Strong conceptual completeness for first-order logic

By a strong conceptual completeness (XC) statement for a logical doctrine we mean an assertion that says that any theory of the doctrine can be recovered from an appropriate structure formed by the models of the theory. The expression ‘logical doctrine’ is used here as in [9] to mean a specific selection of logical operations giving rise to a notion of theory, in the form of a structured category. (Coherent) first-order logic, e.g., is obtained when the selected operations are finite limits and certain finite colimits. The resulting notion of theory is that of pretopos; see [9] or [19]. Strong conceptual completeness is most familiar in (classical) propositional logic, where it takes the form of the Stone duality theorem. A variant, due to Lambek and Rattray [12], of the Stone theory is explained in detail in [15]. This variant uses the notion of codensity [5], and it expresses the full content of the Stone duality theorem within the category of Boolean algebras, whereas in the original topological variant a comparison takes place between the category of Boolean algebras and the category of Stone spaces. (The statement in the second sentence of the last paragraph on page 182 of [15] is wrong; the conclusion that the codensity variant is stronger than the topological variant, in a specific sense explained in [15], remains true, however.) The condensity formulation is also nice because it has the form suggesting that the two-element Boolean algebra ‘generates’, in a certain sense, all Boolean algebras (in fact, all Boolean algebras become canonical limits of (large) diagrams of powers of the two-element Boolean algebra). It is fair to say that the subject of this paper, as well as of its predecessors, is the working out of an analogy between the role of Set, the category of small sets, in first-order logic on the one hand, and the role of the two-element Boolean algebra in propositional logic on the other.