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Substructural fuzzy logics

Published online by Cambridge University Press:  12 March 2014

George Metcalfe  and
Franco Montagna
George Metcalfe
Affiliation:
Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240, USA. E-mail: metcalfe@logic.at
Franco Montagna
Affiliation:
Department of Mathematics, University of Siena, Via Del Capitano 15, 53100 Siena, Italy. E-mail: montagna@unisi.it
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Abstract

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) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX 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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 834 - 864
Copyright
Copyright © Association for Symbolic Logic 2007

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