•  56
    Intentional mathematics (edited book)
    Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co.. 1985.
    Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.
  • Philosophy of Mathematics: Structure and Ontology
    Philosophical Quarterly 50 (198): 120-123. 2000.
  •  190
    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
  •  2
    Simple truth, contradiction, and consistency
    In Graham Priest, J. C. Beall & Bradley Armour-Garb (eds.), The Law of Non-Contradiction, Oxford University Press. 2004.
  •  209
    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.
  •  205
    Mathematical structuralism
    Philosophia Mathematica 4 (2): 81-82. 1996.
    STEWART SHAPIRO; Mathematical Structuralism, Philosophia Mathematica, Volume 4, Issue 2, 1 May 1996, Pages 81–82, https://doi.org/10.1093/philmat/4.2.81.
  •  63
    Structure and Ontology
    Philosophical Topics 17 (2): 145-171. 1989.
  •  71
    Classical Logic
    In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University. 2018.
    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.
  •  80
    Varieties of Logic
    Oxford University Press. 2014.
    Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.
  •  133
    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
  •  42
    Understanding the Infinite
    Philosophical Review 105 (2): 256. 1996.
    Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.
  •  1
    Kit Fine Precis. Discussion
    Philosophical Studies 122 (3). 2005.
  •  98
    Prolegomenon To Any Future Neo‐Logicist Set Theory: Abstraction And Indefinite Extensibility
    British Journal for the Philosophy of Science 54 (1): 59-91. 2003.
    The purpose of this paper is to assess the prospects for a neo‐logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): ∀P∀Q[Ext(P) = Ext(Q) ≡ [(BAD(P) & BAD(Q)) ∨ ∀x(Px ≡ Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’.1 Background: what and why?2…Read more
  •  2
    Understanding the Infinite
    with Shaughan Lavine
    Studia Logica 63 (1): 123-128. 1994.
  •  15
    The articles in this volume represent a part of the philosophical literature on higher-order logic and the Skolem paradox. They ask the question what is second-order logic? and examine various interpretations of the Lowenheim-Skolem theorem.
  •  140
    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
  •  951
    What is mathematical logic?
    Philosophia 8 (1): 79-94. 1978.
    This review concludes that if the authors know what mathematical logic is they have not shared their knowledge with the readers. This highly praised book is replete with errors and incoherency.
  • Thinking about Mathematics: The Philosophy of Mathematics
    Philosophical Quarterly 52 (207): 272-274. 2002.
  •  147
    Frege Meets Aristotle: Points as Abstracts
    Philosophia Mathematica. 2015.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at sta…Read more
  •  52
    Remarks on the development of computability
    History and Philosophy of Logic 4 (1-2): 203-220. 1983.
    The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized …Read more
  •  82
    Second-order logic, foundations, and rules
    Journal of Philosophy 87 (5): 234-261. 1990.
  •  227
    There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or …Read more