The Philosophy of Bertrand Russell

Bertrand Arthur William Russell, third Earl Russell (1872–1970 ce), was born into an aristocratic English family with considerable political tradition and influence. Both his parents died before he turned four; he was brought up by his paternal grandmother, who seems to have been a rigid and domineering character with a powerful sense of duty. He went up to Trinity College Cambridge in 1890 and studied mathematics for three years before taking up philosophy. The outbreak of the First World War aroused Russell’s vehement opposition; his anti-war work led to his dismissal from his position as lecturer at Trinity College in 1916, and to his being jailed in 1918. He was reappointed by Trinity in 1920 but soon resigned. Thereafter he was financially dependent upon sales of books and essays; energy which might have gone into academic philosophy thus went into popular writings. After the Second World War he received the Order of Merit (1949) and the Nobel Prize for literature (1950); he nevertheless devoted much of his time to political activism, in opposition to the establishment. He was motivated by an understanding of the dangers posed by nuclear weapons and, later, by his opposition to the involvement of the United States in Vietnam; in his nineties he again became well known as an anti-war activist.


Russell wrote voluminously, and with astonishing facility, over an immense range of both genres and subjects. It is, however, his philosophical work on logic, metaphysics, epistemology, and related issues which is of lasting value. His writings on these topics from the first two decades of the twentieth century played a large role in setting the tone and framing the questions for what came to be known as “analytic philosophy”; the thought of Wittgenstein and of Carnap, and thus also of many others, is unimaginable without this work of Russell’s. This chapter thus concentrates on that period. His work of the 1920s, 1930s, and 1940s, while perhaps less enduring, made important contributions to debates within analytic philosophy, especially to epistemology and the philosophy of science.

In late nineteenth-century Britain the prevailing philosophical tone was set by attempts to assimilate the work of Kant and, especially, Hegel. These attempts resulted in a variety of views generally grouped under the heading “British idealism”; F. H. Bradley was a leading figure in this movement. A fundamental idea lying behind various versions of idealism is that our knowledge is mediated by conceptual structures and that the (knowable) world is thus in some sense mind-dependent. Another conclusion adopted by many, though not all, idealists was a strong form of holism: that the world is not made of up of independent objects standing in relation to one another but is, instead, a single system whose parts can be isolated only at the price of some distortion. According to this view, our knowledge is never of the whole, so nothing we know is fully true – we know merely partial truths. At first Russell accepted the broad outlines of the idealist position. He was, however, far more interested in science, and especially mathematics, than most of those influenced by idealism. He had immensely ambitious plans for a philosophical treatment of all scientific knowledge from a Hegelian point of view; his first philosophical book, An Assay on the Foundations of Geometry, was intended to be a part of this project.

Two major shifts in Russell’s thought occurred around the turn of the century, one metaphysical and one logical. In metaphysics, he and his younger contemporary at Trinity, G. E. Moore, broke with idealism around 1898, and began to articulate an extreme version of realism. In opposition to idealism, they asserted that the world is made up of objects, each of which is fully real and is completely independent both of our minds and of all other objects – objects are not affected by their relations to other objects, but are merely externally related to one another. (The view is thus a form of atomism; equally it advocates the philosophical method of analysis, which seeks to understand complex wholes in terms of their simple parts, rather than vice versa.) And they postulated that we have a direct cognitive relation, which Russell later called acquaintance, to various objects – not only those perceivable by the senses, such as tables and trees, but also, and especially, abstract objects, such as goodness and numbers. Philosophy, as they conceived it, is wholly independent of psychology, and has no particular concern with the human mind.

In logic, Russell encountered Peano at a conference in Paris in August 1900, and set out to understand his logical work. In an astonishingly short time he had not only mastered it but extended it to handle what Peirce and Schröder had called the logic of relations. The result was a system of logic dealing not only with inferences involving one-place predicates (such as “. . . is mortal”) but also with those involving relations, i.e. predicates of two or more places (such as “. . . loves . . .”, or “. . . is between . . . and . . .”). Russell’s extension of Peano lacked the clarity and simplicity of the logic which Frege had (unbeknownst to Russell) produced in 1879; one way or another, however, this part of Russell’s logic had the power of what is today called first-order quantification theory, or predicate logic (this is logic which uses variables to generalize about objects, but not about properties of objects). Russell made much of the fact that the new logic deals with relations, which many idealists had not accepted as fully real.


