AGAINST CUMULATIVE TYPE THEORY

Abstract

Standard Type Theory, tells us that is well-formed iff. However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, which has more relaxed type-restrictions: According to, is well-formed iff \alpha $ ]]>. In this paper, we set ourselves against. We begin our case by arguing against Linnebo and Rayo's claim that sheds new philosophical light on set theory. We then argue that, while 's type-restrictions are unjustifiable, the type-restrictions imposed by 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. 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.