Most of my work has focused on the philosophies of Gottlob Frege and Bertrand Russell, especially their philosophical logics and their import for contemporary discussions in philosophy of language, intensional logic and the philosophy of mathematics. I am also interested in informal logic, ethics, and the history of logic and analytic philosophy. I am currently writing a book on Russell.
Most of my publications are available below. If not, email me and I’ll send you a copy. Questions and comments welcome.

The Paradoxes and Russell’s Theory of Incomplete Symbols Abstract PDF
Forthcoming in Philosophical Studies.
Abstract:
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 thought of as a single thing, neither can the meaning/intension of any expression capable of singling out one collection (class) of things as opposed to another. If this is right, it shows that Russell’s method of solving the logical paradoxes is wholly incompatible with anything like a Fregean dualism between sense and reference or meaning and denotation. I also discuss how this realization lead to modifications in his understanding of propositions and propositional functions, and suggest that Russell’s confrontation with these issues may be instructive for ongoing research.

Grundgesetze and the Sense/Reference Distinction Abstract PDF
Forthcoming in Essays on Frege’s Grundgesetze der Arithmetik, eds. M. Rossberg and P. Ebert.
Abstract:
Frege developed the theory of sense and reference while composing his Grundgesetze and considering its philosophical implications. The Grundgesetze is thus the most important test case for the application of this theory of meaning. I argue that evidence internal and external to the Grundgesetze suggests that he thought of senses as having a structure isomorphic to the Grundgesetze expressions that would be used to express them, which entails a theory about the identity conditions of senses that is relatively finegrained, though still coarser than some other commentators have suggested. While this interpretation does not make Frege’s ontological commitment to the denizens of a “third realm” as profligate as some have alleged, it is sufficiently bloated to lead to Cantorian paradoxes and diagonal contradictions independent of his Basic Law V.

The Constituents of the Propositions of Logic Abstract PDF
Forthcoming in Acquaintance, Knowledge, and Logic: New Essays on Bertrand Russell’s The Problems of Philosophy, eds. D. Wishon and B. Linsky.
Abstract:
In The Problems of Philosophy and other works of the same period, Russell claims that every proposition must contain at least one universal. Even fully general propositions of logic are claimed to contain “abstract logical universals”, and our knowledge of logical truths claimed to be a species of a priori knowledge of universals. However, these views are in considerable tension with Russell’s own philosophy of logic and mathematics as presented in Principia Mathematica. Universals generally are qualities and relations, but if, for example, PM’s disjunction (∨) is a relation, what is it a relation between? There is no obvious answer to this given Russell’s other philosophical commitments at this time, although hints are left in some of the prePM manuscripts. In this paper, I explore this tension in Russell's philosophy and relate it to developments both before and after Problems.

Universals as Individuals: Reply to Levine Abstract PDF
Draft. Comments welcome.
Abstract:
In forthcoming work, James Levine has argued, contrary to past writings by myself, Gregory Landini and Graham Stevens, that even in the period of Principia Mathematica, Russell held a typed metaphysics according to which universals would not count as individuals, but are nonetheless genuine entities in Russell’s metaphysics. In this reply, I attempt to clarify what Russell’s conception of an individual was in his work prior to the influence of Wittgenstein, and argue that Russell only began to take seriously the possibility that universals might not be individuals in the 1913 Theory of Knowledge manuscript.

A Generic Russellian Elimination of Abstract Objects Abstract PDF
Draft. Comments welcome.
Abstract:
Recently, there’s been interest in “abstractionist” forms of logicism which attempt to deduce mathematical theories from abstraction principles, those of the form:
f(x) = f(y) iff Rxy
where R is an equivalence relation, and f is a functor mapping entities x and y in the same logical type to the same abstract object just in case R holds between them. In this paper I explore (but do not definitely endorse) a method for eliminating the need for principles postulating abstract objects by treating the terms in such an abstraction principles as “incomplete symbols”, using Russell’s noclasses theory as a template from which to generalize. I defend views of this stripe against certain objections, most notably Richard Heck’s charge that syntactic forms of nominalism cannot correctly deal with nonfirstorderizable quantification over apparent abstracta.

Early Russell on Types and Plurals Abstract PDF
Journal for the History of Analytical Philosophy 2/6 (2014): 1–21.
Abstract:
In 1903, in The Principles of Mathematics (PoM), Russell endorsed an account of classes whereupon a class fundamentally is to be considered many things, and not one, and used this thesis to explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction between singular and plural. The view, however, was shortlived—rejected before PoM even appeared in print. However, aside from mentions of a few misgivings, there is little evidence about why he abandoned this view. In this paper, I attempt to clarify Russell’s early views about plurality, arguing that they did not involve countenancing special kinds of plural things distinct from individuals. I also clarify what his misgivings about these views were, making it clear that while the plural understanding of classes helped solve certain forms of Russell’s paradox, certain other Cantorian paradoxes remained. Finally, I aim to show that Russell’s abandonment of something like plural logic is understandable given his own conception of logic and philosophical aims when compared to the views and approaches taken by contemporary advocates of plural logic.

