### Bertrand Russell's Philosophy

BERTRAND RUSSELL

As a philosopher, mathematician, educator, social critic and political activist, Bertrand Russell authored over 70 books and thousands of essays and letters addressing a myriad of topics. Awarded the Nobel Prize in Literature in 1950, Russell was a fine literary stylist, one of the foremost logicians ever, and a gadfly for improving the lives of men and women. Born in 1872 into the British aristocracy and educated at Cambridge University, Russell gave away much of his inherited wealth. But in 1931 he inherited and kept an earldom. His multifaceted career centered on work as a philosophy professor, writer, and public lecturer. (Here is a detailed chronology of Russell's life, an overview of his analytic philosophy, and a complete bibliography of all his publications.) Russell was an author of diverse scope. His first books were German Social Democracy, An Essay on the Foundations of Geometry, and A Critical Exposition of the Philosophy of Leibniz. His last books were War Crimes in Vietnam and The Autobiography of Bertrand Russell. Other noteworthy books include Principles of Mathematics, Principia Mathematica (with A.N. Whitehead), Anti-Suffragist Anxieties, The Problems of Philosophy, Introduction to Mathematical Philosophy, Sceptical Essays, Why I Am Not a Christian, and A History of Western Philosophy. He was arguably the greatest philosopher of the 20th century and the greatest logician since Aristotle. Analytic philosophy, the dominant philosophy of the twentieth century, owes its existence more to Russell than to any other philosopher. And the system of logic developed by Russell and A.N. Whitehead, based on earlier work by Dedekind, Cantor, Frege, and Peano, broke logic out of its Aristotelian straitjacket. He was also one of the century's leading public intellectuals and won the Nobel Prize for Literature in 1950 "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought." Russell was involved, often passionately, in numerous social and political controversies of his time. For example, he supported suffragists, free thought in religion and morals, and world government; he opposed World War I and the Vietnam War, nationalism, and political persecution. He was jailed in 1918 for anti-war views and in 1961 for his anti-nuclear weapons stance. He was married 4 times and had 3 children. With Dora Russell, he founded the experimental Beacon Hill School. He knew or worked with many of the most prominent figures in late 19th and 20th century philosophy, mathematics, science, literature, and politics. Active as a political and social critic until his end, Russell died in 1970 at the age of 97.

Bertrand Russell’s Analytic Philosophy - An Overview*

Bertrand Russell (1872-1970) was the greatest philosopher of the twentieth century and the greatest logician since Aristotle. He wrote original philosophy on dozens of subjects, but his most important work was in logic, mathematical philosophy, and analytic philosophy. Russell is responsible more than anyone else for the creation and development of the modern logic of relations – the single greatest advance in logic since Aristotle. He then used the new logic as the basis of his mathematical philosophy called logicism.
Logicism is the view that all mathematical concepts can be defined in terms of logical concepts and that all mathematical truths can be derived from logical truths to show that mathematics is nothing but logic. In his work on logicism, Russell developed forms of analysis in order to analyze quantifiers in logic (words like “all” and “some”) and numbers in mathematics, but he was soon using them to analyze points in space, moments of time, matter, mind, morality, knowledge, and language itself in what was the beginning of analytic philosophy.
This introduction presents an overview of Russell’s technical work in logic, logicism, and analysis, and then of his broader inquiries
of analytic philosophy in metaphysics, knowledge, and language. A more detailed treatment of them can be found
in the texts listed at the bottom of the page.

