PUBLICATIONS

Books:

Vagueness in context, Oxford, Oxford University Press, 2006; reissued in paperback, March, 2008.

Thinking about mathematics: The philosophy of mathematics, Oxford, Oxford University Press, 2000.

Philosophy of mathematics: Structure and ontology, Oxford, Oxford University Press, 1997; reissued in paperback, Autumn 2000.

Foundations without foundationalism: A case for second-order logic, Oxford Logic Guides 17, Oxford, Oxford University Press, 1991, reissued in paperback, Summer 2000.

Editorships:

Oxford handbook of the philosophy of logic and mathematics, Oxford University Press, 2005.

Special issue of History and Philosophy of Logic 20 (2000), a Festschrift for John Corcoran (edited with Michael Scanlan).

Special issue of Philosophia Mathematica (3) 8 (2000), devoted to abstraction and neo-logicism.

Two special issues of Philosophia Mathematica (3) 7 (1999), 9 (2001), devoted to the proceedings of a conference in memory of George Boolos, held at Notre Dame, during the April of 1998.

The limits of logic: Second-order logic and the Skolem paradox, The international research library of philosophy, Dartmouth Publishing Company, 1996.

Special issue of Philosophia Mathematica (3) 4 (1996), devoted to structuralism. Contributors: P. Benacerraf, G. Hellman, B. Hale, C. Parsons, M. Resnik, S. Shapiro.

Intensional Mathematics, Studies in Logic and the Foundations of Mathematics 113, Amsterdam, North Holland Publishing Company, 1985, Contributors: S. Shapiro, J. Myhill, N. D. Goodman, A. Scedrov, V. Lifschitz, R. Flagg, R. Smullyan. Articles:

“Towards a point-free account of the continuous” (with Geoffrey Hellman), Iyyun 61, 2012, 263- 287.

“Revising logic in light of paradox”, in Insolubles and consequences: essays in honor of Stephen Read, Tributes 18, edited by Catarina Dutilh and Ole Hjortland, College Publications, 2012, 213-237.

“An ‘i’ for an i: singular terms, uniqueness, and reference, Review of Symbolic Logic 5 (2012), 380-415.

“On Richard’s When truth gives out” (with Kevin Scharp), Philosophical Studies 160 (2012), 455-463.

“Higher-order logic and set theory: a false dilemma”, Philosophia Mathematica (3) 20 (2012), 305-323.

“Objectivity, explanation, and cognitive shortfall”, in Mind, meaning, and knowledge: themes from the philosophy of Crispin Wright, edited by Annalisa Coliva, Oxford, Oxford University Press, 2012, 211-237.

“The company kept by cut abstraction (and its relatives)”, Philosophia Mathematica (3) 19, (2011), 107-138.

“Mathematics and objectivity”, in Meaning in mathematics, edited by John Polkinghorne, Oxford, Oxford University Press, 2011, 97-108.

“Reply to Gideon Rosen”, in Meaning in mathematics, edited by John Polkinghorne, Oxford, Oxford University Press, 2011, 112.

“Addendum on Peter Lipton’s ‘Mathematical understanding’”, in Meaning in mathematics, edited by John Polkinghorne, Oxford, Oxford University Press, 2011, 55-59.

“Theology and the actual infinite: Burley and Cantor”, Theology and Science 9 (2011), 101-108.

“Foundations: structures, sets, and categories”, in Foundational theories of classical and constructive mathematics”, Western Ontario Series in the Philosophy of Science, edited by Giovanni Soimmaruga, Dordrecht, Springer, 2011, 97-110.

“Vagueness and logic”, in Vagueness, a guide, edited by Giuseppina Ronzitti, Dordrecht, Springer, 2011, 55-81.

“Varieties of pluralism and relativism for logic”, in A companion to relativism, edited by Steven D. Hales, Oxford, Wiley-Blackwell, 2011, 526-555. “Epistemology of mathematics: What are the questions? What count as answers?”, Philosophical Quarterly 61 (2011), 130-150

“Vagueness, metaphysics, and objectivity”, Cuts and clouds: vagueness, its nature, and its logic, edited by Richard Dietz and Sebastiano Moruzzi, Oxford, Oxford University Press, 2010, 149-162.

“So truth is safe from paradox: now what?”, Philosophical Studies 147 (2010), 445-455.

“Life on the ship of Neurath: mathematics in the philosophy of mathematics”, Croatian Journal of Philosophy 9 (2009), 149-166; reprinted in Between logic and reality, edited by Majda Trobok, Nenad Mišcevic and Berislav Žarnic, Dodrecht, Springer, 2012, 11-27.

“The good, the bad, and the ugly” (with Philip Ebert), Synthese 170 (2009), 415-441.

