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...

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...

This paper is a contribution to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and ∆-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate ...

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...

Among various approaches to fuzzy logics, we have chosen two of them, which are built up in a similar way. Although starting from diierent basic logical connectives, they both use interpretations based on Frank t-norms. Diierent interpretations of the implication lead to diierent axiomatizations, but most logics studied here are complete. We compare the properties, advantages and disadvantages ...

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...

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...

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...