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Evento: 'Palestra: A Characterization of Classic-Like Fuzzy Semantics'

Eventos PESC (Palestras, Seminários, etc.)
Palestras, Seminários, etc. do PESC/COPPE/UFRJ.
Data: Wednesday, May 27, 2009 At 14:00
Duração: 2 Horas

Palestra do Prof. Benjamin Bedregal, da UFRN.
Dia 27/05/2009, às 14 horas
Local: PESC

Titulo: A Characterization of Classic-Like Fuzzy Semantics

Resumo:
The fuzzy set theory introduced by Lofti Zadeh in [40] has as main feature the consideration of a truth degree, i.e. a real value in [0,1], to indicate how much an element belongs, or not, to a given set. This theory is appropriate to deal with concepts (and therefore with sets) not very precise such as the speed, intelligence, high temperatures, etc. In this way, fuzzy logic(the subjacent logic of fuzzy set theory) becomes an important tool to deal with uncertainty of knowledge and to represent the uncertainty of human reasoning. Two main directions can be distinguished in fuzzy logic [41]: 1) Fuzzy logic in the broad sense where the main goal is the development of computational systems based on fuzzy reasoning, such as fuzzy control systems and 2) Fuzzy logic in the narrow sense where fuzzy logic is seen as a symbolic logic and therefore questions like formal theories are studied. Lately, considerable progress has been made in strictly mathematical (formal and symbolic) aspects of fuzzy logic as logic with a comparative notion of truth [23]. Triangular norms were originally introduced by Menger in [30] to model the distance in probabilistic metric spaces. But the axiomatic definition of t-norm used today was given by Schweizer and Sklar in [35]. Nevertheless, Alsina, Trillas and Valverde in [1] showed that t-norms and their dual notion (t-conorm) can be used to model conjunction and disjunction in fuzzy logics generalizing several definitions for those connectives provided by Lotfi Zadeh in [40], Bellman and Zadeh in [7, 8] and Yager in [37] (which define a general class of interpretations), etc. The other usual propositional connectives can also be fuzzy extended from a t-norm or defined given an independent axiomatization [3, 9, 16, 25, 31]. A connective for which there is not a fuzzy extension with a reasonable axiomatization is the bi-implication, but in this paper we provide one. The fuzzy extension of propositional connective will be called propositional fuzzy semantics or simply fuzzy semantic. Thus, each fuzzy semantics determines a different set of true formulas (1-tautologies) and false formulas (0-contradictions) and therefore different (fuzzy) logics.

Clearly each 1-tautology for any fuzzy semantics is a classical tautology. Then a natural question is the following: Is there a fuzzy logic with a greatest quantity of tautologies? In a positive case, how near is this set of the classical tautologies set? In this paper we prove that there are uncountable classes of fuzzy logics with their set of 1-tautologies being the classical one. Moreover we provide sufficient and necessary condition for a fuzzy semantics to have as tautologies the classical one. This kind of fuzzy semantics we call “classic-like fuzzy semantics”. We also introduce a crisp notion of the logical consequence relation for fuzzy semantics in a similar way to the uses in [10] for many values logics, and prove that this notion coincides with the classic logical consequence relation for the classic-like fuzzy semantics.



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