Modal logics for incidence geometries

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Abstract
Incidence geometry is based on two-sorted structures consisting of ‘points’ and ‘lines’ together with an intersort binary relation called incidence. We introduce an equivalent one-sorted geometrical structure, called incidence frame, which is suitable for modal considerations. Incidence frames constitute the semantical basis of MIG, the modal logic of incidence geometry. A completeness theorem for MIG is proved: a modal formula is a theorem of MIG if and only if it is valid in all incidence frames. Extensions to projective and affine geometries are also considered.

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