Journal of Symbolic Logic

Substructural fuzzy logics

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) ∨ ((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].

Article information

Source
J. Symbolic Logic Volume 72, Issue 3 (2007), 834-864.

Dates
First available: 2 October 2007

http://projecteuclid.org/euclid.jsl/1191333844

Digital Object Identifier
doi:10.2178/jsl/1191333844

Mathematical Reviews number (MathSciNet)
MR2354903

Zentralblatt MATH identifier
1139.03017

Citation

Metcalfe, George; Montagna, Franco. Substructural fuzzy logics. Journal of Symbolic Logic 72 (2007), no. 3, 834--864. doi:10.2178/jsl/1191333844. http://projecteuclid.org/euclid.jsl/1191333844.