Collection frames for distributive substructural logics
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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 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.
Restall , G & Standefer , S 2022 , ' Collection frames for distributive substructural logics ' , The Review of Symbolic Logic , vol. FirstView . https://doi.org/10.1017/S1755020322000272
The Review of Symbolic Logic
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
DescriptionThis research was supported by the Australian Research Council, Discovery Grant dp150103801.
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