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120Occam'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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6Saul 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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15European and American PhilosophersIn Robert L. Arrington (ed.), A Companion to the Philosophers, Blackwell. 2017.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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5Set TheoryIn Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell. 2017.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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19Lewis on Mereology and Set TheoryIn Barry Loewer & Jonathan Schaffer (eds.), A Companion to David Lewis, Wiley. 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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10Quine'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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67Neil 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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145When 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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99Supervaluations 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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6Shapiro 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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32Review: 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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79Phenomenal 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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89Error theories and valuesAustralasian Journal of Philosophy 76 (4). 1998.This Article does not have an abstract
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180Computability 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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210The 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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184Vague 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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80Natural deduction rules for a logic of vaguenessErkenntnis 27 (2): 197-229. 1987.Extant semantic theories for languages containing vague expressions violate intuition by delivering the same verdict on two principles of classical propositional logic: the law of noncontradiction and the law of excluded middle. Supervaluational treatments render both valid; many-Valued treatments, Neither. The core of this paper presents a natural deduction system, Sound and complete with respect to a 'mixed' semantics which validates the law of noncontradiction but not the law of excluded midd…Read more
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69In defense of an indeterminist theory of vaguenessThe Monist 81 (1): 233--52. 1998.Regardless of the theory of vagueness we adhere to, we all agree that no facts, known or practically knowable, suffice to determine the location of precise boundaries for vague concepts. According to the epistemic theory of vagueness, this ignorance is entirely an epistemic matter—vague concepts have sharp boundaries but we can never know their exact locations. Opposed to epistemicism is a view—or family of views—I shall call indeterminism. The indeterminist agrees with the epistemicist that we …Read more
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81A Subject with no ObjectPhilosophical Review 108 (1): 106. 1999.This is the first systematic survey of modern nominalistic reconstructions of mathematics, and for this reason alone it should be read by everyone interested in the philosophy of mathematics and, more generally, in questions concerning abstract entities. In the bulk of the book, the authors sketch a common formal framework for nominalistic reconstructions, outline three major strategies such reconstructions can follow, and locate proposals in the literature with respect to these strategies. The …Read more
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On 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.
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Logic, Mathematics, Science. Quine's Philosophy of Logic and MathematicsIn Gilbert Harman & Ernest LePore (eds.), A Companion to W. V. O. Quine, Wiley-blackwell. 2013.
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38Reconciling Anti-Nominalism and Anti-Platonism in Philosophy of MathematicsDisputatio 11 (20). 2022.The author reviews and summarizes, in as jargon-free way as he is capable of, the form of anti-platonist anti-nominalism he has previously developed in works since the 1980s, and considers what additions and amendments are called for in the light of such recently much-discussed views on the existence and nature of mathematical objects as those known as hyperintensional metaphysics, natural language ontology, and mathematical structuralism.
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1066What is Mathematical Rigor?Aphex 25 1-17. 2022.Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.
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