In a 1999 paper, Yde Venema proposed to model projective planes by two-sorted Kripke frames, with a sort for points, another sort for lines, and an 'incidence' accessibility relation between them. Among other things, he showed how to axiomatise the modal logic of projective planes. Affine planes can be similarly modeled, using an additional accessibility relation of 'parallel' - two lines are parallel if and only if they are equal or have no points in common. I will discuss recent joint work with Altaf Hussain on axiomatising the modal logic of affine planes, which, surprisingly, turns out to be more difficult than in the projective case.

slides at http://www.doc.ic.ac.uk/~imh/papers/leics.pdf