•  80
    Reducing Arithmetic to Set Theory
    In Øystein Linnebo & Otavio Bueno (eds.), New Waves in Philosophy of Mathematics, Palgrave Macmillan. pp. 35-55. 2009.
    The revival of the philosophy of mathematics in the 60s following its post-1931 slump left us with two conflicting positions on arithmetic’s ontological relationship to set theory. W.V. Quine’s view, presented in 'Word and Object' (1960), was that numbers are sets. The opposing view was advanced in another milestone of twentieth-century philosophy of mathematics, Paul Benacerraf’s 'What Numbers Could Not Be' (1965): one of the things numbers could not be, it explained, was sets; the other thing …Read more
  •  78
    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
  •  76
    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
  •  73
    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
  •  73
    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
  •  72
    Justin Clarke-Doane* Morality and Mathematics
    with Michael Bevan
    Philosophia Mathematica 28 (3): 442-446. 2020.
    _Justin Clarke-Doane* * Morality and Mathematics. _ Oxford University Press, 2020. Pp. xx + 208. ISBN: 978-0-19-882366-7 ; 978-0-19-2556806.† †
  •  71
    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
  •  70
    Capturing Consequence
    Review of Symbolic Logic 12 (2): 271-295. 2019.
    First-order 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 well-known logics—these two in particular—to formally capture valid and invalid arguments. We show that there is a precise and important sense in which first-order 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
  •  69
    Did Frege commit a cardinal sin?
    Analysis 75 (3): 379-386. 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 first-order concepts to the smaller collection of objects. This article explains what is right and what is wrong with this diagnosis
  •  69
    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
  •  68
    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
  •  65
    Scientific Platonism
    In Mary Leng, Alexander Paseau & Michael Potter (eds.), Mathematical Knowledge, Oxford University Press. pp. 123-149. 2007.
    Does natural science give us reason to believe that mathematical statements are true? And does natural science give us reason to believe in some particular metaphysics of mathematics? These two questions should be firmly distinguished. My argument in this chapter is that a negative answer to the second question is compatible with an affirmative answer to the first. Loosely put, even if science settles the truth of mathematics, it does not settle its metaphysics.
  •  63
    Philosophy of the Matrix
    Philosophia Mathematica 25 (2): 246-267. 2017.
    A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional 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.
  •  62
    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
  •  57
    One true logic: a monist manifesto
    with Owen Griffiths
    Oxford University Press. 2022.
    Logical monism is the claim that there is a single correct logic, the 'one true logic' of our title. The view has evident appeal, as it reflects assumptions made in ordinary reasoning as well as in mathematics, the sciences, and the law. In all these spheres, we tend to believe that there aredeterminate facts about the validity of arguments. Despite its evident appeal, however, logical monism must meet two challenges. The first is the challenge from logical pluralism, according to which there is…Read more
  •  56
    JOHN P. BURGESS Rigor and Structure
    British Journal for the Philosophy of Science 67 (4): 1185-1187. 2016.
  •  55
    Justin Clarke-Doane*Morality and Mathematics
    with Michael Bevan
    Philosophia Mathematica. forthcoming.
  •  53
    Propositionalism
    Journal of Philosophy 118 (8): 430-449. 2021.
    Propositionalism is the claim that all logical relations can be captured by propositional logic. It is usually regarded as obviously false, because propositional logic seems too weak to capture the rich logical structure of language. I show that there is a clear sense in which propositional logic can match first-order logic, by producing formalizations that are valid iff their first-order counterparts are, and also respect grammatical form as the propositionalist construes it. I explain the real…Read more
  •  52
    Deductivism in the Philosophy of Mathematics
    Stanford Encyclopedia of Philosophy 2023. 2023.
    Deductivism says that a mathematical sentence s should be understood as expressing the claim that s deductively follows from appropriate axioms. For instance, deductivists might construe “2+2=4” as “the sentence ‘2+2=4’ deductively follows from the axioms of arithmetic”. Deductivism promises a number of benefits. It captures the fairly common idea that mathematics is about “what can be deduced from the axioms”; it avoids an ontology of abstract mathematical objects; and it maintains that our acc…Read more
  •  50
    On an application of categoricity
    Proceedings of the Aristotelian Society 105 (3). 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 truth-value. My note corrects Walmsley’s formulation of his claim, and shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping that there is a unique …Read more
  •  47
    A focussed issue of The Reasoner on the topic of 'Infinitary Reasoning'. Owen Griffiths and A.C. Paseau were the guest editors.
  •  45
  •  45
    Non-metric Propositional Similarity
    Erkenntnis 87 (5): 2307-2328. 2022.
    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
  •  43
    The Laws of Belief: Ranking Theory & its Philosophical Applications, by SpohnWolfgang. New York: Oxford University Press, 2012. Pp. xv + 598.
  •  40
    One Logic, Or Many?
    Philosophy Now 154 8-9. 2023.
  •  39
    Is English consequence compact?
    Thought: A Journal of Philosophy 10 (3): 188-198. 2021.
    Thought: A Journal of Philosophy, Volume 10, Issue 3, Page 188-198, September 2021.
  •  36
    Erratum to: A measure of inferential-role preservation
    Synthese 194 (4): 1425-1425. 2017.
    Erratum to: Synthese DOI 10.1007/s11229-015-0705-5In 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
  •  35
    Ancestral Links
    The Reasoner 16 (7): 55-56. 2022.
    This short article discusses the fact that the word ‘ancestor’ features in certain arguments that a) are apparently logically valid, b) contain infinitely many premises, and c) are such that none of their finite sub-arguments are logically valid. The article's aim is to motivate, within its brief compass, the study of infinitary logics.
  •  34
    Compactness Theorem
    with Robert Leek
    Internet Encyclopedia of Philosophy. 2022.
    The Compactness Theorem The compactness theorem is a fundamental theorem for the model theory of classical propositional and first-order logic. As well as having importance in several areas of mathematics, such as algebra and combinatorics, it also helps to pinpoint the strength of these logics, which are the standard ones used in mathematics and arguably … Continue reading Compactness Theorem →