1 Logic and logicism: Basic concepts

We start with some basic logical concepts. A sentence is a group of words whose meaning is a complete thought. A declarative sentence has a meaning that is either true or false. A proposition is said to be the meaning expressed by a declarative sentence, such as the true proposition “The earth is round” or the false proposition “The earth is flat.” So propositions are either true or false. The declarative sentences that express them are also said to be true or false.
The subject of a proposition is who or what the proposition is about. “The earth is flat” is about the earth. So the earth is the subject of that proposition. The predicate is what is said about, or attributed to, the subject. Here, the proposition attributes flatness to the earth, so “___ is flat” is the predicate. Logicians write predicates using variables like x, y, or z, instead of blank spaces, to indicate where the subject goes in relation to the predicate. Bertrand Russell called predicates propositional functions. In this book, we use the terms interchangeably.
The predicate “x is flat” is a one-place predicate, because it only has one place where a subject can go – it attributes a property to one thing. Two-place predicates are relations like that in “Indiana is flatter than Ohio.” Here, the subjects are “Indiana” and “Ohio” and the predicate is “x is flatter than y.” (In grammar, the first is the subject and the second is the object; in logic, they are both subjects.) Common two-place relations in mathematics are x = y, x > y, and x < y. There are also three-place relations like that in “Ohio is between Indiana and Pennsylvania,” where the predicate is “x is between y and z,” which is often used in geometry. There are also four-place relations, and so on.
Before Russell’s logic of relations, logic consisted principally of the Aristotelian logic of one-place predicates. This simple logic can analyze sentences that use one-place predicates to attribute properties to objects like “Tom is tall” or “The sky is blue.” It can also analyze slightly more complex sentences like “All humans are animals” (if someone is human, that person is an animal) and “Some humans are thoughtful” (at least one person is both human and thoughtful) and from these two sentences infer that “Some animals are thoughtful.” You can’t get too far with such a simple logic and you certainly can’t analyze many mathematical or scientific statements with it.
It was Russell’s first great achievement to develop the more powerful logic of relations to describe concepts expressed by two-place predicates, such as “x is taller than y” used in propositions like “Tom is taller than Bob,” which you can’t say with a one-place predicate like “x is tall.” This allowed Russell to describe propositions containing two-place mathematical relations like x = y or x > y (needed for arithmetic and algebra), three-place relations like “x is between a and b” (needed for geometry), and the like. With it, all of the concepts of pure mathematics can be expressed, which can’t be done with the logic that came before it.
Russell’s logic includes set theory. This is because his logic contains predicates and every predicate defines a set. For example,
the predicate “x is human” defines the set of all things that can replace the x to make “x is human” true, i.e., it defines the class
of humans. The comprehension axiom is the assumption that every predicate defines a class. It is an assumption of Russell’s logic.
So Russell’s logic contains sets and a theory of sets, as well as one-place predicates and two-place relations. Russell refers to
sets as “classes” and set theory as “the theory of classes.” We will use both ways of speaking indifferently and without distinction.

