Collection frames for distributive substructural logics
MetadataShow full item record
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 2023 , ' Collection frames for distributive substructural logics ' , The Review of Symbolic Logic , vol. 16 , no. 4 , pp. 1120-1157 . https://doi.org/10.1017/S1755020322000272
The Review of Symbolic Logic
DescriptionFunding: This research was supported by the Australian Research Council, Discovery Grant dp150103801.
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.
Showing items related by title, author, creator and subject.
Skinner, James (University of St Andrews, 2021-06-21) - ThesisHow many correct logics are there? For much of logic’s history it was widely assumed that there was exactly one correct logic, a position known as logical monism. However, the monist’s hegemony has recently become increasingly ...
Berto, Franz; Restall, Greg (2019-04-22) - Journal articleWe present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing ...
Restall, Greg (2023-12-01) - Journal articleNatural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion ...