**PUBLICATIONS**

**Size and Function**

To appear in *Erkenntnis*.

#### [abstract]

Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections (i.e. one-to-one correspondences) between them. These results (which I of course do not dispute) settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the paper, however, is to argue that this question is in fact wide open: to argue that are not in a position to know the answer, because we are not in one to know the principle. The aim, that is, is to argue that for all we know there is only one size of infinity.

**Self-referential Propositions**

*Synthese* 194 (2017): 5023–37.

#### [abstract]

Are there ‘self-referential’ propositions? That is, propositions that say of themselves that they have a certain property, such as that of being false. There can seem reason to doubt that there are. At the same time, there are a number of reasons why it matters. For suppose that there are indeed no such propositions. One might then hope that while paradoxes such as the Liar show that many plausible principles about sentences must be given up, no such fate will befall principles about propositions. But the existence of self-referential propositions would dash such hopes. Further, the existence of such propositions would also seem to challenge the widespread claim that Liar sentences fail to express propositions. The aim of this paper is thus to settle the question—at least given an assumption. In particular, I argue that if propositions are structured, then self-referential propositions exist.

**Truth, Hierarchy and Incoherence**

In B. Armour-Garb (ed.), *Reflections on the Liar* (Oxford University Press, 2017).

#### [abstract]

Approaches to truth and the Liar paradox seem invariably to face a dilemma: either appeal to some sort of hierarchy, or declare apparently perfectly coherent concepts incoherent. But since both options lead to severe expressive restrictions, neither seems satisfactory. The aim of this paper is a new approach, which avoids the dilemma and the resulting expressive restrictions. Previous approaches tend to appeal to some new sort of semantic value for the truth predicate to take. I argue that such approaches inevitably lead to the dilemma in question. In contrast, the present proposal sticks with classical semantic values, but allows the compositional rules associated with these to admit of exceptions. I show how such an approach can be developed systematically.

**Proving Unprovability**

*Review of Symbolic Logic* 10 (2017): 92–115.

#### [abstract]

This paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e. S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g. 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness theorem, and its significance, would lead us to believe that the answer is ‘no’. But the present paper explores a positive answer. The general approach is to follow the lead of recent (and not so recent) approaches to truth and the Liar paradox.

**Hierarchical Propositions**

*Journal of Philosophical Logic* 46 (2017): 215–31.

#### [abstract]

The notion of a proposition is central to philosophy. But it is subject to paradoxes. A natural response is a hierarchical account and, ever since Russell proposed his theory of types in 1908, this has been the strategy of choice. But in this paper I raise a problem for such accounts. While this does not seem to have been recognized before, it would seem to render existing such accounts inadequate. The main purpose of the paper, however, is to provide a new hierarchical account that solves the problem.

**Reply to Vann McGee’s ‘Whittle’s Assault on Cantor’s Paradise’**

*Oxford Studies in Metaphysics: Volume 9* (Oxford University Press, 2015): 33–41.

#### [abstract]

This is a reply to Vann McGee’s response to my paper, ‘On Infinite Size’.

**On Infinite Size**

*Oxford Studies in Metaphysics: Volume 9* (Oxford University Press, 2015): 3–19.

#### [abstract]

Cantor showed that there are infinite sets that do not have one-to-one correspondences between them. The standard understanding of this result is that it shows that there are different sizes of infinity. This paper challenges this standard understanding, and argues, more generally, that we do not have any reason to think that there are different sizes of infinity. Two arguments are given against the claim that Cantor established that there are different such sizes: one involves an analogy between Cantor’s result and Russell’s paradox, another is more direct.

**Belief, Information and Reasoning**

*Philosophical Perspectives* 26 (2012): 431–46.

#### [abstract]

Here are two plausible ideas about belief. First: beliefs are our means of storing information. Second: if we believe something, then we are willing to use it in reasoning. But in this paper I introduce a puzzle that seems to show that these cannot both be right. The solution, I argue, is a new picture, on which there is a kind of belief for each idea. An account of these two kinds of belief is offered in terms of two components: a relatively stable one, which represents our information; and a more variable one, which represents what we are taking seriously. I also consider the possibility of solving the puzzle by less radical means; and an alternative argument for the proposed account of belief in terms of considerations from desire.

**There are Brute Necessities**

*Philosophical Quarterly* 60 (2010): 149–59.

#### [abstract]

A necessarily true sentence is ‘brute’ if it does not rigidly refer to anything and if it cannot be reduced to a logical truth. The question of whether there are brute necessities is an extremely natural one. Cian Dorr has recently argued for far‐reaching metaphysical claims on the basis of the principle that there are no brute necessities: he initially argued that there are no non‐symmetric relations, and later that there are no abstract objects at all. I argue that there are nominalistically acceptable brute necessities, and that Dorr's arguments thus fail. My argument is an application of Gödel’s first incompleteness theorem.

**Epistemically Possible Worlds and Propositions**

*Noûs* 43 (2009): 265–85.

#### [abstract]

Metaphysically possible worlds have many uses. Epistemically possible worlds promise to be similarly useful, especially in connection with propositions and propositional attitudes. However, I argue that there is a serious threat to the natural accounts of epistemically possible worlds, from a version of Russell’s paradox. I contrast this threat with David Kaplan’s problem for metaphysical possible world semantics: Kaplan’s problem can be straightforwardly rebutted, the problems I raise cannot. I argue that although there may be coherent accounts of epistemically possible worlds with fruitful applications, any such an account must fundamentally compromise the basic idea behind epistemic possibility.

**Dialetheism, Logical Consequence and Hierarchy**

*Analysis* 64 (2004): 318–26.

#### [abstract]

I argue that dialetheists have a problem with the concept of logical consequence. The upshot of this problem is that dialetheists must appeal to a hierarchy of concepts of logical consequence. Since this hierarchy is akin to those invoked by more orthodox resolutions of the semantic paradoxes, its emergence would appear to seriously undermine the dialetheic treatments of these paradoxes. And since these are central to the case for dialetheism, this would represent a significant blow to the position itself.

### WORK IN PROGRESS

**Ontological Pluralism and Notational Variance **

#### [abstract]

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are.

**Truth and Generalized Quantification **

#### [abstract]

The central theory of Kripke [1975], based on Kleene’s strong evaluation scheme, is probably the most important and influential that has yet been given—at least since Tarski. However, it has been argued that this theory has a problem with generalized quantifiers such as All(F,G), i.e. all Fs are G, or Most(F,G). Specifically, it has been argued that such quantifiers preclude the existence of just the sort of language that Kripke aims to deliver, that is, one that contains its own truth predicate. In this paper I solve the problem: by showing how Kleene’s strong scheme, and Kripke’s theory that is based on it, can in a natural way be extended to accommodate the full range of generalized quantifiers.

**Exceptional Logic **

#### [abstract]

The aim of the paper is to argue that all—or *almost* all—logical rules have exceptions. In particular, it is argued that this is a moral that we should draw from the semantic paradoxes. The idea that we should respond to the paradoxes by revising logic in some way is familiar. But previous proposals advocate the replacement of classical logic with some alternative logic. That is, some alternative system of rules, where it is taken for granted that these hold without exception. The present proposal is quite different. According to this, there is no such alternative logic. Rather, classical logic retains the status of the ‘one true logic’, but this status must be reconceived so as to be compatible with (almost) all of its rules admitting of exceptions. This would seem to have significant repercussions for a range of widely held views about logic: e.g. that it is a priori, or that it is necessary. Indeed, the arguments of the paper would seem to overturn such views.