Mereological foundations of point-free geometry via multi-valued logic

Cristina Coppola, Giangiacomo Gerla



We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naïve and ‘qualitative’ approach to geometry to the ‘quantitative’ approach of advanced science.


point-free geometry; multi-valued logic; fuzzy logic; continuous logic; metric geometry; mereology; naïve science

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ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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