This new logic is crucial for an account of mathematics. To take a simple but important example, the infinitude of natural numbers follows from the following facts: that for every number there is a larger number; that there is at least one number; that no number is larger than itself; and that “larger than” is a transitive relation (i.e. for any numbers, a, b, c, if a is larger than b, and b is larger than c, then a is larger than c). The inference from these premises to the conclusion that there are infinitely many natural numbers is obviously correct; the achievement of Frege and of Russell was to treat it within a rigourous system of logic, thereby dispelling some of the mystery surrounding the infinite.

A treatment of the sorts of inferences typical of mathematics does not by itself afford a complete account of that subject. Kant had put forward a view of mathematics as dependent on the forms of our intuition and had thus made it, at least by Russell’s lights, mind-dependent. This was precisely the sort of view that Russell wished to combat, and he did so by arguing that the truths of mathematics can all be stated in logical terms and, when so stated, can all be proved by logical means (here too Russell was anticipated by Frege, but his work seems to have been independent of Frege’s). This view, known as logicism, requires a logic that includes a theory of classes, or some other theory more or less equivalent to what we now call set theory.

In the course of developing a theory of classes, Russell came across what has become known as Russell’s paradox. It is a natural assumption that there is a class corresponding to each one place predicate (corresponding to the predicate “. . . is mortal” is the class whose members are exactly those things which are mortal, and so on). Since we have the notion of being a member of a class, we also have the predicate “is not a member of itself”; hence, given the natural assumption, we have the class of things which are not members of themselves. But is this class a member of itself? That is, is the class of things which are not members of themselves a member of itself? Either answer leads to its opposite, resulting in paradox: if it is a member of itself then it is a self-member, and so not a member of the class of non-self-members, i.e. not a member of itself after all; but if it is not a member of itself then it is a non-self-member and so is a member of itself. To avoid this paradox, and others which he saw as related, Russell developed the theory of types. According to this theory, entities are of fundamentally different types; what can be said of an entity of one type results in nonsense if we attempt to say it of an entity of another type. In particular, an entity can be a member only of classes immediately higher in type than it is. Sentences which appear to assert or deny self-membership are thus nonsensical. This theory was tentatively put forward in an appendix to The Principles of Mathematics, and was developed in “Mathematical Logic as Based on the Theory of Types.” It reached its full form in Whitehead and Russell’s monumental Principia Mathematica, which made out the technical case for logicism in great detail. This work had a profound influence on the progress of mathematical logic, in the hands of Gödel and others; one application of these advances led in turn to the development of computers, and has thus had incalculable practical influence.

Generality, standardly conveyed by variables, is essential to logic and to mathematics. Russell initially hoped to explain generality, and the use of variables, by a theory of what he called denoting concepts. According to that theory, a sentence containing a description, i.e. a phrase formed with one of the words “all”, “any”, “some”, “a”, and “the”, expresses a proposition containing a concept which denotes an object or objects not contained in the proposition. Thus the sentence “All people are mortal” expresses a proposition which contains a denoting concept, all people, which in turn denotes all people; the proposition does not itself contain all people. This theory is supposed to explain how the sentence is about all people: it is about them because it contains a denoting concept which denotes them. (The theory is also supposed to explain how a definite description, such as “the man who broke the bank at Monte Carlo,” can be part of a sentence which makes sense, even though there is no object which uniquely answers to the description. For the sentence to make sense we need only be sure that there is a proposition which it expresses; the proposition contains the denoting concept, not the supposed man, so its status is unaffected if there is in fact no such man.) Russell’s attempt to explain generality in terms of denoting concepts, however, failed, as he himself came to see. The theory of denoting concepts, moreover, proved to be exceedingly complex, and led to formidable difficulties.

Russell thus had every reason to abandon the theory of denoting concepts and to take the use of variables as fundamental, as Frege did. The one obstacle to his doing so was the case of definite descriptions, descriptions formed with “the” which seem to refer to exactly one object. (Frege had taken such phrases as logically primitive.) At some point in 1905 Russell saw how to analyze definite descriptions; this enabled him to discard the theory of denoting concepts completely. The result was his celebrated “theory of descriptions,” which analyzes sentences of the form “The F is G” (e.g. “The King of France is bald”) as saying: there is an object x which is F, and for every object y, if y is F then y = x, and x is G. More briefly: there is one and only one object which is F, and it is G.

