In classically based modal logic, there are three common conceptions of necessity, the universal conception, the equivalence relation conception, and the axiomatic conception. They provide distinct presentations of the modal logic S5, all of which coincide in the basic modal language. We explore these different conceptions in the context of the relevant logic R, demonstrating where they come apart. This reveals that there are many options for being an S5-ish extension of R. It further reveals a divide between the universal (...) conception of necessity on the one hand, and the axiomatic conception on the other: The latter is consistent with motivations for relevant logics while the former is not. For the committed relevant logician, necessity cannot be the truth in all possible worlds. (shrink)

In this paper, we build on earlier work by Standefer (Logic J IGPL 27(4):543–569, 2019) in investigating extensions of substructural logics, particularly relevant logics, with the machinery of justification logics. We strengthen a negative result from the earlier work showing a limitation with the canonical model method of proving completeness. We then show how to enrich the language with an additional operator for implicit commitment to circumvent these problems. We then extend the logics with axioms for D, 4, and 5, (...) which requires additional justification term operators, following the work of Pacuit (in proceedings of the fifth panhellenic logic symposium, 2005) and Rubtsova (in Grigoriev D, Harrison J (eds) Computer science—theory and applications, CSR 2006; J Logic Comput 16(5):671–684, 2006), and present the required modifications to the frame semantics. We present a simplification of the neighborhood frames from the earlier work and we close by investigating the distinctive contribution of the + operator to the logic. (shrink)

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. (shrink)