Collection Frames for Distributive Substructural Logics

We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley-Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for the basic positive distributive substructural logic, that collection frames on multisets are sound and complete for (the relevant logic, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic. The completeness of set frames for is, currently, an open question.