Modal Logics for Parallelism, Orthogonality, and Affine Geometries

Journal of Applied Non-Classical Logics 12 (3-4):365-397 (2002)
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We introduce and study a variety of modal logics of parallelism, orthogonality, and affine geometries, for which we establish several completeness, decidability and complexity results and state a number of related open, and apparently difficult problems. We also demonstrate that lack of the finite model property of modal logics for sufficiently rich affine or projective geometries (incl. the real affine and projective planes) is a rather common phenomenon.
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Foundations of Geometry.Hilbert, David & Bernays, Paul
A ModalWalk Through Space.Aiello, Marco & van Benthem, Johan

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Line-Based Affine Reasoning in Euclidean Plane.Balbiani, Philippe & Tinchev, Tinko
Logic for Physical Space: From Antiquity to Present Days.Aiello, Marco; Bezhanishvili, Guram; Bloch, Isabelle & Goranko, Valentin

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