Review of Symbolic Logic:1-43 (forthcoming)

Authors
Tim Button
University College London
Robert Trueman
University of York
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 ${\textrm {STT}}$.
Keywords type theory  standard type theory  cumulative type theory
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DOI 10.1017/s1755020321000435
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References found in this work BETA

Objects of Thought.Arthur Norman Prior - 1971 - Oxford, England: Clarendon Press.
Philosophy and Model Theory.Sean Walsh & Tim Button - 2018 - Oxford, UK: Oxford University Press.
Plural Logic.Alex Oliver & Timothy John Smiley - 2013 - Oxford, England: Oxford University Press UK.
Quality and Concept.George Bealer - 1982 - Oxford, England: Oxford University Press.

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Citations of this work BETA

Higher‐Order Metaphysics.Lukas Skiba - 2021 - Philosophy Compass 16 (10):1-11.
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