A simple type theory with partial functions and subtypes11Supported by the MITRE-Sponsored Research program. Presented at the 9th International Congress of Logic, Methodology and Philosophy of Science held in Uppsala, Sweden, August 7-14, 1991.

Simple type theory is a higher-order predicate logic for reasoning about truth values, individuals, and simply typed total functions. We present in this paper a version of simple type theory, called PF*, in which functions may be partial and types may have subtypes. We define both a Henkin-style general models semantics and an axiomatic system for PF*, and we prove that the axiomatic system is complete with respect to the general models semantics. We also define a notion of an interpretation of one PF* theory in another. PF* is intended as a foundation for mechanized mathematics. It is the basis for the logic of IMPS, an Interactive Mathematical Proof System developed at The MITRE Corporation.