Four Imprints of Belnap’s Useful Four-Valued Logic in Computer Science

Abstract We review four areas of theoretical computer science which share technical or philosophical ideas with the work of Belnap on his useful four-valued logic. Perhaps surprisingly, the inspiration by Belnap–Dunn logic is acknowledged only in the study of d-frames. The connections of Belnap’s work and linear logic, Blame Calculus or the study of LVars are not openly admitted. The key to three of these connections with Belnap’s work go via the twist-product representation of bilattices. On the one hand, it allows us to view the class of models of linear logic built using the Chu construction as based on Belnap–Dunn logic. On the other hand, twist-product representation theorems are essential in the theory of d-frames and, also, the key theorem of Blame Calculus is essentially a twist-product representation theorem too, albeit with a strong proof-theoretic flavour.