- Logical Form and the Development of Russell’s Logicism
In Origins and Varieties of Logicism, edited by F. Boccuni and A. Sereni. (Routledge 2022), pp. 147–166.
Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth; the other a logical truth. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases.
- Grundgesetze and the Sense/Reference Distinction
In Essays on Frege’s Basic Laws of Arithmetic, edited by P. Ebert and M. Rossberg. (Oxford University Press 2019), pp. 142–66.
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 fine-grained, 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.
- New Logic and the Seeds of Analytic Philosophy: Boole, Frege
In A Companion to Nineteenth-Century Philosophy, edited by J. Shand. (Wiley Blackwell 2019), pp. 454–79.
This contribution surveys the revolution in logic that took place in the 19th Century and its influence on philosophy, with particular emphasis on the writings of George Boole and Gottlob Frege. Boole invented the algebraic treatment of logic, making a mathematical analysis of the Aristotelian syllogistic possible, and more. Frege invented modern quantificational logic and presented a fully axiomatized second-order function calculus. Frege’s logic was developed as part of his logicism, his aim to derive arithmetic from a logical basis, but unfortunately his precise method involved the use of an inconsistent theory of extensions of concepts. Frege’s very influential views on meaning and truth, including his distinction between sense and reference, and between objects and concepts of different levels, are also discussed. The influence of Frege, and the 19th Century revolution in logic generally, on later philosophy, especially analytic philosophy, is also outlined.
- Russell’s Logicism
In The Bloomsbury Companion to Bertrand Russell, edited by R. Wahl. (Bloomsbury Academic 2019), pp. 151–78.
Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions of arithmetical concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth.
- G. E. Moore’s Unpublished Review of The Principles of Mathematics
Russell n.s. 38 (2018–19): 131–64.
This file contains an introduction along with the text of Moore’s review.
Several interesting themes emerge from G. E. Moore’s previously unpublished review of The Principles of Mathematics. These include a worry concerning whether mathematical notions are identical to purely logical ones, even if coextensive logical ones exist. Another involves a conception of infinity based on endless series neglected in the Principles but arguably involved in Zeno’s paradox of Achilles and the Tortoise. Moore also questions the scope of Russell’s notion of material implication, and other aspects of Russell’s claim that mathematics reduces to logic.
- Russell on Ontological Fundamentality and Existence
In The Philosophy of Logical Atomism: A Centenary Reappraisal, edited by L. Elkind and G. Landini. (Palgrave Macmillan 2018), pp. 155–79.
Russell is often taken as a forerunner of the Quinean position that “to be is to be the value of a bound variable”, whereupon the ontological commitment of a theory is given by what it quantifies over. Among other reasons, Russell was among the first to suggest that all existence statements should be analyzed by means of existential quantification. That there was more to Russell’s metaphysics than what existential quantifications come out as true is obvious in the earlier period where Russell still made a distinction between existence and being/subsistence. But even the later Russell, including that of the Logical Atomism lectures period, would not have understood ontological questions to be first and foremost questions of quantification. He would take fundamentality to be important too, which explains in part his assertions to the effect the the values of individual variables have a reality not attributable to values of higher-order variables, even ineliminable higher-order variables.
- A Generic Russellian Elimination of Abstract Objects
Philosophia Mathematica 25/1 (2017): 91–115.
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 Heck’s charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system treating apparent terms for numbers using these definitions.
- Three Unpublished Manuscripts from 1903
Russell n.s. 36 (2016): 5–44.
I present and discuss three previously unpublished manuscripts written by Bertrand Russell in 1903, not included with similar manuscripts in Volume 4 of his Collected Papers. One is a one-page list of basic principles for his “functional theory” of May 1903, in which Russell partly anticipated the later Lambda Calculus. The next, catalogued under the title “Proof That No Function Takes All Values”, largely explores the status of Cantor's proof that there is no greatest cardinal number in the variation of the functional theory holding that only some but not all complexes can be analyzed into function and argument. The final manuscript, “Meaning and Denotation”, examines how his pre-1905 distinction between meaning and denotation is to be understood with respect to functions and their arguments. In them, Russell seems to endorse an extensional view of functions not endorsed in other works prior to the 1920s. All three manuscripts illustrate the close connection between his work on the logical paradoxes and his work on the theory of meaning.
