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 Henkin-style 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 e-prints
- Pub Date:
- June 2014
- DOI:
- 10.48550/arXiv.1406.7492
- 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
- E-Print:
- This research was supported by NSERC. arXiv admin note: text overlap with arXiv:1406.6706