Space, with its manifold layers of structure, has been an inexhaustible source of intellectual fascination since Antiquity. The science that began with the empirical discoveries of the Egyptian ‘rope-stretchers’, and that has inspired many of the greatest developments in mathematics over the centuries, now comprises such topics as spatial databases, automated geometrical reasoning and digital image processing. In this long intellectual history, however, one relatively recent, yet crucial, event stands out: the rise of the logical stance in geometry. Fundamental to this development is the analysis of geometrical structures in relation to the formal languages used to describe them, and the recognition of the special mathematical challenges—and opportunities—which such an analysis presents. The interplay between logic and geometry is the subject of this book.
By a spatial logic, we mean any formal language for describing geometrical entities and configurations, where ‘geometrical’ is understood in a broad sense. Unlike their well-studied temporal counterparts, spatial logics have been curiously neglected in the literature on mathematical logic, despite some early pioneering work by Tarski and others on the foundations of geometry and topology in the middle years of the previous century. Only in the last decade have spatial logics attracted renewed attention from logicians, partly as a response to work in such diverse fields as artificial intelligence, database theory, physics and philosophy.