
385Against EthicsEthical Theory and Moral Practice 10 (5): 427439. 2007.This is the verbatim manuscript of a paper which has circulated underground for close to thirty years, reaching a metethical conclusion close to J. L. Mackie’s by a somewhat different route.

319A Subject with No Object: Strategies for Nominalistic Interpretation of MathematicsOxford University Press. 1997.Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obsc…Read more

242Being Explained AwayThe Harvard Review of Philosophy 13 (2): 4156. 2005.When I first began to take an interest in the debate over nominalism in philosophy of mathematics, some twentyodd years ago, the issue had already been under discussion for about a halfcentury. The terms of the debate had been set: W. V. Quine and others had given “abstract,” “nominalism,” “ontology,” and “Platonism” their modern meanings. Nelson Goodman had launched the project of the nominalistic reconstruction of science, or of the mathematics used in science, in which Quine for a time had …Read more

236On a derivation of the necessity of identitySynthese 191 (7): 119. 2014.The source, status, and significance of the derivation of the necessity of identity at the beginning of Kripke’s lecture “Identity and Necessity” is discussed from a logical, philosophical, and historical point of view

228Mathematics and bleak housePhilosophia Mathematica 12 (1): 1836. 2004.The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers

199Which modal models are the right ones (for logical necessity)?Theoria 18 (2): 145158. 2003.Recently it has become almost the received wisdom in certain quarters that Kripke models are appropriate only for something like metaphysical modalities, and not for logical modalities. Here the line of thought leading to Kripke models, and reasons why they are no less appropriate for logical than for other modalities, are explained. It is also indicated where the fallacy in the argument leading to the contrary conclusion lies. The lessons learned are then applied to the question of the status o…Read more

178E pluribus unum: Plural logic and set theoryPhilosophia Mathematica 12 (3): 193221. 2004.A new axiomatization of set theory, to be called BernaysBoolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive settheoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory

171Quine, analyticity and philosophy of mathematicsPhilosophical Quarterly 54 (214). 2004.Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific the…Read more

153Alan Weir. Truth through Proof: A Formalist Foundation for Mathematics. Oxford: Clarendon Press, 2010. ISBN 9780199541492. Pp. xiv+281: Critical Studies/Book Reviews (review)Philosophia Mathematica 19 (2): 213219. 2011.Alan Weir’s new book is, like Darwin’s Origin of Species, ‘one long argument’. The author has devised a new kind of haveitbothways philosophy of mathematics, supposed to allow him to say out of one side of his mouth that the integer 1,000,000 exists and even that the cardinal ℵω exists, while saying out of the other side of his mouth that no numbers exist at all, and the whole book is devoted to an exposition and defense of this new view. The view is presented in the book in a way that can ma…Read more

145The truth is never simpleJournal of Symbolic Logic 51 (3): 663681. 1986.The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

127What is the simplest and most natural axiomatic replacement for the settheoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? Several axiomatic systems, old and new, are examined and ev…Read more

123Charles Parsons. Mathematical thought and its objectsPhilosophia Mathematica 16 (3): 402409. 2008.This longawaited volume is a mustread for anyone with a serious interest in philosophy of mathematics. The book falls into two parts, with the primary focus of the first on ontology and structuralism, and the second on intuition and epistemology, though with many links between them. The style throughout involves unhurried examination from several points of view of each issue addressed, before reaching a guarded conclusion. A wealth of material is set before the reader along the way, but a revi…Read more

121Book Review: Text and Psyche: Experiencing Scripture TodayText and Psyche: Experiencing Scripture Today, byBrownSchuyler. Continuum, New York, 1998. 141pp. $18.95. ISBN 0826411118 (review)Interpretation: A Journal of Bible and Theology 53 (4): 430431. 1999.

