This dissertation concerns a special problem posed by the vagueness of ordinary language for the enterprise of giving a coherent semantic theory of natural language. The problem engendered by vagueness is the lack of sharp boundaries between semantic categories. The first three chapters of this dissertation show how three traditional approaches to this problem fail for similar reasons. The three approaches considered are degree theories, supervaluations, and context dependence theories. I then consider attempts to handle this problem by employing a form of semantic anti-realism called "the modeling approach." Finally, I endorse a judgment-dependent approach to the problem and consider its historical antecedents and one contemporary version of the theory. Only by allowing judgment to play a role in determining truth conditions in context, it is argued, can we hope to provide a plausible solution to the problem posed by sharp boundaries.