Graded consequence relations and fuzzy closure operator. (English) Zbl 0872.03012

Gerla, Giangiacomo

J. Appl. Non-Class. Log. 6, No. 4, 369-379 (1996).

The role of a closure operator in the framework of formal (binary) logics has been recognized very early by Tarski: similarly, many papers have proved the usefulness of a fuzzy closure operator in Zadeh’s logic of approximate reasoning. It is well known that there exists a bijection between the class of consequence relations and the class of closure operators on a set of formulas. In 1988, M. K. Chakraborty [Fuzzy logic in knowledge-based systems, decision and control, 247-257 (1988; Zbl 0666.03022)] extended the concept of a consequence relation to the theory of fuzzy sets by introducing a graded consequence relation.
This paper concentrates on the connections between fuzzy closure operators and graded consequence relations. A slightly modified version of the above-mentioned bijection could be established: there exists a bijection between the class of graded consequence relations and a subclass of the class of fuzzy closure operators namely those that can be represented as a chain of (crisp) closure operators.

Reviewer: E.Kerre (Gent)

MSC:

03B52	Fuzzy logic; logic of vagueness
68T27	Logic in artificial intelligence
03E72	Theory of fuzzy sets, etc.
03B50	Many-valued logic

Keywords:

fuzzy closure operator; graded consequence relation

Citations:

Zbl 0666.03022

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