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Against cumulative type theory

Button, T; Trueman, R; (2021) Against cumulative type theory. Review of Symbolic Logic 10.1017/S1755020321000435. (In press). Green open access

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Abstract

Standard Type Theory, STT, tells us that bn(am) is well-formed iff n = m+1. However, Linnebo and Rayo (2012) have advocated for the use of Cumulative Type Theory, CTT, which has more relaxed type-restrictions: according to CTT, bβ(aα) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT’s type-restrictions are unjustifiable, the type-restrictions imposed by STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones (2021); we argue that their theory is best seen as a misleadingly formulated version of STT.

Type: Article
Title: Against cumulative type theory
Open access status: An open access version is available from UCL Discovery
DOI: 10.1017/S1755020321000435
Publisher version: https://doi.org/10.1017/S1755020321000435
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL SLASH
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of Arts and Humanities
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of Arts and Humanities > Dept of Philosophy
URI: https://discovery.ucl.ac.uk/id/eprint/10132767
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