
2II—Patrick GreenoughSupplement to the Proceedings of the Aristotelian Society 79 (1): 167190. 2005.

1Review: Constructibility and mathematical existence by Charles Chihara (review)Mind 101 361364. 1992.

Antirealism and modalityIn J. Czermak (ed.), Philosophy of Mathematics, Hölderpichlertempsky. pp. 269287. 1993.

110The 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

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

171The classical continuum without pointsReview of Symbolic Logic 6 (3): 488512. 2013.We develop a pointfree construction of the classical one dimensional continuum, with an interval structure based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle,although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually ex…Read more

Thinking about Mathematics: The Philosophy of MathematicsPhilosophical Quarterly 52 (207): 272274. 2002.

108Foundations Without Foundationalism: A Case for SecondOrder LogicOxford University Press. 1991.The central contention of this book is that secondorder logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higherorder logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of secondorder concepts in mathematics and the extent to which mathematical ideas can be formulated in higherorder logic. He also shows how firstorder languages ar…Read more

4Webb Judson Chambers. Mechanism, mentalism, and metamathematics. An essay on finitism. Synthese library, vol. 137. D. Reidel Publishing Company, Dordrecht, Boston, and London, 1980, xiii + 277 pp (review)Journal of Symbolic Logic 51 (2): 472476. 1986.

21Mechanism, Mentalism and Metamathematics: An Essay on FinitismJournal of Symbolic Logic 51 (2): 472. 1980.

170Where in the (world wide) web of belief is the law of noncontradiction?Noûs 41 (2). 2007.It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analyticsynthetic distinction. This logicfriendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so…Read more

153Do not claim too much: Secondorder logic and firstorder logicPhilosophia Mathematica 7 (1): 4264. 1999.The purpose of this article is to delimit what can and cannot be claimed on behalf of secondorder logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.

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

31Life on the Ship of NeurathCroatian Journal of Philosophy 9 (2): 149166. 2009.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

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

56Typically, a logic consists of a formal or informal language together with a deductive system and/or a modeltheoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truthconditions, or possible truth conditions, for at least part of the language.

2ReviewsPhilosophy of Mathematics: Structure and OntologyBritish Journal for the Philosophy of Science 49 (4): 652. 1998.

107All sets great and small: And I do mean ALLPhilosophical Perspectives 17 (1). 2003.A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the relativist if …Read more

37The status of logicIn Paul Boghossian & Christopher Peacocke (eds.), New Essays on the a Priori, Oxford University Press. pp. 333338. 2000.
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 