“Geometric Models for Relevant Logics”, p. 223–239 in Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs, Edited by Ivo Duntsch and Edwin Mares, Springer.
Alasdair Urquhart’s work on models for relevant logics is distinctive in a number of different ways. One key theme, present in both his undecidability proof for the relevant logic R, and his proof of the failure of interpolation in R, is the use of techniques from geometry. In this paper, inspired by Urquhart’s work, I explore ways to generate natural models of R from geometries, and different constraints that an accessibility relation in such a model might satisfy. I end by showing that a set of natural conditions on an accessibility relation, motivated by geometric considerations, is jointly unsatisfiable.