•  274
    Defining ultimate ontological basis and the fundamental layer
    Philosophical Quarterly 60 (238): 169-175. 2010.
    I explain why Ross Cameron's definition of ultimate ontological basis is incorrect, and propose a different definition in terms of ontological dependence, as well as a definition of reality's fundamental layer. These new definitions cover the conceptual possibility that self-dependent entities exist. They also apply to different conceptions of the relation of ontological dependence.
  •  270
    Naturalism in mathematics and the authority of philosophy
    British Journal for the Philosophy of Science 56 (2): 377-396. 2005.
    Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism …Read more
  •  175
    Proofs of the Compactness Theorem
    History and Philosophy of Logic 31 (1): 73-98. 2010.
    In this study, several proofs of the compactness theorem for propositional logic with countably many atomic sentences are compared. Thereby some steps are taken towards a systematic philosophical study of the compactness theorem. In addition, some related data and morals for the theory of mathematical explanation are presented
  •  157
    Resemblance theories of properties
    Philosophical Studies 157 (3): 361-382. 2012.
    The paper aims to develop a resemblance theory of properties that technically improves on past versions. The theory is based on a comparative resemblance predicate. In combination with other resources, it solves the various technical problems besetting resemblance nominalism. The paper’s second main aim is to indicate that previously proposed resemblance theories that solve the technical problems, including the comparative theory, are nominalistically unacceptable and have controversial philosop…Read more
  •  146
    Knowledge of Mathematics without Proof
    British Journal for the Philosophy of Science 66 (4): 775-799. 2015.
    Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathemat…Read more
  •  146
    Science and mathematics: the scope and limits of mathematical fictionalism Content Type Journal Article Category Book Symposium Pages 1-26 DOI 10.1007/s11016-011-9640-3 Authors Christopher Pincock, University of Missouri, 438 Strickland Hall, Columbia, MO 65211-4160, USA Alan Baker, Department of Philosophy, Swarthmore College, Swarthmore, PA 19081, USA Alexander Paseau, Wadham College, Oxford, OX1 3PN UK Mary Leng, Department of Philosophy, University of York, Heslington, York, YO10 5DD UK Jour…Read more
  •  132
    Proving Induction
    Australasian Journal of Logic 10 1-17. 2011.
    The hard problem of induction is to argue without begging the question that inductive inference, applied properly in the proper circumstances, is conducive to truth. A recent theorem seems to show that the hard problem has a deductive solution. The theorem, provable in zfc, states that a predictive function M exists with the following property: whatever world we live in, M correctly predicts the world’s present state given its previous states at all times apart from a well-ordered subset. On the…Read more
  •  104
    Review: Logical Pluralism (review)
    Mind 116 (462): 391-396. 2007.
  •  103
    Boolos on the justification of set theory
    Philosophia Mathematica 15 (1): 30-53. 2007.
    George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.
  •  102
    An exact measure of paradox
    Analysis 73 (1): 17-26. 2013.
    We take seriously the idea that paradoxes come in quantifiable degree by offering an exact measure of paradox. We consider three factors relevant to the degree of paradox, which are a function of the degree of belief in each of the individual propositions in the paradox set and the degree of belief in the set as a whole. We illustrate the proposal with a particular measure, and conclude the discussion with some critical remarks
  •  90
    Why the subtraction argument does not add up
    Analysis 62 (1): 73-75. 2002.
    Gonzalo Rodriguez-Pereyra (1997) has refined an argument due to Thomas Baldwin (1996), which claims to prove nihilism, the thesis that there could have been no concrete objects, and which apparently does so without reliance on any heavy-duty metaphysics of modality. This note will show that on either reading of its key premiss, the subtraction argument Rodriguez-Pereyra proposes is invalid. [A sequel to this paper, 'The Subtraction Argument(s)', was published in Dialectica in 2006.]
  •  87
    Genuine modal realism and completeness
    Mind 115 (459): 721-730. 2006.
    John Divers and Joseph Melia have argued that Lewis's modal realism is extensionally inadequate. This paper explains why their argument does not succeed.
  •  86
    Mathematical instrumentalism, Gödel’s theorem, and inductive evidence
    Studies in History and Philosophy of Science Part A 42 (1): 140-149. 2011.
    Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the element…Read more
  •  85
    Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological…Read more
  •  85
    Pure Second-Order Logic with Second-Order Identity
    Notre Dame Journal of Formal Logic 51 (3): 351-360. 2010.
    Pure second-order logic is second-order logic without functional or first-order variables. In "Pure Second-Order Logic," Denyer shows that pure second-order logic is compact and that its notion of logical truth is decidable. However, his argument does not extend to pure second-order logic with second-order identity. We give a more general argument, based on elimination of quantifiers, which shows that any formula of pure second-order logic with second-order identity is equivalent to a member of …Read more
  •  78
    Motivating reductionism about sets
    Australasian Journal of Philosophy 86 (2). 2008.
