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30Proofs of the Compactness TheoremHistory and Philosophy of Logic 32 (4): 407-407. 2011.In this study, the author compares several proofs of the compactness theorem for propositional logic with countably many atomic sentences. He thereby takes some steps towards a systematic philosophical study of the compactness theorem. He also presents some data and morals for the theory of mathematical explanation. [The author is not responsible for the horrific mathematical typo in the second sentence.]
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228Knowledge of Mathematics without ProofBritish 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
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83Fitch's Argument and Typing KnowledgeNotre 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
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92Should 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
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60On an application of categoricityProceedings 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
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110Genuine modal realism and completenessMind 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.
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113Pure Second-Order Logic with Second-Order IdentityNotre 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
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231Mathematical instrumentalism, Gödel’s theorem, and inductive evidenceStudies 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
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99A puzzle about naturalismMetaphilosophy 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
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27Should the Logic of Set Theory be Intuitionistic?: Graduate Papers from the Joint Session 2000Proceedings of the Aristotelian Society 101 (3): 369-378. 2001.The paper critically examines whether the open-endedness of the set concepts mandates the use of intuitionistic logic in set theory, as some philosophers think. [The sequel to this paper is ‘The Open-Endedness of the Set Concept and the Semantics of Set Theory' published in Synthese in 2003.]
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334Proving InductionAustralasian 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 ncorrectly predicts the world’s present state given its previous states at all times apart from a well-ordered subset. On th…Read more
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78How to type: Reply to HalbachAnalysis 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
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317Defining ultimate ontological basis and the fundamental layerPhilosophical 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.
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205Resemblance theories of propertiesPhilosophical 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
Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Epistemology |
Metaphysics |
Philosophy of Religion |