Russell, Bertrand Arthur William (1872–1970)

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Bertrand Arthur William Russell, the British philosopher, mathematician, and social reformer, was born in Trelleck, Wales. He was the grandson of Lord John Russell, who introduced the Reform Bill of 1832 and later twice served as prime minister under Queen Victoria. John Stuart Mill, a close friend of Russell's parents, was his godfather in an informal sense. Russell's parents died when he was a little child. Both of them had been freethinkers, and his father's will had provided that he and his brother were to have as their guardians friends of his father's who shared the latter's unorthodox opinions. As the result of litigation the will was set aside by the Court of Chancery and the two boys were placed in the care of their paternal grandparents. Lord John Russell died two years later, and it was the boys' grandmother who determined the manner of their upbringing. Russell was not sent to school but received his early education from a number of Swiss and German governesses and, finally, English tutors. He entered Cambridge University in October 1890 and studied mathematics and philosophy at Trinity College from 1890 to 1894. He was a fellow of Trinity College from 1895 to 1901 and lecturer in philosophy there from 1910 to 1916. In 1916 Russell was dismissed by Trinity College because of his pacifist activities. He was reinstated in 1919 but resigned before taking up his duties.

What is generally considered Russell's most important work in philosophy was done between 1900 and the outbreak of the first world war. From 1916 until the late 1930s Russell did not hold any academic position and supported himself by writing and public lecturing. During this period he wrote some of his most influential books on social questions, including Marriage and Morals (London, 1929) and his two books on educationOn Education, Especially in Early Childhood (London, 1926) and Education and the Social Order (London, 1932). These views were put into practice in Russell's experimental school, the Beacon Hill School, which he started with his second wife, Dora, in 1927. Russell left the school in 1934 after he and Dora were divorced (the school itself continued until 1943). Russell returned to more concentrated work in philosophy around 1936. He moved to the United States in 1938, teaching first at the University of Chicago and then at the University of California at Los Angeles. In 1940 he accepted an invitation from the Board of Higher Education of New York City to join the department of philosophy at City College. However, he never had an opportunity to take up this appointment, having been found unfit for this position in a remarkable opinion by a judge who felt he had to protect "public health, safety and morals." From 1941 until 1943 Russell lectured at the Barnes Foundation in Philadelphia (these lectures were later expanded into A History of Western Philosophy ). Dr. Albert Barnes, the head of this foundation, dismissed Russell in January 1943, on three days' notice. In this instance Russell successfully brought action for wrongful dismissal. In 1944 he returned to Cambridge where he had been reelected to a fellowship at Trinity College.

Russell was a candidate for Parliament on three occasions and was defeated each time: In 1907 he ran at Wimbledon as a candidate of the National Union of Women's Suffrage Societies, in 1922 and 1923 he stood as the Labour Party candidate for Chelsea. Russell was twice jailedin 1918 for six months on a count of an allegedly libelous article in a pacifist journal and in 1961, at the age of eighty-nine, for one week, in connection with his campaign for nuclear disarmament.

In 1908 Russell was elected a fellow of the Royal Society. He became an honorary fellow of the British Academy in 1949, and in the same year he was awarded the Order of Merit. Russell twice served as president of the Aristotelian Society and was for many years president of the Rationalist Press Association. In 1950 he received the Nobel Prize for literature. In making the award, the committee described him as "one of our time's most brilliant spokesmen of rationality and humanity, and a fearless champion of free speech and free thought in the West."

Russell had three children and was married four times. In 1931, upon the death of his brother, he became the third earl Russell.

Writing in 1935 the German historian Rudolf Metz referred to Russell as "the only British thinker of the age who enjoys world-wide repute." At that time his works could not circulate in Germany, Italy, or Russia. Now they are available in every major and a great number of minor languages (a truncated version of A History of Western Philosophy was allowed to circulate even in the Soviet Union). It is safe to say that not since Voltaire has there been a philosopher with such an enormous audience. Russell also shares with Voltaire a glittering and graceful prose style and a delicious sense of humor. It is perhaps Russell's humorous irreverence as much as the substance of his heretical opinions that has so deeply offended several generations of moralists and religious conservatives.

In the following section we shall briefly recount some of the highlights and formative influences in Russell's eventful life and sketch his views on political and social issues. Although these views are certainly logically independent of his more technical work as a philosopher, they deal with questions that have traditionally been discussed by philosophers, and they also help one to understand the basic motives inspiring Russell's thought.

Life and Social Theories

Russell's childhood and adolescence were unhappy. The atmosphere in his grandmother's house was one of puritan piety and austerity, and his loneliness, he recalls, was almost unbearable. Only virtue was prized"virtue at the expense of intellect, health, happiness, and every mundane good." At the age of five Russell reflected that if he lived to be seventy, he had endured only a fourteenth part of his life, and he felt the long-spread-out boredom ahead of him to be unendurable. In adolescence, he remarks, he was continually on the verge of suicide, from which, however, he was "restrained by the desire to know more mathematics." His grandmother had gradually moved from Scottish Presbyterianism to Unitarianism. As a child Russell was taken on alternate Sundays to the parish church and to the Presbyterian Church, while at home he was taught the tenets of Unitarianism. When he was fourteen he began to question theological doctrines and in the course of four years abandoned successively belief in free will, immortality, and God, the last as the result of reading John Stuart Mill's Autobiography. For some time, however, Russell had metaphysical attachments that served as substitutes for religion, and it was not until the end of the first world war that he became a militant opponent of all forms of supernaturalism.

early platonism and hegelianism

Under the influence of J. M. E. McTaggart and F. H. Bradley, Russell came, in his early years at Cambridge, to believe "more or less" in the Absolute and the rest of the apparatus of British Hegelianism. "There was a curious pleasure," Russell wrote in retrospect, "in making oneself believe that time and space are unreal, that matter is an illusion, and that the world really consists of nothing but mind." In a "rash moment," however, he turned "from the disciples to the Master." G. W. F. Hegel's remarks in the philosophy of mathematics he found "both ignorant and stupid," and in other ways Hegel's work appeared a "farrago of confusions." After that Russell was converted by G. E. Moore to a "watered down" version of Plato's theory of Ideas, regarding the subject matter of mathematics as eternal and unchanging entities whose exactness and perfection is not duplicated anywhere in the world of material objects. Eventually Russell abandoned this "mathematical mysticism" as "nonsense." Following Ludwig Wittgenstein he came to believe "very reluctantly" that mathematics consists of tautologies. As to the timelessness of mathematics, Russell now regarded this as resulting from nothing more than that the pure mathematician is not talking about time. Aside from this, it became emotionally difficult for him to remain attached to "a world of abstraction" in the midst of the slaughter of the Great War. "All the high-flown thoughts that I had had about the abstract world of ideas," he wrote later, "seemed to me thin and rather trivial in view of the vast suffering that surrounded me." The nonhuman world, he added, "remained as an occasional refuge, but not as a country in which to build one's permanent habitation." After his abandonment of Platonism, Russell wrote, he was not able to find religious satisfaction in any philosophical doctrine that he could accept.


Russell was interested in social questions throughout his life. He was an early member of the Fabian Society and for some time in the 1890s, under the influence of Sidney and Beatrice Webb, championed imperialism and supported the Boer War. In 1901 he had a quasi-religious experience. He became "suddenly and vividly aware of the loneliness in which most people live" and felt the need to find ways of "diminishing this tragic isolation." In the course of a few minutes he changed his mind about the Boer War, about harshness in the education of children and in the administration of the criminal law, as well as about fierceness in personal relations. This experience led him to write his famous essay "A Free Man's Worship" (1903). Although Russell became a pacifist right then, for another ten years or more he was preoccupied with work in mathematical logic and theory of knowledge. It was not until the war that he became passionately concerned about social issues. It is probable, he observed later, that "I should have remained mainly academic and abstract but for the War." The war, however, "shook him" out of many prejudices and made him reexamine a number of fundamental questions. He recalled:

I had watched with growing anxiety the policies of all the European Great Powers in the years before 1914, and was quite unable to accept the superficial melodramatic explanations of the catastrophe which were promulgated by all the belligerent governments. The attitude of ordinary men and women during the first months amazed me, particularly the fact that they found a kind of pleasure in the excitement. (Selected Papers of Bertrand Russell, p. xi)

He decided that he had been quite mistaken in believing the claims of pacifists that wars were the work of devious tyrants who forced them on reluctant populations. Although he was not then familiar with the theories of psychoanalysis, Russell concluded that the majority of human beings in our culture were filled with destructive and perverse impulses and that no scheme for reform would achieve any substantial improvement in human affairs unless the psychological structure of the average person was suitably transformed.

Russell recalls that his decision to oppose the war was made particularly difficult by his passionate love of England. Nevertheless, he had no doubt as to what he had to do. "When the war came I felt as if I heard the voice of God. I knew that it was my business to protest, however futile protest might be. My whole nature was involved. As a lover of truth, the national propaganda of all the belligerent nations sickened me. As a lover of civilisation, the return to barbarism appalled me" (Portraits from Memory, p. 27). Russell remarks that he never believed much tangible good would come from opposition to the war, but he felt that "for the honor of human nature," those who "were not swept off their feet" should stand their ground. He patiently argued in lectures and books that the slaughter of millions of men was not justified by any of the possible gains of a defeat of the Central Powers. Russell's pacifism was not mystical. It was not then and had not been his contention at any time that the use of force is always wrong, that war can never possibly be justified. He maintained that this war in these circumstances was not worth all the pain and misery, and the lying of all the parties. Consistently with his general position, Russell favored the Allies during World War II on the ground that the defeat of the Nazis was essential if human life was to remain tolerable. The Kaiser's Germany by contrast was "only swashbuckling and a little absurd," allowing a good deal of freedom and democracy.

Prior to the war there had been strong pacifist sentiment in all the major Western countries, especially among the intellectuals and the powerful socialist and liberal parties. When war came only a tiny minority of these pacifists remained true to its convictions. Overwhelmed by their need to conform and in many cases by what Russell would have regarded as their own primitive impulses, many of them became the most violent jingoists. Russell was bitterly attacked for his pacifist activities not only, as one might have expected, by conservatives and professional patriots but also by many of his erstwhile friends. H. G. Wells, for example, publicly heaped abuse on Russell when he was already in trouble with the authorities. Russell's political philosophy, according to Wells, amounted to a "tepid voluntaryism," and he (unlike Wells) had no right to speak for British socialism. Wells even abused Russell's work as a mathematical philosopher. Russell, he wrote, is that "awe-inspiring" man who "objected to Euclid upon grounds no one could possibly understand, in books no one could possibly read" (preface to P. H. Loyson, The Gods in the Battle, London, 1917).

At Cambridge, Russell's teacher and friend McTaggart led a move for his ouster. Meetings addressed by Russell were broken up by violent mobs without any police interference. Eventually he was prosecuted by the government. For writing a pamphlet on the case of a conscientious objector he was fined £100. When he would not pay the fine the government sold parts of his library, including rare books on mathematics that Russell was never able to recover. In 1918 he was sentenced to six months' imprisonment for an article in the Tribunal, a pacifist weekly, in which he had written that "unless peace comes soon the American garrison, which will by that time be occupying England and France, will no doubt be capable of intimidating strikers, an occupation to which the American army is accustomed when at home." In a fierce denunciation which accompanied the sentence, the magistrate, Sir John Dickinson, referred to Russell's offense as "a very despicable one" and added that Russell "seems to have lost all sense of decency." It should be added that as the result of the intervention of Arthur Balfour, Russell was treated with consideration while in prisonhe finished there his Introduction to Mathematical Philosophy and began work on The Analysis of Mind.

Attitude toward the Soviet Union

Russell's isolation was not ended with the return of peace. This was due to his failure to support the Bolshevist regime in Russia. Like many Western socialists he at first welcomed the news of the revolution, but, wanting to see things for himself, he visited Russia in 1920 and came back totally disillusioned. Some of Russell's friends argued that any criticism of the revolution would only play into the hands of the reactionaries who wanted to reestablish the old order. After some hesitation Russell decided to publish the truth as he saw it. Russia, he later wrote, "seemed to me one vast prison in which the jailors were cruel bigots. When I found my friends applauding these men as liberators and regarding the regime that they were creating as a paradise, I wondered in a bewildered manner whether it was my friends or I that were mad."

The little book in which he recorded his views of the Soviet Union, The Theory and Practise of Bolshevism (1920), was remarkable for, among other things, its prescience. Long before most Westerners had heard of Joseph Stalin, Russell predicted, point by point, the reactionary features that came to characterize the Soviet system under Stalinits militarism and nationalism, the hostility to free art and science, its puritanism, and the gradual ascendancy of bureaucrats and sycophants over the early idealists. Russell was able to reprint the book in 1947 without a single alteration. His isolation after his return from Russia was even greater than during the war. The patriots had not yet forgiven him his opposition to the war, while the majority of his former political friends denounced him for his opposition to the Soviet regime. But Russell has never played to the galleries. As on many other occasions he acted in accordance with his favorite biblical text"Thou shalt not follow a multitude to do evil."

Education and sexual morality

Probably the most controversial of Russell's opinions are those relating to education and sexual morality. These were closely connected with his observations of the joy people took in the fighting and killing during the war. Russell wrote that he thought he saw the inward and outward defeats that led to cruelty and admiration of violence and that these defeats were, in turn, largely the outcome of what had happened to people when they were very young. A peaceful and happy world could not be achieved without drastic changes in education. In sexual matters, although not only in these, irrational prohibitions and dishonesty were exceedingly harmful. "I believe," he wrote in Marriage and Morals, "that nine out of ten who have had a conventional upbringing in their early years have become in some degree incapable of a decent and sane attitude towards marriage and sex generally" (p. 249). Conventional education was judged to be at fault in a great many other ways as well. Its general tendency was to cramp creative impulses and to discourage a spirit of critical inquiry. While a certain amount of discipline is necessary, very much of the coercion traditionally employed cannot be justified. The child who is coerced "tends to respond with hatred, and if, as is usual, he is not able to give free vent to his hatred, it festers inwardly, and may sink into the unconscious with all sorts of strange consequences throughout the rest of life."

Although puritanical moralists were or professed to be violently shocked by Russell's views on sex and education, it is worth emphasizing that his recommendations are not extreme and that unlike his opponents he stated his position temperately and without recourse to personal abuse. Russell may be characterized as a "libertarian" in education, but he was strongly opposed to the view of other educational pioneers who played down the importance of intellectual training and encouraged originality without insisting on the acquisition of technical skill. Similarly, although he may quite fairly be called a champion of free love, it is grossly misleading to describe Russell as an advocate of "wild living." On the contrary, he disavowed any such intentions. He wrote:

The morality which I should advocate does not consist simply in saying to grown-up people or adolescents: "follow your impulses and do as you like." There has to be consistency in life; there has to be continuous effort directed to ends that are not immediately beneficial and not at every moment attractive; there has to be consideration for others; and there should be certain standards of rectitude. (Marriage and Morals, p. 243)

But this does not mean that we should be "dominated by fears which modern discoveries have made irrational." Russell could see nothing wrong in sexual relations before marriage, and he advocated temporary, childless marriages for most university students. This, he wrote, "would afford a solution to the sexual urge neither restless nor surreptitious, neither mercenary nor casual, and of such a nature that it need not take up time which ought to be given to work" (Education in the Modern World, pp. 119120). It would be wrong to regard Russell as an enemy of the institution of marriage. He did indeed object to keeping a marriage going when no love is left, and, what shocked people a great deal, he remarked that a "permanent marriage" need not exclude "temporary episodes," but he also emphatically affirmed that "marriage is the best and most important relation that can exist between two human beings something more serious than the pleasure of two people in each other's company" (Marriage and Morals, p. 115).

