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Nominalism ReconsideredIn Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press. 2005.Nominalism is the view that mathematical objects do not exist. This chapter delimits several types of nominalistic projects: revolutionary programs that attempt to change mathematics and hermeneutic programs that attempt to interpret mathematics. Some programs accord with naturalism, and some oppose naturalism. Steven Yablo’s fictionalism is brought into the fold and discussed at some length.
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1Logic and Philosophical MethodologyIn Herman Cappelen, Tamar Gendler & John Hawthorne (eds.), The Oxford Handbook of Philosophical Methodology, Oxford University Press. 2016.This article explores the role of logic in philosophical methodology, as well as its application in philosophy. The discussion gives a roughly equal coverage to the seven branches of logic: elementary logic, set theory, model theory, recursion theory, proof theory, extraclassical logics, and anticlassical logics. Mathematical logic comprises set theory, model theory, recursion theory, and proof theory. Philosophical logic in the relevant sense is divided into the study of extensions of classical…Read more
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Introduction To Part IIn Richard Jeffrey (ed.), Logic, Logic, and Logic, Harvard University Press. pp. 3-11. 1998.
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12Reflections on" Wang's paradox"Teorema: International Journal of Philosophy 32 (1): 125-139. 2013.
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14This long-awaited volume is a must-read for anyone with a serious interest in\nphilosophy of mathematics. The book falls into two parts, with the primary focus of\nthe first on ontology and structuralism, and the second on intuition and\nepistemology, though with many links between them. The style throughout involves\nunhurried examination from several points of view of each issue addressed, before\nreaching a guarded conclusion. A wealth of material is set before the reader along\nthe way, but …Read more
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35On the outside looking in : a caution about conservativenessIn Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: essays for his centennial, Association For Symbolic Logic. 2010.My contribution to the symposium on Goedel’s philosophy of mathematics at the spring 2006 Association for Symbolic Logic meeting in Montreal. Provisional version: references remain to be added. To appear in an ASL volume of proceedings of the Goedel sessions at that meeting.
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36A revision of a sermon on the evils of calling model theory “semantics”, preached at Notre Dame on Saint Patrick’s Day, 2005. Provisional version: references remain to be added. To appear in Mathematics, Modality, and Models: Selected Philosophical Papers, coming from Cambridge University Press.
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13Occam's razor and scientific methodIn Matthias Schirn (ed.), The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993, Clarendon Press. pp. 195--214. 1998.
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Kripke on modalityIn Otávio Bueno & Scott A. Shalkowski (eds.), The Routledge Handbook of Modality, Routledge. 2018.
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3Saul KripkeIn John Shand (ed.), Central Works of Philosophy, Vol. 5: The Twentieth Century: Quine and After, Acumen Publishing. pp. 166-186. 2006.
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25European and American PhilosophersIn Robert L. Arrington (ed.), A Companion to the Philosophers, Wiley-blackwell. 1991.Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categ…Read more
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4Set TheoryIn Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Wiley-blackwell. 2001.Set theory is the branch of mathematics concerned with the general properties of aggregates of points, numbers, or arbitrary elements. It was created in the late nineteenth century, mainly by Georg Cantor. After the discovery of certain contradictions euphemistically called paradoxes, it was reduced to axiomatic form in the early twentieth century, mainly by Ernst Zermelo and Abraham Fraenkel. Thereafter it became widely accepted as a framework ‐ or ‘foundation’ ‐ for the development of the othe…Read more
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17Lewis on Mereology and Set TheoryIn Barry Loewer & Jonathan Schaffer (eds.), A companion to David Lewis, Wiley-blackwell. 2015.David Lewis in the short monograph Parts of Classes (PC) undertakes a fundamental re‐examination of the relationship between mereology, the general theory of parts, and set theory, the general theory of collections. Given Lewis's theses, to be an element of a set or member of class is just to have a singleton that is a part thereof. Lewis in PC adds a claim of kind of ontological innocence, comparable to that of first‐order logic, for mereology. The only substantive assumption of plethynticology…Read more
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6Quine's Philosophy of Logic and MathematicsIn Ernie Lepore & Gilbert Harman (eds.), A Companion to W. V. O. Quine, Wiley-blackwell. 2013.Thomas Kelly, “Quine and Epistemology”: For Quine, as for many canonical philosophers since Descartes, epistemology stands at the very center of philosophy. In this chapter, I discuss some central themes in Quine's epistemology. I attempt to provide some historical context for Quine's views, in order to make clear why they were seen as such radical challenges to then prevailing orthodoxies within analytic philosophy. I also highlight aspects of his views that I take to be particularly relevant t…Read more
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16Neil Tennant. The Taming of the True. Oxford: Clarendon Press, 1997. Pp. xviii + 466. ISBN 0-19-823717-0 (cloth), 0-19-925160-6 (paper) (review)Philosophia Mathematica 13 (2): 202-215. 2005.
