Two Manuscripts, One by Routley, One by Meyer: The Origins of the Routley-Meyer Semantics for Relevance Logics

Bimbó, Katalin; Dunn, J. Michael; Ferenz, Nicholas

doi:10.26686/ajl.v15i2.4066

Cited by 4 publications

(4 citation statements)

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“…where we have the (unordered) pair of x and y on the one hand, and the z on the other. The fact that this is an unordered pair, and not a set is important, because when we consider Rxxz what we have is [x, x]Rz, 9 Mares' monograph Relevant Logic [31, p. 210] gives a definition of frames for R + using this generalisation of the ternary relation. This shifted perspective on R comes with advantages.…”

Section: [X Y]rzmentioning

confidence: 99%

“…This generalisation into an arbitrary n-ary relation, where n ≥ 3 is extremely natural, and conditions on R 2 and still higher orders of R play a role in the specification of various substructural logics. 9 Our attempt to understand the phenomenon of higher order accessibility relationsand how they relate to each other-is the starting point for a new, simpler characterisation of frame semantics for substructural logics. In the next section we will start with one case, frames for the logics RW + and R + .…”

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confidence: 99%

“…Scott’s contributions are discussed by Brian Chellas, in a 1975 article [10, see p. 143 and notes 17 and 18]. Thanks to Lloyd Humberstone for bringing this reference to our attention. For recent discussions of Routley and Meyer’s early work on the ternary relational semantics, see papers by Bimbó and Dunn [9] and Ferenz [19].…”

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confidence: 99%

“…For recent discussions of Routley and Meyer’s early work on the ternary relational semantics, see papers by Bimbó and Dunn [9] and Ferenz [19].…”

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confidence: 99%

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Collection Frames for Distributive Substructural Logics

Restall¹,

Standefer²

2022

The Review of Symbolic Logic

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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 $\mathsf {B}^+$ , that collection frames on multisets are sound and complete for $\mathsf {RW}^+$ (the relevant logic $\mathsf {R}^+$ , 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 $\mathsf {R}^+$ . The completeness of set frames for $\mathsf {R}^+$ is, currently, an open question.

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Section: [X Y]rzmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

“…For recent discussions of Routley and Meyer’s early work on the ternary relational semantics, see papers by Bimbó and Dunn [9] and Ferenz [19].…”

mentioning

confidence: 99%

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