  This paper introduces the logic $QLET_{F}$, a quantified extension of the
logic of evidence and truth $LET_{F}$, together with a corresponding sound and
complete first-order non-deterministic valuation semantics. $LET_{F}$ is a
paraconsistent and paracomplete sentential logic that extends the logic of
first-degree entailment ($FDE$) with a classicality operator ${\circ}$ and a
non-classicality operator $\bullet$, dual to each other: while ${\circ} A$
entails that $A$ behaves classically, ${\bullet} A$ follows from $A$'s
violating some classically valid inferences. The semantics of $QLET_{F}$
combines structures that interpret negated predicates in terms of
anti-extensions with first-order non-deterministic valuations, and completeness
is obtained through a generalization of Henkin's method. By providing sound and
complete semantics for first-order extensions of $FDE$, $K3$, and $LP$, we show
how these tools, which we call here the method of ``anti-extensions +
valuations'', can be naturally applied to a number of non-classical logics.