Russell's views on sexual morality featured prominently in the New York City case of 1940. When his appointment was announced, Bishop Manning of the Episcopal Church wrote an inflammatory letter to all New York City newspapers in which he denounced Russell's subjectivism in ethics and his position on religion and morality. It was unthinkable that "a man who is a recognized propagandist against both religion and morality, and who specifically defends adultery" should be held up "before our youth as a responsible teacher of philosophy." The bishop's letter was the beginning of a campaign of vilification and intimidation unsurpassed in a democratic nation in recent times. The ecclesiastical journals, the Hearst press, and numerous Democratic politicians joined in the chorus of abuse. Russell was described as "the Devil's minister to men," as an advocate of "the nationalization of women," as "the mastermind of free love and of hatred for parents," and also, needless to say, as an exponent of communism.

The climax of the campaign was a taxpayer's suit by a Mrs. Jean Kay of Brooklyn demanding that Russell's appointment be annulled. The case was heard before Justice McGeehan, who had previously shown his notions of tolerance by trying to have a portrait of Martin Luther removed from a courthouse mural illustrating legal history. In a startling decision, which was bitterly criticized by legal experts as in many respects grossly improper, McGeehan voided Russell's appointment on three grounds: First, Russell had not been given a competitive examination; second, he was an alien and there was no reason to suppose that the post in question could not be competently filled by an American citizen; and, finally, the appointment would establish "a chair of indecency." Elaborate arguments were adduced in behalf of this last claim. Among other things it was maintained that Russell's doctrines would tend to bring his students "and in some cases their parents and guardians in conflict with the Penal Law." In some fashion not explained by the judge, Russell's appointment would lead to "abduction" and rape. Russell's opposition to the laws that make homosexuality a crime was misread as advocacy of a "damnable felony which warrants imprisonment for not more than 20 years in New York State." Evasive actions of the mayor of New York, Fiorello La Guardia, prevented any effective appeal against this monstrous decision, and Russell was never able to take up his position at City College. In 1950, shortly after receiving the Nobel Prize, he returned to New York to deliver the Machette Lectures at Columbia University. He received a rousing reception that those who were present were not likely to forget. It was compared with the acclaim given Voltaire in 1784 on his return to Paris, the place where he had been imprisoned and from which he had later been banished. As for McGeehan, it is safe to say that he will go down in history as a minor inquisitor who used his one brief moment in the limelight to besmirch and injure a great and honest man.

McGeehan did not pass judgment on Russell's competence as a philosopher, but other opponents of the appointment were not so restrained. Thus, Joseph Goldstein, attorney for Mrs. Kay, described Russell as "lecherous, libidinous, lustful, venerous, erotomaniac, aphrodisiac, irreverent, narrow-minded, untruthful, and bereft of moral fiber." After a few gratuitous lies about Russell's private life, he concluded:

He is not a philosopher in the accepted meaning of the word; not a lover of wisdom; not a searcher after wisdom; not an explorer of that universal science which aims at the explanation of all phenomena of the universe by ultimate causes all his alleged doctrines which he calls philosophy are just cheap, tawdry, worn-out, patched-up fetishes and propositions, devised for the purpose of misleading the people.

In the present encyclopedia a somewhat different view is taken of the value of Russell's philosophy. Some of his most important theories in epistemology and metaphysics will be discussed in the next section, his contributions to logic and the foundations of mathematics will be covered in the following section, and his views on ethics and religion will be dealt with in the last section. However, a number of Russell's most interesting ideas are not at all or only briefly discussed in the present entry. Many of these are treated elsewhere in the encyclopedia.

Epistemology and Metaphysics

Russell exercised an influence on the course of Anglo American philosophy in the twentieth century second to that of no other individual. Yet, unlike many influential thinkers, he neither founded nor attached himself to any definite movement. Although he wanted above all to be empirical, he always had reservations of one sort or another to the proposition that all acceptable beliefs can be derived from purely empirical premises, and although his stress on analysis as the proper philosophical method is one of the chief sources of the analytical bent that philosophy currently has in English-speaking countries, he never accepted the view that philosophy is nothing but analysis.

early realism

Russell's first distinctive philosophical work was colored by a violent reaction against the absolute idealism then dominant in England, which was ultimately based on the thought of G. W. F. Hegel and whose outstanding British exponent was F. H. Bradley. According to Bradley if we try to think through the implications of any fact whatever, we will inevitably be forced to conclude that everything that there is constitutes a single, immediate unity of consciousness. In Russell's view the main weapon used to bludgeon people into submission to this result was the "doctrine of internal relations," according to which any relational factfor example, that x is above y is really a fact about the natures of the terms involved. This doctrine in effect refuses to take relations as ultimate.

It follows from this position that whenever x and y are related, each "enters into the nature of the other." For when x is above y, then being above y is part of the nature of x and being below x is part of the nature of y. Hence, y is part of the nature of x and x is part of the nature of y. Since everything is related to everything else in one way or another, it follows that everything else enters into the nature of any given thing, which is just another way of saying that there is no "other thing" relative to a given thing. In other words, the only thing that exists is one all-comprehensive entity. From the related principle that when we are aware of something, that something enters into the nature of the awareness or of the mind which has the awareness, it follows that it is impossible to conceive of anything which is not included within consciousness. Thus, the one all-comprehensive entity is a unity of consciousness.

Although in his youth Russell, with most of his philosophical contemporaries, was caught up in this philosophy, he and G. E. Moore became disenchanted with it shortly before the turn of the twentieth century. Russell came to hold that in sense perception we are as immediately aware of the relations between things as of the things themselves and therefore that any philosophy which denied ultimate reality to relations must be mistaken. Moreover, he came to think that mathematics would be impossible if we held that every relation enters into the nature of its terms; for in mathematics we must understand what our units are before we can know anything about their relations to other units. Russell therefore argued for a "doctrine of external relations," according to which relations have a reality over and above the terms they relate and do not enter into the definition of the terms they relate. This led him to a kind of philosophical atomism that thenceforth was characteristic of his philosophy. We may think of the basic core of atomism, which runs through all the shifts in Russell's later philosophizing, as being constituted by the following principles:

(1) There are nonmental facts that are what they are whether or not any mind ever becomes aware of them. This does not follow from the doctrine of external relations, but that doctrine enabled Russell to reject the idealistic argument based on the doctrine of internal relations and thus left him free to hold his native realist convictions with a good conscience.

(2) A particular proposition (for example, that my car is in the garage) can be unqualifiedly true "in isolation." This follows from the thesis that facts are "atomic" in the sense that any given fact could hold, whatever is the case with the rest of the world, together with the correspondence theory of truththat what makes a true proposition true is its correspondence with an objective fact. Hegelians, on the other hand, had argued that since one could not adequately think about any particular fact without inflating it into the absolute totality of being, whenever one is saying something short of everything, what he is saying is not quite true in any absolute sense.

(3) An important corollary of (2) is the usefulness of analysis as a method in philosophy. If it is possible to get an adequate grasp of the parts of a totality without considering their place in the whole, then it is possible to give an illuminating account of something complex by showing how its simple parts are related to form the whole. Hegelians had argued that analysis cannot get started because we cannot understand what any part is without already seeing how it fits into the whole, which means already knowing everything about the whole. The conviction that analysis is the proper method of philosophy has remained the most prominent strand in Russell's thought.

Intoxicated by his release from idealism, Russell, as he later put it, tended to accept as objectively real anything that the absolute idealists had not succeeded in showing to be unreal. Numbers, points of space, general properties like roundness, physical objects as they appear to sense perception, were all regarded as having an independent existence. Under the influence of Alexius Meinong this extreme realism was reinforced by an extreme form of the referential theory of meaning, the view that in order for a linguistic expression to have a meaning there must be something that it means, something to which it refers. In this stage of Russell's thought, represented most fully by The Principles of Mathematics and to a lesser extent by The Problems of Philosophy, Russell was inclined to think that the meaningfulness of the sentence "The car is in the garage" required that there be objectively existing referents not only for the words car, garage, and in but even for the words the and is. An objectively existing "isness" soon proved to be too much for Russell's self-proclaimed "robust sense of reality." He came to think that terms belonging to the logical framework of sentences, such as "the," "is," "or," could perform their function without each being correlated with extralinguistic referents. Nevertheless, a modified form of the referential theory of meaning continued to dominate Russell's thinking.

logical constructionism

Russell's decisive shift away from the full-blooded realism of The Principles of Mathematics came with the development of logical constructionism. The theory can be generally stated as follows. We start with a body of knowledge or supposed knowledge which we feel strongly inclined to accept but which has the following drawbacks: (1) the knowledge claims do not seem to be adequately justified, (2) there are unresolved problems about the natures of the entities involved, and (3) we feel uncomfortable about committing ourselves to the existence of such entities. If we can show that this body of knowledge could be formulated in terms of relations between simpler, more intelligible, more undeniable entities and that when so formulated there is a decisive justification for it, we will have made a philosophical advance. We will have converted the problematic to the unproblematic, the obscure to the clear, the uncertain to the certain. Russell called this technique logical constructionism because the problematic entities were said, in a possibly misleading metaphor, to be "constructed" out of the simpler ones.

Reduction of mathematics to logic

The technique of logical constructionism was first employed in the theory of mathematics worked out by Russell and A. N. Whitehead and published in Principia Mathematica (3 vols., 19101913). In the Principia the authors set out to show that all of pure mathematics can be stated in terms of logic, using no undefined terms other than those required for logic in generalfor example, implication, disjunction, class membership, and class inclusion. In the course of carrying out this reduction, various more or less problematic mathematical entities were "constructed" out of what were thought to be less problematic entities. Thus, numbers were defined as classes of classes: Zero is the class of all empty classes. The number 1 is the class of all classes each of which is such that any member is identical with any other member. The number 2 is the class of all classes each of which is such that it includes a member not identical with another member and such that any member is identical with one or the other of these. If one is puzzled about what sort of entity a number is (it does not seem to be in space or time and is not perceivable by the senses) or is uncomfortable about assuming that such queer entities exist, he will presumably be reassured by the discovery that he can think of numbers as classes of classes of familiar, unproblematic entities. Of course analogous problems may arise with respect to the entities made use of in this first reductionfor example, classes. And in fact various difficulties in doing mathematics in terms of classes led Russell to try to "construct" classes out of "prepositional functions." (See the section on logic and mathematics, below.) Starting from a given point we may well have to perform a series of reductions before we get down to maximally intelligible, indubitable entities.

Construction of physical objects

After Principia Mathematica, Russell applied the technique of logical constructionism to our knowledge of physical objects, both in physical science and in common sense. Physical theories are formulated in terms of a variety of unperceivable entitieselectromagnetic fields, protons, energy quanta, forces exerted at a point, and so on. There are serious problems in the philosophy of science both about the content of our concepts of such entities and about the basis for our accepting their existence. We can try to show that such entities can be inferred from what we know about perceivable entities, but how could we get an empirical basis for a principle correlating observed and unobserved entities? Or we can try to show that unobserved entities have to be postulated in order to give an adequate explanation of observed happenings, but it seems impossible to show conclusively that no adequate explanation could be given purely in terms of observables. If we apply the constructionist principle, "Whenever possible, substitute constructions out of known entities for inferences to unknown entities," to this problem, we shall try to show that electromagnetic fields can be construed as complexes of less problematic entities related in various ways. Russell devoted a large proportion of his philosophical energy to trying to show that scientific entities can be constructed out of undeniable data of perception. But it will be easier to illustrate this kind of analysis by taking ordinary physical objects like trees and buildings, for Russell thought that they raise analogous problems, although in less obvious ways.

There is a long tradition, dominant since the time of René Descartes, according to which common sense is mistaken in supposing that we directly perceive physical objects. According to this tradition what we are directly and indubitably aware of in sense perception is something private to the individual observer. There are several sources of this view, the most important of which are, first, the fact that the content of one's perception can change with, for example, changes in perspective, lighting, and physiological condition of the observer, without there being any change in the physical object which, according to common sense, one is perceiving, and, second, the fact that in dreams and hallucinations one can have experiences which are intrinsically indistinguishable from those one has when one is "really" seeing a tree, but in these cases no tree is present. In dreams and hallucinations one is really aware of something that is not a physical object and is not perceivable by anyone else. And since these experiences are intrinsically just like those in which a physical object is present, one must be perceiving these private objects in the latter cases as well. This consideration is reinforced by the first, which is designed to show that even where a physical object admittedly is involved, I am often aware of different things without the physical object's undergoing any change.

The conclusion of these arguments is that the colors, shapes, sounds, and so on, of which we are directly aware in sense perception (sense data) are private objects that must be distinguished from the entities in the physical world (if any) which we suppose ourselves to be perceiving. This conclusion inevitably gives rise to the question how, if at all, I can start from the private objects of whose existence I can be certain and show that public, physical objects like trees exist. No generally accepted solution to this problem has emerged in several centuries of discussion. Here again Russell tries to avoid the necessity for an inference by showing that the public physical objects can be construed as a complex structure of data of immediate experience. At first Russell aimed at a solipsistic reduction in which a given physical object would be constructed out of the actually experienced data of a single observer, but he soon came to lower his aspiration and to admit into the construction data experienced by others, as well as data which would have been experienced by others if they had been in a certain place. The view, then, is that a tree can be regarded as a system of all the actual and possible sense experiences that would be regarded as figuring in perceptions of that tree. This is a form of the position known as phenomenalism, and it is subject to the difficulties to which that position is notoriously subject, particularly the apparent impossibility of specifying which experiences go into defining a particular physical object without referring to that physical object or others in the specification.

Construction of mind

Until about 1920 Russell was a mind-matter dualist. As we have just seen, physical objects were regarded as complex structures of data of the sort given in sense perception. Now, although the mind might be partly constituted by data which are given to "inner sense"that is, things which are the objects of introspective awareness, such as images and feelingsit seemed to Russell, as it had to most philosophers, that in any act of awareness, be it directed to the external or to the internal world, there is in addition to the data of which one is aware a subject or self which has the experience or which performs the act of awareness. But as the spirit of logical constructionism took increasing hold of Russell, he came to feel that there was no real warrant for believing in a subject of awareness which performs acts. He became convinced that one cannot really find any such constituent of the experience; its apparent obviousness is a reflection of the grammar of the sentences in which we speak about such matterswe say "I saw a flash of light" rather than "A flash of light occurred." As it presents itself, a minimal piece of consciousness does not involve a relation between two components. It is a unitary whole. Only the flash of light is given. The "I" and the "saw" are added interpretations. If we have no real basis for accepting a subject or mind as an ultimate entity, then the logical constructionist will try to show that it can be exhibited as a complex of entities of which we are directly assured by our experience. Here Russell followed the lead of William James, who had earlier formulated a view known as neutral monism, according to which both mind and matter consisted of the data of immediate experience, the difference between them lying in the grouping of the constituents.

