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182So truth is safe from paradox: now what?Philosophical Studies 147 (3): 445-455. 2010.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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16Do Not Claim Too Much: Second-order Logic and First-order LogicPhilosophia Mathematica 6 (3): 42-64. 1998.The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.
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111Why anti-realists and classical mathematicians cannot get alongTopoi 20 (1): 53-63. 2001.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
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6Classical logic II: Higher-order logicIn Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell. pp. 33--54. 2001.A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐…Read more
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2Vagueness and ConversationIn J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox, Clarendon Press. 2004.
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10II—Patrick Greenough: Contextualism about Vagueness and Higher‐order VaguenessAristotelian 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
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1Review: Constructibility and mathematical existence by Charles Chihara (review)Mind 101 361-364. 1992.
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278New V, ZF and AbstractionPhilosophia Mathematica 7 (3): 293-321. 1999.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 …Read more
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86Translating Logical TermsTopoi 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
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16The Lindenbaum construction and decidabilityNotre Dame Journal of Formal Logic 29 (2): 208-213. 1988.
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27Expressive completeness and decidabilityNotre Dame Journal of Formal Logic 31 (4): 576-579. 1990.
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58The Company Kept by Cut Abstraction (and its Relatives)Philosophia Mathematica 19 (2): 107-138. 2011.This article concerns the ongoing neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections . The overall plan is to develop branches of established mathematics using abstraction principles in the form: Formula where a and b are variables of a g…Read more
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102Frege meets dedekind: A neologicist treatment of real analysisNotre 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
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66Review of T. Franzen, Godel's theorem: An incomplete guide to its use and abuse (review)Philosophia Mathematica 14 (2): 262-264. 2006.This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience with the rigor of mathemat…Read more
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34Mechanism, Mentalism and Metamathematics: An Essay on FinitismJournal of Symbolic Logic 51 (2): 472. 1980.
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40Deflation and conservationIn Volker Halbach & Leon Horsten (eds.), Principles of Truth, Dr. Hänsel-hohenhausen. pp. 103-128. 2002.
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1Vagueness, Metaphysics, and ObjectivityIn Richard Dietz & Sebastiano Moruzzi (eds.), Cuts and Clouds: Vaguenesss, its Nature and its Logic, Oxford University Press. 2010.
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34Life on the Ship of NeurathCroatian Journal of Philosophy 9 (2): 149-166. 2009.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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140Structure and identityIn Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 34--69. 2006.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 crit…Read more
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1""Bertrand Russell," On Denoting"(1905) and" Mathematical Logic as Based on the Theory of Types"(1908)In Jorge J. E. Gracia, Gregory M. Reichberg & Bernard N. Schumacher (eds.), The Classics of Western Philosophy: A Reader's Guide, Blackwell. pp. 460. 2003.
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162Incompleteness, mechanism, and optimismBulletin of Symbolic Logic 4 (3): 273-302. 1998.§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” wou…Read more
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7Review of Michael D. Resnik: Mathematics as a Science of Patterns_; Stewart Shapiro: _Philosophy of Mathematics: Structure and Ontology (review)British Journal for the Philosophy of Science 49 (4): 652-656. 1998.
Columbus, Ohio, United States of America
Areas of Specialization
Philosophy of Language |
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Areas of Interest
Philosophy of Language |
Logic and Philosophy of Logic |
Philosophy of Mathematics |