Against Cumulative Type Theory
Review of Symbolic Logic:1-43 (2021)
Abstract
Standard Type Theory, ${\textrm {STT}}$, tells us that $b^n$ is well-formed iff $n=m+1$. However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$, which has more relaxed type-restrictions: according to $\textrm {CTT}$, $b^\beta $ is well-formed iff $\beta>\alpha $. In this paper, we set ourselves against $\textrm {CTT}$. We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$ ’s type-restrictions are unjustifiable, the type-restrictions imposed by ${\textrm {STT}}$ 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 $\textrm {CTT}$. We end by examining an alternative approach to cumulative types due to Florio and Jones [10]; we argue that their theory is best seen as a misleadingly formulated version of ${\textrm {STT}}$.Author Profiles
DOI
10.1017/s1755020321000435
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Added to PP
2021-08-27
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2021-08-27
Downloads
210 (#39,629)
6 months
87 (#10,510)
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