 Supervaluations employ what Bencivenga (1986) calls a “counterfactual theory” of truth: an empty-termed statement is true if it would be true on any assignment of referents to its empty terms. This has struck many critics as simply false. Moreover, the logic itself leaves much to be desired. For one thing, supervaluational consequence is too strong. Thus, for example, although the formula \(Pt \rightarrow \rE!t\) is (quite properly) not valid on a supervaluational semantics, nevertheless since \(\rE!t\) is true on every supervaluation on which \(Pt\) is true, the sequent (derivability statement) \(Pt \vdash \rE!t\) is improperly semantically valid. Therefore, although PFL is sound on supervaluational semantics and every semantically valid formula is a theorem of PFL, not all semantically valid sequents are provable in PFL. In fact, supervaluational consequence is not axiomatizable by any extension of free logic. This follows from a result of Woodruff (1984), who has shown that supervaluational semantics has many of the undesirable properties of second-order semantics. Jerry A. Fodor and Ernest Lapore (1996) argue, furthermore, that the completions needed to construct supervaluations are not meaning-preserving. Hence, they conclude, two alleged advantages of supervaluations—that they explain the meaningfulness of sentences with truth value gaps and that they allow us to preserve classical logic—are illusory. Finally, since supervaluations are built from completions that are in effect positive dual-domain models, we may wonder whether the detour through supervaluations is worth the trouble, since positive dual-domain models alone are simpler and more adequate to PFL.