Abstract
As one of the important indicators to describe the degree of similarity of two fuzzy sets, the similarity measure is the basic theoretical foundation in pattern recognition of fuzzy information. Based on the concept principle, each concept should be compatible, independent and complete. Following the concept principle and meanwhile using counterexamples in Cantor sets with finite elements, the similarity measures in fuzzy mathematics are questioned. Firstly, the definition of the similarity measure between Cantor sets and clear sets both with finite elements are given based on Concept Principle. Then, the incompleteness of existing similarity measures between two fuzzy sets is proved by some counterexamples in Cantor sets with finite elements. Two useful conclusions are presented as follows: Lattice-similarity and Hamming-similarity are questionable. The properties in the axiom definition of similarity measures are necessary but not sufficient conditions in fuzzy mathematics. Finally, we propose to use a new mathematic tool, the similarity measures of clear sets to solve the pattern recognition of fuzzy information.
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