  This paper is a study of first-order coherent logic from the point of view of
duality and categorical logic. We prove a duality theorem between coherent
hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply
to prove completeness, omitting types, and Craig interpolation theorems for
coherent or intuitionistic logic. Our approach emphasizes the role of
interpolation and openness properties, and allows for a modular, syntax-free
treatment of these model-theoretic results. As further applications of the same
method, we prove completeness theorems for constant domain and G\"odel-Dummett
intuitionistic predicate logics.