“Reference to indiscernible objects”, The Logica yearbook 2008, edited by Michal Peliš, London, College Publications, 2008, 223-235.

“We hold these truths to be self evident: But what do we mean by that?”, Review of Symbolic Logic 2 (2009), 175-207.

“The measure of Scottish neo-logicism”, Logicism, intuitionism, and formalism, edited by Sten Lindström et al., Synthese Library 341, Springer, 2009, 69-90.

“Reasoning with slippery predicates”, Studia Logica 90 (2008), 313-336.

“Identity, indiscernibility, and ante rem structuralism: the tale of i and -i”, Philosophia Mathematica (3) 16 (2008), 285-309.

“Frege meets Zermelo: a perspective on ineffability and reflection” (with Gabriel Uzquiano), Review of Symbolic Logic 1 (2008), 241-266.

“Mathematical objects”, in Proof and other dilemmas: mathematics and philosophy, edited by Bonnie Gold and Roger A. Simons, Washington, D.C., The Mathematical Association of America, 2008, 131-156.

“The objectivity of mathematics”, Synthese 156 (2007), 337-381.

“Where in the (world wide) web of belief is the law of non-contradiction?” (with Jack Arnold), Nous 41 (2007) , 276–297.

“Burali-Forti’s revenge, Revenge of the liar, edited by J. C. Beall, Oxford, Oxford University Press, 2007, 320-344.

“Corcoran the mathematician”, Current topics in logic and analytic philosophy, edited by Concha Martinez, José L. Falguera, and José M. Sagüillo, Santiago de Compostela, Universidade de Santiago de Compastela, 2007, 271-288.

“Philosophy of mathematics: 5 questions”, in Philosophy of mathematics: 5 questions, edited by Vincent F. Hendricks and Hannes Leitgeb, Automatic Press, 2008, 219-232.

“Faith and reason: the perpetual war”, in Philosophers without gods, 2007, edited by Louise Antony, Oxford, Oxford University Press, 3-16.

“All things indefinitely extensible” (with Crispin Wright), in Absolute generality, edited by Agustín Rayo, Oxford, Oxford University Press, 2006, 255-304.

“Structure and identity” in Modality and identity, edited by Fraser MacBride, Oxford University Press, 2006, 109-145.

“The governance of identity”, in Modality and identity, edited by Fraser MacBride, Oxford University Press, 2006, 164-173.

“Computability, proof, and open-texture”, in Church’s thesis after 70 years, edited by Adam Olszewski, Jan Woleñski, and Robert Janusz, Frankfurt, Ontos Verlag, 2006, 420-455.

“Effectiveness”, The age of alternative logics: assessing philosophy of logic and mathematics today, edited by Johan van Benthem, Gerhard Heinzmann, Manuel Rebuschi, and Henk Visser, Dordrecht, The Netherlands, Springer, 2006, 37-49.

“Externalism, anti-realism, and the KK-thesis”, in A logical approach to philosophy: essays in honour of Graham Solomon, edited by David DeVidi and Tim Kenyon, Dordrecht, The Netherlands, Springer, 2005, 22-35.

“Context, conversation, and so-called ‘higher-order vagueness’”, Proceedings of the Aristotelian Society, Supplemental Volume 79 (2005), 147-165.

“Sets and abstracts”, Philosophical Studies 122 (2005), 315-332.

“Gurus, logical consequence, and truth-bearers: What is it that is true?”, in Deflationary truth, edited by Bradley Armour-Garb and J.C. Beall, Open Court Publishers, 2005, 153-170.

“Categories, structures, and the Frege-Hilbert controversy: the status of meta-metamathematics”, Philosophia Mathematica (3) 13 (2005), 61-77; reprinted in Logicism, intuitionism, and formalism, edited by Sten Lindström et al., Synthese Library 341, Springer, 2009, 435- 448.

“Logical consequence, proof theory, and model theory”, Oxford handbook for the philosophy of mathematics and logic, edited by Stewart Shapiro, Oxford, Oxford University Press, 2005, 651-670.

“Higher-order logic”, Oxford handbook for the philosophy of mathematics and logic, edited by Stewart Shapiro, Oxford, Oxford University Press, 2005, 751-780.

“Philosophy of mathematic and its logic”, Oxford handbook for the philosophy of mathematics and logic, edited by Stewart Shapiro, Oxford, Oxford University Press, 2005, 3-27.

“Simple truth, contradiction, and consistency”, The law of non-contradiction, edited by Graham Priest and J. C. Beall, Oxford, Oxford University Press, 2004, 336-354.

“The nature and limits of abstraction”, Philosophical Quarterly 54 (2004), 166-174.

“Foundations of mathematics: metaphysics, epistemology, structure”, Philosophical Quarterly 54 (2004), 16-37.

“All sets great and small: and I do mean ALL”, Philosophical Perspectives 17 (2003), 467-490.

