Bi-intermediate logics of trees and co-trees

A bi-Heyting algebra validates the Gödel-Dummett axiom (p → q) ∨ (q → p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension bi-GD of bi-intuit...

ver descrição completa

Detalhes bibliográficos
Autores: Bezhanishvili, Guram, Martins, Miguel, Moraschini, Tommaso
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2024
País:España
Recursos:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:dnet:ubarcelona__::e7f2fecb559506ab7ace8f01c239b05d
Acesso em linha:https://hdl.handle.net/2445/228837
Access Level:acceso abierto
Palavra-chave:Semàntica (Filosofia)
Lògica
Varietats algebraiques
Intuïció
Tabulatures
Semantics (Philosophy)
Logic
Algebraic varieties
Intuition
Tablatures
id ES_ac05b019ba99d263b73fed2fd207bc93
oai_identifier_str oai:dnet:ubarcelona__::e7f2fecb559506ab7ace8f01c239b05d
network_acronym_str ES
network_name_str España
repository_id_str
spelling Bi-intermediate logics of trees and co-treesBezhanishvili, GuramMartins, MiguelMoraschini, TommasoSemàntica (Filosofia)LògicaVarietats algebraiquesIntuïcióTabulaturesSemantics (Philosophy)LogicAlgebraic varietiesIntuitionTablaturesA bi-Heyting algebra validates the Gödel-Dummett axiom (p → q) ∨ (q → p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension bi-GD of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice Λ(bi-GD) of extensions of bi-GD. We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of bi-GD. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of bi-GD. We introduce a sequence of co-trees, called the finite combs, and show that a logic in Λ(bi-GD) is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of bi-GD and consequently, a unique pre-locally tabular extension of bi-GD. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.Elsevier B.V.2024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/228837Articles publicats en revistes (Filosofia)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésReproducció del document publicat a: https://doi.org/10.1016/j.apal.2024.103490Annals of Pure and Applied Logic, 2024https://doi.org/10.1016/j.apal.2024.103490cc-by-nc-nd (c) Bezhanishvili, Guram et al., 2024http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:dnet:ubarcelona__::e7f2fecb559506ab7ace8f01c239b05d2026-05-27T06:46:51Z
dc.title.none.fl_str_mv Bi-intermediate logics of trees and co-trees
title Bi-intermediate logics of trees and co-trees
spellingShingle Bi-intermediate logics of trees and co-trees
Bezhanishvili, Guram
Semàntica (Filosofia)
Lògica
Varietats algebraiques
Intuïció
Tabulatures
Semantics (Philosophy)
Logic
Algebraic varieties
Intuition
Tablatures
title_short Bi-intermediate logics of trees and co-trees
title_full Bi-intermediate logics of trees and co-trees
title_fullStr Bi-intermediate logics of trees and co-trees
title_full_unstemmed Bi-intermediate logics of trees and co-trees
title_sort Bi-intermediate logics of trees and co-trees
dc.creator.none.fl_str_mv Bezhanishvili, Guram
Martins, Miguel
Moraschini, Tommaso
author Bezhanishvili, Guram
author_facet Bezhanishvili, Guram
Martins, Miguel
Moraschini, Tommaso
author_role author
author2 Martins, Miguel
Moraschini, Tommaso
author2_role author
author
dc.subject.none.fl_str_mv Semàntica (Filosofia)
Lògica
Varietats algebraiques
Intuïció
Tabulatures
Semantics (Philosophy)
Logic
Algebraic varieties
Intuition
Tablatures
topic Semàntica (Filosofia)
Lògica
Varietats algebraiques
Intuïció
Tabulatures
Semantics (Philosophy)
Logic
Algebraic varieties
Intuition
Tablatures
description A bi-Heyting algebra validates the Gödel-Dummett axiom (p → q) ∨ (q → p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension bi-GD of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we initiate the study of the lattice Λ(bi-GD) of extensions of bi-GD. We develop the methods of Jankov-style formulas for bi-Gödel algebras and use them to prove that there are exactly continuum many extensions of bi-GD. We also show that all these extensions can be uniformly axiomatized by canonical formulas. Our main result is a characterization of the locally tabular extensions of bi-GD. We introduce a sequence of co-trees, called the finite combs, and show that a logic in Λ(bi-GD) is locally tabular iff it contains at least one of the Jankov formulas associated with the finite combs. It follows that there exists the greatest nonlocally tabular extension of bi-GD and consequently, a unique pre-locally tabular extension of bi-GD. These results contrast with the case of the intermediate logic axiomatized by the Gödel-Dummett axiom, which is known to have only countably many extensions, all of which are locally tabular.
publishDate 2024
dc.date.none.fl_str_mv 2024
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/228837
url https://hdl.handle.net/2445/228837
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/10.1016/j.apal.2024.103490
Annals of Pure and Applied Logic, 2024
https://doi.org/10.1016/j.apal.2024.103490
dc.rights.none.fl_str_mv cc-by-nc-nd (c) Bezhanishvili, Guram et al., 2024
http://creativecommons.org/licenses/by-nc-nd/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc-by-nc-nd (c) Bezhanishvili, Guram et al., 2024
http://creativecommons.org/licenses/by-nc-nd/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier B.V.
publisher.none.fl_str_mv Elsevier B.V.
dc.source.none.fl_str_mv Articles publicats en revistes (Filosofia)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869416319729795072
score 15,812429

Registos relacionados