BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Tipo de documento: | Artigo |
Idioma: | por |
Título da fonte: | Manuscrito (Online) |
Texto Completo: | https://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8643907 |
Resumo: | In [9] it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz (see [4]). The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic (see [1]). We call commutative B-C-K logic, briefly cBCK, to the extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups (see [10], [5]) . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups (see [4]). |
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BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGICB-C-K-logic. BCK-algebras. MV-algebras. Lukasiewicz logicIn [9] it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz (see [4]). The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic (see [1]). We call commutative B-C-K logic, briefly cBCK, to the extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups (see [10], [5]) . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups (see [4]).Universidade Estadual de Campinas2016-03-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8643907Manuscrito: Revista Internacional de Filosofia; v. 28 n. 2 (2005): Jul./Dec.; 575-583Manuscrito: International Journal of Philosophy; Vol. 28 No. 2 (2005): Jul./Dec.; 575-583Manuscrito: Revista Internacional de Filosofía; Vol. 28 Núm. 2 (2005): Jul./Dec.; 575-5832317-630Xreponame:Manuscrito (Online)instname:Universidade Estadual de Campinas (UNICAMP)instacron:UNICAMPporhttps://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8643907/11375Copyright (c) 2005 Manuscritoinfo:eu-repo/semantics/openAccessSagastume, Marta2016-03-08T11:27:17Zoai:ojs.periodicos.sbu.unicamp.br:article/8643907Revistahttps://periodicos.sbu.unicamp.br/ojs/index.php/manuscritoPUBhttps://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/oaimwrigley@cle.unicamp.br|| dascal@spinoza.tau.ac.il||publicacoes@cle.unicamp.br2317-630X0100-6045opendoar:2016-03-08T11:27:17Manuscrito (Online) - Universidade Estadual de Campinas (UNICAMP)false |
dc.title.none.fl_str_mv |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC |
title |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC |
spellingShingle |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC Sagastume, Marta B-C-K-logic. BCK-algebras. MV-algebras. Lukasiewicz logic |
title_short |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC |
title_full |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC |
title_fullStr |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC |
title_full_unstemmed |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC |
title_sort |
BOUNDED COMMUTATIVE B-C-K LOGIC AND LUKASIEWICZ LOGIC |
author |
Sagastume, Marta |
author_facet |
Sagastume, Marta |
author_role |
author |
dc.contributor.author.fl_str_mv |
Sagastume, Marta |
dc.subject.por.fl_str_mv |
B-C-K-logic. BCK-algebras. MV-algebras. Lukasiewicz logic |
topic |
B-C-K-logic. BCK-algebras. MV-algebras. Lukasiewicz logic |
description |
In [9] it is proved the categorical isomorphism of two varieties: bounded commutative BCK-algebras and MV -algebras. The class of MV -algebras is the algebraic counterpart of the infinite valued propositional calculus L of Lukasiewicz (see [4]). The main objective of the present paper is to study that isomorphism from the perspective of logic. The B-C-K logic is algebraizable and the quasivariety of BCKalgebras is the equivalent algebraic semantics for that logic (see [1]). We call commutative B-C-K logic, briefly cBCK, to the extension of B-C-K logic associated to the variety of commutative BCK–algebras. Moreover, we present the extension Boc of cBCK obtained by adding the axiom of “boundness”. We prove that the deductive system Boc is equivalent to L. We observe that cBCK admits two interesting extensions: the logic Boc, treated in this paper, which is equivalent to the system L of Lukasiewicz, and the logic Co that is naturally associated to the system Balo of `-groups (see [10], [5]) . This constructions establish a link between L and Balo , that would be a logical approach to the categorical relationship between MV–algebras and `-groups (see [4]). |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016-03-04 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8643907 |
url |
https://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8643907 |
dc.language.iso.fl_str_mv |
por |
language |
por |
dc.relation.none.fl_str_mv |
https://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8643907/11375 |
dc.rights.driver.fl_str_mv |
Copyright (c) 2005 Manuscrito info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Copyright (c) 2005 Manuscrito |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Universidade Estadual de Campinas |
publisher.none.fl_str_mv |
Universidade Estadual de Campinas |
dc.source.none.fl_str_mv |
Manuscrito: Revista Internacional de Filosofia; v. 28 n. 2 (2005): Jul./Dec.; 575-583 Manuscrito: International Journal of Philosophy; Vol. 28 No. 2 (2005): Jul./Dec.; 575-583 Manuscrito: Revista Internacional de Filosofía; Vol. 28 Núm. 2 (2005): Jul./Dec.; 575-583 2317-630X reponame:Manuscrito (Online) instname:Universidade Estadual de Campinas (UNICAMP) instacron:UNICAMP |
instname_str |
Universidade Estadual de Campinas (UNICAMP) |
instacron_str |
UNICAMP |
institution |
UNICAMP |
reponame_str |
Manuscrito (Online) |
collection |
Manuscrito (Online) |
repository.name.fl_str_mv |
Manuscrito (Online) - Universidade Estadual de Campinas (UNICAMP) |
repository.mail.fl_str_mv |
mwrigley@cle.unicamp.br|| dascal@spinoza.tau.ac.il||publicacoes@cle.unicamp.br |
_version_ |
1800216566134996992 |