WORKS IN PROGRESS

Please do not cite without my permission.

• Russell’s Logicism and Neo-Logicism
  
  (coming soon)
• Frege’s Changing Conception of Number
  
  (coming soon)
• The Functions of Russell’s Having No Class
  
  (coming soon)
• Russell, His Paradoxes and Cantor’s Theorem: Part I (DRAFT) and Russell, His Paradoxes and Cantor's Theorem: Part II (DRAFT);
  
  invited submissions to Philosopy Compass: forthcoming?
• Gottlob Frege
  
  forthcoming in The Routledge Companion to Nineteenth Century Philosophy, ed. Dean Moyar.
• The Senses of Functions in the Logic of Sense and Denotation
  
  To appear.

(In addition to these, I’m working on a book on Russell, and a number of smaller projects.)

PUBLISHED WORKS

(Selected)

Book

• Frege and the Logic of Sense and Reference
  
  Routledge: New York: 2002. Available free online—click the link above.

Articles and Book Chapters

• A Cantorian Argument Against Frege’s and Early Russell’s Theories of Descriptions
  
  In The Legacy of “On Denoting”, eds. Nicholas Griffin and Dale Jacquette. New York: Routledge, 2008.
• Russell’s Logical Atomism
  
  In The Stanford Encyclopedia of Philosophy (plato.stanford.edu).
• The Origins of the Propositional Functions Version of Russell’s Paradox
  
  Russell, n.s. 24 (Winter 2004–05): 101–32.
• Does Frege Have Too Many Thoughts? A Cantorian Problem Revisited
  
  Analysis 65:1 (2005): 44–49.
• Putting Form Before Function: Logical Grammar in Frege, Russell and Wittgenstein
  
  Philosopher’s Imprint 4:2 (2004): 1–47.
• Russell’s 1903–05 Anticipation of the Lambda Calculus
  
  History and Philosophy of Logic 24 (2003): 15–37.
• The Numbers of Senses
  
  Erkenntnis 58 (2003): 302–323.
• Russell on “Disambiguating With the Grain”
  
  Russell n.s. 21 (Winter 2001–02): 101–27.
• When is Genetic Reasoning not Fallacious?
  
  Argumentation 16 (2002): 383–400.
• Russell’s Paradox in Appendix B of the Principles of Mathematics: Was Frege’s Response Adequate?
  
  History and Philosophy of Logic 22 (2001): 13–28.
• Frege, Gottlob, Propositional Logic, Paradox, Russell’s, Paradox, Russell-Myhill, Argument, Induction and Deduction, Validity and Soundness, and Square of Opposition
  
  In The Internet Encyclopedia of Philosophy (http://www.iep.utm.edu/).
• Is Pacifism Irrational?
  
  Peace Review 11:1 (March 1999): 65–70.

Reviews and Review Essays

• A New Century in the Life of a Paradox
  
  review essay covering Godehard Link, ed. One Hundred Years of Russell’s Paradox (de Gruyter 2004), Review of Modern Logic, forthcoming.
• Review of G. Imaguire and B. Linsky, “On Denoting” 1905–2005
  
  (Philosophia 2005): Notre Dame Philosophical Reviews (ndpr.nd.edu), 2006.
• Review of Nicholas Griffin, ed. The Cambridge Companion to Bertrand Russell
  
  (Cambridge 2003); Review of Modern Logic 10 #1–2 (2005): 161–70.
• Review of Richard Mendelsohn, The Philosophy of Gottlob Frege
  
  (Cambridge 2005); Notre Dame Philosophical Reviews (ndpr.nd.edu), 2005.
• A Faithful Companion
  
  review essay covering Nicholas Griffin, ed. The Cambridge Companion to Bertrand Russell (Cambridge 2003); The Bertrand Russell Society Quarterly no. 121 (February 2004): 25–41.
• Review of A. Sullivan, ed. Logicism and the Philosophy of Language: Selections from Frege and Russell
  
  (Broadview 2003) in The Bertrand Russell Society Quarterly no. 120 (Nov. 2003): 39–43.
• Review of Erich Reck, ed. From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy
  
  (Oxford 2002); in The Review of Metaphysics 57 (2003): 177–178.

If you have questions or comments about any of the above, please email me at klement <at> philos <dot> umass <dot> edu!

ABSTRACT:
Most advocates of the so-called “neologicist” 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 lights of its assumption of axioms of infinity, reducibiity and so on). In this paper I have three aims: firstly, to identify more clearly the primary metaontological and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that Neo-Russellian forms of neologicism remain viable positions for current philosophers of mathematics.

ABSTRACT:
I trace changes to Frege’s understanding of numbers, arguing in particular that the view of arithmetic based in geometry developed at the end of his life (1924–1925) was not as radical a deviation from his views during the logicist period as some have suggested. Indeed, by looking at his earlier views regarding the connection between numbers and second-level concepts, his understanding of extensions of concepts, and the changes to his views, firstly, in between Grundlagen and Grundgesetze, and, later, after learning of Russell’s paradox, this position is natural position for him to have retreated to, when properly understood.

ABSTRACT:
In a recent discussion piece, Scott Soames argues that Russell’s “no-classes theory,” in which apparent reference to classes or sets is eliminated using higher-order quantification was “not a genuine achievement,” noting the obscurity surrounding Russell’s understanding of so-called “propositional functions.” Soames claims that nominalist understandings of propositional functions are problematic, and that realist readings—though, in Soames's mind, philosophically superior—do not succeed in escaping commitment to sets or classes. I argue that, contrary to Soames, Russell did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I also argue that a reading taking propositional functions to be open formulas is possible which does not succumb to the problems Soames alleges, and which does not, pace Soames, amount to a reduction of classes to language.

