A Reinterpretation of the Semilattice Semantics with Applications

In the early 1970s, Alasdair Urquhart proposed a semilattice semantics for relevance logic which he provided with an influential informational interpretation. In this article, I propose a BHK-inspired reinterpretation of the semantics which is related to Kit Fine's truthmaker semantics. I discuss and compare Urquhart's and Fine's semantics and show how simple modifications of Urquhart's semantics can be used to characterize both full propositional intuitionistic logic and Jankov's logic. I then present (quasi-)relevant companions for both of these systems. Finally, I provide sound and complete labelled sequent calculi for all of the systems discussed.

Automated summary

The article discusses Alasdair Urquhart's semilattice semantics for relevance logic and proposes a BHK-inspired reinterpretation related to Kit Fine's truthmaker semantics. It compares and shows how modifications of Urquhart's semantics can characterize different logic systems, such as full propositional intuitionistic logic and Jankov's logic. Additionally, relevant companions for these systems are presented along with sound and complete labelled sequent calculi.