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177PM's Circumflex, Syntax and Philosophy of TypesIn Kenneth Blackwell, Nicholas Griffin & Bernard Linsky (eds.), Principia mathematica at 100, Bertrand Russell Research Centre. pp. 218-246. 2011.Along with offering an historically-oriented interpretive reconstruction of the syntax of PM ( rst ed.), I argue for a certain understanding of its use of propositional function abstracts formed by placing a circum ex on a variable. I argue that this notation is used in PM only when de nitions are stated schematically in the metalanguage, and in argument-position when higher-type variables are involved. My aim throughout is to explain how the usage of function abstracts as “terms” (loosely speak…Read more
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43Deductive and inductive argumentsInternet Encyclopedia of Philosophy. 2003.A simple summary of the difference between induction and deduction.
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230The Origins of the Propositional Functions Version of Russell's ParadoxRussell: The Journal of Bertrand Russell Studies 24 (2). 2004.Russell discovered the classes version of Russell's Paradox in spring 1901, and the predicates version near the same time. There is a problem, however, in dating the discovery of the propositional functions version. In 1906, Russell claimed he discovered it after May 1903, but this conflicts with the widespread belief that the functions version appears in _The Principles of Mathematics_, finished in late 1902. I argue that Russell's dating was accurate, and that the functions version does not ap…Read more
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21Review of Logicism and the Philosophy of Language, Arthur Sullivan (review)The Bertrand Russell Society Quarterly 120 39-43. 2003.
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27Review: Gregory Landini, Russell. London and New York, Routledge 2011 (review)Journal for the History of Analytical Philosophy 1 (2). 2012.This essay reviews Gregory Landini's book Russell
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119Gottlob FregeInternet Encyclopedia of Philosophy. 2001.Gottlob Frege (1848-1925) was a German logician, mathematician and philosopher who played a crucial role in the emergence of modern logic and analytic philosophy. Frege's logical works were revolutionary, and are often taken to represent the fundamental break between contemporary approaches and the older, Aristotelian tradition. He invented modern quantificational logic, and created the first fully axiomatic system for logic, which was complete in its treatment of propositional and first-order l…Read more
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30A faithful companion (review)The Bertrand Russell Society Quarterly (120): 25-41. 2004.We can at last release our breath: the long awaited Russell volume in the popular Cambridge Companion series has finally arrived. It contains fifteen chapters written by well known Russell scholars dealing with a wide array of Russelliana, along with a quite extensive introductory essay by the volume editor. It is not difficult to see what took so long. Russell’s corpus, even considering only his philosophical writings, outstrips in both breadth and volume almost all the other figures covered in…Read more
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375Russell's Paradox in Appendix B of the Principles of Mathematics : Was Frege's response adequate?History and Philosophy of Logic 22 (1): 13-28. 2001.In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy due to his dist…Read more
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258Russell's Logicism through Kantian Spectacles [review of Anssi Korhonen, Logic as Universal Science: Russell’s Early Logicism and Its Philosophical Context ]Russell: The Journal of Bertrand Russell Studies 34 (1): 79-84. 2014.Review of Logic as Universal Science: Russell’s Early Logicism and Its Philosophical Context, by Anssi Korhonen (Palgrave Macmillan 2013).
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362Putting form before function: Logical grammar in Frege, Russell, and WittgensteinPhilosophers' Imprint 4 1-47. 2004.The positions of Frege, Russell and Wittgenstein on the priority of complexes over (propositional) functions are sketched, challenging those who take the "judgment centered" aspects of the Tractatus to be inherited from Frege not Russell. Frege's views on the priority of judgments are problematic, and unlike Wittgenstein's. Russell's views on these matters, and their development, are discussed in detail, and shown to be more sophisticated than usually supposed. Certain misreadings of Russell, in…Read more
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421Does Frege have too many thoughts? A Cantorian problem revisited.Analysis 65 (1). 2005.This paper continues a thread in Analysis begun by Adam Rieger and Nicholas Denyer. Rieger argued that Frege’s theory of thoughts violates Cantor’s theorem by postulating as many thoughts as concepts. Denyer countered that Rieger’s construction could not show that the thoughts generated are always distinct for distinct concepts. By focusing on universally quantified thoughts, rather than thoughts that attribute a concept to an individual, I give a different construction that avoids Denyer’s prob…Read more
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218The paradoxes and Russell's theory of incomplete symbolsPhilosophical Studies 169 (2): 183-207. 2014.Russell claims in his autobiography and elsewhere that he discovered his 1905 theory of descriptions while attempting to solve the logical and semantic paradoxes plaguing his work on the foundations of mathematics. In this paper, I hope to make the connection between his work on the paradoxes and the theory of descriptions and his theory of incomplete symbols generally clearer. In particular, I argue that the theory of descriptions arose from the realization that not only can a class not be thou…Read more
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57Review of Richard L. Mendelsohn, The Philosophy of Gottlob Frege (review)Notre Dame Philosophical Reviews 2005 (11). 2005.
