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356Response to my criticsAnálisis Filosófico 32 (1): 69-97. 2012.During the Winter of 2011 I visited SADAF and gave a series of talks based on the central chapters of my manuscript on the Yablo paradox. The following year, I visited again, and was pleased and honored to find out that Eduardo Barrio and six of his students had written ‘responses’ that addressed the claims and arguments found in the manuscript, as well as explored new directions in which to take the ideas and themes found there. These comments reflect my thoughts on these responses (also collec…Read more
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111Alethic pluralism, generic truth, and mixed conjunctionsPhilosophical Quarterly 61 (244): 624-629. 2011.A difficulty for alethic pluralism has been the idea that semantic evaluation of conjunctions whose conjuncts come from discourses with distinct truth properties requires a third notion of truth which applies to both of the original discourses. But this line of reasoning does not entail that there exists a single generic truth property that applies to all statements and all discourses, unless it is supplemented with additional, controversial, premises. So the problem of mixed conjunctions, while…Read more
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1New waves on an old beach: Fregean philosophy of mathematics todayIn Ø. Linnebo O. Bueno (ed.), New Waves in Philosophy of Mathematics, . 2009.
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141Hume’s Big Brother: counting concepts and the bad company objectionSynthese 170 (3). 2009.A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of …Read more
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200Curry, Yablo and dualityAnalysis 69 (4): 612-620. 2009.The Liar paradox is the directly self-referential Liar statement: This statement is false.or : " Λ: ∼ T 1" The argument that proceeds from the Liar statement and the relevant instance of the T-schema: " T ↔ Λ" to a contradiction is familiar. In recent years, a number of variations on the Liar paradox have arisen in the literature on semantic paradox. The two that will concern us here are the Curry paradox, 2 and the Yablo paradox. 3The Curry paradox demonstrates that neither negation nor a falsi…Read more
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171Vagueness and mathematical precisionMind 111 (442): 225-247. 2002.One of the main reasons for providing formal semantics for languages is that the mathematical precision afforded by such semantics allows us to study and manipulate the formalization much more easily than if we were to study the relevant natural languages directly. Michael Tye and R. M. Sainsbury have argued that traditional set-theoretic semantics for vague languages are all but useless, however, since this mathematical precision eliminates the very phenomenon (vagueness) that we are trying to …Read more
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11There Are Non-circular Paradoxes (But Yablo’s Isn't One of Them!)The Monist 89 (1): 118-149. 2006.
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56Canonicity and Normativity in Massive, Serialized, Collaborative FictionJournal of Aesthetics and Art Criticism 71 (3): 271-276. 2013.
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327Patterns of paradoxJournal of Symbolic Logic 69 (3): 767-774. 2004.We begin with a prepositional languageLpcontaining conjunction (Λ), a class of sentence names {Sα}αϵA, and a falsity predicateF. We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names {Sβ}βϵB,is a well-formed formula (we will useWFFto denote the set of well-formed formulae).The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation functi…Read more
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487Abstraction and identityDialectica 59 (2). 2005.A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.
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468Let a thousand flowers Bloom: A tour of logical pluralismPhilosophy Compass 5 (6): 492-504. 2010.Logical pluralism is the view that there is more than one correct logic. In this article, I explore what logical pluralism is, and what it entails, by: (i) distinguishing clearly between relativism about a particular domain and pluralism about that domain; (ii) distinguishing between a number of forms logical pluralism might take; (iii) attempting to distinguish between those versions of pluralism that are clearly true and those that are might be controversial; and (iv) surveying three prominent…Read more
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93Frege's Cardinals and Neo-LogicismPhilosophia Mathematica 24 (1): 60-90. 2016.Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternativ…Read more
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238What’s Wrong with TonkJournal of Philosophical Logic 34 (2). 2005.In “The Runabout Inference Ticket” AN Prior (1960) examines the idea that logical connectives can be given a meaning solely in virtue of the stipulation of a set of rules governing them, and thus that logical truth/consequence
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126Comments on Patricia Blanchette's Book: Frege's Conception of Logic (review)Journal for the History of Analytical Philosophy 3 (7). 2015.All contributions included in the present issue were originally presented at an ‘Author Meets Critics’ session organised by Richard Zach at the Pacific Meeting of the American Philosophical Association in San Diego in the Spring of 2014.
