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22Review of P. Benacerraf and H. Putnam (eds.) Philosophy of Mathematics (review)Philosophy of Science 52 (3): 488-. 1985.
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212Mathematical structuralismPhilosophia 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.
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2Simple truth, contradiction, and consistencyIn Graham Priest, J. C. Beall & Bradley Armour-Garb (eds.), The Law of Non-Contradiction, Oxford University Press. 2004.
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219Do not claim too much: Second-order logic and first-order logicPhilosophia 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.
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80Classical LogicIn Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, The Metaphysics Research Lab. 2014.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.
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86Varieties of LogicOxford 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.
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12Review: Stephen C. Kleene, Origins of Recursive Function Theory; Martin Davis, Why Godel Didn't have Church's Thesis; Stephen C. Kleene, Reflections on Church's Thesis (review)Journal of Symbolic Logic 55 (1): 348-350. 1990.
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136All 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
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47Understanding the InfinitePhilosophical 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.
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101Prolegomenon To Any Future Neo‐Logicist Set Theory: Abstraction And Indefinite ExtensibilityBritish 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
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17The 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.
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152Incompleteness and inconsistencyMind 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
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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 |