Reasoning with Higher Order Partial Functions

. In this paper we introduce the logic PHOL, which embodies higher-order functions through a simply-typed -calculus and deals with partial objects by using partially ordered domains and three truth values. We define a refutationally complete tableaux method for PHOL and we show how to derive a sound and complete cut free sequent calculus through a systematic analysis of the rules for tableaux construction. 1 Introduction The formal methodologies used for the specification and verification of software (and also hardware) systems have motivated during the last years the development of so called partial logics, where partial functions can be used to argue about errors, diverging computations, and similar phenomena. Following some earlier precedents, such as [16], [21] and [5], quite many recent papers (as e.g. [2], [24], [17], [14], [19], [13]) have proposed partial versions of first order predicate logic aiming at this field of application. Partiality has been also investigated within t..