Semantics of Higher-Order Logic (ESSLLI 2006 Course Notes)

Definition 2 A typed family of sets D is a collection of sets Dα indexed by types α ∈ T . A typed family of nonempty sets D is a typed family of sets such that Dα is nonempty for each α ∈ T . Given two typed families of sets D and D, a typed function f : D → D is a collection of functions fα : Dα → D α indexed by types α ∈ T . We use FT (D,D ) to denote the set of typed functions from a typed family of sets D to a typed family of sets D.