## On global requirements for implication operators in fuzzy modus ponens.(English)Zbl 0931.68117

Summary: When specifying the properties required of an operation used to manipulate fuzzy sets consideration must be given to the meaning or semantics of the whole granule resulting from these operations. Is the fuzzy granule resulting from the operation as a whole meaningful? This requires us to go beyond consideration of only pointwise performance properties of fuzzy operations and to include consideration of properties based upon the global or granular results of these operations. In this spirit we look at the process of fuzzy modus ponens and consider global requirements on the definition of the implication operator used in this process. One global requirement investigated here is implicit in the information boundedness principle, this principle requires that the information contained in a fuzzy granule resulting from an inference must be no greater than the information contained in the consequent of the if then proposition and that the information contained in inferences under two different datum should be ordered by the degree f matching of the datum and the antecedent of the if-then proposition. Using thiis principle we show that any t-conorm $$S$$ used in defining implication operators must satisfy the condition of moderate growth, for $$a> b$$, $$S(a,v)- S(b,v)$$ must be an nonincreasing function of $$v$$.

### MSC:

 68T27 Logic in artificial intelligence 03B52 Fuzzy logic; logic of vagueness

### Keywords:

fuzzy modus ponens
Full Text:

### References:

 [1] Dubois, D.; Prade, H., A class of fuzzy measures based on triangular norms, Int. J. general systems, 8, 43-61, (1982) · Zbl 0473.94023 [2] Dubois, D.; Prade, H., Fuzzy sets in approximate reasoning part I: inference with possibility distributions, Fuzzy sets and systems, 40, 143-202, (1991) · Zbl 0722.03017 [3] Hohle, U., Probabilistic uniformization of fuzzy topologies, Fuzzy sets and systems, 1, 1978, (1978) [4] Kimberling, C., On a class of associative functions, Publ. math. debrecen., 20, 21-39, (1973) · Zbl 0276.26011 [5] Klir, G.J.; Bo, Y., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall Englewood Cliffs, NJ · Zbl 0915.03001 [6] Sugeno, M., Fuzzy measures and fuzzy integrals: a survey, (), 89-102 [7] Yager, R.R., Default knowledge and measures of specificity, Inform. sci., 61, 1-44, (1992) · Zbl 0738.68075 [8] Yager, R.R., On the specificity of a possibility distribution, Fuzzy sets and systems, 50, 279-292, (1992) · Zbl 0783.94035 [9] R.R. Yager, On measures of specificity, in: O. Kaynak, L.A. Zadeh, B. Turksen, I.J. Rudas (Eds.), Computational Intelligence: Soft Computing and Fuzzy-Neuro Integration with Applications, Springer, Berlin, to appear · Zbl 0931.94060 [10] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002 [11] Zadeh, L.A., A theory of approximate reasoning, (), 149-194 [12] Zadeh, L.A., Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy sets and systems, 90, 111-127, (1997) · Zbl 0988.03040
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.