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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