The fundamental element in Russell’s philosophy is his logic. His views on metaphysics and ethics, on the nature and relations of matter and mind, changed profoundly in the course of his life, but these changes all proceeded from successively deeper applications of his logical method. He, therefore, preferred to classify his philosophy not as a species of idealism or realism but as “logical atomism”, since what distinguishes the whole of his work is his use of logical analysis as a method and his belief that by it we can arrive at ultimate “atomic facts” logically independent both of one another and of being known.
First Russell tried to free logical analysis from the domination of ordinary grammar by showing that the grammatical form of a sentence often fails to reflect the logical form of its meaning. In his Principles of Mathematics he insisted that relations could not be reduced to qualities of their terms and that relational facts were not of the subject-predicate forms, but he still thought that any descriptive phrase which could be made the subject of a sentence must stand for a term which had being, even if like “the round square” it were self-contradictory. In his article “On Denoting” and in subsequent writings, he put forward his theory of descriptions, which is perhaps the most important and influential of his innovations in logic. According to this theory, “the present king of
” is not a name for a nonexistent entity but an “incomplete symbol” which only has meaning in connection with a context. The meaning of such a statement as “the present king of France is bald” is first that there is someone who is at present both king of France and bald, and secondly that there are not at present two kings of France; and when such statements are analyzed in this way the need to believe in entities such as “the present king of France” (which are said by some philosophers to have “being” but not “existence”) is altogether removed. Similarly when it is said that “unicorns are not real”, this does not mean that certain animals, namely unicorns, lack the characteristics of reality but that there are no horse-like animals with one horn. France
Russell applied similar methods to classes and to numbers, and argued that each of these categories consists of what he called “logical constructions”. In saying, for example, that classes are logical constructions, he did not mean that they are entities constructed by the human mind, but that when we express facts by sentences which have for subject such a phrase as “the class of men”, the true analysis of the fact does not correspond to the grammatical analysis of the sentence. When, for instance, we say “the class of men includes the class of criminals”, the fact asserted by us is really about the characteristics of being a man and a criminal and not about any such entities as classes at all. This notion of a logical construction was much employed by Russell in his work in mathematical logic, and he also used it extensively in the philosophy of matter and mind, and even adopted as a fundamental principle that constructions (in his special sense of the word) are to be substituted for inferred entities wherever possible.
By applying this method Russell was led to a view of the world in which the ultimate constituents of mind and matter are of the same type, the difference between minds and bodies lying in their structure and not in the elements of which they are composed. A man’s mind is composed of sensations and images, which are identified by Russell with physical events in his brain, and the difference between physics and psychology lies not in the events which they study but in the kind of laws about these events which they seek to establish, physics being concerned with structure, and psychology with quality. This theory was worked out by Russell in connection with physics in The Analysis of Matter.
In the theory of knowledge, Russell’s earlier rationalism was considerably modified in a pragmatist or behaviourist direction, and in The Analysis of Mind he rejected consciousness as a fundamental characteristic of mind and adopted a form of “neutral monism” about perception, which he combined with representationism in regard to memory and judgment.
Russell’s logical atomism was the starting-point for the Tractatus Logico-Philosophicus (1921) of his pupil L. Wittgenstein and so one of the sources of logical positivism. Then, after a period between World Wars I and II when it dominated the philosophy of the English-speaking world, his programme was brought into doubt by the later teaching of Wittgenstein, according to which philosophical difficulties arise not from any inadequacy of ordinary language but from failure to respect the limits of normal usage. In his own later writings, Russell showed some misgivings about logical atomism, but for different reasons. He came to think, for example, that there might be necessary connections between distinct events.
Russell maintained that mathematics and formal logic are one and that the whole of pure mathematics can be rigorously deduced from a small number of logical axioms. He argued this in outline in Principles of Mathematics and then tried to give a detailed demonstration of his thesis in Principia Mathematica, written with A.N. Whitehead. In this colossal work the deduction is carried so far as to include all the essential parts of the theory of aggregates and real numbers. Besides this, the great advances made by Russell in the analysis of logical concepts allowed the deductions to be carried not only much farther forward but also much farther backward toward first principles. Above all, he appeared to solve the notorious paradoxes of the theory of aggregates by means of the theory of types. In this connection, however, he found it necessary to introduce an “axiom of reducibility” which has never won general acceptance, so that his work cannot be regarded as a final solution of the problem.