{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T10:13:41Z","timestamp":1775038421607,"version":"3.50.1"},"reference-count":54,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2022,7,13]],"date-time":"2022-07-13T00:00:00Z","timestamp":1657670400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a <jats:italic>generalisation<\/jats:italic> of Routley\u2013Meyer ternary frames and a <jats:italic>simplification<\/jats:italic> of them. The key innovation of this semantics is the use of a single accessibility relation to relate <jats:italic>collections<\/jats:italic> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000272_inline1.png\"\/><jats:tex-math>\n$\\mathsf {B}^+$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, that collection frames on multisets are sound and complete for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000272_inline2.png\"\/><jats:tex-math>\n$\\mathsf {RW}^+$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (the relevant logic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000272_inline3.png\"\/><jats:tex-math>\n$\\mathsf {R}^+$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000272_inline4.png\"\/><jats:tex-math>\n$\\mathsf {R}^+$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. The completeness of set frames for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000272_inline5.png\"\/><jats:tex-math>\n$\\mathsf {R}^+$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is, currently, an open question.<\/jats:p>","DOI":"10.1017\/s1755020322000272","type":"journal-article","created":{"date-parts":[[2022,7,13]],"date-time":"2022-07-13T05:07:31Z","timestamp":1657688851000},"page":"1120-1157","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":6,"title":["COLLECTION FRAMES FOR DISTRIBUTIVE SUBSTRUCTURAL 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