
11Matftematical ObjectsIn Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Association of America. pp. 157. 2008.

104Structures and Logics: A Case for (a) RelativismErkenntnis 79 (S2): 309329. 2014.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 …Read more

2Classical logic II: Higherorder logicIn Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell. pp. 3354. 2001.

24Vagueness in ContextOxford University Press UK. 2006.Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, modeltheoretic 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 of S…Read more

43Life on the Ship of Neurath: Mathematics in the Philosophy of MathematicsIn Majda Trobok Nenad Miščević & Berislav Žarnić (eds.), Croatian Journal of Philosophy, Springer. pp. 1127. 2012.Some central philosophical issues concern the use of mathematics in putatively nonmathematical 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 countersupport to various philosophical programs concerning the foundations of mathematics

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11I—Stewart ShapiroSupplement to the Proceedings of the Aristotelian Society 79 (1): 147165. 2005.

179New V, ZF and abstractiontPhilosophia Mathematica 7 (3): 293321. 1999.We examine George Boolos's proposed abstraction principle for extensions based on the limitationofsize conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neologicist 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

82Incompleteness and inconsistencyMind 111 (444): 817832. 2002.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 i…Read more

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62Regionsbased two dimensional continua: The Euclidean caseLogic and Logical Philosophy 24 (4). 2015.

59Frege meets dedekind: A neologicist treatment of real analysisNotre Dame Journal of Formal Logic 41 (4): 335364. 2000.This paper uses neoFregeanstyle 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 firstorder 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 ratio…Read more

18Review of P. Benacerraf and H. Putnam (eds.) Philosophy of Mathematics (review)Philosophy of Science 52 (3): 488. 1985.

101‘Neologicist‘ logic is not epistemically innocentPhilosophia Mathematica 8 (2): 160189. 2000.The neologicist 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 secondorder axiom of comprehension applied to noninstantiated properties and the standard firstorder 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 epistemic…Read more

25Deflation and conservationIn Volker Halbach & Leon Horsten (eds.), Principles of Truth, Dr. Hänselhohenhausen. pp. 103128. 2002.

85Why antirealists and classical mathematicians cannot get alongTopoi 20 (1): 5363. 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), 205229) suggested that a Dummettian antirealist can accept the law of excluded middle as a synthetic, a priori principle groun…Read more

100Mathematics and philosophy of mathematicsPhilosophia Mathematica 2 (2): 148160. 1994.The purpose of this note is to examine the relationship between the practice of mathematics and the philosophy of mathematics, ontology in particular. One conclusion is that the enterprises are (or should be) closely related, with neither one dominating the other. One cannot 'read off' the correct way to do mathematics from the true ontology, for example, nor can one ‘read off’ the true ontology from mathematics as practiced.

103Reasoning with Slippery PredicatesStudia Logica 90 (3): 313336. 2008.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 antiextensions of vague predicates are tracked by what David Lewis calls the “conversational score”, and are regu…Read more

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.

1Vagueness and ConversationIn J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox, Clarendon Press. 2004.

39Logical consequence, proof theory, and model theoryIn Oxford Handbook of Philosophy of Mathematics and Logic, Oxford University Press. pp. 651670. 2005.This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between prooftheoretic notion of consequence, in terms of deduction, and a modeltheoretic approach, in terms of truthconditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.

80An “I” for an I: Singular terms, uniqueness, and referenceReview of Symbolic Logic 5 (3): 380415. 2012.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 lingu…Read more

45Translating Logical TermsTopoi 38 (2): 291303. 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
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 