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

DOI10.26686/AJL.V18I4.6829arXiv2012.15077MaRDI QIDQ6357206

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Publication date: 30 December 2020

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.

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