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24Proofs about Proofs: a defense of classical logic. Part I: the aims of classical logicIn Michael Detlefsen (ed.), Proof, Logic and Formalization, Routledge. 1992.
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76Thomas McKay. Plural predicationPhilosophia Mathematica 16 (1): 133-140. 2008.This work, the first book-length study of its topic, is an important contribution to the literature of philosophical logic and philosophy of language, with implications for other branches of philosophy, including philosophy of mathematics. However, five of the book's ten chapters , including many of the author's most original contributions, are devoted to issues about natural language, and lie pretty well outside the scope of this journal, not to mention that of the reviewer's competence. For th…Read more
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290Mathematics and bleak housePhilosophia Mathematica 12 (1): 18-36. 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
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Read, Stephen, "Relevant Logic: A Philosophical Examination of Inference" (review)Mind 99 (n/a): 140. 1990.
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11Chapter Three. Modal LogicIn J. W. Davis (ed.), Philosophical logic, D. Reidel. pp. 40-70. 1969.
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39The completeness of intuitionistic propositional calculus for its intended interpretationNotre Dame Journal of Formal Logic 22 (1): 17-28. 1981.
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175Alan Weir. Truth through Proof: A Formalist Foundation for Mathematics. Oxford: Clarendon Press, 2010. ISBN 978-0-19-954149-2. Pp. xiv+281: Critical Studies/Book Reviews (review)Philosophia Mathematica 19 (2): 213-219. 2011.Alan Weir’s new book is, like Darwin’s Origin of Species, ‘one long argument’. The author has devised a new kind of have-it-both-ways 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
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69One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.
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17Careful choices---a last word on Borel selectorsNotre Dame Journal of Formal Logic 22 (3): 219-226. 1981.
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36No requirement of relevanceIn Stewart Shapiro (ed.), Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press. pp. 727--750. 2005.There are schools of logicians who claim that an argument is not valid unless the conclusion is relevant to the premises. In particular, relevance logicians reject the classical theses that anything follows from a contradiction and that a logical truth follows from everything. This chapter critically evaluates several different motivations for relevance logic, and several systems of relevance logic, finding them all wanting.
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32Synthetic mechanics revisitedJournal of Philosophical Logic 20 (2). 1991.Earlier results on eliminating numerical objects from physical theories are extended to results on eliminating geometrical objects
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47KripkePolity. 2012.Saul Kripke has been a major influence on analytic philosophy and allied fields for a half-century and more. His early masterpiece, _Naming and Necessity_, reversed the pattern of two centuries of philosophizing about the necessary and the contingent. Although much of his work remains unpublished, several major essays have now appeared in print, most recently in his long-awaited collection _Philosophical Troubles_. In this book Kripke’s long-time colleague, the logician and philosopher John P. B…Read more
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48Review of B. Hale and A. Hoffmann (eds.), Modality: Metaphysics, Logic, and Epistemology (review)Notre Dame Philosophical Reviews 2010 (10). 2010.
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231E pluribus unum: Plural logic and set theoryPhilosophia Mathematica 12 (3): 193-221. 2004.A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic 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
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135Charles Parsons. Mathematical thought and its objectsPhilosophia Mathematica 16 (3): 402-409. 2008.This long-awaited volume is a must-read 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
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106Quick completeness proofs for some logics of conditionalsNotre Dame Journal of Formal Logic 22 (1): 76-84. 1981.
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77Probability logicJournal of Symbolic Logic 34 (2): 264-274. 1969.In this paper we introduce a system S5U, formed by adding to the modal system S5 a new connective U, Up being read “probably”. A few theorems are derived in S5U, and the system is provided with a decision procedure. Several decidable extensions of S5U are discussed, and probability logic is related to plurality quantification.
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12It is shown that for invariance under the action of special groups the statements "Every invariant PCA is decomposable into (1 invariant Borel sets" and "Every pair of invariant PCA is reducible by a pair of invariant PCA sets" are independent of the axioms of set theory.
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42Sets and Point-Sets: Five Grades of Set-Theoretic Involvement in GeometryPSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988. 1988.The consequences for the theory of sets of points of the assumption of sets of sets of points, sets of sets of sets of points, and so on, are surveyed, as more generally are the differences among the geometric theories of points, of finite point-sets, of point-sets, of point-set-sets, and of sets of all ranks.
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75Hintikka et Sandu versus Frege in re Arbitrary FunctionsPhilosophia Mathematica 1 (1): 50-65. 1993.Hintikka and Sandu have recently claimed that Frege's notion of function was substantially narrower than that prevailing in real analysis today. In the present note, their textual evidence for this claim is examined in the light of relevant historical and biographical background and judged insufficient.
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17Chapter Two. Temporal LogicIn J. W. Davis (ed.), Philosophical logic, D. Reidel. pp. 13-39. 1969.
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11Review: Hao Wang, From Mathematics to Philosophy (review)Journal of Symbolic Logic 42 (4): 579-580. 1977.
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35The decision problem for linear temporal logicNotre Dame Journal of Formal Logic 26 (2): 115-128. 1985.
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226Quine, 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
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