    The paper raises some difficulties for the typical motivations behind set reductionism, the view that sets are reducible to entities identified independently of set theory.
  •  78
    A puzzle about naturalism
    Metaphilosophy 41 (5): 642-648. 2010.
    Abstract: This article presents and solves a puzzle about methodological naturalism. Trumping naturalism is the thesis that we must accept p if science sanctions p, and biconditional naturalism the apparently stronger thesis that we must accept p if and only if science sanctions p. The puzzle is generated by an apparently cogent argument to the effect that trumping naturalism is equivalent to biconditional naturalism. It turns out that the argument for this equivalence is subtly question-begging…Read more
  •  68
    Should the logic of set theory be intuitionistic?
    Proceedings of the Aristotelian Society 101 (3). 2001.
    It is commonly assumed that classical logic is the embodiment of a realist ontology. In “Sets and Semantics”, however, Jonathan Lear challenged this assumption in the particular case of set theory, arguing that even if one is a set-theoretic Platonist, due attention to a special feature of set theory leads to the conclusion that the correct logic for it is intuitionistic. The feature of set theory Lear appeals to is the open-endedness of the concept of set. This article advances reasons internal…Read more
  •  65
    How to type: Reply to Halbach
    Analysis 69 (2): 280-286. 2009.
    In my paper , I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘ K’, but infinitely many, ‘ K 1’, ‘ K 2’, … , then the type rules prevent application of the predicate ‘ K i’ to sentences containing ‘ K i’ such as ‘ p ∧¬ K i⌜ p⌝’. This provides a motivated response to Fitch's argument so long as knowledge typing is itself motivated. It was the burden of my p…Read more
  •  65
    A measure of inferential-role preservation
    Synthese 196 (7): 2621-2642. 2019.
    The point of formalisation is to model various aspects of natural language. Perhaps the main use to which formalisation is put is to model and explain inferential relations between different sentences. Judged solely by this objective, a formalisation is successful in modelling the inferential network of natural language sentences to the extent that it mirrors this network. There is surprisingly little literature on the criteria of good formalisation, and even less on the question of what it is f…Read more
  •  60
    The subtraction argument(s)
    Dialectica 60 (2). 2006.
    The subtraction argument aims to show that there is an empty world, in the sense of a possible world with no concrete objects. The argument has been endorsed by several philosophers. I show that there are currently two versions of the argument around, and that only one of them is valid. I then sketch the main problem for the valid version of the argument
  •  59
    Fairness and Aggregation
    Utilitas 27 (4): 460-469. 2015.
    Sometimes, two unfair distributions cancel out in aggregate. Paradoxically, two distributions each of which is fair in isolation may give rise to aggregate unfairness. When assessing the fairness of distributions, it therefore matters whether we assess transactions piecemeal or focus only on the overall result. This piece illustrates these difficulties for two leading theories of fairness before offering a formal proof that no non-trivial theory guarantees aggregativity. This is not intended as …Read more
  •  58
    What’s the Point of Complete Rigour?
    Mind 125 (497): 177-207. 2016.
    Complete inferential rigour is achieved by breaking down arguments into steps that are as small as possible: inferential ‘atoms’. For example, a mathematical or philosophical argument may be made completely inferentially rigorous by decomposing its inferential steps into the type of step found in a natural deduction system. It is commonly thought that atomization, paradigmatically in mathematics but also more generally, is pro tanto epistemically valuable. The paper considers some plausible cand…Read more
  •  54
    The overgeneration argument attempts to show that accepting second-order validity as a sound formal counterpart of logical truth has the unacceptable consequence that the Continuum Hypothesis is either a logical truth or a logical falsehood. The argument was presented and vigorously defended in John Etchemendy’s The Concept of Logical Consequence and it has many proponents to this day. Yet it is nothing but a seductive fallacy. I demonstrate this by considering five versions of the argument; as …Read more
  •  52
    Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a c…Read more
  •  50
    Isomorphism invariance and overgeneration
    Bulletin of Symbolic Logic 22 (4): 482-503. 2016.
    The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrec…Read more
  •  47
    The overgeneration argument attempts to show that accepting second-order validity as a sound formal counterpart of logical truth has the unacceptable consequence that the Continuum Hypothesis is either a logical truth or a logical falsehood. The argument was presented and vigorously defended in John Etchemendy’s The Concept of Logical Consequence and it has many proponents to this day. Yet it is nothing but a seductive fallacy. I demonstrate this by considering five versions of the argument; as …Read more
  •  47
    Fitch's Argument and Typing Knowledge
    Notre Dame Journal of Formal Logic 49 (2): 153-176. 2008.
    Fitch's argument purports to show that if all truths are knowable then all truths are known. The argument exploits the fact that the knowledge predicate or operator is untyped and may thus apply to sentences containing itself. This article outlines a response to Fitch's argument based on the idea that knowledge is typed. The first part of the article outlines the philosophical motivation for the view, comparing it to the motivation behind typing truth. The second, formal part presents a logic in…Read more
  •  46
    Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilem…Read more