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Stewart Shapiro
Ohio State University
  1. Foundations Without Foundationalism: A Case for Second-Order Logic.Stewart Shapiro - 1991 - Oxford University Press.
    The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...)
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  2. Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 1997 - Oxford University Press.
    Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...)
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  3.  47
    Varieties of Logic.Stewart Shapiro - 2014 - Oxford University Press.
    Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.
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  4. Thinking About Mathematics: The Philosophy of Mathematics.Stewart Shapiro - 2000 - Oxford University Press.
    This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), (...)
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  5. Vagueness in Context.Stewart Shapiro - 2006 - 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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  6.  31
    Vagueness in Context. [REVIEW]Stewart Shapiro - 2008 - Philosophy and Phenomenological Research 76 (2):471-483.
  7. Actual and Potential Infinity.Øystein Linnebo & Stewart Shapiro - 2019 - Noûs 53 (1):160-191.
    The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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  8.  34
    Vagueness in Context.Stewart Shapiro - 2006 - Oxford University Press UK.
    Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic 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 with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professional basketball players. The main feature (...)
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  9. Identity, Indiscernibility, and Ante Rem Structuralism: The Tale of I and –I.Stewart Shapiro - 2008 - 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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  10. Proof and Truth.Stewart Shapiro - 1998 - Journal of Philosophy 95 (10):493-521.
  11. Oxford Handbook of Philosophy of Mathematics and Logic.Stewart Shapiro (ed.) - 2005 - Oxford University Press.
    This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book includes a comprehensive (...)
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  12. Conservativeness and Incompleteness.Stewart Shapiro - 1983 - Journal of Philosophy 80 (9):521-531.
  13.  28
    Philosophy of Mathematics.Stewart Shapiro - 2003 - In Peter Clark & Katherine Hawley (eds.), Philosophy of Science Today. Clarendon Press.
    Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.
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  14. Mathematics and Reality.Stewart Shapiro - 1983 - Philosophy of Science 50 (4):523-548.
    The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies (...)
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  15.  83
    Logical Consequence, Proof Theory, and Model Theory.Stewart Shapiro - 2005 - In Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 651--670.
    This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.
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  16. Logical Consequence: Models and Modality.Stewart Shapiro - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. pp. 131--156.
  17.  59
    Classical Logic.Stewart Shapiro & Teresa Kouri Kissel - 2018 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
    Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language.
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  18. Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-Mathematics.Stewart Shapiro - 2005 - 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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  19.  34
    Proof and Truth: Through Thick and Thin.Stewart Shapiro - 1998 - Journal of Philosophy 95 (10):493-521.
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  20. New V, ZF and Abstraction.Stewart Shapiro & Alan Weir - 1999 - Philosophia Mathematica 7 (3):293-321.
    We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...)
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  21. We Hold These Truths to Be Self-Evident: But What Do We Mean by That?: We Hold These Truths to Be Self-Evident.Stewart Shapiro - 2009 - Review of Symbolic Logic 2 (1):175-207.
    At the beginning of Die Grundlagen der Arithmetik [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 both (...)
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  22. All Things Indefinitely Extensible.Stewart Shapiro & Crispin Wright - 2006 - In Agustín Rayo & Gabriel Uzquiano (eds.), ¸ Iterayo&Uzquiano:Ag. Clarendon Press. pp. 255--304.
  23.  90
    Structure and Identity.Stewart Shapiro - 2006 - In Fraser MacBride (ed.), Identity and Modality. Oxford University Press. pp. 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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  24. Modality and Ontology.Stewart Shapiro - 1993 - Mind 102 (407):455-481.
  25. The Classical Continuum Without Points.Geoffrey Hellman & Stewart Shapiro - 2013 - 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 logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence (...)
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  26.  72
    Frege Meets Dedekind: A Neologicist Treatment of Real Analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of (...)
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  27.  75
    Set Theory and Its Philosophy: A Critical Introduction.Stewart Shapiro - 2005 - Mind 114 (455):764-767.
  28.  32
    Friedrich Waismann: The Open Texture of Analytic Philosophy.Dejan Makovec & Stewart Shapiro (eds.) - 2019 - Palgrave Macmillan.
    This edited collection covers Friedrich Waismann's most influential contributions to twentieth-century philosophy of language: his concepts of open texture and language strata, his early criticism of verificationism and the analytic-synthetic distinction, as well as their significance for experimental and legal philosophy. -/- In addition, Waismann's original papers in ethics, metaphysics, epistemology and the philosophy of mathematics are here evaluated. They introduce Waismann's theory of action along with his groundbreaking work on fiction, proper names and Kafka's Trial. -/- Waismann is known (...)
