In this paper, we investigate neighbourhood semantics for modal extensions of relevant logics. In particular, we combine the neighbourhood interpretation of the relevant implication (and related connectives) with a neighbourhood interpretation of modal operators. We prove completeness for a range of systems and investigate the relations between neighbourhood models and relational models, setting out a range of augmentation conditions for the various relations and operations.

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...) “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)