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) 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].

From Axioms to Analytic Rules in Nonclassical Logics

“…Note that our interpretation of "|" differs from that of [11] due to lack of weakening and linearity.…”

“…Since then hypersequent calculi have been discovered for a wide range of nonclassical logics, e.g. [2,3,4,5,11]. This is traditionally done by (i) looking for the "right" inference rule(s) formalizing the particular properties of the logic under consideration (e.g., Avron introduced the rule (com), corresponding to the prelinearity axiom) and (ii) proving cut-elimination (or cut-admissibility) to show that the resulting calculus is analytic.…”

“…This holds even when adding the same rules to different base calculi (e.g. IL + (com) dealt with in [2], IL + (com) -contraction in [4], and IL + (com) -weakening -contraction in [11]), which might cause a combinatorial explosion on the number of the papers to be produced.…”

A Logical Study of Local and Global Graded Similarities

“…This characteristic implies that these kinds of operations admit in general a good compensating behaviour. As future work we plan to study the FCT based in UL , the logic of uninorms [Metcalfe and Montagna, 2007], in order to deal with similarities in this context. We want to explore the plausibility of the results obtained in the present paper in the more general context of uninorms.…”

Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions

“…Our proofs follow the strategy of [14,8]; namely, we construct embeddings of L-chains into dense L-chains and then into standard L-algebras. However, we remark that there exists also an alternative "proof-theoretic" strategy, used to prove standard completeness for non-modal fuzzy logics in [16] (see also [6,18]). In this approach, a density rule is first added to the logic which guarantees completeness with respect to dense L-chains, and then eliminated in similar fashion to cut elimination from proofs in hypersequent calculi.…”