Partiality and Recursion in Higher-Order Logic

We present an illative system Is of classical higher-order logic with subtyping and basic inductive types. The system Is allows for direct definitions of partial and general recursive functions, and provides means for handling functions whose termination has not been proven. We give examples of how properties of some recursive functions may be established in our system. In a technical appendix to the paper we prove consistency of Is. The proof is by model construction. We then use this construction to show conservativity of Is over classical first-order logic. Conservativity over higher-order logic is conjectured, but not proven. Note: This paper is an extended technical report based on a conference paper with the same title published by Springer-Verlag in the proceedings of FoSSaCS 2013. The final publication is available at springerlink.com.