PM’s Circumflex, Syntax and Philosophy of Types Abstract PDF
In The Palgrave Centenary Companion to Principia Mathematica, eds. N. Griffin and B. Linsky. New York: Palgrave Macmillian, 2013, pp. 218–246.
Abstract:
Along with offering an historicallyoriented interpretive reconstruction of the syntax of Principia Mathematica (first ed.), I argue for a certain understanding of its use of propositional function abstracts formed by placing a circumflex on a variable. I argue that this notation is used in PM only when definitions are stated schematically in the metalanguage, and in argumentposition when highertype variables are involved. My aim throughout is to explain how the usage of function abstracts as “terms” (loosely speaking) is not inconsistent with a philosophy of types that does not think of propositional functions as mind and languageindependent objects, and adopts a nominalist/substitutional semantics instead. I contrast PM’s approach here both to function abstraction found in the typed λcalculus, and also to Frege’s notation for functions of various levels that forgoes abstracts altogether, between which it is a kind of intermediary.

Neologicism and Russell’s Logicism Abstract PDF
Russell n.s. 32 (Winter 2012–13): 127–59.
Abstract:
Most advocates of the socalled “neologicist” movement in the philosophy of mathematics identify themselves as “NeoFregeans” (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 neologicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that NeoRussellian forms of neologicism remain viable positions for current philosophers of mathematics.

Frege’s Changing Conception of Number Abstract PDF
Theoria 78 (2012): 146–167.
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 secondlevel 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.

The Functions of Russell’s No Class Theory Abstract PDF
Review of Symbolic Logic 3/4 (2010): 633–664.
Abstract:
Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higherorder quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of socalled “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 meet the demands placed on classes by mathematics. I show that Russell did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extralinguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language.

The Senses of Functions in the Logic of Sense and Denotation Abstract PDF
Bulletin of Symbolic Logic 16/2 (2010): 153–188.
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 “sensefunctions“, functions from sense to sense. However, the conditions he lays out under which a sensefunction is to be regarded as a sense presenting another function as denotation allow for certain undesirable results given certain unusual or “deviant” sensefunctions. 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.

Gottlob Frege Abstract PDF
In The Routledge Companion to Nineteenth Century Philosophy, ed. Dean Moyar. Abingdon: Routledge, 2010, pp. 858–886.
Abstract:
A summary of the philosophical career and intellectual contributions of Gottlob Frege (1848–1925), including his invention of first and secondorder quantified logic, his logicist understanding of arithmetic and numbers, the theory of sense (Sinn) and reference (Bedeutung) of language, the thirdrealm metaphysics of “thoughts”, his arguments against rival views, and other topics.

Russell, His Paradoxes and Cantor’s Theorem: Part I Abstract PDF
and
Russell, His Paradoxes and Cantor’s Theorem: Part II Abstract PDF
Philosophy Compass 5/1 (2010): 16–28 and 29–41.
Abstract:
In these articles, I describe Cantor’s powerclass 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, classintensions 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 in 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.

A Cantorian Argument Against Frege’s and Early Russell’s Theories of Descriptions Abstract PDF
In Russell vs. Meinong: The Legacy of “On Denoting”, eds. Nicholas Griffin and Dale Jacquette. New York: Routledge, 2008, pp. 65‒77.
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 Russellstyle 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.

Russell’s Logical Atomism Abstract Link
In The Stanford Encyclopedia of Philosophy (plato.stanford.edu).
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.

The Origins of the Propositional Functions Version of Russell’s Paradox Abstract PDF
Russell, n.s. 24 (2004–05): 101–32.
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.

Does Frege Have Too Many Thoughts? A Cantorian Problem Revisited Abstract PDF
Analysis 65:1 (2005): 44–49.
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.

Putting Form Before Function: Logical Grammar in Frege, Russell and Wittgenstein Abstract PDF
Philosopher’s Imprint 4/2 (2004): 1–47.
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.

Russell’s 1903–05 Anticipation of the Lambda Calculus Abstract PDF
History and Philosophy of Logic 24 (2003): 15–37.
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 “valueranges”. 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.

The Number of Senses Abstract PDF
Erkenntnis 58 (2003): 302–323.
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 ramifiedorders in order to avoid antinomy, but that the philosophical justification of such orders is, as yet, still somewhat problematic.

Russell on ‘Disambiguating With the Grain’ Abstract PDF
Russell n.s. 21 (Winter 2001–02): 101–27.
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 neoFregeans (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.

When is Genetic Reasoning not Fallacious? Abstract PDF
Argumentation 16 (2002): 383–400.
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.

Russell’s Paradox in Appendix B of the Principles of Mathematics: Was Frege’s Response Adequate? Abstract PDF
History and Philosophy of Logic 22 (2001): 13–28.
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 illformed 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.

Paradox, RussellMyhill Link
Induction and Deduction Link
Validity and Soundness Link
Square of Opposition Link
In The Internet Encyclopedia of Philosophy (http://www.iep.utm.edu/).

Is Pacifism Irrational? Abstract PDF
Peace Review 11/1 (March 1999): 65–70.
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.