Negation in relevant logics. (How I stopped worrying and learned to love the Routley star) (Q2715516)

The relational semantics for relevant logic is now well established, at least from a formal point of view. Questions remain, however, on how to interpret the components of relevant frames, including both the triadic relation \(R\) used to interpret \(\to\) formulas, and especially the Routley star * for negation. This paper presents insights not only into the latter and the role of negation in relevant logic, its ostensible topic, but also into the understanding of \(R\), and indeed the nature of the consequence relation itself. Key to its interpretation of the frame semantics is taking the points of a frame to be `states' rather than the popular `possible worlds'. States are more fine-grained than worlds; they can be incomplete and inconsistent while worlds cannot. Interpreting negation with respect to states, it is appropriate to treat it as incompatibility, and the Routley star provides a natural way to do that. Worlds are not excluded from this picture, however; indeed they have a place as states of a certain kind, those that are complete and consistent. With respect to worlds the truth conditions for negations are classical even when they are not with respect to states. With respect to worlds the disjunction syllogism is valid; with respect to states it is not. One might then think of logical consequence in two ways. As truth preservation over states, it leads to relevant logic; as truth preservation over worlds, to classical logic.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00012].