Thus, if I am looking at a tree the visual datum (an irregularly shaped green splotch) of which I am directly aware is both part of my mind and part of the tree. When grouped together with other experiences from this and other perspectives that would be said to be experiences of that tree, it goes to make up a tree; when grouped with other data bound together in a single conscious field, along with other data related to these by memory, it goes to make up a mind. If this theory is acceptable, traditional puzzles about the mind-body relation are dissolved. We are faced not with two radically distinct kinds of stuff but with two different kinds of arrangement of the same elementary components. (That is, some of the components are the same. Russell considers images to be peculiar to mind.) It is in the light of this theory that one should consider Russell's notorious view that what one perceives is always his own brain. Whenever I have any sense perception whatever, I do so because a certain kind of physical activity is going on in my brain. This activity, as a physical process, is to be regarded, like all physical processes, as a construction out of the sort of data given in immediate experience. And since whatever may be the case otherwise, my brain is always active when I perceive, the data of which I am aware enter into the constitution of my brain, whatever other entities they may enter into. Hence the paradoxical view that whenever one is conscious he is aware of his own brain.

When Russell abandoned the subject of experience as an ultimate constituent of the world he rejected sense data and thenceforth spoke simply of sense experiences. But he would have represented his view more clearly by saying that he had given up belief in anything other than sense data. For in the old paradigm of subject aware of sense data, it was the subject exercising awareness that was abandoned. In The Analysis of Mind Russell set out to construct the conscious mind out of sensations and images. (Insofar as facts regarded as mental do not consist of consciousness, Russell's strategy is to give a behavioristic analysis. Thus, desire, belief, and emotions can be regarded as made up, at least in part, of dispositions to behave in one way rather than another in certain circumstances.) The results are admittedly equivocal. Russell has always been too honest to overlook glaring deficiencies in his analyses. One that has particularly bothered Russell is this: On a commonsense basis it seems clear that one must distinguish between simply having a sensation and taking that sensation as an indication of a tree, and there seems to be an important difference between simply having an image and employing that image in, for example, thinking about a forthcoming election. If this analysis of mind is to be made to work, one must give an account of the reference of perception and thought in terms of the interrelations of data. Thus, we might hold that to take a sensation as an indication of a tree is to be disposed to have the sensation of surprise if certain other sensations were to follow. But apart from difficulties about the nature of these dispositions, which are themselves neither images nor sensations, this is all extremely difficult to work out in detail, and it is equally difficult to make sure that one has shown that it can be done.

It is clear that logical constructionism is based on a tendency opposite to that of the realism briefly espoused by Russell in his youth. Logical constructionism wields Ockham's razor with a heavy hand. We begin with those entities whose existence is indubitable because they are given in immediate experience, and we then try to show that anything we might wish to say about anything else can be stated in terms of relations between these indubitable entities. In other words, anything we want to say about something else is not really about something else. Thus, we try to represent all our knowledge as having to do with as few kinds of entities as possible, thereby reducing the possibility of error.

logical atomism

Thus far we have concentrated on the epistemological side of logical constructionism, its concern with reducing the number of assumptions we make and with exhibiting clearly the basis for what we claim to know. But it also has a metaphysical side, although Russell wavers about this. Sometimes he talks as if his constructionism is metaphysically neutral. At such times he says that in showing that minds can be constructed out of sensations and images we do not show that there is no ultimate, irreducible subject of awareness; we show merely that everything we know about minds can be expressed without assuming the existence of such an entity. At other times, however, he claims that by showing that minds can be constructed out of sensations and images we have shown what minds really arewe have revealed their metaphysical status. And by carrying through constructions of everything that can be constructed out of simpler entities we will have developed a complete metaphysical scheme.

Ideal language

The most systematic presentation of this metaphysical side of logical constructionism is found in the set of lectures The Philosophy of Logical Atomism, which Russell gave in 1918. Here Russell makes explicit the principle on which a metaphysical interpretation of logical constructionism dependsnamely, isomorphism of the structure of an ideal language and of the structure of reality. If we can determine in outline how the world would be described in an ideal language, we will have, in outline, an account of what the world is like. The restriction to an "ideal" language is essential. Since there are alternative ways of stating the same body of facts, it could not be the case that all these ways reflect the real structure of the world. In this approach to metaphysics the basic metaphysical commitment is to the identity of structure between reality and an ideal language, and one shows one's hand metaphysically by choosing one rather than another set of criteria for an ideal language.

For Russell the most important requirement for an ideal language is an empiricist one, formulated in the "principle of acquaintance": "Every proposition which we can understand must be composed wholly of constituents with which we are acquainted." In other words, we can understand a linguistic expression only if it either refers to something we have experienced or is defined by other expressions which are so used. This principle plays a part in the constructions we have been surveying, as do the considerations we have already made explicit. That is, Russell holds not only that if physical objects were not defined in terms of sense experiences we would have no way of knowing anything about them but alsoand even more importantwe would not be able to understand talk about them. In logical atomism this principle is reflected in the requirement that the expressions which figure in the "atomic" sentences in terms of which everything is to be expressed must get their meaning through direct correlation with experience. They will, therefore, be names of particular sense data and terms for properties of sense data and relations between sense data. Russell is forced to exclude the logical framework of sentences from this requirement ("is," "the," etc.), but he is recurrently uneasy about this exclusion and recurrently disturbed by the question how, in that case, we can understand them.

In addition to the need for its undefined terms getting their meaning through correlation with immediately experienced items, the ideal language will have to satisfy some more strictly logical requirements. These will include the absence of vagueness and having one and only one expression for each meaning. But the most important restriction concerns the form of the basic sentences. An atomic sentence will be one that contains a single predicate or relational term and one or more than one name, the whole sentence asserting that the entity named has the indicated property ("This is white") or that the entities named stand in the indicated relation ("This is above that"). If a sentence (1) has this form, (2) contains only terms that get their meaning through correlation with experienced items, and (3) has to do with entities that cannot be analyzed into anything simpler, then it is an atomic sentence. It is clear that for Russell the sentences which satisfy these requirements will all state a minimal fact about a momentary content of sense experience.

Logical atomism can then be presented as the thesis that all knowledge can be stated in terms of atomic sentences and their truth-functional compounds. A truth-functional compound of two sentences is one whose truth or falsity is a determinate function of the truth or falsity of the components. Thus, "I am leaving and you are staying" is a truth-functional compound of "I am leaving" and "You are staying." For the compound is true if and only if both its components are true. There is an empiricist motivation for maintaining this thesis. Atomic sentences, in the sense specified above, can be conclusively verified or falsified by a single experience, and as long as we are dealing only with truth-functional compounds of these no further problem can arise concerning their truth or falsity. Consider a "contrary-to-fact conditional," such as "If I had offered him more money, he would have accepted the job." As it stands this sentence is not a truth-functional compound of its constituents. For in saying it we are presupposing that both its constituents are false, yet this does not settle the question whether the whole statement is true or false. There is a corresponding puzzle about what empirical evidence would settle the question. Obviously I cannot go back in time and offer him more money and see what he will do. If we could find some way to restate this as a (very complicated) truth-functional compound of atomic sentences, it would become clear which experiences would verify or falsify it.

Pluralism and knowledge by acquaintance

The metaphysical correlate of this sketch of the ideal language brings together two of Russell's deepest convictions, the logical independence of particular facts (pluralism) and the dependence of knowledge on the data of immediate experience. In this view reality consists of a plurality of facts, each of which is the sort of fact which could be infallibly discerned in a single moment of experience and each of which could conceivably be what it is even if nothing else were in existence. All the familiar and seemingly relatively simple objects in the world of common sense are really extremely complicated complexes of atomic facts of these sorts.

Russell was well aware that logical atomism in this extreme form was untenable. For example, he insisted that generalizations could not be truth-functional compounds of atomic sentences. The most promising way of so construing them would be to take, for example, "All lemons are yellow" as a conjunction of a large number of atomic sentences of the form "This lemon is yellow," "That lemon is yellow," . But as Russell points out, even if it were possible to list all the lemons, the conjunction would say the same thing as the original universal generalization only if we added the conjunct "and that is all the lemons there are." And this last addition is not an atomic sentence. Moreover, Russell had doubts about so-called intensional contexts, such as "Smith believes that the White Sox will win," where the truth or falsity of the compound is clearly independent of the truth or falsity of the components. Whether Smith has this belief does not in any way depend on whether the White Sox win. Russell has always hoped that neutral monism would help him to get out of this difficulty. If we could construct beliefs out of sensations and images we might be able to restate this fact as some truth-functional derivative of atomic sentences.

Later doubts

In the mid-twentieth century Russell came to have more fundamental doubts about logical atomism, including doubts concerning the very notion of a logical atom. How can we ever be sure that we are dealing with something that cannot be further analyzed into parts? How can one be sure that yellowness is an absolutely simple property? More basically, what makes a property logically simple? Does the fact that one can explain the word yellow to someone by saying "Something is yellow if it has the same color as the walls of your room" show that being the same color as the walls of your room is logically a part of yellowness? If so, then yellowness is not absolutely simple. If not, what does count against logical simplicity? Moreover, if there are alternative minimum vocabularies, then a simple, undefined term in one mode of formulation may turn out to be definable in another. Thus, on one systematization "pleasure" might be defined as the satisfaction of desire, whereas on a different systematization "desire" would be defined as the belief that something is pleasant. Russell gave up the belief that we can know that we have gotten down to ultimate simples and even the belief that there must be absolute simples. He became disposed to think, in more relativistic terms, of a class of things that can be taken as simple at a given stage of analysis. In those terms he still tended to fall back on sense experiences that are as apparently simple as anything we can find. Such experiences, even if not absolutely simple, can be regarded as being independent of anything except their possible components.

later developments

Despite Russell's frank admissions that logical atomism does not work as a depiction of the structure of an ideally adequate language, he did not develope an alternative metaphysics. On the principle of isomorphism, if one cannot represent general statements as functions of atomic statements, then one must admit general facts as ultimate constituents of the world. This metaphysical implication did not seem to bother Russell as it once had. This is partly because he became less preoccupied with metaphysics in his later years and partly because the principle of isomorphism became so heavily qualified as to remove most of the cutting edge. In his major philosophical works of the 1940s, An Inquiry Into Meaning and Truth and Human Knowledge, he is more concerned with the nature of atomic facts thought of as the ultimate pieces of empirical data and the kinds of inferences required to get from these to the rest of what one wants to count as knowledge than he is with inferring a metaphysical structure from the logical form which an adequate statement of our knowledge would assume.

In these works there is a major shift in his view of the structure of atomic facts. Russell had earlier interpreted the word this in "This is red" as referring to a particular, something which has qualities and stands in relations but is not itself a quality or relation or set of qualities or relations. This is the traditional concept of substance as the substratum of properties, which was still alive in the realm of sense data even after physical objects and minds were no longer taken to be substances. But eventually the sense datum as substratum of properties went the way of physical objects, minds, and numbers. Here, too, Russell became convinced that there is no empirical warrant for assuming the existence of any such thing. In sense experience I am aware of a variety of qualities and their interrelations, but I am not also aware of something which has qualities. The bearer of qualities turns out to be the shadow of the usual grammatical form of the sentences used to report atomic facts. (There is a subject of the sentencefor example, "this"which does not refer to any quality.) Russell's latest position was that the subject of qualities is simply a construction out of a set of compresent qualities. Thus, in the ideal language "This is red" would be restated as "Red is compresent with ," where in place of the dots we have a specification of the other properties involved in that experience, for example, being round, being in the middle of the visual field, having ragged edges. It might be thought that this necessarily involves giving up the idea of absolute simples, for what takes the place of things in this view is bundles of qualities. But in this theory qualities themselves are regarded as the ultimate particulars (possibly simple) of which the world consists. Thus, in "Red is compresent with ," "red" does not refer to a particular exemplification of redness. If we took that line we would have to suppose that there is something which distinguishes this exemplification from other exemplifications of just the same color, and that would have to be something as unempirical as a substratum. Instead, it is taken to refer to the color conceived as a "scattered particular," something which can exist in a number of different places at the same time. And such a particular might well be simple.

Russell continued to think of commonsense physical objects and the entities of physics as constructions out of entities of the sort that are given in sense experience. But he came to require less similarity to sense data in the elements of these constructions. His later view was that although all ultimate entities have basic structural similarities to sense experiences, they need not involve only qualities that are given in sense experience. They may have qualities that it is impossible for us to be aware of. This uncertainty does not carry with it any serious gap in our knowledge, since for physical science it is the structure of external events that is important. In the 1940s Russell became increasingly concerned with the principles that are required to justify inferences from sense experience to unexperienced events and complexes of unexperienced events. The simplest form this takes is, for example, the inference that my desk has continued to exist in my office throughout the night, when no one was observing it. On Russell's view this is an inference from certain sense experiences to structurally similar events spatiotemporally connected with them in certain ways. He felt that the principle of induction by simple enumeration (the more often one has observed A and B to be associated, the more it is likely that they are invariably correlated) is insufficient to justify such inferences. What is needed, he thought, is a set of assumptions having to do with spatiotemporal connections of events of like structures. In Human Knowledge he presents a set of such assumptions. He does not claim that they can be known to be true. His point is a Kantian one: We must accept these assumptions if we are to accept the inferences to unobserved events that we all do accept in the course of our daily life.

Russell's entire philosophical career was dominated by the quest for certainty. In the middle decades of the twentieth century he was driven to admit that it is less attainable than he had hoped, but nevertheless the desire to approximate it as much as possible continued to shape his thinking about knowledge and the nature of the world. Because of this desire he was continually preoccupied with the problem of how to formulate those pieces of knowledge that are rendered indubitable by experience. And because of it he consistently attempted to analyze anything that appears dubitable into constituents about which there can be no doubt. Even where he was forced to admit that inferences beyond the immediately given are inevitable, he strove to reduce the principles of such inferences to the minimum. Russell is distinguished from other seekers after absolute certainty chiefly by the ingenuity of his constructions and by the candor with which he admits the failures of the quest.

Logic and Mathematics

reduction of mathematics to logic

Russell's main work in logic and mathematics was concerned with the problem of bringing the two together and with the interpretation of mathematicsarithmetic in particularas a simple extension of logic, involving no undefined ideas and no unproved propositions except purely logical ones. Russell achieved this synthesis at the beginning of the twentieth century, a little later than Gottlob Frege, but independently of him; in working it out in detail he had the collaboration of A. N. Whitehead. By current standards Russell's work lacks rigor, and in this respect it compares unfavorably with that of Frege; at an early stage, however, Russell did notice a difficulty that had escaped Frege's attention, the paradox about the self-membership of classes, which will be examined later. Because of its complexity it will be best to treat Russell's picture of the logical foundations of mathematics systematically rather than historically, with occasional comments about the actual development of his thought. We shall also separate from the outset two elements of Russell's treatment of his and other paradoxes, the theory of "types" and the theory of "orders," which Russell himself ran together, and thereby give a slightly clearer picture of his intention than his own writings immediately furnish.

Definition of "similarity"

Russell took over from Giuseppe Peano the reduction of all other arithmetical notions to complications of the three arithmetically undefined ideas of "zero," "number," and "successor" and defined these in terms of the theory of logical relations between classes or sets. In particular, he defined a number as a class of classes with the same number of members; for example, he defined the number 2 as the class of pairs. This procedure may seem unnatural (do we really mean by "2" the class of two-membered classes?) and circular. To the charge of unnaturalness Russell's answer was that his definition (together with the definitions of addition, etc.) gives all the ordinary results (2 + 2 = 4, for example) and that for a pure mathematician this is enough; another answer can be given only after it has been made clearer what Russell means by a class. With regard to the charge of circularity, Russell defines the complex "having the same number of members," or "similarity," as he calls it, not in terms of "number" (or of his definition of "number") but in other terms altogether.

