Positive, Negative, and Reliable Information in a First-Order Logic of Evidence and Truth

TL;DR

This paper introduces QLETF+, a first-order logic that handles evidence and truth, with properties like normal forms and soundness in a six-valued semantics.

Contribution

It presents the first first-order logic of evidence and truth with properties like replacement and normal forms, extending previous propositional work.

Findings

QLETF+ satisfies the replacement property.

It admits conjunctive, disjunctive, and prenex normal forms.

The deductive system is sound and complete with respect to six-valued semantics.

Abstract

In this paper we present the first-order logic QLETF+, a quantified version of the logic LETF+, introduced in Coniglio and Rodrigues (Studia Logica 112:561-606, 2024). QLETF+ exhibits several properties that are not always enjoyed by logics equipped with classicality operators. We show that it satisfies the replacement property and admits conjunctive, disjunctive, and prenex normal forms. Alongside extensions and anti-extensions, as in the previously studied first-order semantics for LETs, we make use here of what we call o-extensions: given an n-ary predicate symbol P, the o-extension of P is the set of n-tuples of individuals that satisfy the predicate oP. We prove the soundness and completeness of the deductive system of QLETF+ with respect to the six-valued first-order semantics.