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Profile: Stewart Shapiro (Ohio State University)
  1. Salvatore Florio & Stewart Shapiro (forthcoming). Set Theory, Type Theory, and Absolute Generality. Mind:fzu039.
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or that neither (...)
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  2. Stewart Shapiro (2014). Structures and Logics: A Case for (a) Relativism. Erkenntnis 79 (2):309-329.
    In this paper, I use the cases of intuitionistic arithmetic with Church’s thesis, intuitionistic analysis, and smooth infinitesimal analysis to argue for a sort of pluralism or relativism about logic. The thesis is that logic is relative to a structure. There are classical structures, intuitionistic structures, and (possibly) paraconsistent structures. Each such structure is a legitimate branch of mathematics, and there does not seem to be an interesting logic that is common to all of them. One main theme of my (...)
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  3. Geoffrey Hellman & Stewart Shapiro (2013). The Classical Continuum Without Points. Review of Symbolic Logic 6 (3):488-512.
    We develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary . Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishopgunky lineindecomposabilityCantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, (...)
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  4. Stewart Shapiro (2013). Tarski's Theorem and the Extensionality of Truth. Erkenntnis 78 (5):1197-1204.
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  5. Stewart Shapiro (2013). Vagueness, Open-Texture, and Retrievability. Inquiry 56 (2-3):307-326.
    Just about every theorist holds that vague terms are context-sensitive to some extent. What counts as ?tall?, ?rich?, and ?bald? depends on the ambient comparison class, paradigm cases, and/or the like. To take a stock example, a given person might be tall with respect to European entrepreneurs and downright short with respect to professional basketball players. It is also generally agreed that vagueness remains even after comparison class, paradigm cases, etc. are fixed, and so this context sensitivity does not solve (...)
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  6. Kevin Scharp & Stewart Shapiro (2012). On Richard's When Truth Gives Out. [REVIEW] Philosophical Studies 160 (3):455-463.
    On Richard’s When Truth Gives Out Content Type Journal Article Pages 1-9 DOI 10.1007/s11098-011-9796-0 Authors Kevin Scharp, Department of Philosophy, The Ohio State University, 350 University Hall, 230 North Oval Mall, Columbus, OH 43210, USA Stewart Shapiro, Department of Philosophy, The Ohio State University, 350 University Hall, 230 North Oval Mall, Columbus, OH 43210, USA Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.
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  7. Stewart Shapiro (2012). An “I” for an I: Singular Terms, Uniqueness, and Reference. Review of Symbolic Logic 5 (3):380-415.
    There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and (...)
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  8. Stewart Shapiro (2012). Life on the Ship of Neurath: Mathematics in the Philosophy of Mathematics. In. In Majda Trobok Nenad Miščević & Berislav Žarnić (eds.), Between Logic and Reality. Springer. 11--27.
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  9. Stewart Shapiro (2012). Objectivity, Explanation, and Cognitive Shortfall. In Crispin Wright & Annalisa Coliva (eds.), Mind, Meaning, and Knowledge: Themes From the Philosophy of Crispin Wright. Oxford University Press.
     
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  10. Stewart Shapiro (2011). Epistemology of Mathematics: What Are the Questions? What Count as Answers? Philosophical Quarterly 61 (242):130-150.
    A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietism. For this purpose the notion of entitlement (...)
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  11. Stewart Shapiro (2011). Mathematics and Objectivity. In John Polkinghorne (ed.), Meaning in Mathematics. Oup Oxford.
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  12. Stewart Shapiro (2011). Vagueness and Logic. In Giuseppina Ronzitti (ed.), Vagueness: A Guide. Springer Verlag. 55--81.
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  13. Kenneth Easwaran, Philip Ehrlich, David Ross, Christopher Hitchcock, Peter Spirtes, Roy T. Cook, Jean-Pierre Marquis, Stewart Shapiro & Royt Cook (2010). The Palmer House Hilton Hotel, Chicago, Illinois February 18–20, 2010. Bulletin of Symbolic Logic 16 (3).
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  14. Stewart Shapiro, Mathematical Structuralism. Internet Encyclopedia of Philosophy.
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  15. Stewart Shapiro (2010). So Truth is Safe From Paradox: Now What? [REVIEW] Philosophical Studies 147 (3):445 - 455.
    The article is part of a symposium on Hartry Field’s “Saving truth from paradox”. The book is one of the most significant intellectual achievements of the past decades, but it is not clear what, exactly, it accomplishes. I explore some alternatives, relating the developed view to the intuitive, pre-theoretic notion of truth.
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  16. Stewart Shapiro (2010). Vagueness, Metaphysics, and Objectivity. In Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and Clouds: Vaguenesss, its Nature and its Logic. Oup Oxford.
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  17. Stewart Shapiro (2010). 2010 Winter Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 16 (3):438-444.
     
