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

We suggest possible approaches to point-free geometry based on multi- valued logic. The idea is to assume as primitives the notion of region together with suitable ‘vague’ predicates whose meaning is geometrical in nature. For example, predicates as ‘close’, ‘small’, ‘contained’. In accordance, 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 in account that very interesting metrical approaches to geometry exist (see for example Blumenthal’s book [2]), 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 furnishes a tool to illustrate the passage from a naive and ‘qualitative’ approach to geometry to a ‘quantitative’ one of advanced science.