•  11
    Matftematical Objects
    In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy, Mathematical Association of America. pp. 157. 2008.
  •  104
    Structures and Logics: A Case for (a) Relativism
    Erkenntnis 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
  •  24
    Vagueness in Context
    Oxford University Press UK. 2006.
    Stewart Shapiro's aim in Vagueness in Context is to develop both a philosophical and a formal, model-theoretic 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 with such contextual factors as the comparison class and paradigm cases. A person can be tall with respect to male accountants and not tall with respect to professional basketball players. The main feature of S…Read more
  •  43
    Life on the Ship of Neurath: Mathematics in the Philosophy of Mathematics
    In Majda Trobok Nenad Miščević & Berislav Žarnić (eds.), Croatian Journal of Philosophy, Springer. pp. 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
  •  16
    Review: Sets and Abstracts: Discussion (review)
    Philosophical Studies 122 (3). 2005.
  •  10
    Book reviews (review)
    Mind 101 (402): 225-250. 1992.
  •  95
    Tarski's Theorem and the Extensionality of Truth
    Erkenntnis 78 (5): 1197-1204. 2013.
  •  11
    I—Stewart Shapiro
    Supplement to the Proceedings of the Aristotelian Society 79 (1): 147-165. 2005.
  •  179
    New V, ZF and abstractiont
    with Alan Weir
    Philosophia Mathematica 7 (3): 293-321. 1999.
    We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New …Read more
  •  71
    The George Boolos memorial symposium II
    Philosophia Mathematica 9 (1): 3-4. 2001.
  •  82
    Incompleteness and inconsistency
    Mind 111 (444): 817-832. 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
  •  77
    Principles of reflection and second-order logic
    Journal of Philosophical Logic 16 (3). 1987.
  •  59
    Frege meets dedekind: A neologicist treatment of real analysis
    Notre Dame Journal of Formal Logic 41 (4): 335--364. 2000.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of ratio…Read more
  •  101
    ‘Neo-logicist‘ logic is not epistemically innocent
    with Alan Weir
    Philosophia Mathematica 8 (2): 160--189. 2000.
    The neo-logicist 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 second-order axiom of comprehension applied to non-instantiated properties and the standard first-order 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
  •  25
    Deflation and conservation
    In Volker Halbach & Leon Horsten (eds.), Principles of Truth, Dr. Hänsel-hohenhausen. pp. 103-128. 2002.
  •  85
    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), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle groun…Read more
  •  100
    Mathematics and philosophy of mathematics
    Philosophia Mathematica 2 (2): 148-160. 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.
  •  103
    Reasoning with Slippery Predicates
    Studia Logica 90 (3): 313-336. 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 anti-extensions of vague predicates are tracked by what David Lewis calls the “conversational score”, and are regu…Read more
  •  39
    This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.
  •  80
    An “I” for an I: Singular terms, uniqueness, and reference
    Review of Symbolic Logic 5 (3): 380-415. 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
  •  45
    Translating Logical Terms
    Topoi 38 (2): 291-303. 2019.
    The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the not…Read more