacio UNED
dc.subject.none.fl_str_mv 72 Filosofía
11 Lógica
topic 72 Filosofía
11 Lógica
description This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. -algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-10-01
2022
2022-10-01
2025
2025-10-29
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14468/30670
url https://hdl.handle.net/20.500.14468/30670
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
http://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Oxford University Press
publisher.none.fl_str_mv Oxford University Press
dc.source.none.fl_str_mv reponame:e-spacio. Repositorio Institucional de la UNED
instname:Universidad Nacional de Educación a Distancia
instname_str Universidad Nacional de Educación a Distancia
reponame_str e-spacio. Repositorio Institucional de la UNED
collection e-spacio. Repositorio Institucional de la UNED
repository.name.fl_str_mv
repository.mail.fl_str_mv
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