- The Constituents of the Propositions of Logic
In Acquaintance, Knowledge, and Logic: New Essays on Bertrand Russell’s The Problems of Philosophy, edited by D. Wishon and B. Linsky. (CSLI Publications 2015), pp. 189–229.
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 pre-PM manuscripts. In this paper, I explore this tension in Russell's philosophy and relate it to developments both before and after Problems.
- The Russell–Dummett Correspondence on Frege and his Nachlaß
The Bertrand Russell Society Bulletin no. 150 (2014): 25–29.
Russell corresponded with Sir Michael Dummett (1925–2011) between 1953 and 1963 while the latter was working on a book on Frege, eventually published as Frege: Philosophy of Language (1973). In their letters they discuss Russell’s correspondence with Frege, translating it into English, as well as Frege’s attempted solution to Russell’s paradox in the appendix to vol. 2 of his Grundgesetze der Arithmetik. After Dummett visited the University of Münster to view Frege’s Nachlaß, he sent reports back to Russell concerning both the philosophical materials Frege left behind, as well as information from Frege’s journal revealing his anti-semitic political opinions. Their interaction contains interpretive conjectures and insights on Dummett’s side, and some dark humor on Russell’s.
- The Paradoxes and Russell’s Theory of Incomplete Symbols
Philosophical Studies 169/2 (2014): 183–207.
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.
- Early Russell on Types and Plurals
Journal for the History of Analytical Philosophy 2/6 (2014): 1–21.
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 short-lived—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
In The Palgrave Centenary Companion to Principia Mathematica, edited by N. Griffin and B. Linsky. (Palgrave Macmillian 2013), pp. 218–46.
Along with offering an historically-oriented 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 argument-position when higher-type 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 language-independent objects, and adopts a nominalist/substitutional semantics instead. I contrast PM’s approach here both to function abstraction found in the typed λ-calculus, and also to Frege’s notation for functions of various levels that forgoes abstracts altogether, between which it is a kind of intermediary.
- Neo-logicism and Russell’s Logicism
Russell n.s. 32 (2012–13): 127–59.
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 embarrassment 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.
- Frege’s Changing Conception of Number
Theoria 78 (2012): 146–67.
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.
- The Functions of Russell’s No Class Theory
Review of Symbolic Logic 3/4 (2010): 633–64.
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 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 extra-linguistic 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
Bulletin of Symbolic Logic 16/2 (2010): 153–88.
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.
- Gottlob Frege
In The Routledge Companion to Nineteenth Century Philosophy, edited by Dean Moyar. (Routledge 2010), pp. 858–86.
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.
- Russell, His Paradoxes and Cantor’s Theorem [Parts I–II]
Philosophy Compass 5/1 (2010): 16–28 and 29–41.
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 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
In Russell vs. Meinong: The Legacy of “On Denoting”, edited by N. Griffin and D. Jacquette. (Routledge 2008), pp. 65–77.
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.
- The Origins of the Propositional Functions Version of Russell’s Paradox
Russell n.s. 24 (2004–05): 101–32.
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
Analysis 65/1 (2005): 44–49.
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
Philosopher’s Imprint 4/2 (2004): 1–47.
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
History and Philosophy of Logic 24 (2003): 15–37.
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.
- The Number of Senses
Erkenntnis 58 (2003): 302–23.
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.
- Russell on ‘Disambiguating With the Grain’
Russell n.s. 21 (2001–02): 101–27.
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.
- When is Genetic Reasoning not Fallacious?
Argumentation 16 (2002): 383–400.
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?
History and Philosophy of Logic 22 (2001): 13–28.
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.
- Is Pacifism Irrational?
Peace Review 11/1 (1999): 65–70.
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.
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