  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.
  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.
