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93Early Analytic Philosophy: Frege, Russell, Wittgenstein : Essays in Honor of Leonard Linsky (edited book)Open Court. 1996.These essays present new analyses of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.
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78Review: H. G. Rice, On Completely Recursively Enumerable Classes and Their Key Arrays (review)Journal of Symbolic Logic 23 (1): 48-48. 1958.
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76The Hilton New York Hotel New York, NY December 27–29, 2005Bulletin of Symbolic Logic 12 (3). 2006.
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339Gödel's reformulation of Gentzen's first consistency proof for arithmetic: The no-counterexample interpretationBulletin of Symbolic Logic 11 (2): 225-238. 2005.The last section of “Lecture at Zilsel’s” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Godel's reformulation, and discuss the relation between the two proofs.
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74William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gödel.
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45Extensional Equality in the Classical Theory of TypesVienna Circle Institute Yearbook 3 219-234. 1995.The classical theory of types in question is essentially the theory of Martin-Löf [1] but with the law of double negation elimination. I am ultimately interested in the theory of types as a framework for the foundations of mathematics and, for this purpose, we need to consider extensions of the theory obtained by adding ‘well-ordered types,’ for example the type N of the finite ordinals; but the unextended theory will suffice to illustrate the treatment of extensional equality
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46A Nonconstructive Proof of Gentzen's Hauptsatz for Second Order Predicate LogicJournal of Symbolic Logic 33 (2): 289-290. 1968.
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309The completeness of Heyting first-order logicJournal of Symbolic Logic 68 (3): 751-763. 2003.Restricted to first-order formulas, the rules of inference in the Curry-Howard type theory are equivalent to those of first-order predicate logic as formalized by Heyting, with one exception: ∃-elimination in the Curry-Howard theory, where ∃x : A.F (x) is understood as disjoint union, are the projections, and these do not preserve firstorderedness. This note shows, however, that the Curry-Howard theory is conservative over Heyting’s system.
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235Proof-theoretic Semantics for Classical MathematicsSynthese 148 (3): 603-622. 2006.We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equalit…Read more
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98Kurt Godel. Collected Works. Volume IV: Selected Correspondence AG; Volume V: Selected Correspondence HZPhilosophia Mathematica 14 (1): 76. 2006.
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4Zermelo's Conception of Set Theory and Reflection PrinciplesIn Matthias Schirn (ed.), The Philosophy of Mathematics Today, Clarendon Press. 2003.
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260Gödel's Correspondence on Proof Theory and Constructive Mathematics †Charles Parsons read part of an early draft of this review and made important corrections and suggestionsPhilosophia Mathematica 14 (1): 76-111. 2006.
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160Curtis Franks The Autonomy of Mathematical Knowledge: Hilbert's Program RevisitedHistory and Philosophy of Logic 32 (2). 2011.History and Philosophy of Logic, Volume 32, Issue 2, Page 177-183, May 2011
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76The Palmer House Hilton Hotel, Chicago, Illinois April 19–21, 2007Bulletin of Symbolic Logic 13 (4). 2007.
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145The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.
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To appear in the Proceedings of Logic Colloquium 2006. (32 pages).
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268Intensional interpretations of functionals of finite type IJournal of Symbolic Logic 32 (2): 198-212. 1967.
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To appear in the Proceedings of Logic Colloquium 2006. (28 pages).
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153Meeting of the association for symbolic logic: Biloxi, 1979Journal of Symbolic Logic 46 (1): 191-198. 1981.
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67Grzegorczyk A.. Some proofs of undecidability of arithmetic. Fundamenta mathematicae, vol. 43 , pp. 166–177Journal of Symbolic Logic 23 (1): 46-47. 1958.
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104Kleene S. C.. Extension of an effectively generated class of functions by enumeration. Colloquium mathematicum, vol. 6 , pp. 68–78Journal of Symbolic Logic 25 (3): 279-280. 1960.
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