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110Should Anti-Realists be Anti-Realists About Anti-Realism?Erkenntnis 79 (S2): 233-258. 2014.On the Dummettian understanding, anti-realism regarding a particular discourse amounts to (or at the very least, involves) a refusal to accept the determinacy of the subject matter of that discourse and a corresponding refusal to assert at least some instances of excluded middle (which can be understood as expressing this determinacy of subject matter). In short: one is an anti-realist about a discourse if and only if one accepts intuitionistic logic as correct for that discourse. On careful exa…Read more
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49B. Jack Copeland, Carl J. Posy, and Oron Shagrir, eds, Computability: Turing, Gödel, Church, and Beyond. Cambridge, Mass.: MIT Press, 2013. ISBN 978-0-262-01899-9. Pp. x + 362 (review)Philosophia Mathematica 22 (3): 412-413. 2014.
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24ParadoxesPolity. 2013.Paradoxes are arguments that lead from apparently true premises, via apparently uncontroversial reasoning, to a false or even contradictory conclusion. Paradoxes threaten our basic understanding of central concepts such as space, time, motion, infinity, truth, knowledge, and belief. In this volume Roy T Cook provides a sophisticated, yet accessible and entertaining, introduction to the study of paradoxes, one that includes a detailed examination of a wide variety of paradoxes. The book is organi…Read more
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110If A then B: How the World Discovered LogicHistory and Philosophy of Logic 35 (3): 301-303. 2014.If A then B: How the World Discovered Logic is a historically oriented introduction to the basic notions of logic. In particular, and in the words of the authors, it is focused on the idea that ‘lo...
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126Do Comics Require Pictures? Or Why Batman #663 Is a ComicJournal of Aesthetics and Art Criticism 69 (3): 285-296. 2011.
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34Vagueness and MeaningIn Giuseppina Ronzitti (ed.), Vagueness: A Guide, Springer Verlag. pp. 83--106. 2011.
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30The Arché Papers on the Mathematics of Abstraction (edited book)Springer. 2007.Unique in presenting a thoroughgoing examination of the mathematical aspects of the neo-logicist project (and the particular philosophical issues arising from these technical concerns).
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94Hintikka's Revolution: The Priciples of Mathematics Revisited (review)British Journal for the Philosophy of Science 49 (2): 309-316. 1998.
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3Appendix: How to read GrundgesetzeIn Gottlob Frege (ed.), Basic Laws of Arithmetic, Oxford University Press. 1893.This appendix is intended to assist the reader in becoming comfortable with the notations, rules, and definitions of Frege's Grundgesetze
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48Mathematics, Models, and ModalityHistory and Philosophy of Logic 31 (3): 287-289. 2010.John P. Burgess, Mathematics, Models, and Modality: Selected Philosophical Essays. Cambridge: Cambridge University Press, 2008. xiii + 301 pp. $90.00, £50.00. ISBN 978-0-521-88034-3. Adobe eBook, $...
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79Frege's RecipeJournal of Philosophy 113 (7): 309-345. 2016.In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and explain how it differs f…Read more
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118The state of the economy: Neo-logicism and inflationPhilosophia Mathematica 10 (1): 43-66. 2002.In this paper I examine the prospects for a successful neo–logicist reconstruction of the real numbers, focusing on Bob Hale's use of a cut-abstraction principle. There is a serious problem plaguing Hale's project. Natural generalizations of this principle imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. I also in…Read more
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96Conservativeness, Stability, and AbstractionBritish Journal for the Philosophy of Science 63 (3): 673-696. 2012.One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principl…Read more
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University of St. Andrews3- Year Post-doctoral Fellow
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University of MinnesotaTenured
Ohio State University
PhD, 2000
St Andrews, United Kingdom of Great Britain and Northern Ireland
Areas of Specialization
Science, Logic, and Mathematics |
PhilPapers Editorships
Theories of Mathematics |