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82Natural 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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72In 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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78A 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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39Reconciling 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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1221What 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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39The middle chapters of Soames’s The World Philosophy Made are briefly summarized and examined. There are some local slips, but globally the work displays an impressive knowledge of and a distinctive viewpoint on a wide range of important intellectual disciplines and their original roots in and continuing connections with philosophy.
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Set TheoryCambridge University Press. 2022.Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, con…Read more
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63Measurable Selections: A Bridge Between Large Cardinals and Scientific Applications?†Philosophia Mathematica 29 (3): 353-365. 2021.There is no prospect of discovering measurable cardinals by radio astronomy, but this does not mean that higher set theory is entirely irrelevant to applied mathematics broadly construed. By way of example, the bearing of some celebrated descriptive-set-theoretic consequences of large cardinals on measurable-selection theory, a body of results originating with a key lemma in von Neumann’s work on the mathematical foundations of quantum theory, and further developed in connection with problems of…Read more
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Cats, Dogs, and So OnIn Dean Zimmerman (ed.), Oxford Studies in Metaphysics: Volume 4, Oxford University Press Uk. 2008.
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51Luca Incurvati* Conceptions of Set and the Foundations of MathematicsPhilosophia Mathematica 28 (3): 395-403. 2020.
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70Platonism and Anti-Platonism in MathematicsPhilosophical Review 110 (1): 79. 2001.Mathematics tells us there exist infinitely many prime numbers. Nominalist philosophy, introduced by Goodman and Quine, tells us there exist no numbers at all, and so no prime numbers. Nominalists are aware that the assertion of the existence of prime numbers is warranted by the standards of mathematical science; they simply reject scientific standards of warrant.
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4Christ and Culture RevisitedJournal of the Society of Christian Ethics 31 (2): 55-74. 2011.WESTERN SCHOLARS HAVE POINTED OUT BOTH THE USEFULNESS AND limitations of H. Richard Niebuhr's Christ and Culture. This essay relates Niebuhr's five types to discussions of church and culture in contemporary Russian Orthodoxy. I propose a sixth type, Christ in culture, that best illuminates the Church's current program of votserkovlenie. To its Russian representatives, "Christ in culture" enabled the Christian faith to survive communist efforts to destroy the Church, and this cultural legacy cont…Read more
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6Making the Best of It: Following Christ in the Real WorldJournal of the Society of Christian Ethics 31 (1): 225-227. 2011.
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67Truth and the Absence of FactPhilosophical Review 111 (4): 602-604. 2002.This volume reprints a dozen of the author’s papers, most with substantial postscripts, and adds one new one. The bulk of the material is on topics in philosophy of language, but there are also two papers on philosophy of mathematics written after the appearance of the author’s collected papers on that subject, and one on epistemology. As to the substance of Field’s contributions, limitations of space preclude doing much more below than indicating the range of issues addressed, and the general o…Read more
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32Platonism and anti-platonism in mathematicsPhilosophical Review 110 (1): 79-82. 2001.Mathematics tells us there exist infinitely many prime numbers. Nominalist philosophy, introduced by Goodman and Quine, tells us there exist no numbers at all, and so no prime numbers. Nominalists are aware that the assertion of the existence of prime numbers is warranted by the standards of mathematical science; they simply reject scientific standards of warrant.
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34Charles Parsons, Mathematics in Philosophy: Selected Essays. Ithaca, NY: Cornell University Press (2005), 368 pp., $35.00 (paper) (review)Philosophy of Science 74 (4): 549-552. 2007.
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24C. L. Hamblin. The modal “probably.”Mind, n.s. vol. 68 , pp. 234–240Journal of Symbolic Logic 35 (4): 582-583. 1970.
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14Robert Vaught. Descriptive set theory in Lω,1ω, Cambridge summer school in mathematical logic, held in Cambridge/England, August 1–21, 1971, edited by A.R.D. Mathias and H. Rogers, Lecture notes in mathematics, vol. 337, Springer-Verlag, Berlin, Heidelberg, and New York, 1973, pp. 574–598. - Robert Vaught. Invariant sets in topology and logic. Fundamenta mathematicae, vol. 82 no. 3 , pp. 269–294 (review)Journal of Symbolic Logic 47 (1): 217-218. 1982.
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28George Boolos. To be is to be a value of a variable . The journal of philosophy, vol. 81 , pp. 430–449. - George Boolos. Nominalist Platonism, The philosophical review, vol. 94 , pp. 327–344 (review)Journal of Symbolic Logic 54 (2): 616-617. 1989.
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15Hailperin Theodore. Sentential probability logic. Origins, development, current status, and technical applications. Lehigh University Press, Bethlehem, Pennsylvania, and Associated University Presses, London, 1996, 304 pp (review)Journal of Symbolic Logic 62 (3): 1040-1041. 1997.
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23Jonathan Bennett. A philosophical guide to conditionals. Clarendon Press, Oxford, 2003, viii + 388 pp (review)Bulletin of Symbolic Logic 10 (4): 565-570. 2004.
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12Jody Azzouni. Deflating existential consequence: a case for nominalism. Oxford University Press, Oxford, 2004, viii + 342 pp (review)Bulletin of Symbolic Logic 10 (4): 573-577. 2004.
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