On specifications, theories, and models with higher types

We discuss the mathematical foundations of specifications, theories, and models with higher types. Higher type theories are presented by specifications either using the language of cartesian closure or a typed λ-calculus. We prove equivalence of both the specification methods, the main result being the equivalence of cartesian closure and a typed λ-calculus. Then we investigate “intensional” and extensional” models (the distinction is similar to that between λ-algebras and (λ)-models). We prove completeness of higher type theories with regard to intensional models as well as existence of free intensional models. For extensional models we prove that completeness and existence of an initial models implies that the theory itself already is the initial model. As a consequence intensional models seem to be better suited for the purposes of data type specification.