A herbrandized functional interpretation of classical first-order logic

Detalhes bibliográficos
Autor(a) principal: Ferreira, Fernando
Data de Publicação: 2017
Outros Autores: Ferreira, Gilda
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10400.2/7089
Resumo: We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.
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spelling A herbrandized functional interpretation of classical first-order logicMathematical logicFunctional interpretationsFirst-order logicStar combinatory calculusFinite setsTautologiesHerbrand’s theoremWe introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.SpringerRepositório AbertoFerreira, FernandoFerreira, Gilda2018-09-30T00:30:23Z20172017-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.2/7089engFerreira, Fernando; Ferreira, Gilda - A herbrandized functional interpretation of classical first-order logic. "Archive for Mathematical Logic" [Em linha]. ISSN 0933-5846 (Print) 1432-0665 (Online). Vol. 56, nº 5-6 (2017), p. 523-5390933-5846 (Print)10.1007/s00153-017-0555-6info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-07-07T16:06:21ZPortal AgregadorONG
dc.title.none.fl_str_mv A herbrandized functional interpretation of classical first-order logic
title A herbrandized functional interpretation of classical first-order logic
spellingShingle A herbrandized functional interpretation of classical first-order logic
Ferreira, Fernando
Mathematical logic
Functional interpretations
First-order logic
Star combinatory calculus
Finite sets
Tautologies
Herbrand’s theorem
title_short A herbrandized functional interpretation of classical first-order logic
title_full A herbrandized functional interpretation of classical first-order logic
title_fullStr A herbrandized functional interpretation of classical first-order logic
title_full_unstemmed A herbrandized functional interpretation of classical first-order logic
title_sort A herbrandized functional interpretation of classical first-order logic
author Ferreira, Fernando
author_facet Ferreira, Fernando
Ferreira, Gilda
author_role author
author2 Ferreira, Gilda
author2_role author
dc.contributor.none.fl_str_mv Repositório Aberto
dc.contributor.author.fl_str_mv Ferreira, Fernando
Ferreira, Gilda
dc.subject.por.fl_str_mv Mathematical logic
Functional interpretations
First-order logic
Star combinatory calculus
Finite sets
Tautologies
Herbrand’s theorem
topic Mathematical logic
Functional interpretations
First-order logic
Star combinatory calculus
Finite sets
Tautologies
Herbrand’s theorem
description We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.
publishDate 2017
dc.date.none.fl_str_mv 2017
2017-01-01T00:00:00Z
2018-09-30T00:30:23Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10400.2/7089
url http://hdl.handle.net/10400.2/7089
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Ferreira, Fernando; Ferreira, Gilda - A herbrandized functional interpretation of classical first-order logic. "Archive for Mathematical Logic" [Em linha]. ISSN 0933-5846 (Print) 1432-0665 (Online). Vol. 56, nº 5-6 (2017), p. 523-539
0933-5846 (Print)
10.1007/s00153-017-0555-6
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str RCAAP
institution RCAAP
reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
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repository.mail.fl_str_mv
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