Abstract
AbstractStandard Type Theory,${\textrm {STT}}$, tells us that$b^n(a^m)$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$\textrm {CTT}$,$b^\beta (a^\alpha )$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}}$.
Highlights
Standard Type Theory, STT, tells us that ( ) is well-formed iff = + 1
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
We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory: in §2 we highlight some important mathematical differences between CTT and set theory, and in §3 we explore the philosophical consequences of these differences
Summary
We start by outlining the formalisms of STT and CTT. For simplicity of exposition, in this paper we focus on monadic type theories. (We only consider un-ramified type theories.). For all types , the following inferences are licensed, provided that (i) all expressions are well-formed, and (ii) does not occur in any undischarged assumptions on which ( ) depends:. Linnebo and Rayo (2012) ask us to consider an alternative, cumulative, type theory, CTT. (Again, we only outline the rules for ∀.) For all types ≥ , the following inferences are licensed, provided that (i) all expressions are well-formed, and (ii) does not occur in any undischarged assumption on which ( ) depends: These rules are intuitively sound, given the idea of cumulation: every type entity is a type ≥ entity too; so if holds of every type entity, holds of each type entity. In moving from STT to CTT, we are asked to relax STT’s syntax: ( ) is well-formed iff >. We refer to the cumulative type theories in general as ‘CTT’, using ‘CTT ’ with the superscript when it is important to pay attention to the bound
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