Points, lines and diamonds: A two-sorted modal logic for projective planes

We introduce a modal language for talking about projective planes. This language is two-sorted, containing formulas to be evaluated at points and at lines, respectively. The language has two diamonds whose intended accessibility relations are the two directions of the incidence relation between points and lines. We provide a Sound and complete axiomatization for the formulas that are valid in the class of projective planes. We also show that iris decidable whether a given formula is satisfiable in a projective plane, and we characterize the computational complexity of this satisfaction problem.