frege and russell on logic and language
Kevin C. Klement
Review of Logicism and the Philosophy of Language: Selections from Frege and Russell, Arthur Sullivan, ed. Toronto: Broadview Press, 2003. 298 pp. $24.95 paperback.
This new anthology brings together 15 pieces by the most prominent defenders of logicism: Russell, and his German predecessor Gottlob Frege (1848-1925). Logicism is the position in the philosophy of mathematics that mathematical truth is a species of logical truth. According to logicists, when properly analyzed, the truths of mathematics reveal themselves to be expressible in the vocabulary of logic alone, and deducible from purely logical premises. This position dates back to the 17th century, and was first championed by Leibniz, but prior to the late 19th century, the study of logic had not advanced sufficiently for this thesis to be fully tested. Frege, one of the chief innovators in the turn-of-the-century advance in logic, was the first to develop a thoroughly axiomatic calculus for logic. Therein, he hoped to show that the truths of arithmetic could be proved using axioms of logic alone. Working independently, Russell developed views remarkably similar to Frege's, and although Russell later discovered problems with Frege's logical system, he went on to develop his own extensive attempt to reduce mathematics to logic.
Their works during the years 1879-1925 not only represent contributions to logic and the philosophy of mathematics, but as the title of the anthology suggests, have also had a considerable impact on the philosophy of language. Building upon what they had learned in developing their logical calculi and attempting to analyze the statements of arithmetic therein, both Frege and Russell faulted traditional Aristotelian logic for taking the subject/predicate analysis of grammar as a guide in understanding logical form. Frege went on to argue for a sense/reference dualism in meaning, and for analyzing all language in terms of the notions of function and argument typically only applied to mathematical formulae. Although he rejected Frege's sense/reference distinction, Russell too argued that the apparent grammatical form of statements was systematically misleading about logical form. For instance, with his influential Theory of Descriptions, Russell argued that statements of the form "the so-and-so is such-and-such" must actually be analyzed as complicated existentially quantified propositions.
The anthology contains nine works by Frege. They include firstly an excerpt from his 1879 classic Conceptual Notation, in which Frege first presented his logical system, followed by two additional papers from the early 1880s in which he informally explains the advantages to his function calculus over rival systems. The next item is the introduction to Frege's 1884 Foundations of Arithmetic, in which he lays out some methodological principles used in his philosophy of mathematics. Next, the anthology includes three pieces from the early 1890s, together considered to be Frege's most important contributions to metaphysics and philosophy of language: "Function and Concept," "On Concept and Object," and "On Sense and Reference." Here Frege describes his function/argument analysis of both natural and logical languages, and describes his views on meaning. The last two pieces by Frege are 1904's "What is a Function?", in which he clarifies his understanding of the nature of functions, and highlights some misunderstandings in the work of some of his contemporaries, and 1919's "The Thought", in which Frege discusses his views on the nature of truth, and argues for a "third realm" of abstract senses and thoughts, distinct from both the physical and the mental realms.
The anthology only contains six works by Russell. First is the 1901 essay "Mathematics and the Metaphysicians," in which he describes how recent work by mathematicians has helped to solve some longstanding philosophical puzzles about the nature of number and infinity. Next is the classic 1905 paper "On Denoting," in which Russell first outlined the theory of descriptions and argued against the rival positions of Frege and Meinong. This is followed by the 1911 essay, "Knowledge by Acquaintance and Knowledge by Description," which describes some epistemological and other philosophical developments related to the theory of descriptions. The next entry is a chapter entitled "Logic and the Essence of Philosophy," taken from 1914's Our Knowledge of the External World, in which Russell explains how past misunderstandings in logical analysis have lead to philosophical mistakes. The final two contributions are from Russell's 1919 Introduction to Mathematical Philosophy: first, the chapter on descriptions, and second, the final chapter in which Russell explores some still undecided questions about the nature of logic itself.
