•  144
    Modality and ontology
    Mind 102 (407): 455-481. 1993.
  •  1
    Structure and Ontology
    Philosophical Topics 17 (2): 145-171. 1989.
  •  15
    Review: The Nature and Limits of Abstraction (review)
    Philosophical Quarterly 54 (214). 2004.
  • Book Reviews (review)
    Mind 101 (402): 361-364. 1992.
  •  79
    Understanding church's thesis
    Journal of Philosophical Logic 10 (3): 353--65. 1981.
  •  2
    II—Patrick Greenough
    Supplement to the Proceedings of the Aristotelian Society 79 (1): 167-190. 2005.
  •  77
  • Anti-realism and modality
    In J. Czermak (ed.), Philosophy of Mathematics, Hölder-pichler-tempsky. pp. 269--287. 1993.
  •  110
    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
  •  98
    Foundations of Mathematics: Metaphysics, Epistemology, Structure
    Philosophical 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
  •  331
  •  171
    The classical continuum without points
    Review of Symbolic Logic 6 (3): 488-512. 2013.
    We develop a point-free 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 Mathematics
    Philosophical Quarterly 52 (207): 272-274. 2002.
  •  108
    The central contention of this book is that second-order 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 higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages ar…Read more
  •  170
    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 analytic-synthetic distinction. This logic-friendly 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
  •  76
    Second-order logic, foundations, and rules
    Journal of Philosophy 87 (5): 234-261. 1990.
  •  153
    Do not claim too much: Second-order logic and first-order logic
    Philosophia Mathematica 7 (1): 42-64. 1999.
    The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.
  •  103
    Vagueness in Context
    Oxford 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
  •  31
    Life on the Ship of Neurath
    Croatian Journal of Philosophy 9 (2): 149-166. 2009.
    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
  •  74
    Structure and identity
    In Fraser MacBride (ed.), Identity and Modality, Oxford University Press. pp. 34--69. 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
  •  56
    Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic 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 truth-conditions, or possible truth conditions, for at least part of the language.
  •  2
    Reviews-Philosophy of Mathematics: Structure and Ontology
    British Journal for the Philosophy of Science 49 (4): 652. 1998.
  •  107
    All sets great and small: And I do mean ALL
    Philosophical 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
  •  37
    The status of logic
    In Paul Boghossian & Christopher Peacocke (eds.), New Essays on the a Priori, Oxford University Press. pp. 333--338. 2000.