There is a Standard Objection to the idea that concepts might be prototypes (or exemplars, or stereotypes): Because they are productive, concepts must be compositional. Prototypes aren't compositional, so concepts can't be prototypes. However, two recent papers (Osherson and Smith, 1988; Kamp and Partee, 1995) reconsider this consensus. They suggest that, although the Standard Objection is probably right in the long run, the cases where prototypes fail to exhibit compositionality are relatively exotic and involve phenomena which any account of compositionality is likely to find hard to deal with; for example, the effects of quantifiers, indexicals, contextual constraints, etc. In this paper, we argue that the Standard Objection to prototype theory was right after all: The problems about compositionality are insuperable in even the most trivial sorts of examples; it is therefore as near to certain as anything in cognitive science ever gets that the structure of concepts is not statistical. Theories of categorization, concept acquisition, lexical meaning and the like, which assume the contrary simply don't work.