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243What’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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100Frege'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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163Comments 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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52B. 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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111Should 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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25ParadoxesPolity. 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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115If 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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34Vagueness and MeaningIn Giuseppina Ronzitti (ed.), Vagueness: A Guide, Springer Verlag. pp. 83--106. 2011.
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133Do Comics Require Pictures? Or Why Batman #663 Is a ComicJournal of Aesthetics and Art Criticism 69 (3): 285-296. 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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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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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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51Mathematics, 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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80Frege'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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99Conservativeness, 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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158Still counterintuitive: A reply to KremerAnalysis 63 (3). 2003.In (2002) I argued that Gupta and Belnap’s Revision Theory of Truth (1993) has counterintuitive consequences. In particular, the pair of sentences: (S1) At least one of S1 and S2 is false. (S2) Both of S1 and S2 are false.1 is pathological on the Revision account. There is one, and only one, assignment of truth values to {(S1), (S2)} that make the corresponding Tarski..
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38Yablo Paradox. 2015.The Yablo Paradox The Yablo Paradox implies there is no way to coherently assign a truth value to any of the sentences in the countably infinite sequence of sentences, each of the form, “All of the subsequent sentences are false.” Specifically, the Yablo Paradox arises when we consider the following infinite sequence of sentences: The … Continue reading Yablo Paradox →.
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110Patricia A. Blanchette. Frege's Conception of Logic. Oxford University Press, 2012. ISBN 978-0-19-926925-9 (hbk). Pp. xv + 256 (review)Philosophia Mathematica (1). 2013.
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52Iteration one more timeNotre Dame Journal of Formal Logic 44 (2): 63--92. 2003.A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that ther…Read more
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192What is a Truth Value And How Many Are There?Studia Logica 92 (2): 183-201. 2009.Truth values are, properly understood, merely proxies for the various relations that can hold between language and the world. Once truth values are understood in this way, consideration of the Liar paradox and the revenge problem shows that our language is indefinitely extensible, as is the class of truth values that statements of our language can take – in short, there is a proper class of such truth values. As a result, important and unexpected connections emerge between the semantic paradoxes…Read more
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89Charles E. Rickart. Structuralism and Structures: A Mathematical Perspective. Singapore: World Scientific Publishing, 1995. pp. xiii + 219. ISBN 981-02-1860-5 (review)Philosophia Mathematica 6 (2): 227-231. 1998.
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148There is No Paradox of Logical ValidityLogica Universalis 8 (3-4): 447-467. 2014.A number of authors have argued that Peano Arithmetic supplemented with a logical validity predicate is inconsistent in much the same manner as is PA supplemented with an unrestricted truth predicate. In this paper I show that, on the contrary, there is no genuine paradox of logical validity—a completely general logical validity predicate can be coherently added to PA, and the resulting system is consistent. In addition, this observation lead to a number of novel, and important, insights into th…Read more
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82Aristotelian logic, axioms, and abstractionPhilosophia Mathematica 11 (2): 195-202. 2003.Stewart Shapiro and Alan Weir have argued that a crucial part of the demonstration of Frege's Theorem (specifically, that Hume's Principle implies that there are infinitely many objects) fails if the Neo-logicist cannot assume the existence of the empty property, i.e., is restricted to so-called Aristotelian Logic. Nevertheless, even in the context of Aristotelian Logic, Hume's Principle implies much of the content of Peano Arithmetic. In addition, their results do not constitute an objection to…Read more
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71RICHARD G. HECK, Jr. Frege's Theorem. Oxford: Clarendon Press, 2011. ISBN 978-0-19-969564-5. Pp. xiv + 307Philosophia Mathematica 20 (3): 346-359. 2012.
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64Necessity, Necessitism, and NumbersPhilosophical Forum 47 (3-4): 385-414. 2016.Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to a…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 |