The theory of descriptions was immensely important as an example of logical analysis; F. P. Ramsey, in a description endorsed by Moore, called it a “paradigm of philosophy.” For Russell it also played an important, though indirect, role in his development of the theory of types. He saw it as a particular case of a more general theory of incomplete symbols. Phrases such as “the King of France,” which appear to get their meaning by their relation to some non-linguistic entity (whether a monarch or a denoting concept), may, according to this theory, function in quite a different way. Such expressions, according to Russell, “have no meaning in isolation.” They are “incomplete symbols”: we can explain each sentence in which they occur, but not the phrase itself in isolation. Russell applied this idea to classes, and analyzed expressions which appeared to refer to classes in terms of what he called propositional functions (very roughly, the non-linguistic correlates of expressions containing free variables, such as “x is a prime number”). This helped because he was more willing to suppose that propositional functions are stratified into types than that classes are. According to his new view, we need not assume that there are classes (hence he called it the no-classes theory). Symbols for classes make sense in context, because they are defined, in each context, in terms of propositional functions.


Because of Russell’s concern to block a whole range of paradoxes, and because of the intricacies of his view of propositional functions, his theory of types is far more complex than would be needed simply to block Russell’s paradox. Ramsey showed that a simpler version – now known as simple type theory – would suffice for that task (Ramsey, 1925); since that time, Russell’s full version has been known as ramified type theory. Proving the truths of mathematics from either version of type theory requires the additional assumption, which Russell himself thought could not be justified by logic, that there are infinitely many objects. (Without this assumption we cannot prove that there are infi- nitely many numbers.) The ramified version also necessitates an axiom – the axiom of reducibility – which is hard to justify except on ad hoc grounds. These two points go a long way toward undermining the force of Russell’s logicism, even for a sympathetic commentator.

Russell’s increasing exploitation of the theory of incomplete symbols marked a partial retreat from his earlier extreme realism. He now accepts that there are many phrases which appear to refer to objects but do not in fact do so. An important example is the so-called multiple relation theory of judgment. According to this view, a proposition is not a genuine entity with which the judging mind is acquainted. There are no propositions in that sense; instead judgment involves acquaintance with a number of entities, and it is an act of the mind which unites them into a judgment. (Russell first considered this view in 1906, and had defi- nitely adopted it by 1909.) An advantage of this theory is that it allows truth to be defined in terms of the existence of appropriate facts. (On his earlier view, truth was an indefinable notion.)

After the completion of Principia Mathematica, Russell turned his attention away from logic and mathematics and toward issues raised by our empirical and scientific knowledge. He continued to hold that all knowledge comes via acquaintance, but realized that it is not plausible to think that we are acquainted with the ordinary objects we appear to know. The objects of acquaintance in empirical knowledge, he came to think, must rather be what he called sense-data – a certain color which I perceive, for example, or a sound which I hear. For him these were not mental objects but objective entities, directly given to the mind in sense-perception. How, then, do we get from knowledge of sense-data to knowledge of tables and trees, much less electrons and distant stars? Russell’s attempt to answer this question exploited the same techniques used in his logicism: talk of ordinary objects was to be analyzed and defined in terms of sense-data and classes of sense-data, and classes of classes of sense-data, and so on. (Russell always attributed this technique of logical construction to Whitehead). This idea is most clearly set out in Our Knowledge of the External World and in some of the essays in Mysticism and Logic; for the most obvious signs of its influence, see Carnap (1928).

Russell began this work on empirical knowledge without having fully articulated the implications of his multiple relation theory of propositions, and its relation to his logic and to his underlying metaphysical views. His work on these topics (now published as volume 7 of Collected Papers) was brought to a halt by criticisms from Wittgenstein. Russell never resolved these fundamental problems; his lectures “The Philosophy of Logical Atomism” contain an excellent summary of his views, but leave many problems admittedly unresolved and suggest that their solution may require a more psychological view of meaning. (Wittgenstein’s Tractatus may be thought of as offering a different kind of solution to these problems – but the solution is a drastic one, involving the abandonment of logicism, and a complete rethinking of the nature of logic.)