118Book Review: Reading the Bible with the Dead: What You Can Learn from the History of Exegesis that You Can't Learn from Exegesis AloneReading the Bible with the Dead: What You Can Learn from the History of Exegesis that You Can't Learn from Exegesis AlonebyThompsonJohn L.Eerdmans, Grand Rapids, 2007. 336 pp. $20.00. ISBN 9780802807533 (review)Interpretation: A Journal of Bible and Theology 62 (3): 332332. 2008.

117The unreal futureTheoria 44 (3): 157179. 1978.Perhaps if the future existed, concretely and individually, as something that could be discerned by a better brain, the past would not be so seductive: its demands would he balanced by those of the future. Persons might then straddle the middle stretch of the seesaw when considering this or that object. It might be fun. But the future has no such reality (as the pictured past and the perceived present possess); the future is but a figure of speech, a specter of thought.

116The Development of Modern LogicHistory and Philosophy of Logic 32 (2). 2011.History and Philosophy of Logic, Volume 32, Issue 2, Page 187191, May 2011

115Quinus ab omni naevo vindicatusIn Ali A. Kazmi (ed.), Meaning and Reference, University of Calgary Press. pp. 2566. 1998.

112Philosophical LogicPrinceton University Press. 2009.Philosophical Logic is a clear and concise critical survey of nonclassical logics of philosophical interest written by one of the world's leading authorities on the subject. After giving an overview of classical logic, John Burgess introduces five central branches of nonclassical logic, focusing on the sometimes problematic relationship between formal apparatus and intuitive motivation. Requiring minimal background and arranged to make the more technical material optional, the book offers a choi…Read more

107Book Review: The Word of God for the People of God: An Entryway to the Theological Interpretation of ScriptureThe Word of God for the People of God: An Entryway to the Theological Interpretation of Scripture by BillingsJ. ToddEerdmans, Grand Rapids, 2010. 235 pp. $18.00. ISBN 9780802862358 (review)Interpretation: A Journal of Bible and Theology 65 (3): 328329. 2011.

101Mary Leng. Mathematics and Reality. Oxford: Oxford University Press, 2010. ISBN 9780199280797. Pp. x + 278: Critical Studies/Book Reviews (review)Philosophia Mathematica 18 (3): 337344. 2010.(No abstract is available for this citation)

100Which Modal Logic Is the Right One?Notre Dame Journal of Formal Logic 40 (1): 8193. 1999.The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about modeltheoretic validity, the other about prooftheoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Supecki and Bryll, is presented for the claim that the right demonstrability logi…Read more

77New Foundations for Physical Geometry: The Theory of Linear Structures, by Tim Maudlin: Oxford: Oxford University Press, 2014, pp. x + 363, £50.00 (review)Australasian Journal of Philosophy 93 (1): 187190. 2015.

76Dummett's case for intuitionismHistory and Philosophy of Logic 5 (2): 177194. 1984.Dummett's case against platonism rests on arguments concerning the acquisition and manifestation of knowledge of meaning. Dummett's arguments are here criticized from a viewpoint less Davidsonian than Chomskian. Dummett's case against formalism is obscure because in its prescriptive considerations are not clearly separated from descriptive. Dummett's implicit value judgments are here made explicit and questioned. ?Combat Revisionism!? Chairman Mao

75Quick completeness proofs for some logics of conditionalsNotre Dame Journal of Formal Logic 22 (1): 7684. 1981.

72Book Review: The Ten Commandments: A Preaching CommentaryThe Ten Commandments: A Preaching CommentarybyHolbertJohn C.Abingdon, Nashville, 2002. 143 pp. $18.00. ISBN 0687090482 (review)Interpretation: A Journal of Bible and Theology 57 (4): 452452. 2003.

71TruthPrinceton University Press. 2011.This is a concise, advanced introduction to current philosophical debates about truth. A blend of philosophical and technical material, the book is organized around, but not limited to, the tendency known as deflationism, according to which there is not much to say about the nature of truth. In clear language, Burgess and Burgess cover a wide range of issues, including the nature of truth, the status of truthvalue gaps, the relationship between truth and meaning, relativism and pluralism about …Read more
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