Although all the works in the anthology have been published before, and most are readily available elsewhere, the anthology is the first of its kind to focus exclusively on the works on Frege and Russell together, and therefore may serve to partly fill a void in instructional materials for courses dedicated to these figures. Depending on one's purposes, however, it would very likely need supplementation. For undergraduate students, together these 15 works would serve as a good introduction to Frege's and Russell's views on logical analysis and the philosophy of language. When it comes to logicism itself, they contain very little information about the details of their views on the nature of numbers, or their methodology for reducing mathematics to logic. In fact, many of the works included are polemical pieces in which they attempt to convince readers to read their other works, and/or compare their merits with those of others. At least some familiarity with the details of their programs beyond what the anthology contains would be necessary to draw any informed conclusions about the virtues and/or shortcomings of logicism. For beginners, it could be supplemented with relatively informal full-length treatments such as (the remainders of) Frege's Foundations of Arithmetic and Russell's Introduction to Mathematical Philosophy. For advanced students interested in the details of their logical systems, the difficulties they faced (such as Russell's paradox of sets), their methods of overcoming them (e.g., Russell's Theory of Types), and the details of their logicist arguments, one would need to turn to more technical writings such as Frege's Basic Laws of Arithmetic, Russell's "Mathematical Logic As Based on the Theory of Types," or Russell and Whitehead's Principia Mathematica. Unfortunately, nothing from these technical writings is contained in the collection.
The anthology also includes a 75-page introduction by the editor, which aims to provide an overview to the historical background of their writings, their main philosophical positions and points of disagreement. While the introduction may be helpful to many students, and does a particularly good job at discussing some of the shortcomings of pre-Fregean logic, a number of cautionary notes are in order.
Firstly, certain of the views of Frege, Russell and others are oversimplified. For example, the naďveté of early modern philosophers with regard to philosophical logic is exaggerated; Kant is given too large a place, and also portrayed much more psychologistically than he in fact was. Both Russell's and Frege's views on the nature of logic as an a priori science are distorted, and too closely tied to "inference". The changes in the views of Frege and Russell over time are not mentioned or clarified. For example, both Russell's early metaphysics of propositions and his later fact-based theory are discussed at different points, but it is not mentioned that these views are incompatible, and that, historically, one was succeeded by the other.
A number of the issues discussed in the introduction are presented somewhat sloppily. Distinctions between linguistic items and their meanings are often not kept straight, especially in the discussion of Frege's views on the nature of functions. A logical form is defined as a "sentence-schema", whereas both Russell and Frege took great pains to distinguish the logical forms of objective propositions and thoughts from anything linguistic. The editor often talks about such things as "the meaning of a proposition" or the "meaning of a concept," whereas propositions and concepts are not things with meanings; they themselves are the meanings.
On a number of points, the introduction gets the views of Frege, Russell, or both subtly wrong. For example, it makes such claims as that Frege's quantifiers are limited to a "contextually relevant domain", and that Russell believed that quantifiers are functions from predicates to truth-values, neither of which is true. The editor claims that Frege thought that logical operators were functions from "sentences to sentences", a claim Frege never made. (For Frege, logical connectives refer to functions, but it is doubtful that they themselves are functions.) He claims that both Russell's and Frege's logical systems were extensional, when in actuality only Frege's system is extensional by modern standards. The editor presents the Theory of Types as a hierarchy of different types of sets with different types of members; however, Russell's mature logic actually eschewed commitment to sets as entities altogether, and the Theory of Types was actually one of different ranges of significance for what Russell called "propositional functions", which Russell used to analyze away apparent commitment to classes or sets.
It also oddly claims that Russell, contra Kant, wanted to restore the "analyticity" of arithmetical claims, whereas Russell actually claimed that both logic and mathematics were synthetic a priori. In the Introduction, the editor alleges that Frege's theory that senses exist in a third realm apart from the mental and physical is obscure, and not fully explained. However, he neglects to mention exactly what he finds lacking or unclear about Frege's position, and so the discussion comes off as nothing more than an uncharitable jab. The editor also insinuates that Russell never fully engaged with dualistic theories of meaning (those that draw a distinction between sense or meaning and reference or denotation), which is easily shown false by a study of his 1903-1905 manuscripts.
Finally, he wrongly claims that there is a consensus among experts that logicism has been refuted by Gödel's incompleteness results. Gödel showed that not all arithmetical truths can be captured in a single deductive system. While this shows that the Frege-Russell form of logicism was perhaps somewhat naively strong, it does not touch the core of logicism. Similar results show that not all higher-order logical truths can be captured in a single deductive system, so Gödel's results do not point the way to any difference between logical and mathematical truth.
Department of Philosophy
University of Massachusetts
Amherst, Massachusetts
klement@philos.umass.edu