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318Russell, His Paradoxes, and Cantor's Theorem: Part IPhilosophy Compass 5 (1): 16-28. 2010.In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theo…Read more
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281Jolen Galaugher, Russell’s Philosophy of Logical Analysis: 1897–1905 (review)Journal for the History of Analytical Philosophy 3 (2). 2015.Review of Russell’s Philosophy of Logical Atomism 1897–1905, by Jolen Galaugher (Palgrave Macmillan 2013)
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169A Generic Russellian Elimination of Abstract ObjectsPhilosophia Mathematica 25 (1): 91-115. 2017.In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discus…Read more
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304The functions of Russell’s no class theoryReview of Symbolic Logic 3 (4): 633-664. 2010.Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to…Read more
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182Russell on "Disambiguating with the Grain"Russell: The Journal of Bertrand Russell Studies 21 (2). 2001.Fregeans face the difficulty finding a notation for distinguishing statements about the sense or meaning of an expression as opposed to its reference or denotation. Famously, in "On Denoting", Russell rejected methods that begin with an expression designating its denotation, and then alter it with a "the meaning of" operator to designate the meaning. Such methods attempt an impossible "backward road" from denotation to meaning. Contemporary neo-Fregeans, however, have suggested that we can disam…Read more
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317Propositional logicInternet Encyclopedia of Philosophy. 2004.Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not…Read more
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389Frege and the Logic of Sense and ReferenceRoutledge. 2001.This book aims to develop certain aspects of Gottlob Frege’s theory of meaning, especially those relevant to intensional logic. It offers a new interpretation of the nature of senses, and attempts to devise a logical calculus for the theory of sense and reference that captures as closely as possible the views of the historical Frege. (The approach is contrasted with the less historically-minded Logic of Sense and Denotation of Alonzo Church.) Comparisons of Frege’s theory with those of Russell a…Read more
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242When Is Genetic Reasoning Not Fallacious?Argumentation 16 (4): 383-400. 2002.Attempts to evaluate a belief or argument on the basis of its cause or origin are usually condemned as committing the genetic fallacy. However, I sketch a number of cases in which causal or historical factors are logically relevant to evaluating a belief, including an interesting abductive form that reasons from the best explanation for the existence of a belief to its likely truth. Such arguments are also susceptible to refutation by genetic reasoning that may come very close to the standard ex…Read more
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176A Cantorian argument against Frege's and early Russell's theories of descriptionsIn Nicholas Griffin & Dale Jacquette (eds.), Russell Vs. Meinong: The Legacy of "On Denoting", Routledge. pp. 65-77. 2009.It would be an understatement to say that Russell was interested in Cantorian diagonal paradoxes. His discovery of the various versions of Russell’s paradox—the classes version, the predicates version, the propositional functions version—had a lasting effect on his views in philosophical logic. Similar Cantorian paradoxes regarding propositions—such as that discussed in §500 of The Principles of Mathematics—were surely among the reasons Russell eventually abandoned his ontology of propositions.1…Read more
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141Russell's paradoxInternet Encyclopedia of Philosophy. 2001.Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included i…Read more
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302Russell, His Paradoxes, and Cantor's Theorem: Part IIPhilosophy Compass 5 (1): 29-41. 2010.Sequel to Part I. In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addres…Read more
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487Neo-Logicism and Russell's LogicismRussell: The Journal of Bertrand Russell Studies 32 (2): 127-159. 2012.Abstract:Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more…Read more
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182A New Century in the Life of a ParadoxReview of Modern Logic 11 (2): 7-29. 2008.Review essay covering Godehard Link, ed. One Hundred Years of Russell’s Paradox (de Gruyter 2004).