Logic for physical space

Abstract

Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the major milestones in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces.

Morpho-logic from a Topos Perspective: Application to symbolic AI

“…The proposed definitions and operations will then enhance the reasoning ability of MM, extending previous work on morphological modal logic [5,13]. They will allow, among others, giving morphological semantics to modalities with a topological flavor [60] conventionally used for spatial reasoning [1], and which could not be obtained directly from erosions and dilations; actually, the properties of these modalities are closer to the morphological operators of opening and closing (in their particular form of composition of erosion and dilation) due to a double quantification ∀/∃ in their definition. In [34], MM has been extended to structuring elements based on a notion of neighborhood close to a similar topological notion.…”