Duality for classical first order logic
Dion Coumans (RU)
A logic consists of a collection of axioms and reasoning rules. Many logics may be associated to a class of algebras in such a way that we can obtain information about the logic by studying the class of algebras. We call the class of algebras associated to a logic the 'algebraic semantics' of the logic.
The algebraic semantics for classical propositional logic (CPL) are given by Boolean algebras. The class of Boolean algebras is dually equivalent to the class of Stone spaces. This duality enables us to use topological tools in our study of CPL. The aim of this talk is to show how the duality for CPL may be extended to a duality for classical first order logic (CFOL).
I will first give a short introduction in logic and, in particular, in the use of duality theory in logic. Thereafter we turn to classical first order logic. The algebraic semantics for CFOL are given by Boolean hyperdoctrines. I will explain what these are by abstracting the essential properties of the collection of all first order formulas over a given signature. Thereafter we may identify the dual notion of a Boolean hyperdoctrine and consequently describe a duality for CFOL.