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

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)


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


Zbl 0666.03022
Full Text: DOI


[1] Biacino L., J. Math. Anal. Appl 172 pp 179– (1991) · Zbl 0777.60004
[2] Biacino L., Inform. Sci. 32 pp 181– (1984) · Zbl 0562.06004
[3] Biacino L., International Journal of Intelligent Systems 7 pp 445– (1992) · Zbl 0761.68090
[4] Biacino, L. and Gerla, G. 1993.Closure operators for fuzzy subsets, EUFTT 93, First European Congress on Fuzzy and Intelligent Technologies, Aachen1441–1447.
[5] Castro J. L., Fuzzy Sets and Systems.
[6] Castro J. L., Journal of Applied non-classical Logic.
[7] Chakraborty M. K., Fuzzy Logic in Knowledge-Based Systems, Decision and Control pp 247– (1988)
[8] Chakraborty M. K., Journal of Applied non-classical Logic.
[9] Gerla G., Artificial Intelligence 70 pp 33– (1994) · Zbl 0821.03014
[10] Gerla G., Mathematical Logic Quarterly. 40 pp 357– (1994) · Zbl 0811.03019
[11] Negoita C. V., Kybernetes 4 pp 169– (1975) · Zbl 0352.02044
[12] Pavelka J., Math. Logik Grundlag. Math. 25 pp 45– (1979) · Zbl 0435.03020
[13] Shoesmith D. J., Multiple conclusion logic (1978) · Zbl 0381.03001
[14] Zadeh L. A., Synthese 30 pp 407– (1975) · Zbl 0319.02016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.