Completeness in partial type theory

Abstract

Abstract The present paper provides a completeness proof for a system of higher-order logic framed within partial type theory. The framework is a modification of Tichý’s extension of Church’s simple type theory, equipped with his innovative natural deduction system in sequent style. The system deals with both total and partial (multiargument) functions-as-mappings and also accommodates algorithmic computations arriving at various objects of the framework. The partiality of a function or a failure of a computation is not represented by a postulated null object such as the third truth value. The logical operators of the system are classical. Another welcome feature of this expressive system is that its consequence relation is monotonic.