Beliefs have what I will call 'propositional content'. A belief is always a belief that so-and-
Standard Type Theory, STT, tells us that ( ) is well-formed iff = + 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, ( ) is well-formed iff > . 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.10 This extends Degen and Johannsen's (2000: §4.1) results concerning Z. Linnebo and Rayo (2012: 289, n.28) cover only Z without Foundation. The bound + 2 < is needed as abbreviates ∃ +1 (∀ +2 ( +2 ( +1 ) ↔ +2 ( )) ∧ +1 ( )).11 Linnebo and Rayo (2012: 284, 289) mention differences ( 1) and ( 2) themselves, but they do not mention (3). 65 Zr is equivalent to Potter's (2004) theory Z; this is strictly stronger than Zermelo's Z. 66 Notation: we let ' ⊆ ∈ ℎ' abbreviate '( ⊆ ∧ ∈ ℎ)'; similarly for other infix predicates. 67 See Button (forthcoming: §3) for proofs. 68 Compare these with Degen and Johannsen (2000: 149 Ext, 153 Nullity).
This book articulates and defends Fregean realism, a theory of properties based on Frege's insight that properties are not objects, but rather the satisfaction conditions of predicates. Robert Trueman argues that this approach is the key not only to dissolving a host of longstanding metaphysical puzzles, such as Bradley's Regress and the Problem of Universals, but also to understanding the relationship between states of affairs, propositions, and the truth conditions of sentences. Fregean realism, Trueman suggests, ultimately leads to a version of the identity theory of truth, the theory that true propositions are identical to obtaining states of affairs. In other words, the identity theory collapses the gap between mind and world. This book will be of interest to anyone working in logic, metaphysics, the philosophy of language or the philosophy of mind.
In a recent article, Hofweber (2019) presents a new, and surprising, argument for idealism. His argument is surprising because it starts with an apparently innocent premiss from the philosophy of language: that ‘that’-clauses do not refer. I do not think that Hofweber's argument works, and my first aim in this paper is to explain why. However, I agree with Hofweber that what we say about ‘that’-clauses has important metaphysical consequences. My second aim is to argue that, far from leading us into idealism, denying that ‘that’-clauses refer is the first step toward a kind of direct realism about belief.
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