We define a Kripke semantics for Intuitionistic Higher-Order Logic with constraints formulated within Church's Theory of Types via the addition of a new constraint base type. We then define an executable fragment, hoHH(C), of the logic: a higher-order logic programming language with typed λ-abstraction, implication and universal quantification in goals and constraints, and give a modified model theory for this fragment. Both formal systems are shown sound and complete for their respective semantics.We also solve the impredicativity problem in λProlog semantics, namely how to give a definition of truth without appealing to induction on subformula structure. In the last section we give a simple semantics-based conservative extension proof that the language hoHH(C) satisfies a uniformity property along the lines of [39].