•  77
    Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logic…Read more
  •  86
    Exceptional Logic
    Review of Symbolic Logic 1-37. forthcoming.
    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 proposa…Read more
  •  91
    On infinite size
    Oxford Studies in Metaphysics 9 3-19. 2015.
    This chapter challenges Cantor’s notion of the ‘power’, or ‘cardinality’, of an infinite set. According to Cantor, two infinite sets have the same cardinality if and only if there is a one-to-one correspondence between them. Cantor showed that there are infinite sets that do not have the same cardinality in this sense. Further, he took this result to show that there are infinite sets of different sizes. This has become the standard understanding of the result. The chapter challenges this, arguin…Read more
  •  92
    Truth and Generalized Quantification
    Australasian Journal of Philosophy 97 (2): 340-353. 2019.
    Kripke [1975] gives a formal theory of truth based on Kleene's strong evaluation scheme. It 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—that is, All ϕs are ψ—or Most. Specifically, it has been argued that such quantifiers preclude the existence of just the sort of language that Kripke aims to deliver—one that contains its own truth predicate. In th…Read more
  •  25
    Correction to: Self-referential propositions
    Synthese 196 (6): 2541-2541. 2019.
    Unfortunately, there is a mistake in line 10 of Section 1.2. The correct reference should read: As Kripke pointed out, we can produce one simply by baptizing the string ‘Jack is short’: Jack.
  •  105
    Size and Function
    Erkenntnis 83 (4): 853-873. 2018.
    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 between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for any…Read more
  •  504
    There are brute necessities
    Philosophical Quarterly 60 (238): 149-159. 2010.
    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 a…Read more
  •  55
    This is a reply to Vann McGee’s response to my paper, ‘On Infinite Size’.
  •  44
    Dialetheism, logical consequence and hierarchy
    Analysis 64 (4): 318-326. 2004.
    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 posit…Read more
  •  29
    Truth, Hierarchy and Incoherence
    In Bradley Armour-Garb (ed.), Reflections on the Liar, Oxford University Press. forthcoming.
    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 …Read more
  •  469
    Epistemically possible worlds and propositions
    Noûs 43 (2): 265-285. 2009.
    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…Read more
  •  91
    Hierarchical Propositions
    Journal of Philosophical Logic 46 (2): 215-231. 2017.
    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 p…Read more
  •  145
    Belief, information and reasoning
    Philosophical Perspectives 26 (1): 431-446. 2012.
    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 m…Read more
  •  97
    Self-referential propositions
    Synthese 194 (12): 5023-5037. 2017.
    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 proposition…Read more
  •  52
    Proving Unprovability
    Review of Symbolic Logic 10 (1). 2017.
    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, wou…Read more
  •  278
    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 posit…Read more