
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

103Vagueness in ContextOxford University Press. 2006.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 also var…Read more

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

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

100Mechanism, truth, and Penrose's new argumentJournal of Philosophical Logic 32 (1): 1942. 2003.Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original LucasPenrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). I…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.

98Foundations of Mathematics: Metaphysics, Epistemology, StructurePhilosophical Quarterly 54 (214). 2004.Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to prov…Read more

96Oxford Handbook of Philosophy of Mathematics and Logic (edited book)Oxford University Press. 2005.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 newlycommissioned 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 include…Read more

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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

83Induction and Indefinite Extensibility: The Gödel Sentence is True, but Did Someone Change the Subject?Mind 107 (427): 597624. 1998.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

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

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

76SetTheoretic FoundationsThe Proceedings of the Twentieth World Congress of Philosophy 2000 183196. 2000.Since virtually every mathematical theory can be interpreted in ZermeloFraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higherorder 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

74Structure and identityIn Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 3469. 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

74Book Review: John P. Burgess and Gideon Rose. A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics (review)Notre Dame Journal of Formal Logic 39 (4): 600612. 1998.

72The 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 neologicist 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.

72Prolegomenon To Any Future Neo‐Logicist Set Theory: Abstraction And Indefinite ExtensibilityBritish Journal for the Philosophy of Science 54 (1): 5991. 2003.The purpose of this paper is to assess the prospects for a neologicist 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? 2 Framew…Read more

65Logical pluralism and normativityInquiry: An Interdisciplinary Journal of Philosophy 122. 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

65Introduction II: The George Boolos memorial symposium: Dedicated to the memory of George Boolos (1940 9 41996 5 27)Philosophia Mathematica 7 (3): 244246. 1999.
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 