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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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_09335846_v42_n4_p361_Cignoli |
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todo:paper_09335846_v42_n4_p361_Cignoli2023-10-03T15:48:33Z Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic Cignoli, R. Torrens, A. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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 Cignoli, R. Torrens, A. 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. |
| format |
JOUR |
| author |
Cignoli, R. Torrens, A. |
| author_facet |
Cignoli, R. Torrens, A. |
| author_sort |
Cignoli, R. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_09335846_v42_n4_p361_Cignoli |
| work_keys_str_mv |
AT cignolir hajekbasicfuzzylogicandłukasiewiczinfinitevaluedlogic AT torrensa hajekbasicfuzzylogicandłukasiewiczinfinitevaluedlogic |
| _version_ |
1782030479614017536 |