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

Valuation semantics for first-order logics of evidence and truth (and some related logics)