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

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.