At this point some notions from the logic of relations have to be introduced. A relation is said to be one to one if whatever has that relation to anything has it to one thing only and if whatever has anything standing in that relation to it has one thing only standing in that relation to it. (In strictly monogamous countries, "husband of" is a one-to-one relation in this sense.) Here the phrase "one only" does not presuppose the notion of the number 1. The sentence "x stands in the relation R to one thing only" means "For some y, whatever x stands in the relation R to is identical with y." The domain of a relation is the set of objects that stand in that relation to anything (the domain of "husband of" is the class of all husbands); the relation's converse domain is the set of all objects to which anything stands in that relation (the converse domain of "husband of" is the class of individuals that have husbandsthat is, the class of wives). A class A is similar to (that is, has the same number of members as) another class if there is some one-to-one relation of which the first class is the domain and the second the converse domain.

One can see that in a monogamous country the class of husbands will be similar in this sense to the class of wives, but one might think that two sets of objects could have the same number of members without there being any relation at all that pairs them off in the way that "husband of" does in our example. This, however, is a mistake when the term relation is understood as widely as it is by Russell. A relation in Russell's sense is, roughly, anything that can be expressed by a sentence with two gaps in it where names might go, and this covers not only obvious relating expressions like "______ shaves ( )" or "______ is the husband of ( )" but also ones like "Either ______ is identical with A, or B is identical with ( )." Take any set of two objects C and D. The relation "Either ______ is identical with A and ( ) with C, or ______ is identical with B and ( ) with D " (where all dashes must be replaced by the same name, and similarly with the bracketed blanks) will be a one-to-one relation in which A stands to C alone and B to D alone and in which C has A alone standing to it and D has B alonethat is, it will be a one-to-one relation of which the class with A and B as sole members is the domain and the class with C and D as sole members the converse domain; there are analogous relations in the case of larger classes. (Where the classes are infinitely large these relations will not be expressible in a language with only finite expressions, and perhaps that means that they will not be expressible in any language. Some philosophers would regard this as a serious difficulty; others would not.)

Axiom of infinity

Similarity, then, or having-the-same-number-of-members, is defined in terms of notions from the logic of relations: one to one, domain, and converse domain. The number-of-members of a given class is the class of classes similar to it, and a class of classes is a number (strictly, a cardinal number) if there is some class of which it is the number-of-members. This last step gives rise to another difficulty: Suppose there are (as there might well be) no more than a certain number n of objects in the universe. Then there will be no classes with more than n members and so, by the above definition, no cardinal numbers greater than n. This makes a great part of arithmetic (for example, the principle that every number has a successor different from itself) subject to the hypothesis (sometimes called the axiom of infinity) that there are an infinite number of objects.

Russell came to accept this last consequence of his definitions, but at an earlier stage he had thought that the axiom of infinity was provable, as follows: If we assume that every property demarcates a class, we must admit that some classes are empty (have no members), for example, there are no objects not identical with themselves. (The number 0 is precisely the class of classes with no members.) Thus, even if the universe contains no ordinary objects at all, there will still be at least one object of a more abstract sort, the universe itself considered as an empty class. And if there is this object there will also be two further objects of a still more abstract sort: the class of classes that has the first empty class as its one member and the empty class of classes. That makes three objects, call them A, B, and C. In addition to these there will be four classes of classes of classesthe class with B as its sole member, that with C as its sole member, that containing both B and C as members, and the empty class of classes of classes. And so on ad infinitum.

Russell paradox and the theory of types

Russell was led to abandon the above demonstration (which, as he said, has anyway "an air of hocus-pocus about it") by his discovery of the paradox of self-membership, mentioned earlier. If we can concoct classes with some members that are themselves classes, some that are classes of classes, and so on as we please (if, in other words, we can treat classes, classes of classes, etc., as so many sorts of classifiable "objects"), we can, it seems, argue as follows: The most obvious classes do not contain themselves as membersfor example, the class of men is not itself a man and so is not itself a member of the class of men (that is, of itself). On the other hand, the class of non-men is a non-man (is one of the things that are not men) and thus is a member of itself. We can therefore divide classes into two broad classes of classesthe class of classes that are members of themselves and the class of classes that are not. Now take the class of classes that are not members of themselves: Is it a member of itself or not? If it is, it must possess the defining property of this class to which ex hypothesi it belongsthat is, it must be not-a-member-of-itself. (Thus, if it is a member of itself, it is not a member of itself.) And if it is not a member of itself, ipso facto it possesses its own defining property and so is a member of itself. (If it is not, it is.) Let p be the proposition that our class is a member of itself; it follows even from the attempt to deny it, so it must be truebut it entails its own denial, so it must be false. There is clearly something wrong here.

Russell thought the error lay in treating a class seriously as an object. Perhaps it is an object in a sense, but not in the same sense in which genuine individuals are objectsand classes of classes are different again. They are, as he put it, of different "logical type." In particular, in an intelligible sentence you cannot replace an individual name by a class name or a class name by the name of a class of classes, or vice versa, and still have the sentence make sense. If "Russell is dead" makes sense, "The class of men is dead" does not, and if "The class of men is three-membered" makes sense (even if false), "Russell is three-membered" does not. And where a sentence makes no sense (as opposed to being merely false), its denial makes no sense either. Since "The individual I is a member of the class-of-individuals C " makes sense, "The class-of-individuals C is a member of the class-of-individuals C " does not and neither does "The class-of-individuals C is not a member of the class-of-individuals C "and so on at higher points in the hierarchy. This being granted, the paradox with which we began simply cannot be intelligibly formulated and thus disappears from the system.

At this point it would be wise to remove a possible source of confusion. The relation of class membership is different from the relation of class inclusion. One class is included in another if all the members of the former are members of the latter; for example, the class of men is included in the class of animalsall men are animals. But the class of men is not a member of the class of animals; that is, the class of men is not an animal (or, more strictly, "The class of men is an animal" is nonsense). The class of men is a member, rather, of the class of classes-of-animalsit is a class of animals. And the class of classes of animals is included in (but is not a member of) the class of classes of living thingsany class of animals, in other words, is a class of living things. Inclusion thus relates classes of the same logical type; membership, on the other hand, relates an entity with another entity of the logical type one above its own. The membership of an individual in a class of individuals is membership in a sense different from the membership of a class of individuals in a class of classes, and similarly for inclusionthere is a hierarchy not only of classes but also of membership and inclusion relations.

All this, besides solving a technical problem, is not without some attraction for philosophical common sense. Even apart from paradoxes it seems an artificial "multiplication of entities" to suppose that in addition to the individual objects which form the members of the lowest type of classes there are classes, classes of classes, and so on, and Russell devoted some attention to the problem of showing how what appears to be talk about these rather strange objects is in reality just more and more oblique talk about quite ordinary ones. To see just how he shows this it is necessary to look more closely at what might be called his "straight" language, into which this talk of classes, etc., does not enter and into which, once this talk has been introduced, it can always be "translated back."


From what has been said so far, it is clear that the "logic" to which Russell reduced arithmetic covered, implicitly or explicitly, such subjects as class membership and class inclusion, identity, and some sort of theory of relations. This is that part of logic that we first encounter when we work back to logic from arithmetic. We must now try and work forward to the same point from the fundamentals of logic.

Russell thought of logic as being at bottom "the theory of implication" (to quote the title of one of his early papers). And from the first he considered it important to distinguish implication from inference. He objected to the view that logic is primarily about thinkingconception, judgment, and inference, as some of the traditional logic texts put it. The connection of logic with inference is rather that logic is concerned with that in the real world that makes inference justified, and this is implication. "Where we validly infer one proposition from another," he wrote in 1903, "we do so in virtue of a relation which holds between the two propositions whether we perceive it or not: the mind, in fact, is as purely receptive in inference as common sense supposes it to be in perception of sensible objects" (Principles of Mathematics, p. 33).

Material implication

Even in Russell's purely objective, nonpsychological sense "implication" is ambiguous. Implication may be a relation between complete propositions, in which case it is called "material" implication and holds whenever it is not the case that the implying proposition is true and the implied proposition false. Before enlarging and commenting upon this account, certain grammatical and metaphysical clarifications are in order. Russell originally believed that sentences symbolized abstract objects called "propositions" and that material implication was a relation between these objects in exactly the same sense that marriage might be a relation between two people. He later dropped this view and regarded propositions, like classes, as mere "logical constructions," but he still used the old forms of words (as being, no doubt, accurate enough for practical purposes). In particular, the partly symbolic form "p implies q " (or "p materially implies q ") freely occurs in all his writings, and we ought to be clear about what he means by it. Generally it is simply a variant of "If p then q," or completely symbolically "p q " ("p hook q "), where the phrase "If ______ then ( )"or the hookis not a transitive verb expressing a relation between objects but a conjunction, or, as we now say, a "sentential connective." "If p then q " is thus not a statement about two objects symbolized by "p " and "q " but rather a complex statement about whatever the statements represented by "p " and "q " are about. For example, "If James is going to come, John will stay away" is not about two objects symbolized by "James is going to come" and "John will stay away," nor is it about these subordinate sentences themselves; rather, it links these two sentences to make a more complex statement about James and John. And if we say "That James is going to come implies that John will stay away," this is just a verbal variant of "If James is going to come then John will stay away"; that is, the linking expression "That ______ implies that ( )" has the same meaning as the conjunction "If ______ then ( )." The general form "That p implies that q " thus has the same sense as the form "If p then q " or "p q," and Russell's "p implies q " is thus just a loose way of saying "That p implies that q." In a similar way Russell often uses "p is true" and "p is false" as variants of "It is the case (is true) that p " and "It is not the case (is false) that p "; although sometimes he may really be talking about sentences in such a way that the sentence "John will stay away" may be described as true if and only if John will stay away and as false if and only if he will not, and the sentence "James is going to come" may be said to "imply" the sentence "John will stay away" if and only if the sentence "If James is going to come then John will stay away" is true.

The assertion that an implication is true if and only if it is not the case that the implying statement (antecedent) is true and the implied statement (consequent) false is not intended as a definition of the form "If p then q." It is simply an informal attempt to fix our attention on the relation (or quasi relation) that Russell intends. In his earliest works, like Frege and C. S. Peirce before him, Russell took this relation to be indefinable, and "the discussion of indefinableswhich forms the chief part of philosophical logicis the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple" (Principles of Mathematics, 1st ed., preface; 2nd ed., p. xv). Later he preferred to take as undefined the conjunction "or" and the negative prefix "it is not the case that" (or just "not") and to define "If p then q " as an abbreviation of "Either not p or q "; later still he followed H. M. Sheffer and Jean Nicod in using the stroke form "p | q " (which is true if and only if the component statements are not both true) and defined "if," "not," and "or" in terms of it. But for Russell the central part of logic has always been the study of implication, whether taken as undefined or not.

Since the form "If p then q " as understood by Russell is true as long as it is not the case that the antecedent is true and the consequent false, it is automatically true if the antecedent is false (for then it is not the case that the antecedent is true and thus not the case that the-antecedent-is-true-and-the-consequent-false) or the consequent true (for then it is not the case that the consequent is false and thus not the case that the-antecedent-is-true-and-the-consequent-false). In other words, a false proposition materially implies, and a true one is materially implied by, any proposition whatever. But implication is supposed by Russell to justify inference, and the mere fact that "Grass is pink" is false would not seem to justify us in inferring the 25th proposition of Euclid from it, and the mere fact that Euclid's proposition is true would not seem to justify us in inferring it from "Grass is green"geometry would be much easier if we could do this. Russell's explanation is that the first of these inferences cannot be performed because we cannot get it started (the premise not being true) and that the second inference is justified but we cannot know it to be so unless we already know the conclusion, so that we will not need it. In other words, "Infer a true proposition from anything at all" is a rule with no practical use, but this does not make it logically wrong.

Formal implication and propositional functions

Implications are of practical use when we know their truth without knowing either the falsehood of their antecedents or the truth of their consequents, and this happens most often when a material implication is an instance or particularization of an implication in the second of Russell's senses, a "formal" implication.

Formal implication is not (to use Russell's "realistic" language) a relation between propositions but one between what he calls "propositional functions." One might say roughly that formal implication is a relation between properties and that one property formally implies another if it is never present without the other; for example, being human formally implies being mortal (nothing is human without being mortal). Formal implication is clearly involved in the notion of class inclusionA is included in B if being a member of A formally implies being a member of B. But the notion of a propositional function is wider than that of a property. It is what is meant by an "open sentence," a sentence in which some expressionsay, a namehas been replaced by a variable. "Socrates is a man" expresses a proposition; "x is a man" expresses a propositional function. Sometimes, more simply, Russell uses the term propositional function for the open sentence itself. And the proposition that Socrates is a man may be said to be the value of the propositional function "x is a man" for the value "Socrates" of the argument x. The propositional function "x is a man" formally implies "x is mortal" if x 's being a man materially implies that x is mortal whatever x may bethat is, if we have "For any x, if x is a man then x is mortal." Russell writes this sort of implication as "φx x ψx." At one stage he treated this notion, for systematic purposes, as undefined, but even then he regarded it as complex in meaning, being built up from material implication together with the prefix "for any x," called a quantifier. Writing this last as "(x )," we may spell out the sense of a formal implication by writing it as "(x ) : φx ψx." It should be noted that whereas a propositional function is not a proposition, a formal implication between such functions is a proposition. The propositional function "x is a man" is neither true nor false; only its various values are true or false. But "For any x, if x is human then x is mortal" is as it were complete and is as it happens true. The quantifier is said here to "bind" the variable x, or, in the terminology Russell took over from Peano, x is in this context not a "real" but an "apparent" variable.

A propositional function may also have more than one expression in a proposition replaced by a variable, as in "x shaves y," "x gives y to z," and "If x shaves y then x does not shave z." In such cases the function corresponds to a relation (two-termed or many-termed) rather than to a property, and such functions may again be linked by formal implication, as in "For any x and y, if x is a child of y then x detests y "that is, "All children detest their parents." Symbolically, we have here the form "φxy x,y ψxy," or "(x,y ) : φxy ψxy." Again, formal implication may link a propositional function and a complete proposition, as in "If anything is in that box I'm very much mistaken," which is of the form "For any x, if φx then p " or "φx x p." Moreover, the expression whose place is taken in a propositional function by a variable need not be a name. It might, for example, be a sentence"If p then q " is a propositional function of which "If James is going to be there then John will not come" is the value when "James is going to be there" is the value of the argument p and "John will not come" the value of the argument q. If we prefix quantifiers to forms of this sort we obtain further formal implications, including the laws of propositional logic themselvesfor example, "For any p, q, and r, if p implies q then if q implies r, p implies r," which may be written "(p,q,r ) : (p q ) ((q r ) (p r ))" or "(p q ) p,q,r ((q r ) (p r ))."

