ΠMTL is a schematic extension of the monoidal t-norm based logic (MTL) by the characteristic axioms of product logic. In this paper we prove that ΠMTL satisfies the standard completeness theorem. From the algebraic point of view, we show that the class of ΠMTL-algebras (bounded commutative cancellative residuated l-monoids) in the real unit interval [0, 1] generates the variety of all ΠMTL-alge...

T-norm based fuzzy logics are usually considered as truth preserving, that is, taking 1 as the only truth value to be preserved in inferences. In this paper we study t-norm based fuzzy logics preserving degrees of truth, that is, preserving the lower bounds of the truth degrees of the premises. These logics are axiomatizable by using a restricted form of Modus Ponens together with the rule of A...

We provide general conditions on hypersequent calculi that guarantee standard completeness for the formalized logics. These conditions are implemented in the PROLOG system AxiomCalc that takes as input any suitable axiomatic extension of Monoidal T-norm Logic MTL and outputs a hypersequent calculus for the logic and the result of the check. Our approach subsumes many existing results and allows...

This paper is a humble homage to Enric Trillas. Following his foundational contributions on models of ordinary reasoning in an algebraic setting, we study here elements of these models, like conjectures and hypothesis, in the logical framework of continuous t-norm based fuzzy logics. We consider notions of consistency, conjecture and hypothesis arising from two natural families of consequence o...

For theoretical fuzzy control it is a well known strategy to transform a system of control rules into a system of relation equations. Because these systems of relation equations are not always solvable, solvability criteria and approximate solutions have been discussed. We reconsider some of the results in this field and extend them using more recent results on t-norm based fuzzy logics and on ...

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finit...

We study the computational complexity of some axiomatic extensions of the monoidal t-Norm based logic (MTL), namely NM corresponding to the logic of the so-called nilpotent minimum t-norm (due to Fodor [8]); and SMTL corresponding to left-continuous strict t-norms, introduced by Esteva (and others) in [4] and [5]. In particular, we show that the sets of 1-satisfiable and positively satisfiable ...

Fuzzy description logics (DLs) are a family of logics which allow the representation of (and the reasoning within) structured knowledge affected by vagueness. Although a relatively important amount of work has been carried out in the last years, current fuzzy DLs still present several limitations. In this work we face two problems: the common restriction to Zadeh and Łukasiewicz fuzzy logics an...

We provide general –and automatedly verifiable– sufficient conditions that ensure standard completeness for logics formalized Hilbert-style. Our approach subsumes many existing results and allows for the discovery of new fuzzy logics which extend first-order Monoidal T-norm Logic with propositional axioms.