Substructural Fuzzy Logics

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 Multi plicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom. Axiomatic extensions of UL include known fuzzy logics such as Monoidal i-norm logic MTL and Goedel 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].