A further case of special interest is that in which a variable replaces a verb or equivalent expression, as in "φ (Socrates)," where φ stands indifferently for "is a man," "smokes," "is running," etc. With appropriate quantifiers this function will yield such formal implications as "(φ ) : φa φb," "φa φφb " (roughly, "Whatever a does, b does," or "Whatever goes for a goes for b "). However, Russell says not that "φ (Socrates)" and "If φ (Socrates) then φ (Plato)" are functions of the verb or predicate φ but that they are functions of the function φx or, as he writes it in this type of context, the function φx̂ (the significance of this accenting or "capping" will be indicated later). His aim here is in part to bring out what Peirce and Frege called the "unsaturatedness" of verbs: The function of verbs can be understood only in relation to names and sentences; we use verbs to make statements about objects, not to name a special sort of object. The additional associated variables also enable one to represent unambiguously such complexes as "shaving oneself"if φ is "shaves," shaving oneself is φx̂x̂, as opposed to simply shaving (φx̂ŷ ). But Russell was hampered by not having the word functor to designate what makes a function out of its argument; it is more natural to speak of "Socrates is a man" as a propositional function of "Socrates," of "x is a man" as the same propositional function of "x," and of "is a man" as the functor which forms this function in both cases than to speak of "x̂ is a man" as a propositional function and to treat it in practice as a functor (Frege and W. E. Johnson were more accurate here, although they, too, lacked the term functor ).

This part of Russell's "philosophical grammar" can now be set out fairly straightforwardly: Sentences may be built out of other units in various waysout of other sentences by connectives, as in "p q and out of names by verbs, as in "φx," "φxy," and "φx ψx " (which may be conceived of as constructed out of the subsentences "φx " and "ψx " by the connective "" or out of the name "x " by the complex verb "φx̂ ψ x̂," "ψ' s-if-it-φ' s"). The rest of the hierarchy goes on from herethere are, for example, functors that form sentences out of verbs (that is, out of functors that form sentences out of names) and functors that form sentences out of these again, and so on ad infinitum. Functors may require one or more than one argument to make a sentence (the difference between "is a man" and "shaves," in the transitive sense), and when more arguments than one are required they may or may not be of the same type (for example, "If x̂ is a man then p̂ " requires a name and sentence).


Of functors that form sentences from verbs, the most important are quantifications, such as "(x )ϕ̑x " (which makes a sentence out of the verb whose place in the sentence is kept by "φ "), represented in English by such words as "everything." "Everything" is, or is constructed out of, the universal quantifier; there are many other quantifiers. Russell distinguished one other basic quantifier, "something." "Something is a man" expands in his language to "For some x, x is a man," or symbolically (x ) (x is a man)."

Given the quantifier "something" and negation we can construct the complex "It is not the case that (for some x (x is a man))" or "For no x is x a man," Here we have the philosophical beginnings of the number series. The number 0 makes its appearance as part of a quantification, for we could write the preceding form as (0x )(x is a man)." And the series can be continued. "Some" means "At least one, and "At most one thing is a man" is "For some, x, if anything is human it is identical with x "that is, "For some x : for any y, if y is human y is identical with x." The combination of "At least one thing is a man" with "At most one thing is a man" gives us "Exactly one thing is a man" that is, "(1x )(x is a man)." "(2x )(x is a man)" is, similarly, "At least two things are men, and at most two things are men"that is, "(For some x and for some y, x is a man, y is a man, and y is not identical with x ) and (for some x and for some y : for any z, if z is a man z is either identical with x or identical with y )," Apparent occurrences of numbers as objects can be analyzed away in terms of this primary sense; "1 and 1 is 2" for instance, becomes "For any φ and for any ψ, if exactly one thing φ' s, exactly one thing ψ' s, and nothing does both, then exactly two things either-φ -or-ψ." Numbers are inseparable components of functors of functors of names, or, as Russell would say, functions of functions, but the naturalness of this analysis is disguised in his own work by the fact that before he brings arithmetic into the picture he introduces the language of classes and defines numbers in terms of classes. (The notation "(0x )φx," etc., is not Russell's.)


Before going on to Russell's discussion of classes, we should note that "(x )ϕ̑x," "(x )ϕ̑x," and also "(0x )ϕ̑x," "(2x )ϕ̑x, and so on, are functions of functions of names, not arguments of such functionsthat is, they are not names. "Something," "nothing," exactly one thing," etc., are not names, although, like names, they go with verbs to make sentences. They go, so to speak, on the other side of verbs: They "govern" the verbs; the verbs do not govern them. And although Russell's hierarchy of types of functors or "functions" provides innumerable ways of constructing sentences (and so of constructing functions), it provides no way of constructing genuine names. It is of the essence of the expressions represented by Russell's variables of lowest type (x, y, z, etc.)that is, individual namesthat they are logically structureless; they pick out individuals, and that is all. But in common speech and in mathematics we do seem to construct names, or at least ways of designating objects, out of expressions of other types: For example, "the man who broke the bank at Monte Carlo" seems to function as a name, yet it seems to be constructed from the verb "broke the bank at Monte Carlo." On Russell's view this appearance is illusory, and sentences in which such apparent names occur can always be replaced by paraphrases expressed entirely in Russell's language of structureless names, functions of functions, etc. However, he regarded it as useful for logical symbolism to reproduce at this point, although with greater precision, some of the devices of common speech and to have, as it were, a secondary language imposed on the primary one.

(Some account of Russell's handling of descriptionsthat is, expressions of the form "The so-and-so"and of other points raised below is given in the entry Existence.) "There φ -er," or "The thing that φ' s," when it occurs as the apparent subject of a further verbthat is, in a context of the form "The φ -er ψ' s"is in reality a functor, in some way like a quantifying expression, of which the verbs "φ' s" and "ψ' s" are arguments; in fact "The thing that φ' s ψ' s" amounts precisely to "Exactly one thing φ' s, and whatever φ' s ψ' s," whose first component has been analyzed above and whose second component is a simple formal implication. Expressions of this kind are especially important in mathematics when the contained functor φ is relational in form, as in "The φ -er of y "that is, "The thing that φ' s y," "The square root of y " (the number that yields y when multiplied by itself) is such an expression. Russell called expressions of this kind "descriptive Functions." They include most "functions" in the ordinary mathematical sense. It is a little inaccurate, of course, to use name symbols like "y " for numbers, which on Russell's view are not genuine individuals, but once the devices that yield class language and number language have been worked out, Russell's analysis of descriptive functions can be reproduced at the new level in a transposed form. This language of classes and numbers, to which we shall now turn, is itself a case of a secondary language containing apparent names (like "The class of persons that shave themselves") that disappear from the primary-language paraphrase.

classes, functions, and properties

Russell represented the form "The class of things that φ " as "x̂ (φx )"usually read as "the x 's such that φx and represented "y is a member of the class of φ -ers" as "y x̂ (φx )." Alternatively we may read "x̂ (φx )" simply as "φ -er" and "y x̂ (φx )" as "y is a φ -er." The expression "y is a φ -er" is true if and only if y φ' s. One can in fact simply define "y x̂ (φx )" as "φy." Given this definition, other concepts associated with class theory are easily introduced. For example, as noted earlier, "The class of φ -ers is included in the class of ψ -ers" amounts to the formal implication "For any x, if x is a φ -er then x is a ψ -er."

Classes of classes are related to functions of functions as classes are related to functions. To say that a given classx̂ (φx ), for exampleis a member of the class of two-membered classes (or, as Russell would write it, "x̂ (φx ) 2") is just to say that exactly two things φ i.e., the class of classes that Russell identifies with the number 2 is just the correlate in the class hierarchy of the function of functions (2x )ϕ̑x.

Counting classes

There are two difficulties in Russell's views concerning classes. One is that classes, and, for that matter, numbers, can themselves be counted, as can individuals, but a number of classes would have to be not a class of classes but a class of classes of classes, and a number of numbers would similarly have to be a class of classes of classes of classes. This means that when we say "The number of numbers between 2 and 5 is 2," the first "2" has a sense quite different (belongs to a place quite different in the type hierarchy) from the second; and this seems a little implausible. Russell at this point is content to speak of the "systematic ambiguity"" of the key expressions of his symbolic language. Given the proof of "1 + 1 =2," for instance, considered as a statement about numbers of individuals, an analogous proof can always be constructed for the analogous statements about numbers of classes, numbers of numbers, etc., so that in practice it does not matter at which place in the type hierarchy we are working, provided we keep the types going up in order.

Ludwik Borkowski has suggested what may be a better solution: Suppose we always express quantification by a sign followed by a variable; for Russell's "(x )" we might put "(x )," by analogy with "(x )." We might then use the term quantifier not for this expression as a whole but for the initial sign, which can then be described as a functor that constructs a sentence out of a variable followed by a sentence, usually an "open" sentence in which the variable just mentioned occurs. We might then say that the initial sign "" or " "or in the case of numerical quantifiers "0" or "1" or "2," of the same logical type whatever the type of the variable that comes between it and the sentence following it. For counting properties (and, therefore, classes), we would have prefixes like "(2φ )"for example, "(2φ )φ (Socrates)" would mean "Socrates has exactly two properties" or, better, "Exactly two things are true of Socrates"; and "(2φ )" is different from "(2x )," but the "2" is exactly the same in both contexts.

Counting functions

The other difficulty in Russell's theory is that classes dissolve into functions, but we do not count classes and functions in quite the same way. We would say, for example, that any two-membered class has four subclasses, in the sense that there are four ways of selecting members from such a class (both members, the first only, the second only, and neither). The corresponding theorem about functions would seem to be this: If exactly two things φ, then for exactly four ψ' s, whatever ψ' s φ' s. But in fact there will always be vastly more than four ψ meeting this condition. Suppose, for example, that there are just two men in a roomi.e., (2x )(x is a man in the room)and that one of them wears spectacles, spats, spotted socks, a red tie, and striped trousers; this much alone gives us five ψ' s (namely, "______ is a man in the room wearing spectacles," "______ is a man in the room wearing spats," etc.), such that whatever ψ' s is a man in the room. The key point here is simply that we count classes as being the same when they have the same members, but we do not count propositional functions as being the same merely because they are satisfied by the same arguments, and all the numerical concepts that are built up from the concept of identity must be similarly adjusted. For instance: "At most one class is a sub-class of x̂ (φx )" does not mean "For some ψ : for any χ, if x̂ (χx ) is a subclass of x̂ (φx ), then χ -ing is the same as ψ -ing," but rather it means "For some ψ : for any χ, if x̂ (χx ) is a subclass of x̂ (φx ), then whatever χ 's ψ' s and whatever ψ' s χ 's." It is the same when we move up a type and count numbers themselves. If we write "(0x )(φx )" for "It is not the case that (for some x, φx )" and "(0x )(φx )" for "For any x, if x φ s then x is not identical with itself," we may say that these are different functions of functionsbut whatever function either of them applies to the other applies to also; thus, they determine a single class of classes and a single "number," 0. The class and number language that Russell superimposes on his basic one is such that this is the way these quasi entities are counted.


One very radical way of simplifying this whole problem (one that Russell has considered from time to time) is to say that functions (properties, relations, etc.) are to be counted in just the same way that classes are; that is, that if φx̂ and ψx̂ characterize precisely the same objects (are formally equivalent), they are the same function. This is called the principle or law of extensionality; it in effect simply identifies a function with its "extension"that is, with the class that it determines. The objection to this principle is simply its extreme implausibility in particular cases. For example, it seems obvious that even when two individuals and these two only are the men in a certain room wearing spats and the men in that room wearing spectacles, being a man in the room with spats is something different from being a man in it with spectacles.

Quine's criticism

Logicians such as W. V. Quine, following Ernst Zermelo and John von Neumann, have developed systems in which classes, classes of classes, and so on, are treated not as logical constructions but as genuine objects, and Russell's paradox is dealt with not by saying that "x is (is not) a member of x " is meaningless but by denying that "xφ' s" always implies that x is a member of the class of φ -ers. This account runs into difficulty when we try to handle certain nonmathematical properties of these supposed objects. Russell's view seems to have the advantage of not unnecessarily "multiplying entities," but Quine argues that Russell succeeds in dispensing with classes only by making genuine objects of properties or functions. This is said on the ground that in the course of his treatment of classes and numbers Russell is compelled to quantify over predicate variablesthat is, to employ such quantifiers as "(φ )" (for example, in defining "Exactly as many things ψ as χ as "For some relation, φ whatever φ' s anything ψ 's and vice versa, whatever is φ' d by anything χ' s and vice versa, and whatever φ' s or is φ' d by anything φ' s or is φ' d that thing only"). This, Quine says, is to make properties and relations (like φ -ing) the "values of bound variables," and to do this is to treat them as existing.

This amounts to saying that to generalize an expression by quantifying over it is ipso facto to make it a name of an object; but this claim may be contested. We do not elucidate "He must have killed him somehow" by translating it "There must be some way in which he killed him" (which, taken literally, suggests that there are objects called "ways") but rather vice versa: We understand "somehow" directly as a generalization of qualifications like "with a knife," and the "way" line of talk is merely a variant of this. Even "something" is often to be understood as a generalized adjective rather than as a generalized individual namefor example, when I say "I am something that Jones is notlogical." It seems more plausible to interpret "I have something that Jones has notlogicality" as a verbal variant of the preceding sentence than to say that the latter alone brings out what I am really doing. And the logical rules for such higher-order quantifications are simplewe proceed from the specific case to the generalization, from "I am logical and Jones is not" to "For some A, I am A and Jones is not A," exactly as we do from "I am logical but not intelligent" to "For some individual x, x is logical but not intelligent."

Elimination of abstract terms

Russell might more plausibly be said to "hypostatize" or "reify" abstractions on the ground that there are some contexts from which it seems impossible to eliminate from his basic language his symbols for "abstracts," that is, φx, etc. This part of his system is developed more tidily in Alonzo Church's calculus of λ -conversion, in which the property of φ -ing is represented not by "φx̂ but by "λxφx," and of "ψ -ing if one φ' s" by "λx. φx ψx." The basic rule of this calculus is that the application of λxφx to an object a, symbolized by (λxφx )a, is equivalent to the plain φa, and similarly (λx. φx ψx )a is equivalent to φa ψa, And where we have a function of functions ƒ, we can in general similarly replace ƒ(λxφx ) by ƒ(φ )but not always. For instance, it is an obvious law that any such function ƒ which holds for any φ whatever will hold for χ -ing-if-one-ψ' s, as in formula F :
(ƒ) : (φ ) ƒ(φ ) . . ƒ(λx . ψx χx.
Here the expression with λ seems uneliminable. We cannot replace it with "ψ χ," for this is meaninglessthe hook joins sentence forms, not predicate forms. Where we have a specific ƒ the elimination is again possible; for example, if ƒ is the function "applying to exactly two objects," then ƒ(λx . ψx χx ) will amount to (2y ) : (λx . ψx χx )y and thus to (2y )(ψy χy ). But where the ƒ itself is a variable, as it is in formula F, nothing of this sort is done. We could indeed (following Stanisław Leśniewski) introduce a symbol for the predicate "χ -ing-if-one-φ' s" by a special definition; for example
[ψχ ]x =Df φx ψx
and so replace F with G :
(ƒ) : (φ )ƒ(φ ) . .ƒ([ ψχ ]),
but then it would be impossible to eliminate the defined symbol from G in favor of the symbols by which it is defined, and it seems an odd sort of definition that would be thus limited. (Church's use of λ can in fact be regarded as simply a generalization of Leśniewski's procedure.)

