  Ken Pledger devised a one-sorted approach to the incidence relation of plane
geometries, using structures that also support models of propositional modal
logic. He introduced a modal system 12g that is valid in one-sorted projective
planes, proved that it has finitely many non-equivalent modalities, and
identified all possible modality patterns of its extensions. One of these
extensions 8f is valid in elliptic planes. These results were presented in his
doctoral dissertation [14], which has been reprinted in the Australasian
Journal of Logic, vol. 18, no. 4. https://doi.org/10.26686/ajl.v18i4.6831
  Here we show that 12g and 8f are strongly complete for validity in their
intended one-sorted geometrical interpretations, and have the finite model
property. The proofs apply standard technology of modal logic (canonical
models, filtrations) together with a step-by-step procedure introduced by Yde
Venema for constructing two-sorted projective planes.