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  29. The Good, the Bad and the Ugly.Philip Ebert & Stewart Shapiro - 2009 - 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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  30. Incompleteness and Inconsistency.Stewart Shapiro - 2002 - Mind 111 (444):817-832.
    Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or ‘naïve’ notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or (...)
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  31.  20
    Logical Pluralism and Normativity.Stewart Shapiro & Teresa Kouri Kissel - 2020 - Inquiry: An Interdisciplinary Journal of Philosophy 63 (3-4):389-410.
    We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that (...)
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  32.  47
    The Status of Logic.Stewart Shapiro - 2000 - In Paul Boghossian & Christopher Peacocke (eds.), New Essays on the a Priori. Oxford University Press. pp. 333--338.
  33.  91
    Prolegomenon To Any Future Neo‐logicist Set Theory: Abstraction And Indefinite Extensibility.Stewart Shapiro - 2003 - British Journal for the Philosophy of Science 54 (1):59-91.
    The purpose of this paper is to assess the prospects for a neo-logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): PQ[Ext(P) = Ext(Q) [(BAD(P) & BAD(Q)) x(Px Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’. 1 Background: what and why? (...)
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  34. ‘Neo-Logicist‘ Logic is Not Epistemically Innocent.Stewart Shapiro & Alan Weir - 2000 - Philosophia Mathematica 8 (2):160--189.
    The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...)
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  35. Institutionism, Pluralism, and Cognitive Command.Stewart Shapiro & William W. Taschek - 1996 - Journal of Philosophy 93 (2):74.
  36. Second-Order Languages and Mathematical Practice.Stewart Shapiro - 1985 - Journal of Symbolic Logic 50 (3):714-742.
  37.  53
    Computing with Numbers and Other Non-Syntactic Things: De Re Knowledge of Abstract Objects.Stewart Shapiro - 2017 - Philosophia Mathematica 25 (2):268-281.
    ABSTRACT Michael Rescorla has argued that it makes sense to compute directly with numbers, and he faulted Turing for not giving an analysis of number-theoretic computability. However, in line with a later paper of his, it only makes sense to compute directly with syntactic entities, such as strings on a given alphabet. Computing with numbers goes via notation. This raises broader issues involving de re propositional attitudes towards numbers and other non-syntactic abstract entities.
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  38.  97
    Principles of Reflection and Second-Order Logic.Stewart Shapiro - 1987 - Journal of Philosophical Logic 16 (3):309 - 333.
  39. An “I” for an I: Singular Terms, Uniqueness, and Reference.Stewart Shapiro - 2012 - 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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  40. Simple Truth, Contradiction, and Consistency.Stewart Shapiro - 2004 - In Graham Priest, J. C. Beall & Bradley Armour-Garb (eds.), The Law of Non-Contradiction. Oxford University Press.
  41.  8
    The Limits of Abstraction.Stewart Shapiro - 2004 - Philosophical Quarterly 54 (214):166-174.
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  42.  30
    Mechanism, Mentalism and Metamathematics: An Essay on Finitism.Stewart Shapiro - 1980 - Journal of Symbolic Logic 51 (2):472.
  43. Incompleteness, Mechanism, and Optimism.Stewart Shapiro - 1998 - Bulletin of Symbolic Logic 4 (3):273-302.
  44.  80
    Logical Pluralism and Normativity.Teresa Kouri Kissel & Stewart Shapiro - 2017 - Inquiry: An Interdisciplinary Journal of Philosophy:1-22.
    We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that (...)
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  45.  38
    Understanding the Infinite.Stewart Shapiro - 1996 - Philosophical Review 105 (2):256.
    Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.
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  46. Frege Meets Aristotle: Points as Abstracts.Stewart Shapiro & Geoffrey Hellman - 2015 - Philosophia Mathematica:nkv021.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...)
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  47. Where in the (World Wide) Web of Belief is the Law of Non-Contradiction?Jack Arnold & Stewart Shapiro - 2007 - 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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  48.  11
    Reflections on Kurt Godel.Stewart Shapiro - 1991 - Philosophical Review 100 (1):130.
  49. Aristotelian Continua.Øystein Linnebo, Stewart Shapiro & Geoffrey Hellman - 2016 - Philosophia Mathematica 24 (2):214-246.
    In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, (...)
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  50. Frege Meets Brouwer.Stewart Shapiro & Øystein Linnebo - 2015 - Review of Symbolic Logic 8 (3):540-552.
    We show that, by choosing definitions carefully, a version of Frege's theorem can be proved in intuitionistic logic.
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