Dualities in logic

Luca Reggio

The mathematical theory of Stone duality allows for a rigorous analysis of the connection between syntax and semantics, with powerful applications in logic and computer science. In its basic form, due to mathematician M. H. Stone (1934), it states that the algebraic theory of Boolean algebras, the algebraic counterpart to classical propositional logic, is equivalent to the topological theory of so-called Boolean (Stone) spaces. This allows one to translate properties and questions from one field (of syntactic or algebraic nature) to the other (of semantic or geometric nature), and vice versa. Since then, basic Stone duality has been generalised in several directions, e.g. to take into account additional operations such as modalities, or to capture larger classes of spaces.

In the first part of the tutorial I will provide a gentle introduction to duality theory and its applications in logic. In the second part I will present some more advanced topics, including recent applications of duality in mathematics and computer science. I will conclude by discussing some open problems in the field, and possible directions for future research