Multi-valued Logic for a Point-Free Foundation of Geometry

The paper starts from the observation that in the inclusion-based approach to point-free geometry there are serious difficulties in defining points. These difficulties disappear once we reformulate this approach in the framework of continuous multivalued logic. So, a theory of ‘graded inclusion’ is proposed as a counterpart of the usual ‘crisp inclusion’ of mereology. Again, a second theory is considered in which the graded predicates ‘to be close’ and ‘to be small’ are assumed as primitive. In both cases a suitable notion of abstractive sequence and of equivalence between abstractive sequences enables us to define the points. In the resulting set of points a distance is defined in a natural way and this enables a metrical approach to point-free geometry and therefore to go beyond mereotopology.The general idea is that it is possible to search for mathematical formalizations of the naive theory of the space an ordinary man needs to have in its everyday life. To do this we have to direct our attention not only to regions and the related relation of inclusion as it is usual in point-free geometry, but also to those (vague) properties which are geometrical in nature.