Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic

Using the theory of BL-algebras, it is shown that a prepositional formula φ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ∼∼ φ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ ⇒ ∼ φ)) ⇒ ψ, then φ is deriva...

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Detalles Bibliográficos
Publicado: 2003
Materias:
Łukasiewicz logic
Basic fuzzy logic
BL-algebras
Glivenko's theorem
MV-algebras
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09335846_v42_n4_p361_Cignoli
http://hdl.handle.net/20.500.12110/paper_09335846_v42_n4_p361_Cignoli
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_09335846_v42_n4_p361_Cignoli
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spelling paper:paper_09335846_v42_n4_p361_Cignoli2023-06-08T15:53:12Z Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic Łukasiewicz logic Basic fuzzy logic BL-algebras Glivenko's theorem MV-algebras Using the theory of BL-algebras, it is shown that a prepositional formula φ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ∼∼ φ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ ⇒ ∼ φ)) ⇒ ψ, then φ is derivable in in classical logic if and only if ∼∼ φ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras. 2003 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09335846_v42_n4_p361_Cignoli http://hdl.handle.net/20.500.12110/paper_09335846_v42_n4_p361_Cignoli
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Łukasiewicz logic
Basic fuzzy logic
BL-algebras
Glivenko's theorem
MV-algebras
spellingShingle Łukasiewicz logic
Basic fuzzy logic
BL-algebras
Glivenko's theorem
MV-algebras
Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
topic_facet Łukasiewicz logic
Basic fuzzy logic
BL-algebras
Glivenko's theorem
MV-algebras
description Using the theory of BL-algebras, it is shown that a prepositional formula φ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ∼∼ φ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ ⇒ ∼ φ)) ⇒ ψ, then φ is derivable in in classical logic if and only if ∼∼ φ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras.
title Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
title_short Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
title_full Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
title_fullStr Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
title_full_unstemmed Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic
title_sort hájek basic fuzzy logic and łukasiewicz infinite-valued logic
publishDate 2003
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09335846_v42_n4_p361_Cignoli
http://hdl.handle.net/20.500.12110/paper_09335846_v42_n4_p361_Cignoli
_version_ 1768546308335861760

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