Geometrical structures and modal logic

Abstract

Although, in natural language, space modalities are used as frequently as time modalities, the logic of time is a well-established branch of modal logic whereas the same cannot be said of the logic of space. The reason is probably in the more simple mathematical structure of time: a set of moments together with a relation of precedence. Such a relational structure is suited to a modal treatment. The structure of space is more complex: several sorts of geometrical beings as points and lines together with binary relations as incidence or orthogonality, or only one sort of geometrical beings as points but ternary relations as collinearity or betweeness. In this paper, we define a general framework for axiomatizing modal logics which Kripke semantics is based on geometrical structures: structures of collinearity, projective structures, orthogonal structures.