“Vagueness and Conversation”, in Liars and Heaps, edited by J. C. Beall and Michael Glanzberg, Oxford, Oxford University Press, 2003, 39-72.

“Philosophy of mathematics”, in Philosophy of science today, edited by Peter Clark and Katherine Hawley, Oxford, Oxford University Press, 2003, 181-200.

“Mechanism, truth, and Penrose’s new argument”, Journal of Philosophical Logic 32 (2003), 19- 42.

“Prolegomenon to any future neo-logicist set theory: extensionality and indefinite extensibility”, British Journal for the Philosophy of Science 54 (2003), 59-91 , 160-189; reprinted in The Arché papers on the mathematics of abstraction, edited by Roy T. Cook, Dordrecht, The Netherlands, Springer, 353-382.

“The guru, the logician, and the deflationist”, Noûs 37 (2003), 113-132.

“Incompleteness and inconsistency”, Mind 111 (2002), 817-832.

“Deflation and conservation”, Principles of truth, edited by Volker Halbach and Leon Horsten, Hänsel-Hohenhaus, 2002, 103-128.

“The ‘triumph’ of first-order logic”, Logic, meaning and computation: memorial volume for Alonzo Church, edited by C. Anthony Anderson and M. Zeleny, Kluwer Academic Publishers, 2002, 219-259.

“Modeling and normativity: How much revisionism can we take?”, Agora 20 (2001), 159-173.

“Why anti-realists and classical mathematicians cannot get along”, Topoi 20 (2001), 53-63.

“Frege meets Dedekind: a neo-logicist treatment of real analysis”, Notre Dame Journal of Formal Logic 41 (2000), 335-364 , 160-189; reprinted in The Arché papers on the mathematics of abstraction, edited by Roy T. Cook, Dordrecht, The Netherlands, Springer, 219-252.

“The status of logic”, New essays on the a priori, edited by Paul Boghossian and Christopher Peacocke, Oxford University Press, 2000, 333-366; reprinted in part as “Quine on Logic”, Logica Yearbook 1999, edited by Timothy Childers, Prague, Czech Academy Publishing House, 11-21.

“The work of John Corcoran: An appreciation” (with Michael Scanlan), History and Philosophy of Logic 20 (2000), 149-158.

“Neo-logicist logic is not epistemically innocent” (with Alan Weir), Philosophia Mathematica (3) 8, (2000), 160-189; reprinted in The Arché papers on the mathematics of abstraction, edited by Roy T. Cook, Dordrecht, The Netherlands, Springer, 119-146.

“Set-theoretic foundations”, Analytic Philosophy and Logic: Proceedings of the Twentieth World Congress of Philosophy 6, Philosophy Documentation Center, Bowling Green State University, 2000, 183-196.

“Introduction: abstraction and neo-logicism”, Philosophia Mathematica (3) 8 (2000), 97-99.

“New V, ZF, and Abstraction” (with Alan Weir), Philosophia Mathematica (3) 7 (1999), 293- 321 , 160-189; reprinted in The Arché papers on the mathematics of abstraction, edited by Roy T. Cook, Dordrecht, The Netherlands, Springer, 275-302.

“Introduction II”, Philosophia Mathematica (3) 7 (1999), 244-246.

“Do not claim too much: Second-order logic and first-order logic”, Philosophia Mathematica (3) 7 (1999), 42-64.

“Incompleteness, mechanism, and optimism”, Bulletin of Symbolic Logic 4 (1998), 273-302.

“Proof and truth: Through thick and thin”, Journal of Philosophy 95 (1998), 493-521.

“Logical consequence: Models and modality”, Philosophy of mathematics today: Proceedings of an international conference in Munich, ed. by Mathias Schirn, The Mind Association, Oxford, Oxford University Press, 1998, 131-156.

“Induction and indefinite extensibility: The Gödel sentence is true, but did someone change the subject?”, Mind 107 (1998), 597-624.

“The limits of logic”, introduction to The limits of logic: Second-order logic and the Skolem paradox, The international research library of philosophy (see above), xi-xxii.

“Intuitionism, pluralism, and cognitive command” (with William Taschek), Journal of Philosophy 93 (1996), 74-88. “Space, number, and structure: A tale of two debates”, Philosophia Mathematica (3) 4 (1996), 148-173.

“Introduction”, Philosophia Mathematica (3) 4 (1996), 81-82.

“Understanding Church’s thesis, again”, Acta Analytica 11 (1993), 59-77 (published in August, 1995).

“Reasoning, logic, and computation”, Philosophia Mathematica (3) 3 (1995), 31-51.

“Mathematics and philosophy of mathematics”, Philosophia Mathematica (3) 2 (1994), 148-160.

“Modality and ontology”, Mind 102 (1993), 455-481.

