Higher-order illative combinatory logic

We show a model construction for a system of higher-order illative combinatory logic $\mathcal{I}_\omega$, thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system $\mathcal{I}_0$ of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.