Annals of Pure and Applied Logic 64 (3):211-240 (1993)

Abstract
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, an Interactive Mathematical Proof System developed at The MITRE Corporation.
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DOI 10.1016/0168-0072(93)90144-3
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