Andrews' Type Theory with Undefinedness
Abstract
${\cal Q}_0$ is an elegant version of Church's type theory formulated and extensively studied by Peter B. Andrews. Like other traditional logics, ${\cal Q}_0$ 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. ${\cal Q}^{\rm u}_{0}$ is a modification of Andrews' type theory ${\cal Q}_0$ that directly formalizes the traditional approach to undefinedness. This paper presents ${\cal Q}^{\rm u}_{0}$ and proves that the proof system of ${\cal Q}^{\rm u}_{0}$ is sound and complete with respect to its semantics which is based on Henkinstyle general models. The paper's development of ${\cal Q}^{\rm u}_{0}$ closely follows Andrews' development of ${\cal Q}_0$ to clearly delineate the differences between the two systems.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.7492
 Bibcode:
 2014arXiv1406.7492F
 Keywords:

 Mathematics  Logic;
 Computer Science  Logic in Computer Science;
 03B15 (Primary);
 03B35 (Secondary);
 F.4.1;
 I.2.3
 EPrint:
 This research was supported by NSERC. arXiv admin note: text overlap with arXiv:1406.6706