Rivals to Belnap–Dunn Logic on Interlaced Trilattices

Abstract

The work of Arnon Avron and Ofer Arieli has shown a deep relationship between the theory of bilattices and the Belnap-Dunn logic $\mathsf {E}_{\mathtt {fde}}$. This correspondence has been interpreted as evidence that $\mathsf {E}_{\mathtt {fde}}$ is “the” logic of bilattices, a consideration reinforced by the work of Yaroslav Shramko and Heinrich Wansing in which $\mathsf {E}_{\mathtt {fde}}$ is shown to be similarly entrenched with respect to the theories of trilattices and, more generally, multilattices. In this paper, we export Melvin Fitting’s “cut-down” connectives—propositional connectives that “cut down” available evidence—to the case of multilattices and show that two related first degree systems—Harry Deutsch’s four-valued $\mathsf {S}_{\mathtt {fde}}$ and Richard Angell’s $\mathsf {AC}$—emerge just as elegantly and are as intimately connected to the theories of bilattices and trilattices as the Belnap-Dunn logic.