The uneliminability of "abstracts" from these contexts is an odd and perhaps awkward fact, but it need not be taken to imply that there are abstract objects, for "abstracts" need not be regarded as a kind of name. In expositions of the λ -calculus it is often said that the form λxφx corresponds to the ordinary-language quasi noun "φ -ing," but this is not strictly correct, as may be seen from the fundamental equation (λxφx )a = φ " If "φ " here represents not a name but a verb ("φa " means "a φ' s"), then so must "λxφx " ("(λxφx )a " also means "a φ' s"), so that if ƒ in ƒ(φ ) is a function with not names but predicates as arguments, so it must be in "ƒ(λxφx )."

ramified theory of types

We may now describe the added feature that makes Russell's own presentation of his theory of types more complex than the presentation so far given here. Russell divides functions into types not only according to the types of argument that they take but also according to whether they do or do not involve an internal reference to all functions of (what appear to be) their own type. For example, the function x̂ has all the qualities of a great general" has individual-name arguments, just as "x̂ is brave" does, but unlike "x̂ is brave" it has a "for all φ " within itselfit amounts to "For all φ, if whoever is a great general φ' s, then x̂ φ' s." Russell therefore regards it as of a different type, or, as he often says, of a different order, from "x̂ is brave." Functions that do not thus involve a reference to all functions of (what appear to be) their own type he calls "predicative" functions and symbolizes them by putting an exclamation mark or "shriek" after the symbol, as in "φ !x," Functions cannot in fact (on Russell's view) strictly contain references to all functions of their own type or order but references only to ones of orders below their own. A function of individuals, which contains a reference to all predicative functions of individuals, is not itself predicative and cannot be regarded as being among the functions to which it implicitly refers. Having all the properties of a great general, for example, is not itself a property of a great general, at least not in the same sense of "property"it is a second-order property.

What this means in practice might be illustrated as follows: It seems that if there were no facts about x that is, if for no φ, φx then there would be at least one fact about x, namely the fact that there are no facts about it, and hence it cannot be that there are no facts about x. In symbols, from
(1)      ψx (φ )(φx )
it seems possible to obtain
(2)      (φ )(φx ) . (φ )(φx
by letting ψx̂ in (1) be, in particular, (φ )(φx̂ ); and from (2) it follows by a kind of reductio ad absurdum that for any given x we have (φ )(φx ). But on Russell's view this proof will not do, for (1) ought to have been written
(3)      ψ !x (φ )(φ !x )
and here (φ )(φ !x̂ ), not being itself predicative, is not a permissible substitution for ψ !x̂. It is worth noting, however, that our final conclusion, (φ )(φ !x ), can be proved from (3) in a different wayby letting our ψ !x̂ be χ !x̂ χ !x̂ ("x̂ χ 's-if-it-χ 's"), which is predicative and is true of any x, so that what it implies must be true of any x also. (The new argument is as follows: There is always some fact about x, since at least it is a fact that x is red-if-it-is-red, square-if-it-is-square, etc.)

Axiom of reducibility

Russell lumps together all his type and order restrictions under the general head of avoiding "vicious circles," and the theory of types with the theory of orders worked into it is called the "ramified" theory of types. One trouble with it is that it vitiates certain essential arguments in the higher reaches of mathematics, and to save these Russell introduced an "axiom of reducibility," that to every function of any order there corresponds a predicative function that is formally equivalent to itthat is, which holds for exactly the same arguments as the given function. This means that any argument like our allegedly invalid proof of (φ )(φx ) above, where it is worth saving, can in principle be replaced by one like our second and valid one; the axiom of reducibility does not itself enable us to find this valid argument but entitles us to proceed as if we had it. It is, however, an intuitively dubious principle and can be dispensed with if we can content ourselves with the theory of types in the "simple" form in which it has been stated in earlier sections.

Semantic paradoxes

It was pointed out by F. P. Ramsey that those paradoxes which Russell lists and which cannot be eliminated (as can, for example, the paradox of the class of all classes not members of themselves) by the "simple" theory of types always contain some implicitly or explicitly "semantic" feature; that is, they all have to do with the relation of language to what it is about and all involve conceptions like truth and meaning. A typical example is the paradox of the liar, of the man who says "What I am now saying is false" and says nothing else but this, so that what he says is true if it is false and false if it is true. Such paradoxes are now generally dealt with by assuming not only a hierarchy of "parts of speech" in one's basic language (this is what the simple theory of types amounts to) but also a hierarchy of languagesa basic language, a "metalanguage" in which we discuss the meaning and truth of expressions in the basic language, a "metametalanguage" in which we deal similarly with the metalanguage, and so on.

It is both easy and necessary to criticize Russell's theories concerning the logical and semantic paradoxes, and his work in logic and the foundations of mathematics generally, but he remains, more than any other one person, the founder of modern logic.

Ethics and the Critique of Religion


Much of Russell's life, as we saw in an earlier section, was devoted to the advocacy of certain moral and political ideals. In this sense of the word moralist, in which it has no derogatory implications, Russell was certainly a moralist and frequently a very passionate one at that. Unlike many other moralists he was also concerned with what are now referred to as "metamoral" or "metaethical" issues. He repeatedly addressed himself to questions about the status of moral principleswhat, if anything, they mean, what kind of disagreement there is between people who support opposite moral positions, and whether inferences from nonmoral premises to a moral conclusion can ever be valid. In discussing Russell's ethics, we will be concerned only with his metamoral theories.

Early views

In his first important essay on this subject, "The Elements of Ethics" (1910), Russell defended a position closely akin to that of G. E. Moore in Principia Ethica. "Good and bad," he wrote, "are qualities which belong to objects independently of our opinions, just as much as round and square do; and when two people differ as to whether a thing is good, only one of them can be right, though it may be very hard to know which is right." The goodness or badness of a thing cannot be inferred from any of its other properties. "Knowledge as to what things exist, have existed, or will exist, can throw absolutely no light upon the question as to what things are good." Russell was by no means unaware at this time of the wide appeal of the familiar arguments for subjectivismthe "divergence of opinion" on moral questions and the difficulty of "finding arguments to persuade people who differ from us in such a question" ("The Elements of Ethics," in Readings in Ethical Theory, edited by Wilfrid Sellars and John Hospers, New York, 1952, pp. 67). But he did not then regard these arguments as having any logical force. "Difficulty in discovering the truth," he wrote, "does not prove that there is no truth to be discovered" (p. 6). Like Moore, he argued that if subjectivism were true it would follow that in a moral dispute there is never really any "difference of opinion" between the disputing parties. If when A says x is good and B says x is bad, A and B were really talking about their respective feelings or desires, they might well both be right at the same time and "there would be no subject of debate between them." At that time Russell regarded this as plainly false. "As a matter of fact," he observed, "we consider some tastes better than others: we do not hold merely that some tastes are ours and other tastes are other people's." When "The Elements of Ethics" was reprinted in 1952 in Readings in Ethical Theory, the anthology mentioned above, Russell added a footnote in which he explained that "not long after publishing this paper [he] came to disagree with the theory that it advocates." He explains that the change in his views was originally due to George Santayana's criticisms in his Winds of Doctrine, but he adds that he "found confirmation" for his later position "in many other directions." Russell's later position was first mentioned very briefly in a 1921 preface to a paperback reprint of "A Free Man's Worship"; it was explained in some detail in What I Believe (1925) and in The Outline of Philosophy (1927), and it received its fullest formulations in Religion and Science (1935), Power (1938), "Reply to My Critics" (in P. A. Schilpp, ed., The Philosophy of Bertrand Russell, 1944), and Human Society in Ethics and Politics (1955).

The subjectivity of values

Except on one basic issue, Russell's later position is a point-by-point denial of the earlier theory. "Good" and "bad" are no longer regarded as qualities belonging to objects, and in this respect they are now explicitly contrasted with "square" and "sweet": "If two men differ about values, there is not a disagreement as to any kind of truth, but a difference of taste" (Religion and Science, pp. 237238); "There are no facts of ethics" (Power, p. 257); "I see no property analogous to truth that belongs or does not belong to an ethical judgment" ("Reply to My Critics," p. 723). "Taste" in the first of these passages is used in a very broad sense to cover all kinds of psychological states and attitudes, including desires. Russell does not, of course, deny the plain fact that people regard some tastes as better than others and some desires as higher than other desires, but now he is willing to maintain that this merely means that the tastes or desires are their own. "What we 'ought' to desire is merely what someone else wishes us to desire" (What I Believe, p. 29).

Russell is quite ready to have his later theory classified as a form of "the doctrine of the subjectivity of values" (Religion and Science, p. 237), but it differs in some significant respects from the older theories that have gone by that name. If somebody maintains that pleasure, for example, or the love of God, is intrinsically good, or good "on its own account," this must not be taken to be equivalent to the statement that he approves of it or in some way desires it. Like the advocates of the so-called emotive theory of ethics, Russell maintains that intrinsic moral judgments, grammatical appearances notwithstanding, are not statements or assertions at all but expressions of desire. "A judgment of intrinsic value," he writes in Power, "is to be interpreted, not as an assertion, but as an expression of desire concerning the desires of mankind. When I say 'hatred is bad,' I am really saying: 'would that no one felt hatred.' I make no assertion; I merely express a certain type of wish" (Power, p. 257).

Both here and in his capacity as a reformer Russell places much emphasis on the distinction between purely personal and what he calls "impersonal" desires. A hungry man's desire for food or an ambitious man's desire for fame are examples of the former; a desire for the abolition of the death penalty or the end of racial discrimination, independently of whether the person in question stands to gain from these changes, are examples of the latter. In moral judgments we express certain of our impersonal desires. A king who says, "Monarchy is better than republican forms of government," is using the word better in its properly moral sense if he is expressing not just his desire to remain king but a desire that nations have monarchical systems regardless of his own personal position. Russell occasionally writes as if the desire expressed by moral judgments must be a second-order desirethat is, a desire that everybody have a certain first-order desirebut as several of his own examples make clear, this is not part of his position. What is essential is that the desire be impersonal. In this connection he also observes that the philosophers who stressed the "universality" of moral principles were in a sense quite right. This universality, however, does not consist in any a priori character or logical necessity. What is universal is the object of the desire expressed by a moral judgment. "The wish, as an occurrence, is personal, but what it desires is universal. It is this curious interlocking of the particular and the universal which has caused so much confusion in ethics" (Religion and Science, p. 236).

As we shall see, Russell had a tendency to overestimate the scope of application of his subjectivism, but in a number of places he points out quite explicitly that large classes of everyday moral judgments and disputes do not come within the purview of the theory. "Ethical controversies are very often as to means, not ends" (Power, p. 259). "The framing of moral rules, so long as the ultimate Good is supposed known, is matter for science" (Religion and Science, p. 228). It follows from this that if human beings could agree about ultimate ends, all moral disputes would in principle be decidable by an appeal to facts even though the intrinsic judgments would still be not bona fide propositions but expressions of wishes. In fact, however, Russell insists, there is no such agreement about ends. In "The Elements of Ethics" he had conceded that there were some ultimate ethical differences but had maintained that people in fact "differ very little in their judgments of intrinsic value." Many of the commonly observed differences are wrongly regarded as ultimate because what are really disagreements about means are mistaken for disagreements about ends. In his subjectivist phase Russell seems to think that differences about ends are not at all uncommon. Behind such disputes as, for example, the subjection of women or the persecution of religious minorities, which do involve questions of means, he writes, "there is generally a difference as to ends," and this sometimes becomes "nakedly apparent," as in Friedrich Nietzsche's criticisms of Christian ethics. In Christianity, all men are valued equally, but for Nietzsche the majority exists only as means to the superman. This, Russell maintains, is an example of a dispute about ends, and "it cannot be conducted, like scientific controversies, by appeals to facts" (Power, p. 259).

In "The Elements of Ethics" Russell had quite properly observed that the mere existence of widespread ethical disagreement (if it is indeed widespread) does not establish any form of subjectivism. Although he has evidently come to believe that ethical disagreement is more widespread than he had thought earlier, he does not offer this as evidence for his new theory. What he does offer as evidence is the undecidability of ethical disputes. He writes:

[The chief ground for adopting this view] is the complete impossibility of finding any arguments to prove that this or that has intrinsic value. We cannot prove, to a color-blind man, that grass is green and not red. But there are various ways of proving to him that he lacks a power of discrimination which most men possess, whereas in the case of values there are no such ways since no way can be even imagined for deciding a difference as to values, the conclusion is forced upon us that the difference is one of taste, not one as to any objective truth. (Religion and Science, p. 238)

If three men argue, one saying "The good is pleasure," the second "The good is pleasure for Aryans and pain for Jews," and the third "The good is to praise God and glorify him forever," they cannot, as people engaged in a scientific dispute, "appeal to facts," for facts, it seems obvious, "are not relevant to the dispute" (Power, p. 257).

Russell's later view agrees with the earlier position on only one significant point, its opposition to naturalism. By "naturalism" is here meant the theory that there is a logical connection between some moral judgments and factual premises where the latter are not necessarily confined to empirical statements but may also include metaphysical doctrines. We saw how in "The Elements of Ethics" Russell had insisted that from statements concerning what exists nothing can be inferred about "the goodness of anything." "It is logically impossible," he repeated in the course of expounding his later position, "that there should be evidence for or against" a moral judgment, but now this is maintained because a moral judgment "makes no assertion" and hence possesses neither truth nor falsehood (Religion and Science, pp. 236237).

"Incredibility" of Russell's subjectivism

Rather than attempt a detailed critical evaluation of Russell's subjectivism, we will discuss one objection that has been urged by a number of his critics and which, in one form or another, has been leveled against nearly all forms of subjectivism. It has been argued that a subjectivist cannot consistently make moral judgments. All he can say is that some people have one kind of feeling or attitude while other people feel differently. More specifically, how can Russell's subjectivism be reconciled with his judgments as a moral critic and reformer?

It may be replied that as a matter of pure logic there is no inconsistency between holding that moral judgments are expressions of taste and using moral language to express one's own tastes. Russell, it might be said, would be inconsistent only if he claimed that his moral judgments, unlike those of his opponents, are more than expressions of taste. Then he would indeed be like the man who, in the course of an argument about the value of a piece of music, remarked to his opponent "It is all a matter of taste, except that my taste is better than yours." However, while this answer is valid as far as it goes, it does not meet the heart of the objection. For Russell seems to be sayingor at least he would like to be able to saythat his moral judgments (for example, his judgment that democracy is a better system than totalitarianism or that the sexual code advocated in Marriage and Morals is superior to that associated with orthodox religion) are in some sense rational or right or well-grounded while the judgments of his opponents are irrational, wrong, or unsupported by the evidence.

Russell apparently did not, when he first advanced his subjectivism, see any serious problem here, but in the 1940s and 1950s he repeatedly expressed dissatisfaction with his own theory on this ground. Thus, in "Reply to My Critics" he writes:

What are "good" desires? Are they anything more than desires that you share? Certainly there seems to be something more. Suppose, for example, that some one were to advocate the introduction of bull-fighting in this country. In opposing the proposal, I should feel, not only that I was expressing my desires, but that my desires in the matter are right, whatever that may mean. As a matter of argument, I can, I think, show that I am not guilty of logical inconsistency in holding to the above interpretation of ethics and at the same time expressing strong ethical preferences. But in feeling I am not satisfied. (The Philosophy of Bertrand Russell, p. 724)

To this he adds: "I can only say that, while my own opinions as to ethics do not satisfy me, other people's satisfy me still less." More than a decade later Russell expressed himself even more strongly. In a letter to the Observer (October 6, 1957) he comments on Philip Toynbee's review of Why I Am Not a Christian : "What Mr. Toynbee says in criticism of my views on ethics has my entire sympathy. I find my own views argumentatively irrefutable, but nevertheless incredible. I do not know the solution."

