Bertrand (Arthur William) (1872–1970), British philosopher, logician, social reformer, and man of letters, one of the founders of analytic philosophy. Born into an aristocratic political family, Russell always divided his interests between politics and philosophy. Orphaned at four, he was brought up by his grandmother, who educated him at home with the help of tutors. He studied mathematics at Cambridge from 1890 to 1893, when he turned to philosophy.
At home he had absorbed J. S. Mill’s liberalism, but not his empiricism. At Cambridge he came under the influence of neo-Hegelianism, especially the idealism of McTaggart, Ward (his tutor), and Bradley. His earliest logical views were influenced most by Bradley, especially Bradley’s rejection of psychologism. But, like Ward and McTaggart, he rejected Bradley’s metaphysical monism in favor of pluralism (or monadism). Even as an idealist, he held that scientific knowledge was the best available and that philosophy should be built around it. Through many subsequent changes, this belief about science, his pluralism, and his anti-psychologism remained constant.
In 1895, he conceived the idea of an idealist encyclopedia of the sciences to be developed by the use of transcendental arguments to establish the conditions under which the special sciences are possible. Russell’s first philosophical book, An Essay on the Foundations of Geometry (1897), was part of this project, as were other (mostly unfinished and unpublished) pieces on physics and arithmetic written at this time (see his Collected Papers, vols. 1–2). Russell claimed, in contrast to Kant, to use transcendental arguments in a purely logical way compatible with his anti-psychologism. In this case, however, it should be both possible and preferable to replace them by purely deductive arguments. Another problem arose in connection with asymmetrical relations, which led to contradictions if treated as internal relations, but which were essential for any treatment of mathematics. Russell resolved both problems in 1898 by abandoning idealism (including internal relations and his Kantian methodology). He called this the one real revolution in his philosophy. With his Cambridge contemporary Moore, he adopted an extreme Platonic realism, fully stated in The Principles of Mathematics (1903) though anticipated in A Critical Exposition of the Philosophy of Leibniz (1900).
Russell’s work on the sciences was by then concentrated on pure mathematics, but the new philosophy yielded little progress until, in 1900, he discovered Peano’s symbolic logic, which offered hope that pure mathematics could be treated without Kantian intuitions or transcendental arguments. On this basis Russell propounded logicism, the claim that the whole of pure mathematics could be derived deductively from logical principles, a position he came to independently of Frege, who held a similar but more restricted view but whose work Russell discovered only later. Logicism was announced in The Principles of Mathematics; its development occupied Russell, in collaboration with Whitehead, for the next ten years. Their results were published in Principia Mathematica (1910–13, 3 vols.), in which detailed derivations were given for Cantor’s set theory, finite and transfinite arithmetic, and elementary parts of measure theory. As a demonstration of Russell’s logicism, Principia depends upon much prior arithmetization of mathematics, e.g. of analysis, which is not explicitly treated. Even with these allowances much is still left out: e.g., abstract algebra and statistics. Russell’s unpublished papers (Papers, vols. 4–5), however, contain logical innovations not included in Principia, e.g., anticipations of Church’s lambda-calculus. On Russell’s extreme realism, everything that can be referred to is a term that has being (though not necessarily existence). The combination of terms by means of a relation results in a complex term, which is a proposition. Terms are neither linguistic nor psychological. The first task of philosophy is the theoretical analysis of propositions into their constituents. The propositions of logic are unique in that they remain true when any of their terms (apart from logical constants) are replaced by any other terms. In 1901 Russell discovered that this position fell prey to self-referential paradoxes. For example, if the combination of any number of terms is a new term, the combination of all terms is a term distinct from any term. The most famous such paradox is called Russell’s paradox. Russell’s solution was the theory of types, which banned self-reference by stratifying terms and expressions into complex hierarchies of disjoint subclasses. The expression ‘all terms’, e.g., is then meaningless unless restricted to terms of specified type(s), and the combination of terms of a given type is a term of different type. A simple version of the theory appeared in Principles of Mathematics (appendix A), but did not eliminate all the paradoxes. Russell developed a more elaborate version that did, in ‘Mathematical Logic as Based on the Theory of Types’ (1908) and in Principia. From 1903 to 1908 Russell sought to preserve his earlier account of logic by finding other ways to avoid the paradoxes – including a well-developed substitutional theory of classes and relations (posthumously published in Essays in Analysis, 1974, and Papers, vol. 5). Other costs of type theory for Russell’s logicism included the vastly increased complexity of the resulting system and the admission of the problematic axiom of reducibility.
