Andrews' Type Theory with Undefinedness

Q0 is an elegant version of Church’s type theory formulated and extensively studied by Peter B. Andrews. Like other traditional logics, Q0 does not admit undefined terms. The traditional approach to undefinedness in mathematical practice is to treat undefined terms as legitimate, nondenoting terms that can be components of meaningful statements. Qu0 is a modification of Andrews’ type theory Q0 that directly formalizes the traditional approach to undefinedness. This paper presents Qu0 and proves that the proof system of Qu0 is sound and complete with respect to its semantics which is based on Henkin-style general models. The paper’s development of Q u 0 closely follows Andrews’ development of Q0 to clearly delineate the differences between the two systems.