How good are fuzzy if-then classifiers?

This paper gives some known theoretical results about fuzzy rule-based classifiers and offers a few new ones. The ability of Takagi-Sugeno-Kang (TSK) fuzzy classifiers to match exactly and to approximate classification boundaries is discussed. The lemma by Klawonn and Element about the exact match of a classification boundary in R-2 is extended from monotonous to arbitrary functions. Equivalence between fuzzy rule-based and nonfuzzy classifiers (1-nn and Parzen) is outlined. We specify the conditions under which a class of fuzzy TSK classifiers turn into lookup tables. It is shown that if the rule base consists of all possible rules (all combinations of linguistic labels on the input features), the fuzzy TSK model is a lookup classifier with hyperbox cells, regardless of the type (shape) of the membership functions used. The question why fuzzy? is addressed in the light of these results.