Collection Frames for Distributive Substructural Logics

December 2023

Greg Restall and Shawn Standefer, “Collection Frames for Distributive Substructural Logics,” Review of Symbolic Logic, (2023) 16:4, 1120-1157.

doi:10.1017/S1755020322000272

We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalization 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 substructural logic B⁺, that collection frames on multisets are sound and complete for RW⁺ (the relevant logic R⁺, 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 R⁺. (The completeness of set frames for R⁺ is, currently, an open question.)

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