Recently four different papers have suggested that the supervaluational solution to the Problem of the Many is flawed. Stephen Schiffer (1998, 2000a, 2000b) has argued that the theory cannot account for reports of speech involving vague singular terms. Vann McGee and Brian McLaughlin (2000) say that theory cannot, yet, account for vague singular beliefs. Neil McKinnon (2002) has argued that we cannot provide a plausible
theory of acceptability of precisifications, which the supervaluational theory needs. And Roy Sorensen (2000) argues that supervaluationism is inconsistent with a directly referential theory of names. While McGee and McLaughlin see this problem, correctly, as a cause for further research rather than for abandoning the theory, the other authors all take the problems they raise to be sufficient reason to jettison supervaluationism. I will argue that none of these problems provide such a reason, though that is not to say the arguments are not valuable critiques. In many cases, we must make some adjustments to the supervaluational theory to meet the challenges these theorists pose. The first goal of this paper is to make those adjustments, and show how (almost) all the challenges can be met. Hopefully the adjustments are of some intrinsic interest, beyond their relevance to debates about the Many. The second goal is to show that, even once we suitably adjust
supervaluationism, it is incompatible with directly referential accounts of ordinary names. But this is no problem with supervaluationism. Rather, the directly referential theory of names is incompatible with any plausible theory of vagueness, and so should be abandoned.