Proof theory for tight apartness

The paper provides a cut-free sequent calculus for the theory of tight apartness in a language where both apartness and equality are primitive notions. The result is obtained by aptly modifying the underlying logical calculus for intuitionistic logic and adding rules of inference corresponding to th...

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Detalhes bibliográficos
Autor: Maffezioli, Paolo
Tipo de documento: artigo
Data de publicação:2025
País:España
Recursos:Universidad Complutense de Madrid (UCM)
Repositório:Docta Complutense
Idioma:inglês
OAI Identifier:oai:docta.ucm.es:20.500.14352/124777
Acesso em linha:https://hdl.handle.net/20.500.14352/124777
Access Level:Acceso aberto
Palavra-chave:164
Apartness
Proof theory
Cut elimination
Lógica simbólica y matemática (Filosofía)
1102.08 Lógica Matemática
1102.03 Lógica Formal
1102.11 Teoría de Pruebas
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oai_identifier_str oai:docta.ucm.es:20.500.14352/124777
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spelling Proof theory for tight apartnessMaffezioli, Paolo164ApartnessProof theoryCut eliminationLógica simbólica y matemática (Filosofía)1102.08 Lógica Matemática1102.03 Lógica Formal1102.11 Teoría de PruebasThe paper provides a cut-free sequent calculus for the theory of tight apartness in a language where both apartness and equality are primitive notions. The result is obtained by aptly modifying the underlying logical calculus for intuitionistic logic and adding rules of inference corresponding to the axioms of apartness and the principles governing the mutual deductive relationships between apartness and equality. While the rules for apartness are found directly from the axioms by applying standard proof-theoretic methods, the others, especially the rule corresponding to the principle of tight apartness, are found by exploiting the logical law of consequentia mirabilis. Along the way, we also provide a cut-free sequent calculus for the theory of weak tight apartness, also known as the theory of negated equality, thus answering in the positive to an open problem in the existing literature on the subject.Institute of Philosophy and Sociology Polish Academy of SciencesUniversidad Complutense de Madrid20252025-01-0120252025-01-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/124777reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/1247772026-06-02T12:44:21Z
dc.title.none.fl_str_mv Proof theory for tight apartness
title Proof theory for tight apartness
spellingShingle Proof theory for tight apartness
Maffezioli, Paolo
164
Apartness
Proof theory
Cut elimination
Lógica simbólica y matemática (Filosofía)
1102.08 Lógica Matemática
1102.03 Lógica Formal
1102.11 Teoría de Pruebas
title_short Proof theory for tight apartness
title_full Proof theory for tight apartness
title_fullStr Proof theory for tight apartness
title_full_unstemmed Proof theory for tight apartness
title_sort Proof theory for tight apartness
dc.creator.none.fl_str_mv Maffezioli, Paolo
author Maffezioli, Paolo
author_facet Maffezioli, Paolo
author_role author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 164
Apartness
Proof theory
Cut elimination
Lógica simbólica y matemática (Filosofía)
1102.08 Lógica Matemática
1102.03 Lógica Formal
1102.11 Teoría de Pruebas
topic 164
Apartness
Proof theory
Cut elimination
Lógica simbólica y matemática (Filosofía)
1102.08 Lógica Matemática
1102.03 Lógica Formal
1102.11 Teoría de Pruebas
description The paper provides a cut-free sequent calculus for the theory of tight apartness in a language where both apartness and equality are primitive notions. The result is obtained by aptly modifying the underlying logical calculus for intuitionistic logic and adding rules of inference corresponding to the axioms of apartness and the principles governing the mutual deductive relationships between apartness and equality. While the rules for apartness are found directly from the axioms by applying standard proof-theoretic methods, the others, especially the rule corresponding to the principle of tight apartness, are found by exploiting the logical law of consequentia mirabilis. Along the way, we also provide a cut-free sequent calculus for the theory of weak tight apartness, also known as the theory of negated equality, thus answering in the positive to an open problem in the existing literature on the subject.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025-01-01
2025
2025-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/124777
url https://hdl.handle.net/20.500.14352/124777
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
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Institute of Philosophy and Sociology Polish Academy of Sciences
publisher.none.fl_str_mv Institute of Philosophy and Sociology Polish Academy of Sciences
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.81155

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