A Deterministic Weakening of Belnap–Dunn Logic

Abstract

A deterministic weakening $\mathsf {DW}$ of the Belnap–Dunn four-valued logic $\mathsf {BD}$ is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic $\mathsf {DW}$ is formalized as a sequent calculus. The completeness and decidability of $\mathsf {DW}$ with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for $\mathsf {DW}$ is described, and found to be a subvariety of Berman’s variety $\mathcal {K}_{1,2}$. Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that $\mathsf {DW}$ is the logic of a nine-element distributive lattice with a negation. Moreover, $\mathsf {BD}$ is embedded into $\mathsf {DW}$ by Glivenko’s double-negation translation.