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

TL;DR

This paper develops a first-order non-deterministic valuation semantics for the logic $QLET_{F}$, extending evidence and truth logic with classicality operators, and demonstrates its applicability to various non-classical logics.

Contribution

It introduces a novel semantics combining anti-extensions and valuations for first-order evidence and truth logics, with a completeness proof via a generalized Henkin's method.

Findings

Provides sound and complete semantics for $QLET_{F}$ and related logics

Generalizes Henkin's method for non-deterministic valuations

Shows applicability to $FDE$, $K3$, and $LP$ logics

Abstract

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.