Describes a semantic approach to problems of vagueness and truth value gaps due to Bas van Fraassen (1941– ) in which sentences are evaluated by sets $S$ of classical valuations (called ‘precisifications’). One can say that a formula $\phi$ is supertrue with respect to a set $S$ of classical valuations if every $v\in S$ evaluates $\phi$ as true (and superfalse if every $v\in S$ evaluates it as false). Then borderline or vague cases are those in which two valuations $v\in S$ and $v^{\prime }\in S$ disagree, i.e., those formulae that are not supertrue. Likewise, validity on the supervaluationist account can be defined as preservation of supertruth from premisses to conclusions, that is:

• • $\Delta$ is a consequence of $\text{Γ}$ if for every set $S$ of classical valuations in which each $v\in S$ evaluates all formulae in $\text{Γ}$ as true, every valuation in $S$ evaluates all formulae in $\Delta$ as true as well.

The supervaluationist account of validity is in a weak sense paracomplete insofar as $\left\{\phi ,\neg \phi \right\}$ is not a consequence of all sets $\text{Γ}$, although the set $\left\{\phi \vee \neg \phi \right\}$ is in fact a consequence of all sets $\text{Γ}$.