It is doubtful whether in such comments Russell is really fair to his own subjectivism. Let us recall that the theory was never meant to apply to anything other than what are variously called intrinsic or fundamental value judgments and differences. The questions whether happiness is better than unhappiness and love better than hate are frequently cited as such ultimate moral issues, but it would be hard to find anybody who seriously maintains that suffering is good on its own account or that hate is better than love, although of course people have often maintained that in certain situations and for certain reasons suffering and hate are preferable to enjoyment and love. However, on occasions there do appear to be real value differences of an ultimate kind. Thus, some people would maintain that dignity is "more important" or "nobler" than happiness. Many who do not despise happiness at all would maintain without hesitation that a man who chose to suffer a great deal rather than compromise his integrity (where it is assumed that he would in fact have suffered much less if he had not stood his ground) lived a better life than he would have if he had made the opposite choice. Or, again, there is sometimes disagreement as to whether a person suffering from a fatal illness should be told the truth, although there may be full agreement about the consequences of both telling and not telling him the truth. Russell's subjectivism does apply to this kind of intrinsic moral disagreement, and in such situations he could not, consistently with his theory, claim that the moral judgment he endorses is "more rational" or better supported than that of his opponents.

However, the examples Russell offers when expressing dissatisfaction with his subjectivism are not at all of this ultimate kind, and this applies to all or nearly all the positions he has advocated in his social and political writings. The man who says that the good is pleasure for Aryans and pain for Jews, if he is willing to engage in moral argument at allif he is not, the problem does not arisepresumably does not just say this but proceeds to make all kinds of factual claims about the psychological and physical qualities of Aryans and Jews, respectively, about the laws of heredity, and about various other matters that he regards as justifying his moral position. Similarly, the man who maintains that "the good is to praise God and glorify him forever" presupposes that there is a God, and a God of a certain kind, probably also that he has revealed himself in certain ways, and, if challenged (or perhaps even without being challenged), he will make claims about the hollowness of all earthly satisfactions and the greater reliability, intensity, and duration of the satisfactions derived from glorifying God. Again, a man, who advocates the introduction of bullfighting into the United States would not just advance this proposal but would give reasons having to do, perhaps, with the benefits to be derived from engaging in dangerous sports and the special thrills experienced by the spectators. All these supporting factual claims are discussable, and it may be possible to show that they are mistaken or highly implausible. If so, it might well be possible to regard the case of one side in such a dispute as well supported and the other as unsupported by the evidence. In all cases in which the person is willing to support his moral judgment by factual premises, it is perfectly consistent for Russell to assert that one position is "more rational" than the other, where "more rational" does not merely mean that Russell shares the attitude of the person taking this position.

What seems to be amiss here is not Russell's subjectivism but his view (which is not logically implied by it) that the theory applies to cases like the dispute about bullfighting. In his later period Russell seems to be guilty of a gross overestimate of the prevalence of ultimate moral disagreements. It is true, as he observes in Power, that behind disagreements about means there is frequently disagreement about ends, but it is very doubtful that the ends in question are in most cases ultimate ends. To give a simple illustration of a very common type: Two people may offer conflicting moral judgments about a bill to legalize abortion. The man who opposes the legislation may give as his reason (or as one of his reasons) that it would remove one of the conditions restraining unmarried people from engaging in sexual intercourse, whereas the other man might offer this as his reason (or one of his reasons) for supporting the legislation. Although the disagreement may in the immediate context be properly described as one about an end, it is clearly not about an ultimate end. In all likelihood the parties to the dispute would differ about the effects of a freer sex life on personal happiness, on society at large, on the future of religious institutions, and many other things. It is doubtful that either of them would maintain that suffering as such is better than happiness or that hate is better than love.

Even people who advocate what by most contemporary standards would be regarded as "outlandish" moral positions can usually be seen to share many of the intrinsic value judgments of the rest of humankind. Thus, Arthur Schopenhauer and other champions of asceticism recommend the suppression of desires, including those that to most human beings seem the most natural and the most innocent, but they do so not because in their opinion suppression of these desires would make people unhappy but, on the contrary, because it would enable them to achieve greater happiness or at least because it would reduce suffering to a minimum. In Norman Mailer's bizarre novel An American Dream the main character offers a defense of murder, but this unusual position is justified by the argument that "murder offers the promise of vast relief. It is never unsexual." It is accompanied by "exhilaration" that must come "from possessing such strength." It should be noted that murder is here justified not because it causes suffering but because, according to the character, it leads to "exhilaration." In other writings Mailer tells us that the "modern soul marooned in emptiness, boredom and a flat, dull terror of death" would be well advised to pass through "violence, cannibalism, insanity, perversion" and other states and activities that are usually considered highly undesirable, but these recommendations are offered not for their own sake but because they will lead the person "back to life."

As for the really intrinsic clashes of the kind mentioned earlier, to which Russell's subjectivism would apply, one wonders if the consequences of the theory are there really so paradoxical. No doubt people do in such disputes regard their position as superior to that of their opponentsthe man who admires integrity will feel contempt for the "cowardly" compromiser, and the compromiser will think the man who chooses to suffer a fool. Here, however, unless there are some hidden differences concerning matters of fact, it seems not at all incredible to maintain that calling one position superior simply amounts to expressing one's own preference for it.

None of the above is meant to prove that Russell's subjectivism is a correct account of the logical status of moral judgments, but it would indicate that the favorite objection of his critics can be disposed of without much difficulty.

critique of religion

No such doubts as Russell has expressed about his subjectivism in ethics mark his views on religion. Unlike many academic philosophers whose position is very similar to his, Russell did not hesitated to express his convictions publicly and without equivocation or compromise. Ever since he abandoned the Platonic theory of ideas, Russell was a forthright opponent of religion in more senses than one: He regards the basic doctrines of (supernaturalistic) religions as intellectually indefensible, he argues that religious belief has not on balance been a force for good but quite the opposite, and he hopes and believes that religion will eventually die out. "I am myself," he wrote in 1922, "a dissenter from all known religions, and I hope that every kind of religious belief will die out. I regard religion as belonging to the infancy of human reason and to a stage of development which we are now outgrowing" (Sceptical Essays, p. 101). In a television interview thirty-seven years later he slightly qualified this prediction. If great wars and great oppressions continue so that many people will be leading very unhappy lives, religion will probably go on, but "if people solve their social problems religion will die out" (Bertrand Russell Speaks His Mind, p. 31).


Russell wavered between calling himself an agnostic and describing himself as an atheist. He evidently did not attach too much importance to this distinction, but he had made it clear that if he is to be classified as an agnostic, it would have to be in a sense in which an agnostic and an atheist are "for practical purposes, at one." In the television interview mentioned earlier the interviewer asked Russell, "Do you think it is certain that there is no such thing as God, or simply that it is just not proved?" "No," Russell answered, "I don't think it is certain that there is no such thingI think that it is on exactly the same level as the Olympic gods, or the Norwegian gods; they also may exist, the gods of Olympus and Valhalla. I can't prove they don't, but I think the Christian God has no more likelihood than they had. I think they are a bare possibility" (Bertrand Russell Speaks His Mind, pp. 2425). He explained his views more fully in an interview published in Look magazine in 1953. An agnostic, in any sense in which he can be regarded as one, Russell said, "may hold that the existence of God, though not impossible, is very improbable; he may even hold it so improbable that it is not worth considering in practice" (Leo Rosten, ed., A Guide to the Religions of America, New York, 1955, p. 150).


On survival, Russell's position is similarly negative. All the evidence indicates that what we regard as our mental life is "bound up with brain structure and organized bodily energy." There is every reason to believe that mental life ceases when the body decays. Russell admits that this argument is "only one of probability" but adds that "it is as strong as those upon which most scientific conclusions are based" (Why I Am Not a Christian, p. 51). It is conceivable that evidence from psychical research might change the balance of probability some day, but, writing in 1925, Russell considered such evidence far weaker "than the physiological evidence on the other side." He did not later see any reason to modify this judgment.

Russell's views on the body-mind problem are known as "neutral monism," and it would be inaccurate to call him a materialist. However, he always emphasized that as a theory about man's place in the universe his philosophy is closely akin to materialism. "Emotionally," he wrote in 1928, "the world is pretty much the same as it would be if the materialists were in the right" (In Praise of Idleness, p. 143). The opponents of materialism, he adds, have been actuated by the desire to prove that the mind is immortal and that the "ultimate power" in the universe is mental and not physical. On both these points, Russell makes clear, he agrees with materialism. When he returned to the subject in 1959 he had not changed his opinion at all. "I still think," he wrote then, "that man is cosmically unimportant, and that a Being, if there were one, who could view the universe impartially, without the bias of here and now, would hardly mention man, except perhaps in a footnote at the end of the volume" (My Philosophical Development, p. 213).

Objections to fideism

Although, needless to say, Russell rejected the traditional arguments for the existence of God and immortality, he greatly preferred the rationalistic theology of such philosophers as Thomas Aquinas and Descartes to the fideism of Blaise Pascal, Jean-Jacques Rousseau, Søren Kierkegaard, and their numerous modern followers. "The rejection of reason in favor of the heart," he writes, "was not, to my mind, an advance." He remarks that "no one thought of this device so long as reason appeared to be on the side of religious belief" (A History of Western Philosophy, p. 720). There are two fatal objections to the practice of justifying religious belief by an appeal to the emotions of the heart. To begin with, the heart says different things to different men and to the same man at different times, but even if the heart said the same thing to all men this would still not be evidence for the existence of anything outside our emotions, and the fideists, no less than the rationalistic believers, mean to make claims about objective fact, not merely about their own emotions. At bottom, Russell concludes, the only reason offered for the acceptance of the new theology is "that it allows us to indulge in pleasant dreams. This is an unworthy reason, and if I had to choose between Thomas Aquinas and Rousseau, I should unhesitatingly choose the Saint" (My Philosophical Development, p. 721).

Some unbelievers have gone out of their way to praise the greatness of Jesus and to admit that religious belief, although perhaps not true, is at least of great value to individual believers and to society. Russell makes no such concessions. Although he grants that some of Christ's maxims were indeed admirable (especially those consistently disregarded by Christian dignitaries) he finds much in the teachings of Jesus to be defective, in particular his doctrine of eternal damnation. "Either in the matter of virtue or in the matter of wisdom," Russell concludes, Christ does not "stand as high as some other people known to history"for example, Buddha and Socrates (Why I Am Not a Christian, p. 19).

Harmfulness of religious belief

Russell's views about the nature of the emotions that inspire religious belief ("it is based, primarily and mainly, upon fear") and also about the harmful influence of religious organizations are very similar to those of David Hume, Baron d'Holbach, and other eighteenth-century freethinkers. He did, however, devote rather more attention to the bad effects of the habit of accepting propositions on faithin the absence of or even in opposition to the evidence. It is an error, Russell contends, to suppose that a person who does not form his beliefs on the basis of evidence in one domain can remain open-minded and scientific in another. Furthermore, somebody holding comfortable beliefs on faith dimly realizes that they are myths and "becomes furious when they are disputed." Such a person will therefore do his best to suppress all critics who might remind him of the feeble backing of his beliefs. Russell makes it clear that in this context he is not criticizing Christianity only. "The important thing," he writes, "is not what you believe, but how you believe it." The objections to "faith" do not depend on what the faith in question may be. "You may believe in the verbal inspiration of the Bible or of the Koran or of Marx's Capital. Whichever of these beliefs you entertain, you have to close your mind against evidence; and if you close your mind against evidence in one respect, you will also do so in another, if the temptation is strong." The person who bases his belief on reason will support it by argument rather than by persecution and will abandon his position if the argument goes against him. If, however, his belief is based on faith, he will conclude that argument is useless and will "therefore resort to force either in the form of persecution or by stunting and distorting the minds of the young whenever he has the power to control their education" (Human Society in Ethics and Politics, pp. 207208).

"The world is horrible"

Russell never denied that in some respects a "godless" philosophy like his has to be gloomy. The beginning of wisdom, he teaches, is acceptance of the fact that the universe does not care about our aspirations and that happiness and unhappiness are not meted out in accordance with what people deserve. "The secret of happiness," he observed during a television program commemorating his ninety-second birthday, "is to face the fact that the world is horrible." What Russell meant by this becomes clear from a story related by his biographer, Alan Wood. Wood's wife had expressed her opinion that it seemed horribly unjust that the young men who had been killed in the war should not somehow or somewhere have a second chance to achieve happiness. "But the universe is unjust," Russell replied, "the secret of happiness is to face the fact that the world is horrible, horrible, horrible you must feel it deeply and not brush it aside you must feel it right here"hitting his breast"and then you can start being happy again" (Bertrand Russell: The Passionate Sceptic, p. 237). Once a person has stopped looking at the universe in terms of anthropomorphic demands, he can concentrate on what is attainable and not waste his time in self-pity and cosmic complaints. For those whose philosophy is shaped not by a respect for facts but by their wishes Russell was always scathing in his contempt. He expressed his amazement that courage is praised in all types of situations but not when it comes to forming a view about the world. "Where traditional beliefs about the universe are concerned," he writes, "craven fears are considered praiseworthy, while intellectual courage, unlike courage in battle, is regarded as unfeeling and materialistic." Writing in 1957, he notes that this attitude is perhaps less widespread than it was in his youth, but he adds that it "still inspires vast systems of thought which have their root in unworthy fears." "I cannot believe," he concludes, that there can ever be any good excuse for refusing to face the evidence in favor of something unwelcome. It is not by delusion, however exalted, that mankind can prosper, but only by unswerving courage in the pursuit of truth" (Fact and Fiction, p. 46).

See also Absolute, The; Asceticism; Analysis, Philosophical; Balfour, Arthur James; Bradley, Francis Herbert; British Philosophy; Certainty; Church, Alonzo; Correspondence Theory of Truth; Descartes, René; Epistemology, History of; Ethical Subjectivism; Existence; Frege, Gottlob; Hegel, Georg Wilhelm Friedrich; Hegelianism; Holbach, Paul-Henri Thiry, Baron d'; Hume, David; Infinity in Mathematics and Logic; James, William; Kierkegaard, Søren Aabye; Logical Paradoxes; Logic, History of; Logic, Modern; Logic, Traditional; Luther, Martin; Mathematics, Foundations of; McTaggart, John McTaggart Ellis; Memory; Metaethics; Mill, John Stuart; Mind-Body Problem; Modal Logic; Moore, George Edward; Neumann, John von; Nietzsche, Friedrich; Number; Pascal, Blaise; Peano, Giuseppe; Peirce, Charles Sanders; Plato; Platonism and the Platonic Tradition; Pluralism; Proper Names and Descriptions; Propositions; Quantifiers; Quine, Willard Van Orman; Ramsey, Frank Plumpton; Realism; Rousseau, Jean-Jacques; Santayana, George; Schopenhauer, Arthur; Sellars, Wilfrid; Socrates; Thomas Aquinas, St.; Types, Theory of; Voltaire, François-Marie Arouet de; Whitehead, Alfred North; Wittgenstein, Ludwig Josef Johann.