2 The emergence of logicism

After the logic of relations, Russell’s greatest achievement is his theory of logicism – the view that mathematics is just logic, so that all mathematical concepts can be defined in terms of logical concepts and all mathematical truths can be derived from logical truths. Russell’s logic and his logicist philosophy were first fully described in his 1903 Principles of Mathematics. The actual derivation of mathematics from logic, to prove that mathematics is just logic, occurs in the three-volume 1910-13 Principia Mathematica that Russell wrote with Alfred North Whitehead. Russell also presents logicism simply and informally in his 1919 Introduction to Mathematical Philosophy. Finally, there is a 1925-27 revised second edition of Principia Mathematica.
Logicism comes down to this: In the nineteenth century, mathematicians had shown that all of classical mathematics can be defined in terms of, and derived from, arithmetic. Most importantly, Richard Dedekind had shown in 1872 that the real numbers can be defined in terms of rational numbers. Then rational numbers were defined in terms of natural numbers, thus demonstrating that the real numbers can be derived from natural numbers. This is called the arithmetization of mathematics. The next step was taken by Giuseppe Peano, based on work by Dedekind, who showed in 1890 that arithmetic can be reduced to five axioms and three undefined concepts. This is the axiomatization of arithmetic.
After this, all one has to do to reduce mathematics to logic – since mathematics has already been reduced to arithmetic and arithmetic has been reduced to 3 concepts and 5 axioms – is to define Peano’s 3 concepts in terms of logical concepts, thus expressing Peano’s axioms logically, and then derive Peano’s 5 axioms from logical truths, showing that Peano’s axioms, and thus all the mathematics based on them, are logical truths. Peano’s three undefined concepts are: 0, natural number, and successor. Russell starts by defining natural numbers logically as classes of classes. Specifically, a natural number is the class of all classes containing the same number of things, so that the number 1 is the class of all singletons (classes with one member), 2 is the class of all couples, and so on. With this definition, Russell then defines Peano’s other two basic concepts logically and derives Peano’s axioms from logic.
Put this way, demonstrating logicism is a seemingly simple task. But Russell and Whitehead soon ran into difficulties, namely, contradictions Russell found in the new logic and set theory. The most famous of these is called Russell’s paradox. Some sets are members of themselves, others are not. The set of things that are not red is itself not red, so it is a member of itself, but the set of red things is not red, so it is not a member of itself. Since “x is a set that is not a member of itself” seems to be a meaningful predicate, there must be a set of the sets that are not members of themselves that corresponds to the predicate (by the comprehension axiom). But is that set a member of itself? If it is a member of itself, then it isn’t. But if it isn’t a member of itself, then it is. A contradiction ensues no matter how one answers.
To avoid this and similar paradoxes, Russell’s logic, and the logicism based on it, became quite complex, and the ultimate success of this logicism is still a matter of debate. Many believe that you cannot completely reduce mathematics to logic. Others say the final verdict is not yet in. Still others say it can be done. In any case, it is significant and astonishing how much of mathematics Russell and Whitehead demonstrated can be reduced to logic. And if one is willing to tolerate a few pesky contradictions here and there, it absolutely can be done.
Russell’s original form of logicism, in his 1903 Principles of Mathematics, did not attempt to avoid the paradoxes of the new
logic, and so did not contain the complexities Russell later added to his logic to avoid them. It is a straightforward theory
that contains all the basic elements of logicism without the complexities. We present this basic logicism, which we call naïve
logicism, in Chapter 2. The complex version meant to avoid paradoxes, which occurs in the 1910-13 Principia Mathematica,
we call restricted logicism. We describe that in Chapter 3.

