Valuation Semantics for First-Order Logics of Evidence and Truth

This paper introduces the logic Q L E T F , a quantified extension of the logic of evidence and truth L E T F , together with a corresponding sound and complete first-order non-deterministic valuation semantics. L E T F is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment ( FDE ) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘ A entails that A behaves classically, ∙ A follows from A ’s violating some classically valid inferences. The semantics of Q L E T 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.