ABSTRACT:
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 theorem, its proof, how it can be used to manufacture paradoxes, and several broad categories of strategies for offering solutions to these paradoxes. Part II discusses the origins and impact of these paradoxes on Bertrand Russell’s philosophy in particular, as well as his own favored brand of solution whereupon those purported entities that, if reified, lead to these contradictions, must not be genuine entities, but "logical fictions" or "logical constructions" instead.

ABSTRACT:
A summary of the philosophical career and intellectual contributions of Gottlob Frege (1848–1925), including his invention of first- and second-order quantified logic, his logicist understanding of arithmetic and numbers, the theory of sense (Sinn) and reference (Bedeutung) of language, the third-realm metaphysics of “thoughts”, his arguments against rival views, and other topics.

ABSTRACT:
This paper discusses certain problems arising within the treatment of the senses of functions in Church’s Logic of Sense and Denotation. Church understands such senses themselves to be “sense-functions“, functions from sense to sense. However, the conditions he lays out under which a sense-function is to be regarded as a sense presenting another function as denotation allow for certain undesirable results given certain unusual or “deviant” sense-functions. Certain absurdities result, e.g., an argument can be found for equating any two senses of the same type. An alternative treatment of the senses of functions is discussed, and is thought to do better justice to Frege’s original theory.

ABSTRACT:
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 and others are given. It is in the end shown that developing Frege’s theory in these ways reveals serious problems hitherto largely unnoticed, including those possibly rendering a Fregean intensional logic inconsistent even if his naïve class theory is excluded.

ABSTRACT:
This paper discusses an argument, inspired by Russell, against certain theories of definite descriptions, like Frege’s and those of the pre-“On Denoting” Russell, that posit a sense or meaning for a descriptive phrase of the form “the φ” distinct from its denotation. If one is committed to (1) a liberal ontology of properties, (2) the existence of at least one descriptive sense for each property, (3) certain plausible principles regarding the identity conditions of senses, and (4) an account of descriptive senses whereupon they can themselves be presented by other senses of the same type, a violation of Cantor’s theorem results leading to a Russell-style antinomy. Let something have property H if and only if it is a descriptive sense that does not have its corresponding property. Consider the sense of “the [thing that is] H”. Does it have H? Various strategies for avoiding the problem are discussed and evaluated.

ABSTRACT:
A summary of Russell’s logical atomism, understood to include both a metaphysical view and a certain methodology for doing philosophy. The metaphysical view amounts to the claim that the world consists of a plurality of independently existing things exhibiting qualities and standing in relations. The methodological view recommends a process of analysis, whereby one attempts to define or reconstruct more complex notions or vocabularies in terms of simpler ones. The origins of this theory, and its influence and reception are also discussed.

ABSTRACT:
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 appear in the Principles. I distinguish the functions and predicates versions, give a novel reading of the Principles, section 85, as a paradox dealing with what Russell calls assertions, and show that Russell’s logical notation in 1902 had no way of even formulating the functions version. The propositional functions version had its origins in the summer of 1903, soon after Russell’s notation had changed in such a way as to make a formulation possible.

ABSTRACT:
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 problem. I also note that this problem for Frege’s philosophy was discovered by Bertrand Russell as early as 1902 and has been discussed intermittently since.

ABSTRACT:
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, including those regarding the relationship between propositional functions and universals, are exposed. Wittgenstein’s and Russell’s views on logical grammar are shown to be very similar. Russell’s type theory does not countenance types of genuine entities nor metaphysical truths that cannot be put into words, contrary to conventional wisdom. I relate this to the debate over “inexpressible truths” in the Tractatus. I lastly comment on the changes to Russell’s views brought about by Wittgenstein’s influence.

ABSTRACT:
It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions as value. Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus.

ABSTRACT:
Many philosophers still countenance senses or meanings in the broadly Fregean vein. However, it is difficult to posit the existence of senses without positing quite a lot of them, including at least one presenting every entity in existence. I discuss a number of Cantorian paradoxes that seem to result from an overly large metaphysics of senses, and various possible solutions. Certain more deflationary and nontraditional understanding of senses, and to what extent they fare better in solving the problems, are also discussed. In the end, it is concluded that one must divide senses into various ramified-orders in order to avoid antinomy, but that the philosophical justification of such orders is, as yet, still somewhat problematic.

ABSTRACT:
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 (especially Pavel Tichý), however, have suggested that we can disambiguate with, rather than against, the grain, by using a notation that begins with expressions designating senses or meanings, and then alters them with a “the denotation of” operator to designate the denotation. I show that in his manuscripts of 1903–05 Russell both considered and rejected a similar notation along with the metaphysical suppositions underlying it. This discussion sheds light on the evolution of Russell’s thought, and may yet be instructive for ongoing debates.

ABSTRACT:
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 examples given of supposedly fallacious genetic reasoning.

ABSTRACT:
In their correspondence in 1902 and 1903, after discussing Russell’s 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 distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naïve class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein.

ABSTRACT:
In this paper, I counter arguments to the effect that pacifism must be irrational which cite hypothetical situations in which violence is necessary to prevent a far greater evil. I argue that for persons similar to myself, for whom such scenarios are extremely unlikely, promoting in oneself the disposition to avoid violence in any circumstances is more likely to lead to better results than not cultivating such a disposition just for the sake of such unlikely eventualities.