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Glivenko Theorems for Substructural Logics over FL

Nikolaos Galatos & Hiroakira Ono

Journal of Symbolic Logic 71 (4):1353 - 1384 (2006)

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Glivenko like theorems in natural expansions of BCK‐logic.Roberto Cignoli & Antoni Torrens Torrell - 2004 - Mathematical Logic Quarterly 50 (2):111-125.details The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK‐logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK‐logic with negation by a family of connectives implicitly defined by equations and compatible with BCK‐congruences. Many of the logics in the current literature are natural expansions of BCK‐logic with negation. The validity of (...) Download Export citation Bookmark 14 citations
Algebraization, Parametrized Local Deduction Theorem and Interpolation for Substructural Logics over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - Studia Logica 83 (1-3):279-308.details Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices. Download Export citation Bookmark 32 citations
Foreword. [REVIEW]J. Font, R. Jansana & D. Pigozzi - 2003 - Studia Logica 74 (1-2):3-12.details Download Export citation Bookmark 100 citations
Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic.Roberto Cignoli & Antoni Torrens - 2003 - Archive for Mathematical Logic 42 (4):361-370.details Using the theory of BL-algebras, it is shown that a propositional 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 (...) Download Export citation Bookmark 11 citations
Negative Equivalence of Extensions of Minimal Logic.Sergei P. Odintsov - 2004 - Studia Logica 78 (3):417-442.details Two logics L1 and L2 are negatively equivalent if for any set of formulas X and any negated formula ¬, ¬ can be deduced from the set of hypotheses X in L1 if and only if it can be done in L2. This article is devoted to the investigation of negative equivalence relation in the class of extensions of minimal logic. Download Export citation Bookmark 2 citations
Glivenko like theorems in natural expansions of BCK‐logic.Roberto Cignoli & Antoni Torrens Torrell - 2004 - Mathematical Logic Quarterly 50 (2):111-125.details The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of (...) Download Export citation Bookmark 13 citations

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