Revisiting the Dunn-Belnap logic

TL;DR

This paper introduces a new semantics for Dunn-Belnap logic based on societal acceptance and rejection attitudes, leading to a novel logic $ extbf{E}^B$ with a sound and complete calculus.

Contribution

It proposes a semantics rooted in cognitive attitudes rather than truth-values, and develops the $ extbf{E}^B$ logic with associated entailment and proof system.

Findings

Defined a semantics based on acceptance and rejection attitudes.

Introduced the $ extbf{E}^B$ logic with $B$-entailment.

Provided a sound and complete sequent calculus for $ extbf{E}^B$.

Abstract

In the present work I introduce a semantics based on the cognitive attitudes of acception and rejection entertained by a given society of agents for logics inspired on Dunn and Belnap's First Degree Entailment ($\mathbf{E}$). In contrast to the epistemic situations originally employed by $\mathbf{E}$, the cognitive attitudes do not coincide with truth-values and they seem more suitable to logics that intend to consider the informational content of propositions "said to be true" as well as of propositions "said to be false" as determinant of the notion of logical validity. After analyzing some logics associated to the proposed semantics, we introduce the logic $\mathbf{E}^B$, whose underlying entailment relation -- the $B$-entailment -- is able to express several kinds of reasoning involving the cognitive attitudes of acceptance and rejection. A sound and complete sequent calculus for $\mathbf{E}^B$ is also presented.