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September 2007 Substructural fuzzy logics
George Metcalfe, Franco Montagna
J. Symbolic Logic 72(3): 834-864 (September 2007). DOI: 10.2178/jsl/1191333844


Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0,1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0,1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani’s density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0,1].


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George Metcalfe. Franco Montagna. "Substructural fuzzy logics." J. Symbolic Logic 72 (3) 834 - 864, September 2007.


Published: September 2007
First available in Project Euclid: 2 October 2007

zbMATH: 1139.03017
MathSciNet: MR2354903
Digital Object Identifier: 10.2178/jsl/1191333844

Rights: Copyright © 2007 Association for Symbolic Logic


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Vol.72 • No. 3 • September 2007
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