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125Logical pluralism and normativityInquiry: 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
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30Ontology via semantics? Introduction to the special issue on the semantics of cardinalsLinguistics and Philosophy 40 (4): 321-329. 2017.As introduction to the special issue on the semantics of cardinals, we offer some background on the relevant literature, and an overview of the contributions to this volume. Most of these papers were presented in earlier form at an interdisciplinary workshop on the topic at The Ohio State University, and the contributions to this issue reflect that interdisciplinary character: the authors represent both fields in the title of this journal.
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67Computing with Numbers and Other Non-syntactic Things: De re Knowledge of Abstract ObjectsPhilosophia Mathematica 25 (2): 268-281. 2017.ABSTRACT Michael Rescorla has argued that it makes sense to compute directly with numbers, and he faulted Turing for not giving an analysis of number-theoretic computability. However, in line with a later paper of his, it only makes sense to compute directly with syntactic entities, such as strings on a given alphabet. Computing with numbers goes via notation. This raises broader issues involving de re propositional attitudes towards numbers and other non-syntactic abstract entities.
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Philosophy of Mathematics: Structure and OntologyPhilosophy and Phenomenological Research 65 (2): 467-475. 2002.
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Mathematics as a Science of PatternsBritish Journal for the Philosophy of Science 49 (4): 652-656. 1998.
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30Induction and Indefinite Extensibility: The Gödel Sentence is True, but Did Someone Change the Subject?Mind 107 (427): 597-624. 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
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Review of Kleene 1981, Davis 1982, and Kleene 1987 (review)Journal of Symbolic Logic 55 348-350. 1990.
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81A procedural solution to the unexpected hanging and sorites paradoxesMind 107 (428): 751-762. 1998.The paradox of the Unexpected Hanging, related prediction paradoxes, and the Sorites paradoxes all involve reasoning about ordered collections of entities: days ordered by date in the case of the Unexpected Hanging; men ordered by the number of hairs on their heads the case of the bald man version of the Sorites. The reasoning then assigns each entity a value that depends on the previously assigned value of one of the neighboring entities. The final result is paradoxical because it conflicts wit…Read more
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49Truth, function and paradoxAnalysis 71 (1): 38-44. 2011.Michael Lynch’s Truth as One and Many is a contribution to the large body of philosophical literature on the nature of truth. Within that genre, advocates of truth-as-correspondence, advocates of truth-as-coherence, and the like, all hold that truth has a single underlying metaphysical nature, but they sharply disagree as to what this nature is. Lynch argues that many of these views make good sense of truth attributions for a limited stretch of discourse, but he adds that each of the contenders …Read more
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49Frege Meets Zermelo: A Perspective on Ineffability and ReflectionReview of Symbolic Logic 1 (2): 241-266. 2008.1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is di…Read more
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83The 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.
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38Priest, Graham. An Introduction to Non-classical Logic (review)Review of Metaphysics 56 (3): 670-672. 2003.
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105Space, number and structure: A tale of two debatesPhilosophia 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
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D. GABBAY and F. GUENTHNER "Handbook of philosophical logic. Volume 1: Elements of classical logic"History and Philosophy of Logic 6 (2): 215. 1985.
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13Matftematical ObjectsIn Bonnie Gold & Roger A. Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Association of America. pp. 157. 2008.
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141Structures and Logics: A Case for (a) RelativismErkenntnis 79 (S2): 309-329. 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
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97Vagueness, Open-Texture, and RetrievabilityInquiry: 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
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52Life on the Ship of Neurath: Mathematics in the Philosophy of MathematicsCroatian Journal of Philosophy 26 (2): 11--27. 2012.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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14Arithmetic Sinn and EffectivenessDialectica 38 (1): 3-16. 1984.SummaryAccording to Dummett's understanding of Frege, the sense of a denoting expression is a procedure for determining its denotation. The purpose of this article is to pursue this suggestion and develop a semi‐formal interpretation of Fregean sense for the special case of a first‐order language of arithmetic. In particular, we define the sense of each arithmetic expression to be a hypothetical process to determine the denoted number or truth value. The sense‐process is “hypothetical” in that t…Read more
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30I—Stewart ShapiroSupplement to the Proceedings of the Aristotelian Society 79 (1): 147-165. 2005.
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39Comparing implicit and explicit memory for brand names from advertisementsJournal of Experimental Psychology: Applied 2 (2): 147. 1996.
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137The guru, the logician, and the deflationist: Truth and logical consequenceNoûs 37 (1). 2003.The purpose of this paper is to present a thought experiment and argument that spells trouble for “radical” deflationism concerning meaning and truth such as that advocated by the staunch nominalist Hartry Field. The thought experiment does not sit well with any view that limits a truth predicate to sentences understood by a given speaker or to sentences in (or translatable into) a given language, unless that language is universal. The scenario in question concerns sentences that are not under…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 |