Two other difficulties with Russell’s extreme realism had important consequences: (1) ‘I met Quine’ and ‘I met a man’ are different propositions, even when Quine is the man I met. In the Principles, the first proposition contains a man, while the second contains a denoting concept that denotes the man. Denoting concepts are like Fregean senses; they are meanings and have denotations. When one occurs in a proposition the proposition is not about the concept but its denotation. This theory requires that there be some way in which a denoting concept, rather than its denotation, can be denoted. After much effort, Russell concluded in ‘On Denoting’ (1905) that this was impossible and eliminated denoting concepts as intermediaries between denoting phrases and their denotations by means of his theory of descriptions. Using firstorder predicate logic, Russell showed (in a broad, though not comprehensive range of cases) how denoting phrases could be eliminated in favor of predicates and quantified variables, for which logically proper names could be substituted. (These were names of objects of acquaintance – represented in ordinary language by ‘this’ and ‘that’. Most names, he thought, were disguised definite descriptions.) Similar techniques were applied elsewhere to other kinds of expression (e.g. class names) resulting in the more general theory of incomplete symbols. One important consequence of this was that the ontological commitments of a theory could be reduced by reformulating the theory to remove expressions that apparently denoted problematic entities. (2) The theory of incomplete symbols also helped solve extreme realism’s epistemic problems, namely how to account for knowledge of terms that do not exist, and for the distinction between true and false propositions. First, the theory explained how knowledge of a wide range of items could be achieved by knowledge by acquaintance of a much narrower range. Second, propositional expressions were treated as incomplete symbols and eliminated in favor of their constituents and a propositional attitude by Russell’s multiple relation theory of judgment.
These innovations marked the end of Russell’s extreme realism, though he remained a Platonist in that he included universals among the objects of acquaintance. Russell referred to all his philosophy after 1898 as logical atomism, indicating thereby that certain categories of items were taken as basic and items in other categories were constructed from them by rigorous logical means. It depends therefore upon reduction, which became a key concept in early analytic philosophy. Logical atomism changed as Russell’s logic developed and as more philosophical consequences were drawn from its application, but the label is now most often applied to the modified realism Russell held from 1905 to 1919. Logic was central to Russell’s philosophy from 1900 onward, and much of his fertility and importance as a philosopher came from his application of the new logic to old problems. In 1910 Russell became a lecturer at Cambridge. There his interests turned to epistemology. In writing a popular book, Problems of Philosophy (1912), he first came to appreciate the work of the British empiricists, especially Hume and Berkeley. He held that empirical knowledge is based on direct acquaintance with sense-data, and that matter itself, of which we have only knowledge by description, is postulated as the best explanation of sense-data. He soon became dissatisfied with this idea and proposed instead that matter be logically constructed out of sensedata and unsensed sensibilia, thereby obviating dubious inferences to material objects as the causes of sensations. This proposal was inspired by the successful constructions of mathematical concepts in Principia. He planned a large work, ‘Theory of Knowledge,’ which was to use the multiple relation theory to extend his account from acquaintance to belief and inference (Papers, vol. 7). However, the project was abandoned as incomplete in the face of Wittgenstein’s attacks on the multiple relation theory, and Russell published only those portions dealing with acquaintance. The construction of matter, however, went ahead, at least in outline, in Our Knowledge of the External World (1914), though the only detailed constructions were undertaken later by Carnap. On Russell’s account, material objects are those series of sensibilia that obey the laws of physics. Sensibilia of which a mind is aware (sense-data) provide the experiential basis for that mind’s knowledge of the physical world. This theory is similar, though not identical, to phenomenalism. Russell saw the theory as an application of Ockham’s razor, by which postulated entities were replaced by logical constructions. He devoted much time to understanding modern physics, including relativity and quantum theory, and in The Analysis of Matter (1927) he incorporated the fundamental ideas of those theories into his