The paper presents a form of rendering classical mathematical notions by formal theories over suitable t-norm fuzzy logics in such a way that references to real numbers are eliminated from definitions and theorems, being removed to the standard semantics of fuzzy logic. Several examples demonstrate how this move conceptually simplifies the theory in exchange for non-classical reasoning, facilit...

A t-norm is a binary operation on [0, 1] that is associative, commutative, with identity 1 and non-decreasing in both argument. The notion of t-norm is very important in the context of fuzzy logic, for it is used in modeling a general form of the propositional conjunction. The basic fuzzy logic (BL logic) was introduced by Hájek in [9,10] and it was inspired by a continuous t-norm and its resid...

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant 0, namely ¬φ isφ → 0. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which i...

Fuzzy Description Logics (DLs) with t-norm semantics have been studied as a means for representing and reasoning with vague knowledge. Recent work has shown that even fairly inexpressive fuzzy DLs become undecidable for a wide variety of t-norms. We complement those results by providing a class of t-norms and an expressive fuzzy DL for which ontology consistency is linearly reducible to crisp r...

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...

A discrete t-norm is a binary operation on a finite subset of the real unit interval fulfilling the same algebraic conditions as t-norms. We show that any left-continuous t-norm can, in a natural sense, be approximated by a discrete t-norm with an arbitrary precision.

Monoidal t-norm based logic MTL is the logic of left continuous t-norms. We introduce two analytic calculi for first-order MTL. These are obtained by lifting two sequent calculi for different fragments of this logic to the hypersequent level with subsequent addition of Avron’s communication rule. Our calculi enable to prove the mid(hyper)sequent theorem. As corollaries follow Herbrand’s theorem...

CADIAG-2 is a successful expert system assisting in the differential diagnosis in internal medicine. With its aid, conjectures about present diseases are derived from possibly vague information about a patient’s symptoms. In this paper we provide a mathematical formalisation of the inferential mechanism of CADIAG-2. A Gentzen-style calculus for the resulting logic is introduced and used to comp...