There is a good deal of autobiographical material in Russell's Portraits from Memory and Other Essays (New York: Simon and Schuster, 1956); in Fact and Fiction (New York: Simon and Schuster, 1962); in his introduction to Selected Papers of Bertrand Russell (New York: Modern Library, 1927); in "My Religious Reminiscences," in Rationalist Annual 55 (1938): 38; in "My Mental Development," in The Philosophy of Bertrand Russell, edited by P. A. Schilpp (Evanston and Chicago: Open Court, 1944); and in My Philosophical Development (New York: Simon and Schuster, 1959). Alan Wood, Bertrand Russell: The Passionate Sceptic (London : Allen and Unwin, 1957), is the only full-length biographical study of Russell. H. W. Leggett, Bertrand Russell (New York, 1950), is a short pictorial biography.

G. H. Hardy, Bertrand Russell and Trinity (Cambridge, U.K.: Cambridge University Press, 1942), traces the controversy between Russell and the fellows of Trinity College over his pacifist activities during World War I. Rex versus Bertrand Russell, Report of the Proceedings before the Lord Mayor (London, 1916), gives the text of the first of Russell's trials.

D. H. Lawrence, Letters to Bertrand Russell (New York: Gotham Book Mart, 1948), reproduces Lawrence's letters to Russell during World War I; Russell's letters to Lawrence have not been preserved.

Russell's part in the Beacon Hill School is most fully described in Joe Park, Bertrand Russell on Education (Columbus: Ohio State University Press, 1963). The Park volume also contains a complete list of Russell's writings on educational topics. Details about the City College case of 1940 can be found in The Bertrand Russell Case, edited by John Dewey and Horace M. Kallen (New York: Viking Press, 1941); in a publication by the American Civil Liberties Union titled The Story of the Bertrand Russell CaseThe Enlightening Record of the Obstruction by Courts and Officials of the Appointment of Bertrand Russell to a Professorship at the College of the City of New York (New York: American Civil Liberties Union, 1941); and in Paul Edwards, "How Bertrand Russell Was Prevented from Teaching at City College," which is an appendix to Russell's Why I Am Not a Christian and Other Essays on Religion and Related Subjects (London: Allen and Unwin, and New York: Simon and Schuster, 1957).

epistemology and metaphysics

Principles of Mathematics (Cambridge, U.K.: Cambridge University Press, 1903) was Russell's first major philosophical work. Its position is one of Platonic realism. In the preface to the second edition (1937) Russell sets forth his later disenchantment with this position. For a nonmathematical exposition of Russell's early realism, see "Meinong's Theory of Complexes and Assumptions," in Mind 13 (1904): 204219; 336354; 509524. Russell's criticisms of the idealist theory of truth are to be found in "The Monistic Theory of Truth," in Philosophical Essays (New York: Longman, 1910), a revised version of "The Nature of Truth," in Mind 15 (1906): 528533. Philosophical Essays also contains two influential essays by Russell attacking the pragmatist theory of truth.

The shift from realism to logical constructionism can be followed in a number of articles, the most important of which is "On Denoting," in Mind 14 (1905): 479493. This, together with other important but otherwise largely unavailable essays, is reprinted in Russell's Logic and Knowledge, edited by R. C. Marsh (London: Allen and Unwin, 1956). Russell's "On the Relations of Universals and Particulars," in PAS 12 (19111912): 124, reprinted in Logic and Knowledge, is a classic presentation of the largely Platonic theory of universals Russell still held at that time. Problems of Philosophy (New York: Holt, 1912) gives an excellent semipopular account of the general state of Russell's thinking then. Russell's early attempts to represent physical objects as logical constructions can be seen in Our Knowledge of the External World (Chicago: Open Court, 1914) and in two essays, "The Ultimate Constituents of Matter," in Monist 25 (1915): 399417, and "The Relations of Sense-Data to Physics," in Scientia (4) (1914), both reprinted in Mysticism and Logic (London: Allen and Unwin, 1918). Other important essays in this collection are "On Scientific Method in Philosophy" (1914); "On the Notion of Cause," originally published in PAS 13 (19121913): 126; and "Knowledge by Acquaintance and Knowledge by Description," originally published in PAS 11 (19101911): 108128. See also "The Philosophy of Logical Atomism," in Monist 28 (1918): 495527; 29 (1919): 3263, 190222, and 345380; reprinted in Logic and Knowledge (see above). The analysis of basic concepts and principles of physical science is pushed further in The Analysis of Matter (New York: Harcourt Brace, 1927). Logical constructionism is applied to mental phenomena in The Analysis of Mind (New York: Macmillan, 1921). Russell's increasing concern with psychological aspects of meaning can be traced in "On Propositions, What They Are and How They Mean," in PAS, supp. 2 (1919): 143, reprinted in Logic and Knowledge, in Ch. 10 of The Analysis of Mind ; and in Russell's most extensive work on meaning and empirical data, the rich but chaotic An Inquiry into Meaning and Truth (New York: Norton, 1940). Russell's later thoughts on meaning and various other problems concerning empirical knowledge, particularly in the physical sciences, are given a relatively systematic presentation in Human Knowledge, Its Scope and Limits (New York: Simon and Schuster, 1948).

In several works Russell summarized his philosophy and/or its development. The most important of these are "Logical Atomism," in Contemporary British Philosophy, edited by J. H. Muirhead, first series (London: Allen and Unwin, 1924), reprinted in Logic and Knowledge (see above); "My Mental Development," in The Philosophy of Bertrand Russell, edited by P. A. Schilpp (see above); and the very interesting recent work My Philosophical Development (New York: Simon and Schuster, 1959). The last-named work also contains some of Russell's polemics against Oxford philosophers and their criticisms of his views. Russell's A History of Western Philosophy (New York: Simon and Schuster, 1946) and The Wisdom of the West (New York: Doubleday, 1959), aside from their intrinsic interest, are of great value to students of Russell's thought in showing us his mature evaluations of the great philosophers of past ages.

The critical literature on different aspects of Russell's epistemology and metaphysics is vast. The Philosophy of Bertrand Russell (see above) contains a number of excellent discussions, together with Russell's replies. Special mention should also be made of C. A. Fritz, Bertrand Russell's Construction of the External World (London: Routledge & K. Paul, 1952); Erik Götlind, Bertrand Russell's Theories of Causation (Uppsala: Almqvist and Wiksells, 1952); J. O. Urmson, Philosophical Analysis: Its Development between Two World Wars (Oxford: Clarendon Press, 1956); and G. J. Warnock, English Philosophy since 1900 (London: Oxford University Press, 1958). The books by Urmson and Warnock contain detailed appraisals of Russell's logical atomism. Russell's logical atomism as well as his neutral monism and his theories about truth and induction are sympathetically discussed by D. J. O'Connor in Ch. 26 of his Critical History of Western Philosophy (New York: Free Press of Glencoe, 1964). Rivista critical di storia della filosofia 8 (2) (1953): 101335, and several articles in Philosophy 35 (January 1960): 150, are devoted to Russell's philosophy, including Anthony Quinton's useful sketch of the development of Russell's ideas in epistemology and metaphysics, "Russell's Philosophical Development," 113.

logic and mathematics

Of Russell's own works on logic and mathematics, see Principles of Mathematics (Cambridge, U.K.: Cambridge University Press, 1903; 2nd ed., London, 1937); Principia Mathematica, 3 vols., written with A. N. Whitehead (Cambridge, U.K.: Cambridge University Press, 19101913; 2nd ed., 1927); Introduction to Mathematical Philosophy (London: Allen and Unwin, 1919); and the papers "On Denoting" (1905), "Mathematical Logic as Based on the Theory of Types" (1908), "The Philosophy of Logical Atomism" (1918), and "Logical Atomism" (1924), all of which are reprinted in Logic and Knowledge (see above).

On Frege's parallel work, see his Grundlagen der Arithmetik (Breslau, 1884), translated by J. L. Austin as The Foundations of Arithmetic (Oxford: Blackwell, 1950); and P. T. Geach and Max Black, eds., Translations from the Philosophical Writings of Gottlob Frege (New York: Philosophical Library, 1952).

Important critical discussions of Russell's work occur in W. E. Johnson, Logic, Pt. II (Cambridge, U.K., 1922), Chs. 3 and 6; F. P. Ramsey, The Foundations of Mathematics (London, 1931), papers I and II; W. V. Quine, From a Logical Point of View (Cambridge, MA: Harvard University Press, 1953), essays I, V, and VI; and G. E. Moore, The Commonplace Book of G. E. Moore, 19191953, edited by Casimir Lewy (New York: Humanities Press, 1963), Notebook II, item 4, and Notebook V, item 13.

On formal implication, see A. N. Prior, "The Theory of Implication," in Zeitschrift für mathematische Logik und Grundlagen der Mathematik 9 (1963): 16. On simplifications of type theory, see Alonzo Church, "A Formulation of the Simple Theory of Types," in Journal of Symbolic Logic 5 (1940): 5668, and Ludwik Borkowski, "Reduction of Arithmetic to Logic Based on the Theory of Types," in Studio Logica 8 (1958): 283295.

ethics and religion

Russell's early views on ethics are in "The Elements of Ethics," Ch. 1 of Philosophical Essays (New York: Longman, 1910); it has been reprinted in Readings in Ethical Theory, edited by Wilfrid Sellars and John Hospers (New York: Appleton-Century-Crofts, 1952), pp. 134. The fullest statements of his later position are in Ch. 9 of Religion and Science (New York: Holt, 1935) and in Human Society in Ethics and Politics (New York: Simon and Schuster, 1955). There are critical discussions of Russell's views in Lillian W. Aiken, Bertrand Russell's Philosophy of Morals (New York: Humanities Press, 1963); in Justus Buchler, "Russell and the Principles of Ethics," in The Philosophy of Bertrand Russell (see above); and in D. H. Monro, "Russell's Moral Theories," in Philosophy 35 (1960): 3050.

Russell's earlier views on religion are in "The Essence of Religion," in Hibbert Journal 11 (1912): 4662. His first published discussion of the arguments for the existence of God is contained in Ch. 15 of A Critical Exposition of the Philosophy of Leibniz (Cambridge, U.K.: Cambridge University Press, 1900; 2nd ed., London and New York, 1937). His later views are expounded in several of the essays in Why I Am Not a Christian (see above) and in Pt. II, Ch. 7, of Human Society in Ethics and Politics (see above). The BBC debate with Father F. C. Copleston (1948), "The Existence of God," is available in the British edition, but not in the American edition, of Why I Am Not a Christian, but it has been reprinted in A Modern Introduction to Philosophy, edited by Paul Edwards and Arthur Pap, 2nd ed. (New York, 1965), and in The Existence of God, edited by John Hick (New York: Macmillan, 1964). Several chapters in The Scientific Outlook (London and New York, 1931) and in Religion and Science (see above) contain criticisms of the attempts of certain scientists to derive theological conclusions from physics and biology. Russell's objections to the fideistic position are found in Ch. 12, Bk. 3, of A History of Western Philosophy (see above). His objections to William James's defense of religion are contained in Ch. 29, Bk. 3, of the same work and in Ch. 5 of Philosophical Essays (see above). Russell's views on religion are criticized in H. G. Wood, Why Mr. Bertrand Russell Is Not a Christian (London, 1928); C. H. D. Clark, Christianity and Bertrand Russell (London: Lutterworth Press, 1958); G. S. Montgomery, Why Bertrand Russell Is Not a Christian (New York, 1959); and E. S. Brightman's contribution to the Schilpp volume, "Russell's Philosophy of Religion," pp. 537556.

social and political theory

In addition to the works mentioned in the first section of the present entry, the following among Russell's books dealing with social and political questions have been influential: Principles of Social Reconstruction (London: Allen and Unwin, 1916); Roads to Freedom: Socialism, Anarchism and Syndicalism (London: Allen and Unwin, 1918); The Problems of China (New York: Century, 1922); Power: A New Social Analysis (London: Allen and Unwin, 1938); Authority and the Individual (London: Allen and Unwin, 1949); and New Hopes for a Changing World (London: Allen and Unwin, 1951). Ch. 17 of New Hopes contains a moving discussion of the problems of growing old and facing death. Russell's fullest discussion of Marxism can be found in Freedom and Organization 18141914 (London: Allen and Unwin, 1934; as Freedom versus Organization, New York, 1934), which is in effect a history of the main social and intellectual forces of the nineteenth century.

other writings

Philosophical discussions sooner or later crop up in most of Russell's writings. Some of his most delightful occasional pieces have been collected in Sceptical Essays (London: Allen and Unwin, 1927); in In Praise of Idleness (London: Allen and Unwin, 1935); and in Unpopular Essays (London: Allen and Unwin, 1950). The last of these contains his "Auto-obituary," which was first published in 1936. Bertrand Russell Speaks His Mind (London: Barker, 1960) is a most interesting volume containing the unedited text of a series of television interviews, dealing with a great variety of topics, which took place in the spring of 1959.

The Basic Writings of Bertrand Russell, 19031959, edited by R. E. Egner and L. E. Dennon (New York, 1961), is a very useful anthology of writings by Russell. The Schilpp volume contains an extremely comprehensive bibliography up to 1944.

recent works

Coffa, Alberto. "The Humble Origins of Russell's Paradox." Russell 33 (1979): 3137.

Eames, Elizabeth R. Bertrand Russell's Theory of Knowledge. London: George Allen and Unwin, 1969.

Griffin, Nicholas. Russell's Idealist Apprenticeship. Oxford: Oxford University Press, 1991.

Hylton, Peter W. Russell, Idealism, and the Emergence of Analytic Philosophy. Oxford: Oxford University Press, 1990.

Irvine, A. D. "Epistemic Logicism and Russell's Regressive Method." Philosophical Studies 55 (1989): 303327.

Irvine, A. D., ed. Bertrand Russell: Critical Assessments. 4 vols. London: Routledge, 1999.

Jager, Ronald The Development of Bertrand Russell's Philosophy. London: George Allen and Unwin, 1972.

Kaplan, David. "What Is Russell's Theory of Descriptions?" (1970). In Pears (1972), 227244.

Landini, Gregory. Russell's Hidden Substitutional Theory. New York and Oxford: Oxford University Press, 1998.

Linsky, Bernard. Russell's Metaphysical Logic. Stanford: CSLI, 1999.

Lycan, William. "Logical Atomism and Ontological Atoms." Synthese 46 (1981): 207229.

Pears, David F. Bertrand Russell and the British Tradition in Philosophy. London: Collins, 1967.

Pears, David F., ed. Bertrand Russell: A Collection of Critical Essays. Garden City, NY: Anchor Books, 1972.

Putnam, Hilary. "The Thesis that Mathematics Is Logic." In Schoenman (1967), 273303.

Ryan, Alan. Bertrand Russell: A Political Life. New York: Hill and Wang, 1988.

Sainsbury, R. M. Russell. London: Routledge, 1979.

Savage, C. Wade, and C. Anthony Anderson, eds. Rereading Russell: Essays on Bertrand Russell's Metaphysics and Epistemology. Minneapolis: University of Minnesota Press, 1989.

Schoenman, Ralph, ed. Bertrand Russell: Philosopher of the Century. London: George Allen and Unwin, 1967.

Schultz, Bart. "Bertrand Russell in Ethics and Politics." Ethics 102 (1992): 594634.

Paul Edwards (1967)

(Life and Social Theories, Ethics and Critique of Religion)

William P. Alston (1967)

(Epistemology and Metaphysics)

A. N. Prior (1967)

(Logic and Mathematics)
Bibliography updated by Ernest Sosa (2005)

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Russell, Bertrand Arthur William (1872–1970)

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