3 Logicism and analysis

As well as founding the logic of relations, developing the theory of logicism, and discovering fundamental contradictions in logic and set theory, Russell more than anyone else founded the twentieth-century movement of analytic philosophy that still dominates philosophy today. Analytic philosophy as practiced by Russell logically analyzes concepts, knowledge, and language to say what there is and how we know it. Analysis is a significant part of analytic philosophy and its role in the movement is largely due to Russell. His logical analysis of mathematics is one of the movement’s primary examples of analysis.
Notions of analysis vary from one analytic philosopher to another and from one analysis to another by a single philosopher. This last case is true of Russell himself. Most generally, “analysis” for him means beginning with something that is common knowledge and seeking the fundamental concepts and principles it is based on. This is followed by a synthesis that begins with the basic concepts and principles discovered by analysis and uses them to derive the common knowledge one first analyzed to begin with.
In Russell’s own words (in Introduction to Mathematical Philosophy): “By analyzing we ask … what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced” (p. 1). Similarly, in Principia Mathematica, he says “There are two opposite tasks which have to be concurrently performed. On the one hand, we have to analyze existing mathematics, with a view to discovering what premises are employed…. On the other hand, when we have decided upon our premisses, we have to build up again [i.e., synthesize] as much as may seem necessary of the data previously analyzed” (vol. 1, p. v).
Immanuel Kant uses the same concepts of analysis and synthesis to describe his Prolegomena to Any Future Metaphysics and Critique of Pure Reason. “I offer here,” he says in the Prolegomena, “a plan which is sketched out after an analytical method, while the Critique itself had to be executed in the synthetical style” (p. 8). In the Prolegomena we start with science (mathematics and physics) and by analysis, he says, “proceed to the ground of its possibility,” that is, to its fundamental concepts, while in the Critique, “they [the sciences] must be derived … from [the fundamental] concepts” (p. 24).
Russell’s Introduction to Mathematical Philosophy, which is an informal introduction to Principia’s logicism, is similarly analytic. About it, Russell says: “Starting from the natural numbers, we have first defined cardinal number and shown how to generalize the conception of number, and have then analyzed the conceptions involved in the definition, until we found ourselves dealing with the fundamentals of logic.” About synthesis, he says “In a synthetic, deductive treatment these fundamentals [reached by analysis] come first, and the natural numbers [with which the analysis started] are reached only after a long journey” (p. 195).
And Principia Mathematica is a synthesis: it begins with the logical fundamentals found by analysis elsewhere, and from them deductively builds up the mathematics the analysis started with. As Russell says in Principia, it is “a deductive system” in which “the preliminary labor of analysis does not appear.” Instead, it “merely sets forth the outcome of the analysis … making deductions from our premisses … up to the point where we have proved as much as is true in whatever would ordinarily be taken for granted” (vol. 1, p. v).
Russell’s Introduction to Mathematical Philosophy is thus to Principia Mathematica what Kant’s Prolegomena is to the Critique of Pure Reason – an analysis that takes common knowledge and finds its basic principles, which synthesis then uses as a basis to demonstrate the knowledge previously analyzed. The Introduction to Mathematical Philosophy and Prolegomena are also both informal introductions to the subjects which are presented more rigorously in the synthetic works. But Kant seeks to justify knowledge with the principles uncovered by analysis. Russell does not. For Russell, the logical ideas analysis uncovers are less certain than the arithmetic that is being analyzed.
For Russell, what we analyze – arithmetic – is certain and because it is certain, the fundamental logical principles found by analysis can be said to be inductively justified once synthesis has demonstrated that arithmetic can be deduced from them. (If synthesis shows that logic implies arithmetic, and arithmetic is true, then logic is probably true. The argument is inductive, just as in the empirical sciences when some theory T implies observation O and O turns out to be true: that does not prove that T is true, but it does increase the likelihood for us that it is.) Russell does not think arithmetic is made certain by being deduced from logic, but that logic is made more certain by arithmetic being deduced from it.
For example, Russell says in Principia: “The chief reason in favor of any theory on the principles of mathematics [i.e., in favor of any premisses that imply mathematics] must always be inductive, i.e. it must lie in the fact that the theory in question enables us to deduce ordinary mathematics” (vol. 1, p. v). What is found by analysis is less certain than what is analyzed. Russell does not seek certainty for mathematics from its analysis, but an understanding of the reasons, however uncertain, for accepting what we normally take for granted in mathematics.
Russell continues: “In mathematics,” he says, “the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point…. Hence, the early deductions [of Principia], until they reach this point, give reasons rather for believing the premisses because true consequences follow from them, than for believing the consequences because they follow from the premisses” (p. v-vi). Principia does indeed show that arithmetic follows from logic, and this makes it likely to us that those logical principles are a true account of arithmetic’s nature.