Russell’s later work in technical philosophy, though perhaps less fundamental than the work we have been discussing, was still of great significance. In 1919 he adopted a view known as “neutral monism,” advocated by Mach and by William james. According to this view there are not two kinds of things in the world, the mental and the physical; instead there is one kind of stuff, in itself neither mental nor physical (hence “neutral”). Some arrangements of it are what we call mental, others what we call physical. Thus the mind is not an entity distinct from the rest of the world but is made up of the same fundamental constituents as everything else. Russell’s Analysis of Mind shows the influence both of that view and of behaviorism; it articulates a psychological approach to issues of belief and meaning. More generally, the book shows a shift from a logical and metaphysical framework towards a naturalistic framework – a shift which might be thought to reach full flower with the work of Quine. (Much of the discussion of the mental in Wittgenstein’s later philosophy can usefully be seen as directed against views that Russell puts forward in this work.) Russell’s Analysis of Matter is an attempt to come to terms with logical and epistemological issues which he took to be raised by the new physics, especially the theory of relativity and quantum mechanics; an important feature of this work is the idea that we know the structural features of the world, but not its intrinsic nature. In his late sixties he gave the William James lectures at Harvard, published as An Inquiry into Meaning and Truth, an investigation of issues in the philosophy of language. His last significant philosophical work was Human Knowledge: Its Scope and Limits, which returned to issues of knowledge and its justification. This work is less concerned than Russell’s earlier views of knowledge to show that human knowledge is constructed on a foundation of certainty; all our knowledge is, instead, held to be fallible. Empirical knowledge beyond particular facts depends upon postulates, which cannot themselves be derived from experience. (An example: suppose that in all cases where we have observed an Atype event and a B-type event together, we have reason to think that the two are causally related; then, when we observe an event of one of those types alone, it is probable that an event of the other type also took place, unobserved.) The postulates are justified, if at all, by the overall coherence which they bring to our total system of beliefs.


Writings An Essay on the Foundations of Geometry (Cambridge: Cambridge University Press, 1897).
The Principles of Mathematics (Cambridge: Cambridge University Press, 1903). “On Denoting”, Mind, NS, XIV (1905), 530–8. [Very widely reprinted.] “Mathematical Logic as Based on the Theory of Types”, American Journal of Mathematics, 30 (1908), 222–62.
Principia Mathematica, with A. N. Whitehead, three volumes (Cambridge: Cambridge University Press, 1910–13).
Our Knowledge of the External World as a Field for Scientific Method in Philosophy (London: George Allen & Unwin, 1914).
Mysticism and Logic and Other Essays (New York: Longmans Green & Co., 1918). “Philosophy of Logical Atomism,” Monist, 28 (1918), 495–527; 29 (1919), 32–63, 190–222, 345–80.
The Analysis of Mind (London: George Allen & Unwin, 1921).
The Analysis of Matter (London: Kegan Paul, Trench, Trubner & Co., 1927).
An Inquiry into Meaning and Truth (London: George Allen & Unwin, 1940).
Human Knowledge Its Scope and Limits (New York: Simon and Schuster, 1948).
Logic and Knowledge, ed. R. C. Marsh (London: George Allen & Unwin, 1956) [Contains Russell (1905, 1908, 1918–19), among other essays]
. Collected Papers (1983–). [Produced by the Bertrand Russell Editorial Project, based at McMaster University. The project aims to publish all of Russell’s writings of less than book length. Ten volumes have been published to date, with various editors; some by George Allen & Unwin, some by Routledge, some by Unwin Hyman.] 4

Further reading
Carnap, R.: Die Logische Aufbau der Welt (Berlin-Schlactensee: Weltkreis-Verlag, 1928); trans. R. A. George, The Logical Structure of the World (London: Routledge & Kegan Paul, 1967).
Griffin, N.: Russell’s Idealist Apprenticeship (Oxford, Oxford University Press, 1991).
Hylton, P. W.: Russell, Idealism, and the Emergence of Analytic Philosophy (Oxford: Oxford University Press, 1990).
Pears, D. F.: Bertrand Russell and the British Tradition in Philosophy (London: Fontana, 1967).
Ramsey, F. P.: “The Foundations of Mathematics.” Proceedings of the London Mathematical Society, series 2, 25 (5) (1925), 338–84; reprinted in F. P. Ramsey, the Foundations of Mathematics and Other Logical Essays, ed. R. B. Braithwaite (London: Routledge & Kegan Paul, 1931) and elsewhere.
Wittgenstein, L.: Logische-philosophische Abhandlung (in Annalen der Naturphilosophie, 1921); trans. D. F. Pears and B. F. McGuiness, Tractatus Logico Philosophicus (London: Routledge & Kegan Paul, 1961).

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