“Anti-realism and modality”, Philosophy of Mathematics: Proceedings of the Fifteenth International Wittgenstein Symposium, Vienna, Verlag Hölder-Pichler-Tempsky, 1993, 269-287.

“Foundationalism and foundations of mathematics”, in Proof and knowledge in mathematics, ed. by Michael Detlefsen, London and New York, Routledge, 1992, 171-207.

“Second-order logic, foundations, and rules”, Journal of Philosophy 87 (1990), 234-261.

“Expressive completeness and decidability” (with George Schumm), Notre Dame Journal of Formal Logic 31 (1990), 576-579.

“Structure and ontology”, Philosophical Topics 17 (1989), 145-171.

“Logic, ontology, mathematical practice”, Synthese 79 (1989), 13-50.

“The Lindenbaum construction and decidability”, Notre Dame Journal of Formal Logic 29 (1988), 208-213.

“Principles of reflection and second-order logic”, Journal of Philosophical Logic 16 (1987), 309- 333.

“Turing projectibility” (with Timothy McCarthy), Notre Dame Journal of Formal Logic 28 (1987), 520-535.

“Actuality and essence” (with William Lycan), Midwest Studies in Philosophy 11 (1986), 343-377.

“Second-order languages and mathematical practice”, Journal of Symbolic Logic 50 (1985), 714-742. “Epistemic arithmetic and intuitionistic arithmetic” in Intensional Mathematics (see above), 11-46.

“Introduction” in Intensional Mathematics (see above), 1-10.

“Arithmetic Sinn and effectiveness”, Dialectica 38 (1984), 3-16.

“On an empiricist philosophy of mathematics”, Philosophia 14 (1984), 213-223.

“Mathematics and reality”, Philosophy of Science 50 (1983), 523-548.

“Remarks on the development of computability”, History and Philosophy of Logic 4 (1983), 203-220.

“Conservativeness and incompleteness”, The Journal of Philosophy 80 (1983), 521-531; reprinted in The philosophy of mathematics, edited by W. D. Hart, Oxford, Oxford University Press, 1996.

“Acceptable notation”, Notre Dame Journal of Formal Logic 23 (1982), 14-20.

“Understanding Church’s thesis”, Journal of Philosophical Logic 10 (1981), 353-365.

“L’argomento con la tesi di Church” (“Argument by Church’s thesis”), Nominazione 2 (1981), 195-199.

“On the notion of effectiveness”, History and Philosophy of Logic 1 (1980), 209-230.

“Incomplete translations of complete logics”, Notre Dame Journal of Formal Logic 18 (1977), 248-250. Other publications:

“Realism and Naturalism, Mathematical”, Encyclopedia of Philosophy, Volume 8 , second edition, edited by Donald Borchert, Detroit, Macmillan Reference USA, 2006, 273-279.

“Bertrand Russell’s ‘On denoting’ (1905) and ‘Mathematical logic based on the theory of types’: metaphysics to logic and back”, The classics of western philosophy: a reader’s guide”, edited by J. E. Gracia, Malden, Massachusetts, Blackwell Publishing Company, 2003, 460-466.

“Necessity, meaning, and rationality: the notion of logical consequence”, A companion to philosophical logic, edited by Dale Jacquette, Oxford, Blackwell Publishers, 2002, 227- 240.

“Classical logic II: Higher-order logic”, Blackwell Guide to Philosophical Logic, Oxford, 2001, 33-54.

“Systems between first-order logic and second-order logic”, Handbook of Philosophical Logic 1, second edition, edited by D. M. Gabbay and F. Geunthner, Kluwer Academic Publishers, 2001, 131-187.

“Second-order logic” (824), “Effective procedure” (254), “mathematical structuralism” (543), Cambridge Dictionary of Philosophy, Second Edition, Cambridge, Cambridge University Press, 1999.

“Second-order logic, philosophical issues” and “Church’s thesis”, Routledge Encyclopedia of Philosophy, 1998.

“Theory of types” (p. 884), “Logic, History of” (496-500, with Peter King), “Impredicative definition” (397), “Formalism” (285-286), “Constructivism” (159-160), “Higher-order logic” (354-355), “Number” (629), “Skolem paradox” (827), “Metalogic” (555), “Categoricity” (125), “Independence, Logical” (398-399), “Church’s thesis” (135), “Mathematics, Problems of the philosophy of” (535-539), Oxford Companion to Philosophy, edited by Ted Honderich, Oxford University Press, Oxford, 1995.

“Meta-mathematics and computability”, Encyclopedia of the History and Philosophy of the Mathematical Sciences 1, London, Routledge, 1994, 644-655.

Work in progress:

Logical pluralism/relativism (under contract for Oxford University Press) Logic for philosophers (under contract for Oxford University Press)