4 Logical analysis: The theory of descriptions

These concepts of analysis and synthesis may seem vague, but they will get you a long way in understanding Russell’s Introduction to Mathematical Philosophy and Principia Mathematica. At some point, however, to understand Russell’s work one must learn his more technical, logical kinds of analysis that are his theory of descriptions and incomplete symbols, his “no-class” theory of classes, his theory of logical types, and his logical constructions.
In the theory of descriptions, Russell analyzes descriptions of objects and classes by translating them into his new logic, where we can see that they do not always mean what they seem to mean in ordinary language. That is, Russell analyzes expressions of ordinary language into more careful logical expressions that are their true meaning. His Introduction to Mathematical Philosophy as a whole is the simpler sort of analysis, but within it are several more technical logical analyses using the theory of descriptions.
Russell first published the theory of descriptions in his 1905 article “On Denoting.” The theory figures prominently in Principia Mathematica, where it is given a fairly clear presentation in the “Introduction.” Russell clearest exposition of it is in the 1918 “Philosophy of Logical Atomism,” and another is in his 1919 Introduction to Mathematical Philosophy (Chapter 16), which is the version that most philosophy students read in college.
For Russell, the theory of descriptions shows that the grammar of ordinary language is often misleading. Using it, sentences containing singular definite descriptions – descriptions of the form “the so-and-so” such as “the author of Waverley” in the sentence “Scott was the author of Waverley” – are analyzed so that the description does not occur in the logical analysis of the sentence, but is replaced by a predicate.
For example, “the author of Waverley” in “Scott was the author of Waverley” is replaced with the predicate “x wrote Waverley” and the sentence becomes “There is exactly one thing x such that x is Scott, and x wrote Waverley,” or more briefly, “Scott wrote Waverley.” The description “the author of Waverley” no longer occurs in the logical analysis of the sentence. In particular, the word “the” is gone. That is the whole function of the theory of descriptions.
Why analyze a sentence so that the definite description it contains, and especially the word “the,” disappears? Notice that “the author of Waverley” seems to function like a name and to denote a particular object. However, the expression that replaces it, “x wrote Waverley,” is a predicate, not a name, and by itself it does not denote any such object. Let us pause here to consider this idea that names denote, but predicates do not. It is an important idea to Russell.
The idea that names refer to, or denote, objects should not be controversial. “Napoleon” refers to the commanding French general at the battle of Waterloo, “Einstein” to the man who created the special and general theories of relativity, and so forth. And as Russell points out, names have these references independently of occurring in propositions. Finally, definite descriptions like “the author of Waverley” seem to function like names and refer to particular individuals too, just as “Sir Walter Scott” does.
Predicates, on the other hand, do not name, or refer to, objects. For example, the predicate “x is red” does not name or denote any particular individual by itself independently of occurring in a proposition. It does not specify which object or objects it might be used to apply to. So a predicate is definitely not a name. Because definite descriptions are not names but are predicates, Russell calls them incomplete symbols. They appear to name objects, but they really don’t.
By showing that definite descriptions, which appear to be names of objects, really aren’t, we can see how sentences containing descriptions can be meaningful without the sentence asserting the existence of what is described. For example, we can see how sentences like “The present king of France rolled the round square down the golden mountain” can be meaningful without asserting that any of these things (“the present king of France,” “the round square,” and “the golden mountain”) exist.
This solves a general problem of logic for Russell – how to logically analyze sentences containing definite descriptions true of no objects. More significantly, Russell uses a variation of this theory, called his “no-class” theory of classes, to remove all references to classes in his logic by treating names of classes and descriptions of classes as predicates. Then, since logic, so interpreted, does not assume that sets exist, the Russell paradox of the set of all sets that are not members of themselves cannot occur – as we will see next.

5 Logical analysis: The “no-class” theory of classes

In addition to analyzing singular definite descriptions so that what appear to be names are seen to actually be predicates that do not name anything, Russell sometimes treats proper names the same way, for example, in Principia Mathematica (in *14.21). He argues there that words like “Homer” that appear to be proper names are actually concealed definite descriptions such as “the author of the Homeric poems.” They are then treated like definite descriptions and replaced with predicates too. By 1918, in “The Philosophy of Logical Atomism,” Russell is using this idea aggressively, insisting that all proper names like “Socrates,” “Napoleon,” and “Einstein” are disguised definite descriptions, but in Principia, he only suggests it once.
After singular definite descriptions such as “the author of Waverley” come plural definite descriptions such as “the inhabitants of London.” These too are analyzed so that they are replaced by predicates. The phrase “the inhabitants of London” in the sentence “The inhabitants of London are cosmopolitan,” seems to name a class of objects, namely, the inhabitants of London. But it is replaced by “x lives in London,” which, being a predicate, names no object or objects. The sentence then reads “If anyone lives in London, that person is cosmopolitan.”
In the slightly different sentence “The class of people who inhabit London is large,” the subject is a description that appears to name a single object, the class of people living in London. It is a different example because it includes the word “class.” But as before, we replace the description with the predicate “x lives in London” to get “Many people live in London.” Similarly, when a symbol stands for a set as the name of the set, we treat it like a disguised definite description, just as “Socrates” is treated as the disguised description “the teacher of Plato.” For example, when α = the class of even numbers, we translate “6 ∈ α” (“the number 6 is a member of the class α of even numbers”) by replacing α with the predicate “x is divisible by two” to get “6 is divisible by 2.” Again, we simply replace the class with the predicate that defines it.