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  18. Philip A. Ebert & Stewart Shapiro (2009). The Good, the Bad and the Ugly. Synthese 170 (3):415 - 441.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...)
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  19. Stewart Shapiro (2009). Life on the Ship of Neurath. Croatian Journal of Philosophy 9 (2):149-166.
    Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics.
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  20. Stewart Shapiro (2009). Review of Michael P. Lynch, Truth as One and Many. [REVIEW] Notre Dame Philosophical Reviews 2009 (9).
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  21. Stewart Shapiro (2009). We Hold These Truths to Be Self-Evident: But What Do We Mean by That? Review of Symbolic Logic 2 (1):175-207.
    At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that (...)
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  22. Stewart Shapiro & Patrick Reeder (2009). A Scientific Enterprise?: A Critical Study of P. Maddy, Second Philosophy: A Naturalistic Method. [REVIEW] Philosophia Mathematica 17 (2):247-271.
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  23. Stewart Shapiro (2008). Identity, Indiscernibility, and Ante Rem Structuralism: The Tale of I and –I. Philosophia Mathematica 16 (3):285-309.
    Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...)
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  24. Stewart Shapiro (2008). Matftematical Objects. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. 157.
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  25. Stewart Shapiro (2008). Reasoning with Slippery Predicates. Studia Logica 90 (3):313 - 336.
    It is a commonplace that the extensions of most, perhaps all, vague predicates vary with such features as comparison class and paradigm and contrasting cases. My view proposes another, more pervasive contextual parameter. Vague predicates exhibit what I call open texture: in some circumstances, competent speakers can go either way in the borderline region. The shifting extension and anti-extensions of vague predicates are tracked by what David Lewis calls the “conversational score”, and are regulated by what Kit Fine calls penumbral (...)
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  26. Stewart Shapiro & Gabriel Uzquiano (2008). Frege Meets Zermelo: A Perspective on Ineffability and Reflection. Review of Symbolic Logic 1 (2):241-266.
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  27. Jack Arnold & Stewart Shapiro (2007). Where in the (World Wide) Web of Belief is the Law of Non-Contradiction? Noûs 41 (2):276–297.
    It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analytic-synthetic distinction. This logic-friendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so immune from revision. The other, radical (...)
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  28. Stewart Shapiro (2007). Burali-Forti's Revenge. In J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox. Oxford University Press.
     
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  29. Stewart Shapiro (2007). Computability, Proof, and Open-Texture. In ¸ Iteolszewskietal:Cta. 420--55.
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  30. Stewart Shapiro (2007). ¸ Iteolszewskietal:Cta.
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  31. Stewart Shapiro (2007). The Objectivity of Mathematics. Synthese 156 (2):337 - 381.
    The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.
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  32. Stewart Shapiro (2006). Effectiveness. In. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 37--49.
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  33. Stewart Shapiro (2006). Structure and Identity. In Fraser MacBride (ed.), Identity and Modality. Oxford University Press. 34--69.
    According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea (...)
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  34. Stewart Shapiro (2006). The Governance of Identity. In Fraser MacBride (ed.), Identity and Modality. Oxford University Press. 164--173.
  35. Stewart Shapiro (2006/2008). Vagueness in Context. Oxford University Press.
    Stewart Shapiro's ambition in Vagueness in Context is to develop a comprehensive account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary according to their context: a person can be tall with respect to male accountants and not tall (even short) with respect to professional basketball players. The key feature of Shapiro's account is that the extensions of vague terms (...)
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  36. Stewart Shapiro & Crispin Wright (2006). All Things Indefinitely Extensible. In Agustín Rayo & Gabriel Uzquiano (eds.), ¸ Iterayo&Uzquiano:Ag. Clarendon Press. 255--304.
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  37. Stewart Shapiro & Crispin Wright (2006). ¸ Iterayo&Uzquiano:Ag.
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  38. Stewart Shapiro (2005). Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-Mathematics. Philosophia Mathematica 13 (1):61-77.
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...)
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  39. Stewart Shapiro (2005). Logical Consequence, Proof Theory, and Model Theory. In , The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. 651--670.
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  40. Stewart Shapiro (2005). Review: Sets and Abstracts: Discussion. [REVIEW] Philosophical Studies 122 (3):315 - 332.
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  41. Stewart Shapiro (2005). Sets and Abstracts – Discussion. [REVIEW] Philosophical Studies 122 (3):315 - 332.
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  42. Stewart Shapiro (2005). Thinking About Mathematics1. Signos Filosóficos 7 (13):135-144.
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  43. Stewart Shapiro (ed.) (2005). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press.
    Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in (...)
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  44. Stewart Shapiro & Patrick Greenough (2005). Patrick Greenough. Aristotelian Society Supplementary Volume 79 (1):167-190.
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  45. Stewart Shapiro & Patrick Greenough (2005). Stewart Shapiro. Context, Conversation, and so-Called 'Higher-Order Vagueness'. Aristotelian Society Supplementary Volume 79 (1):147–165.
    After a brief account of the problem of higher-order vagueness, and its seeming intractability, I explore what comes of the issue on a linguistic, contextualist account of vagueness. On the view in question, predicates like ‘borderline red’ and ‘determinately red’ are, or at least can be, vague, but they are different in kind from ‘red’. In particular, ‘borderline red’ and ‘determinately red’ are not colours. These predicates have linguistic components, and invoke notions like ‘competent user of the language’. On my (...)
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  46. Stewart Shapiro, Alan Weir & Jamie Tappenden (2005). Kit Fine Precis. Discussion. Philosophical Studies 122 (3):305 - 395.
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  47. Stewart Shapiro (2004). Review: The Nature and Limits of Abstraction. [REVIEW] Philosophical Quarterly 54 (214):166 - 174.
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  48. Stewart Shapiro (2004). Simple Truth, Contradiction, and Consistency. In G. Priest, J. C. Beall & B. Armour-Garb (eds.), The Law of Non-Contradiction. Oxford University Press.
  49. Stewart Shapiro (2004). Vagueness and Conversation. In J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox. Clarendon Press.
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  50. Stewart Shapiro (2004). Foundations of Mathematics: Metaphysics, Epistemology, Structure. Philosophical Quarterly 54 (214):16 - 37.
    Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena (...)
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