•  112
    On Richard’s When Truth Gives Out (review)
    Philosophical Studies 160 (3): 455-463. 2012.
    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
  •  107
    Logical pluralism and normativity
    Inquiry: An Interdisciplinary Journal of Philosophy 1-22. 2017.
    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 po…Read more
  •  103
    Famously, Michael Dummett argues that considerations concerning the role of language in communication lead to the rejection of classical logic in favor of intuitionistic logic. Potentially, this results in massive revisions of established mathematics. Recently, Neil Tennant (“The law of excluded middle is synthetic a priori, if valid”, Philosophical Topics 24 (1996), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle groun…Read more
  •  103
    Set-Theoretic Foundations
    The Proceedings of the Twentieth World Congress of Philosophy 6 183-196. 2000.
    Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematica…Read more
  •  102
    The answers to the questions in the title depend on the kind of pluralism one is talking about. We will focus here on our own views. The purpose of this article is to trace out some possible connections between these kinds of pluralism. We show how each of them might bear on the other, depending on how certain open questions are resolved.
  •  100
    Space, number and structure: A tale of two debates
    Philosophia Mathematica 4 (2): 148-173. 1996.
    Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates ill…Read more
  •  98
    Prolegomenon To Any Future Neo‐Logicist Set Theory: Abstraction And Indefinite Extensibility
    British Journal for the Philosophy of Science 54 (1): 59-91. 2003.
    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): ∀P∀Q[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?2…Read more
  •  98
    Sets and Abstracts – Discussion
    Philosophical Studies 122 (3): 315-332. 2005.
  •  96
    Understanding church's thesis
    Journal of Philosophical Logic 10 (3): 353--65. 1981.
  •  96
    Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within a broa…Read more
  •  93
    Link's Revenge: A Case Study in Natural Language Mereology
    In Gabriele M. Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics, De Gruyter. pp. 3-36. 2019.
    Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends tha…Read more
  •  93
    Frege meets dedekind: A neologicist treatment of real analysis
    Notre Dame Journal of Formal Logic 41 (4): 335--364. 2000.
    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 ra…Read more
  •  90
    II—Patrick Greenough: Contextualism about Vagueness and Higher‐order Vagueness
    with Patrick Greenough
    Aristotelian Society Supplementary Volume 79 (1): 167-190. 2005.
    To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order vagueness…Read more
  •  88
    Reasoning, logic and computation
    Philosophia Mathematica 3 (1): 31-51. 1995.
    The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism…Read more
  •  87
  •  85
    Vagueness, Open-Texture, and Retrievability
    Inquiry: An Interdisciplinary Journal of Philosophy 56 (2-3): 307-326. 2013.
    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 sen…Read more
  •  83
    One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what foll…Read more
  •  82
    The Nature and Limits of Abstraction (review)
    Philosophical Quarterly 54 (214). 2004.
    This article is an extended critical study of Kit Fine’s The limits of abstraction, which is a sustained attempt to take the measure of the neo-logicist program in the philosophy and foundations of mathematics, founded on abstraction principles like Hume’s principle. The present article covers the philosophical and technical aspects of Fine’s deep and penetrating study.
  •  82
    Second-order logic, foundations, and rules
    Journal of Philosophy 87 (5): 234-261. 1990.
  •  81
    Translating Logical Terms
    Topoi 38 (2): 291-303. 2019.
    The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the not…Read more
  •  80
    The Axiom of Choice is False Intuitionistically (in Most Contexts)
    with Charles Mccarty and Ansten Klev
    Bulletin of Symbolic Logic 29 (1): 71-96. 2023.
    There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems s…Read more
  •  80
    Varieties of Logic
    Oxford University Press. 2014.
    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.
  •  79
    Divergent Potentialism: A Modal Analysis With an Application to Choice Sequences
    with Ethan Brauer and Øystein Linnebo
    Philosophia Mathematica 30 (2): 143-172. 2022.
    Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripk…Read more
  •  73
    The George Boolos memorial symposium II
    Philosophia Mathematica 9 (1): 3-4. 2001.