•  54
    Possibilities, models, and intuitionistic logic: Ian Rumfitt’s The boundary stones of thought
    Inquiry: An Interdisciplinary Journal of Philosophy 62 (7): 812-825. 2019.
    ABSTRACTAIan Rumfitt's new book presents a distinctive and intriguing philosophy of logic, one that ultimately settles on classical logic as the uniquely correct one–or at least rebuts some prominent arguments against classical logic. The purpose of this note is to evaluate Rumfitt's perspective by focusing on some themes that have occupied me for some time: the role and importance of model theory and, in particular, the place of counter-arguments in establishing invalidity, higher-order logic, …Read more
  •  97
    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
  •  169
    For almost twenty years, Penelope Maddy has been one of the most consistent expositors and advocates of naturalism in philosophy, with a special focus on the philosophy of mathematics, set theory in particular. Over that period, however, the term ‘naturalism’ has come to mean many things. Although some take it to be a rejection of the possibility of a priori knowledge, there are philosophers calling themselves ‘naturalists’ who willingly embrace and practice an a priori methodology, not a whole …Read more
  •  120
    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
  •  582
    Hume’s Principle, Bad Company, and the Axiom of Choice
    Review of Symbolic Logic 16 (4): 1158-1176. 2023.
    One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true…Read more
  •  14
    Mereological Singularism and Paradox
    Erkenntnis 88 (1): 215-234. 2021.
    The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting thr…Read more
  •  57
    Open Texture and Mathematics
    Notre Dame Journal of Formal Logic 62 (1): 173-191. 2021.
    The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.
  • The meaning of logical terms
    In Colin R. Caret & Ole T. Hjortland (eds.), Foundations of Logical Consequence, Oxford University Press. 2015.
  •  183
    Cardinals, Ordinals, and the Prospects for a Fregean Foundation
    In Anthony O'Hear (ed.), Metaphysics, Cambridge University Press. 2018.
    There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitm…Read more
  • Modality in mathematics
    In Otávio Bueno & Scott A. Shalkowski (eds.), The Routledge Handbook of Modality, Routledge. 2018.
  •  10
    Vagueness in Context
    Oxford University Press. 2006.
    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 professionalbasketball players. The main feature of Sh…Read more
  •  98
    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
  •  59
    Groups, sets, and paradox
    Linguistics and Philosophy 45 (6): 1277-1313. 2022.
    Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups rese…Read more
  •  83
    Predicativism as a Form of Potentialism
    Review of Symbolic Logic 16 (1): 1-32. 2023.
    In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.
  •  43
    Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of _which number_ is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of _notation_. The purpose of this article is to explore the relationship b…Read more
  •  254
    Hofweber’s Nominalist Naturalism
    with Eric Snyder and Richard Samuels
    In Gianluigi Oliveri, Claudio Ternullo & Stefano Boscolo (eds.), Objects, Structures, and Logics, Springer. pp. 31-62. 2022.
    In this paper, we outline and critically evaluate Thomas Hofweber’s solution to a semantic puzzle he calls Frege’s Other Puzzle. After sketching the Puzzle and two traditional responses to it—the Substantival Strategy and the Adjectival Strategy—we outline Hofweber’s proposed version of Adjectivalism. We argue that two key components—the syntactic and semantic components—of Hofweber’s analysis both suffer from serious empirical difficulties. Ultimately, this suggests that an altogether different…Read more
  •  319
    Computability, Notation, and de re Knowledge of Numbers
    with Eric Snyder and Richard Samuels
    Philosophies 1 (7). 2022.
    Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship betwe…Read more
  •  469
    Resolving Frege’s Other Puzzle
    Philosophica Mathematica 30 (1): 59-87. 2022.
    Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we …Read more
  •  101
    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
  •  26
    Clarke and Beck import certain assumptions about the nature of numbers. Although these are widespread within research on number cognition, they are highly contentious among philosophers of mathematics. In this commentary, we isolate and critically evaluate one core assumption: the identity thesis.
  • Classical First-Order Logic
    Cambridge University Press. 2022.
    One is often said to be reasoning well when they are reasoning logically. Many attempts to say what logical reasoning is have been proposed, but one commonly proposed system is first-order classical logic. This Element will examine the basics of first-order classical logic and discuss some surrounding philosophical issues. The first half of the Element develops a language for the system, as well as a proof theory and model theory. The authors provide theorems about the system they developed, suc…Read more
  •  1
    Book reviews (review)
    with Richard Shusterman, Rudolf Haller, L. Nathan Oaklander, L. E. Goodman, and George N. Schlesinger
    Peer Reviewed.
  •  35
    John Corcoran
    with José M. Sagüillo and Michael Scanlan
    History and Philosophy of Logic 42 (3): 201-223. 2021.
    We present a memorial summary of the professional life and contributions to logic of John Corcoran. We also provide a full list of his many publications.Courtesy of Lynn Corcoran.
  • Simple Truth, Contradiction, and Consistency
    In Graham Priest, Jc Beall & Bradley P. Armour-Garb (eds.), The law of non-contradiction : new philosophical essays, Oxford University Press. 2004.
  •  23
    Mereological Singularism and Paradox
    Erkenntnis 88 (1): 1-20. 2021.
    The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting thr…Read more
  •  154
    Logic and science: science and logic
    Synthese 199 (3-4): 6429-6454. 2021.
    According to Ole Hjortland, Timothy Williamson, Graham Priest, and others, anti-exceptionalism about logic is the view that logic “isn’t special”, but is continuous with the sciences. Logic is revisable, and its truths are neither analytic nor a priori. And logical theories are revised on the same grounds as scientific theories are. What isn’t special, we argue, is anti-exceptionalism about logic. Anti-exceptionalists disagree with one another regarding what logic and, indeed, anti-exceptionalis…Read more
  •  51
    Group nouns and pseudo‐singularity
    Thought: A Journal of Philosophy 10 (1): 66-77. 2021.
    Thought: A Journal of Philosophy, EarlyView.
  •  34
    Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.