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.


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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I’m Greg Restall, and this is my personal website. I am the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews, and the Director of the Arché Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology I like thinking about – and helping other people think about – logic and philosophy and the many different ways they can inform each other.


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