 Supervaluational semantics has been largely applied to explain phenomena of vagueness (Lewis 1970; Fine 1975b; Kamp 1975; see also the entry on  vagueness). A predicate \(P\) is vague if it exhibits borderline cases for which it is not clear whether \(P\) truly applies or not. Prominent examples of vague predicates are relative adjectives like tall or clever, while adjectives like prime as in 3 is a prime number are normally taken not to be vague. The basic insight of any supervaluational account of vagueness is that a vague language admits of several precisifications (formalized as a set \(V\) of classical valuations) and the semantic value of a statement is fixed only insofar as all those precisifications agree: a statement \(\phi\) is true if it is supertrue (\(s_V(\phi)=T\)), that is, true on every admissible precisification (\(v(\phi)=T\), for all \(v\in V\)), and it is false if it is superfalse (\(s_V(\phi)=F\)), that is, false on every admissible precisification (\(v(\phi)=F\), for all \(v\in V\)); otherwise it has no semantic value. A common argument in favor of supervaluationism relies on the observation that while a predicate like tall may be vague, the compound predicates tall or not tall and tall and not tall are not vague, the former being true and the latter being false of all entities in the predicate application domain. Thus, while \(a\) is tall may fail to be true or false (and the same goes for its negation), \(a\) is either tall or not tall and \(a\) is tall and not tall are always true and false respectively, as predicted by a supervaluational semantics. More recent experimental work, however, has cast some doubts on these observations (e.g., Ripley 2011). Supervaluationism has further been criticized for giving rise to counterexamples to familiar inference-patterns (Williamson 1994: 151–152 and 162–163), (see also Keefe 2000; Williams 2008 for further discussion). For example, it is easy to see that in this framework, one loses the disjunction elimination rule, at least in languages with a determinately/supertrue operator \(D\). Given a class \(V\) of classical valuations, we can introduce the operator \(D\) with the stipulation that \(v(D \phi)=1\) iff \(v_1(\phi)=1 \), for all \(v_1\) in \(V\). Defining supervalidity, \(\models_{sv}\), in terms of preservation of supertruth, we have that \(p \models_{sv} D p\), \(\neg p \models_{sv} D \neg p\) and \(\models_{sv} p \vee \neg p\), but \(\not\models_{sv} Dp \vee D \neg p\). So the disjunction elimination rule does not preserve supervalidity in this language (see also Humberstone, chapter 6, pages 830–833). Whether supervalidity is the right notion of validity from a supervaluational perspective is however controversial (e.g., Varzi 2007).