
An Introduction to Nonclassical Logic (review)Review of Metaphysics 56 (3): 670671. 2003.This book is just what its title says: an introduction to nonclassical logic. And it is a very good one. Given the extensive interest in nonclassical logics, in various parts of the philosophical scene, it is a welcome addition to the corpus. Typical courses in logic, at all levels and in both philosophy departments and mathematics departments, focus exclusively on classical logic. Most instructors, and some textbooks, give some mention to some nonclassical systems, but usually few details are p…Read more

Philosophy of MathematicsOxford University Press USA. 1997.Moving beyond both realist and antirealist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

7Mathematics in philosophy, Selected essays, by Charles Parsons, Cornell University Press, Ithaca, N.Y., 1983, 365 pp (review)Journal of Symbolic Logic 53 (1): 320329. 1988.

24Stephen C. Kleene. Origins of recursive function theory. Annals of the history of computing, vol. 3 , pp. 52– 67.  Martin Davis. Why Gödel didn't have Church's thesis. Information and control, vol. 54 , pp. 3– 24.  Stephen C. Kleene. Reflections on Church's thesis. Notre Dame journal of formal logic, vol. 28 , pp. 490– 498 (review)Journal of Symbolic Logic 55 (1): 348350. 1990.

12Perspectives on the history of mathematical logic, edited by Thomas Drucker, Birkhäuser, Boston, Basel, and Berlin, 1991, xxiii + 195 pp.  John W. Dawson Jr. The reception of Gödel's incompleteness theorems. Pp. 84–100 (review)Journal of Symbolic Logic 57 (4): 14871489. 1992.

8Wilfried Sieg. Step by recursive step: Church's analysis of effective calculability. The bulletin of symbolic logic, vol. 3 , pp. 154–180 (review)Journal of Symbolic Logic 64 (1): 398399. 1999.

23Does logical pluralism imply, or suggest, truth pluralism, or vice versa?Synthese 112. forthcoming.The answers to the questions in the title depend on the kind of pluralism one is talking about. We will focus here on our own views. The purpose of this article is to trace out some possible connections between these kinds of pluralism. We show how each of them might bear on the other, depending on how certain open questions are resolved.

Friedrich Waismann: The Open Texture of Analytic Philosophy (edited book)Palgrave Macmillan. forthcoming.

10Ineffability within the limits of abstraction aloneIn Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism: Essays in Philosophy of Mathematics, Oxford University Press. 2016.The purpose of this article is to assess the prospects for a Scottish neologicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neologicist abstraction principle. That puts the enterprise on the neologicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that our abstraction based theory can be mod…Read more

Mathematical StructuralismCambridge University Press. 2018.The present work is a systematic study of five frameworks or perspectives articulating mathematical structuralism, whose core idea is that mathematics is concerned primarily with interrelations in abstraction from the nature of objects. The first two, settheoretic and categorytheoretic, arose within mathematics itself. After exposing a number of problems, the book considers three further perspectives formulated by logicians and philosophers of mathematics: sui generis, treating structures as a…Read more

132Actual and Potential InfinityNoûs 53 (1): 160191. 2019.The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

20Robert Lorne Victor Hale FRSE May 4, 1945 – December 12, 2017Philosophia Mathematica 26 (2): 266274. 2018.

Varieties of Continua: From Regions to Points and BackOxford University Press. 2018.Hellman and Shapiro explore the development of the idea of the continuous, from the Aristotelian view that a true continuum cannot be composed of points to the now standard, entirely punctiform frameworks for analysis and geometry. They then investigate the underlying metaphysical issues concerning the nature of space or spacetime.

94Oxford 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

64Logical 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

11Ontology via semantics? Introduction to the special issue on the semantics of cardinalsLinguistics and Philosophy 40 (4): 321329. 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.

37Computing with Numbers and Other Nonsyntactic Things: De re Knowledge of Abstract ObjectsPhilosophia Mathematica 25 (2): 268281. 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 numbertheoretic 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 nonsyntactic abstract entities.

Philosophy of Mathematics: Structure and OntologyPhilosophy and Phenomenological Research 65 (2): 467475. 2002.

16Thinking about Mathematics. The Philosophy of MathematicsBulletin of Symbolic Logic 8 (1): 8991. 2002.

Mathematics as a Science of PatternsBritish Journal for the Philosophy of Science 49 (4): 652656. 1997.

7The Limits of HigherOrder Logic and the LöwenheimSkolem TheoremErkenntnis 49 (3): 393396. 1998.

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

Review of Kleene 1981, Davis 1982, and Kleene 1987 (review)Journal of Symbolic Logic 55 348350. 1990.

45A procedural solution to the unexpected hanging and sorites paradoxesMind 107 (428): 751762. 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
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 