The study of the Gentzen system Gew determined by the well known sequent calculus FLew [Ono98, Ono03c] is interesting for the study of the substructural aspects of t-norm based logics [Háj98, EG01]. In [BGV05] we studied the 〈∨, ∗,¬, 0, 1〉 and the 〈∨,∧, ∗,¬, 0, 1〉-fragments of this Gentzen system and the same fragments of the logic of residuated lattices IPC∗\c. In this paper we continue the st...

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...

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.

In this paper, we introduce triangle algebras: a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1. We show that these algebras serve as an equational representation of intervalvalued residuated lattices (IVRLs). Furthermore, we present Triangle Logic (TL), a system of many-valued logic capturing the tautologies of IVR...

Fuzzy formal logics were introduced in order to handle graded truth values instead of only ‘true’ and ‘false’. A wide range of such logics were introduced successfully, like Monoidal T-norm based Logic, Basic Logic, Gödel Logic, Lukasiewicz Logic etc. However, in general, fuzzy set theory is not only concerned with vagueness, but also with uncertainty. A possible solution is to use intervals in...

Keywords: mathematical fuzzy logic, basic fuzzy logic, monoidal t-norm logic, hoop logic, non-commutative fuzzy logic. This paper is a companion to my [12]. The latter is a rather technical paper developing a generalized (mathematical) fuzzy logic. Main definitions and results of [12] are reproduced here without any proofs; but this is preceded by a survey on the basic proposi-tional fuzzy logi...