Completeness of Pledger's modal logics of one-sorted projective and elliptic planes

TL;DR

This paper proves the strong completeness and finite model property of certain modal logics related to one-sorted projective and elliptic planes, using standard modal logic techniques and a novel construction method.

Contribution

It establishes strong completeness and finite model properties for Pledger's modal systems 12g and 8f in their geometric interpretations, extending prior results.

Findings

12g and 8f are strongly complete for their intended geometrical semantics.

Both systems have the finite model property.

The proofs utilize canonical models, filtrations, and a new construction procedure.

Abstract

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.