Ian Hodkinson

Some developments in the modal logic of space

This year is the 70th anniversary of the appearance in the Annals of Mathematics of a seminal paper by McKinsey and Tarski, called "The algebra of topology".

In that paper, the authors studied a mathematical model of space. Formally, they considered an arbitrary separable dense-in-itself metric space. Such a space is rather abstract, but also rather general. Many common examples are covered, including the real line or plane, ordinary 3D or higher-dimensional space, the rational line, toruses, and so on.

The authors introduced a modal language for describing such spaces. The language is simple and not hugely expressive, but it can express many basic spatial arguments. In a famous and sophisticated result, the authors showed that the logic of every separable dense-in-itself metric space is S4. This gives us a rigorous foundation for correct spatial reasoning in this language.

Without going into details, I will describe in a fairly general way McKinsey and Tarski's original 1944 result, and a few of the developments since then - including joint work with Goldblatt earlier this year on extensions to the modal mu-calculus and languages with the [d]-operator and the universal modality.