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An Extension Principle for Fuzzy Logics
Author(s) -
Gerla Giangiacomo
Publication year - 1994
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19940400306
Subject(s) - mathematics , extension (predicate logic) , closure (psychology) , t norm fuzzy logics , fuzzy logic , algebra over a field , type 2 fuzzy sets and systems , operator (biology) , conservative extension , fuzzy number , class (philosophy) , discrete mathematics , fuzzy set , calculus (dental) , pure mathematics , computer science , artificial intelligence , medicine , biochemistry , chemistry , dentistry , repressor , economics , transcription factor , market economy , gene , programming language
Let S be a set, P ( S ) the class of all subsets of S and F ( S ) the class of all fuzzy subsets of S. In this paper an “extension principle” for closure operators and, in particular, for deduction systems is proposed and examined. Namely we propose a way to extend any closure operator J defined in P ( S ) into a fuzzy closure operator J* defined in F ( S ). This enables us to give the notion of canonical extension of a deduction system and to give interesting examples of fuzzy logics. In particular, the canonical extension of the classical propositional calculus is defined and it is showed its connection with possibility and necessity measures. Also, the canonical extension of first order logic enables us to extend some basic notions of programming logic, namely to define the fuzzy Herbrand models of a fuzzy program. Finally, we show that the extension principle enables us to obtain fuzzy logics related to fuzzy subalgebra theory and graded consequence relation theory. Mathematics Subject Classification : 03B52.