The method of supervaluations offers an elegant procedure by which semantic theory can come to terms with sentences that, for one reason or another, lack truth-value. I argue, however, that this method rests on a fundamental mistake, and so is unsuitable for semantics. The method of supervaluations, I argue, assigns semantic values to sentences based not on the semantic values of their components, but on the values of other, perhaps homophonic, but nevertheless distinct, expressions. That is because supervaluations are generated (...) from classical valuations which necessarily require reinterpreting the component expressions, but the reinterpretation of an expression is tantamount to the introduction of a new expression, or alternatively, to a shift to an entirely new language. To confuse the expression of the language for which a semantic theory is developed with its reinterpreted counterpart, is to commit a fallacy of equivocation. That is the flaw within the method of supervaluations. We see it manifest in a number of examples. (shrink)