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...
| Autores: | , , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
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| Acceso en línea: | https://hdl.handle.net/2445/228837 |
| Access Level: | acceso abierto |
| Palabra clave: | Semàntica (Filosofia) Lògica Varietats algebraiques Intuïció Tabulatures Semantics (Philosophy) Logic Algebraic varieties Intuition Tablatures |
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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.2026202620242026info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion55 p.application/pdfhttps://hdl.handle.net/2445/228837https://hdl.handle.net/2445/228837Articles publicats en revistes (Filosofia)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésReproducció 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:recercat____::3e2089483361beb4f8ef63ab572ce07c2026-05-29T05:05:01Z |
| 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 2026 2026 2026 |
| 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 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/ |
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openAccess |
| dc.format.none.fl_str_mv |
55 p. 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:Recercat. Dipósit de la Recerca de Catalunya instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
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Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
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Recercat. Dipósit de la Recerca de Catalunya |
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Recercat. Dipósit de la Recerca de Catalunya |
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15,811543 |