Abstract Osherson and Smith (1981, Cognition, 11, 237–262) discuss a number of problems which arise for a prototype-based account of the meanings of simple and complex concepts. Assuming that concept combination in such a theory is to be analyzed in terms of fuzzy logic, they show that some complex concepts inevitably get assigned the wrong meanings. In the present paper we argue that many of the problems O&S discovered are due to difficulties that are intrinsic to fuzzy set theory, and that most of them disappear when fuzzy logic is replaced by supervaluation theory. However, even after this replacement one of O&S's central problems remains: the theory still predicts that the degree to which an object is an instance of, say, “striped apple” must be less than or equal to both the degree to which it is an instance of “striped” and the degree to which it is an instance of “apple”, but this constraint conflicts with O&S's experimental results. The second part of the paper explores ways of solving this and related problems. This leads us to suggest a number of distinctions and principles concerning how prototypicality and other mechanisms interact and which seem important for semantics generally. Prominent among these are (i) the distinction between on the one hand the logical and semantic properties of concepts and on the other the linguistic that between concepts for which the extension is determined by their prototype and concepts for which extension and prototypicality are independent.