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9When is circularity in definitions benign?Philosophical Quarterly 58 (231). 2007.I aim to show how and why some definitions can be benignly circular. According to Lloyd Humberstone, a definition that is analytically circular need not be inferentially circular and so might serve to illuminate the application-conditions for a concept. I begin by tidying up some problems with Humberstone's account. I then show that circular definitions of a kind commonly thought to be benign have inferentially circular truth-conditions and so are malign by Humberstone's test. But his test is to…Read more
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6Supervaluations and the propositional attitude constraintJournal of Philosophical Logic 26 (1): 103-119. 1997.For the sentences of languages that contain operators that express the concepts of definiteness and indefiniteness, there is an unavoidable tension between a truth-theoretic semantics that delivers truth conditions for those sentences that capture their propositional contents and any model-theoretic semantics that has a story to tell about how indetifiniteness in a constituent affects the semantic value of sentences which imbed it. But semantic theories of both kinds play essential roles, so the…Read more
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2Review: Stewart Shapiro, Foundations without Foundationalism. A Case for Second-Order Logic (review)Journal of Symbolic Logic 58 (1): 363-365. 1993.
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1Shapiro Stewart. Foundations without foundationalism. A case for second-order logic. Oxford logic guides, no. 17. Clarendon Press, Oxford University Press, Oxford and New York 1991, xx + 277 pp (review)Journal of Symbolic Logic 58 (1): 363-365. 1993.
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5Phenomenal qualities and the nontransitivity of matchingAustralasian Journal of Philosophy 68 (2): 206-220. 1990.This Article does not have an abstract
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9Fixing FregePrinceton University Press. 2005.The great logician Gottlob Frege attempted to provide a purely logical foundation for mathematics. His system collapsed when Bertrand Russell discovered a contradiction in it. Thereafter, mathematicians and logicians, beginning with Russell himself, turned in other directions to look for a framework for modern abstract mathematics. Over the past couple of decades, however, logicians and philosophers have discovered that much more is salvageable from the rubble of Frege's system than had previous…Read more
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8Error theories and valuesAustralasian Journal of Philosophy 76 (4). 1998.This Article does not have an abstract
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9Computability and LogicCambridge University Press. 1980.Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel's incompleteness theorems, but also a large number of optional topics, from Turing's theory of computability to Ramsey's theorem. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each c…Read more
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7The sorites paradox and higher-order vaguenessSynthese 85 (3): 417-474. 1990.One thousand stones, suitably arranged, might form a heap. If we remove a single stone from a heap of stones we still have a heap; at no point will the removal of just one stone make sufficient difference to transform a heap into something which is not a heap. But, if this is so, we still have a heap, even when we have removed the last stone composing our original structure. So runs the Sorites paradox. Similar paradoxes can be constructed with any predicate which, like 'heap', displays borderli…Read more
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25Vague Identity: Evans MisrepresentedAnalysis 49 (3). 1989.In 'Vague Identity: Evans Misunderstood' David Lewis defends Gareth Evans against a widespread misunderstanding of an argument that appeared in his article 'Can There be Vague Objects?'. Lewis takes himself to be 'defending Evans' and not just correcting a mistake; witness his remark that, 'As misunderstood, Evans is a pitiful figure: a "technical philosopher" out of control of his technicalities, taken in by a fallacious proof of an absurd conclusion'. Let me say at the outset that I take Lewis…Read more
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