
13Nonmetric Propositional SimilarityErkenntnis 122. forthcoming.The idea that sentences can be closer or further apart in meaning is highly intuitive. Not only that, it is also a pillar of logic, semantic theory and the philosophy of science, and follows from other commitments about similarity. The present paper proposes a novel way of comparing the ‘distance’ between two pairs of propositions. We define ‘\ is closer in meaning to \ than \ is to \’ and thereby give a precise account of comparative propositional similarity facts. Notably, our definition esche…Read more

7Justin ClarkeDoaneMorality and MathematicsPhilosophia Mathematica. forthcoming._Justin ClarkeDoane* * Morality and Mathematics. _ Oxford University Press, 2020. Pp. xx + 208. ISBN: 9780198823667 ; 9780192556806.† †

65A measure of inferentialrole preservationSynthese 196 (7): 26212642. 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

45Capturing consequenceReview of Symbolic Logic 12 (2): 271295. 2019.Firstorder formalisations are often preferred to propositional ones because they are thought to underwrite the validity of more arguments. We compare and contrast the ability of some wellknown logics—these two in particular—to formally capture valid and invalid arguments. We show that there is a precise and important sense in which firstorder logic does not improve on propositional logic in this respect. We also prove some generalisations and related results of philosophical interest. The re…Read more

28Isomorphism invariance and overgeneration – corrigendumBulletin of Symbolic Logic 23 (4): 546546. 2017.

12On an application of categoricityProceedings of the Aristotelian Society 105 (1): 395399. 2005.James Walmsley in “Categoricity and Indefinite Extensibility” argues that a realist about some branch of mathematics X (e.g. arithmetic) apparently cannot use the categoricity of an axiomatisation of X to justify her belief that every sentence of the language of X has a truthvalue. My discussion note first corrects Walmsley’s formulation of his claim. It then shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping th…Read more

50Isomorphism invariance and overgenerationBulletin of Symbolic Logic 22 (4): 482503. 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 settheoretic 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

13The Subtraction Argument(s)Dialectica 60 (2): 145156. 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.

34Philosophy of the MatrixPhilosophia Mathematica 25 (2): 246267. 2017.A mathematical matrix is usually defined as a twodimensional array of scalars. And yet, as I explain, matrices are not in fact twodimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.

29The Laws of Belief: Ranking Theory & its Philosophical Applications, by Wolfgang SpohnMind 126 (501): 273278. 2017.The Laws of Belief: Ranking Theory & its Philosophical Applications, by SpohnWolfgang. New York: Oxford University Press, 2012. Pp. xv + 598.

58What’s the Point of Complete Rigour?Mind 125 (497): 177207. 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

1Philosophy of Mathematics (edited book)Routledge. 2016.Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and the epistemological question of how we know ab…Read more

16The OpenEndedness of the Set Concept and the Semantics of Set TheorySynthese 135 (3): 379399. 2003.Some philosophers have argued that the openendedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a nonstandard semantics for set theory that allegedly sanctions a nonclassical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and…Read more

90Why the subtraction argument does not add upAnalysis 62 (1): 7375. 2002.Gonzalo RodriguezPereyra (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 heavyduty metaphysics of modality. This note will show that on either reading of its key premiss, the subtraction argument RodriguezPereyra proposes is invalid. [A sequel to this paper, 'The Subtraction Argument(s)', was published in Dialectica in 2006.]

43JOHN P. BURGESS Rigor and StructureBritish Journal for the Philosophy of Science 67 (4): 11851187. 2016.

59Fairness and AggregationUtilitas 27 (4): 460469. 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 nontrivial theory guarantees aggregativity. This is not intended as …Read more

45Did Frege commit a cardinal sin?Analysis 75 (3): 379386. 2015.Frege’s _Basic Law V_ is inconsistent. The reason often given is that it posits the existence of an injection from the larger collection of firstorder concepts to the smaller collection of objects. This article explains what is right and what is wrong with this diagnosis

24Erratum to: A measure of inferentialrole preservationSynthese 194 (4): 14251425. 2017.Erratum to: Synthese DOI 10.1007/s1122901507055In line 3 of footnote 8 on page 4, ‘allow’ should be ‘disallow’.In line 8 of page 5, \ should be \ and \ should be \. Similarly for lines 1, 2, 3, 7, 8, 13 and 14 of page 6.The entry in row 20 column 6 of the table on page 5 should be 1 rather than 0.The entry \ in row 30 column 5 of the table on page 5 should be \.In line 27 of page 13, ‘it’ should be ‘them’.Four lines from the end of section 12.3 on page 20, ‘premisses’ should be ‘premiss sets’…Read more

54The overgeneration argument(s): A succinct refutationAnalysis 74 (1): 4047. 2014.The overgeneration argument attempts to show that accepting secondorder 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

47The overgeneration argument(s): A succinct refutationAnalysis 74 (1). 2014.The overgeneration argument attempts to show that accepting secondorder 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

102An exact measure of paradoxAnalysis 73 (1): 1726. 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

34James Robert Brown. Platonism, naturalism, and mathematical knowledge. New York and London: Routledge, 2012. Isbn 9780415872669. Pp. X + 182 (review)Philosophia Mathematica 20 (3): 359364. 2012.

52The openendedness of the set concept and the semantics of set theorySynthese 135 (3). 2003.Some philosophers have argued that the openendedness 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 nonstandard semantics for set theory that allegedly sanctions a nonclassical logic. I show that the views about quantification the argument relies on turn out to sanction a c…Read more

65How to type: Reply to HalbachAnalysis 69 (2): 280286. 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

274Defining ultimate ontological basis and the fundamental layerPhilosophical Quarterly 60 (238): 169175. 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 selfdependent entities exist. They also apply to different conceptions of the relation of ontological dependence.

157Resemblance theories of propertiesPhilosophical Studies 157 (3): 361382. 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

78Motivating reductionism about setsAustralasian 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.
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Areas of Interest
Epistemology 
Metaphysics 
Logic and Philosophy of Logic 
Philosophy of Mathematics 