BERTRAND RUSSELL’S LOGIC Andrew D. Irvine Bertrand Russell is generally recognized as one of the most important English- speaking philosophers, logicians and essayists of the twentieth century. Often cited along with G.E. Moore as one of the founders of modern analytic philosophy and along with Kurt G¨odel as one of the most influential logicians of his time, Russell is also widely recognized for his sustained public contributions to many of the most controversial social, political and educational issues of his day. Even so, more than anything else, it is Russell’s work in logic and the foundations of mathematics that serves as his core contribution to intellectual history and that makes Russell the seminal thinker he is. His most significant achievements include 1. his refining and popularizing of Giuseppe Peano’s and Gottlob Frege’s first attempts at developing a modern mathematical logic, 2. his discovery of the paradox that bears his name, 3. his introduction of the theory of types (his way of avoiding the paradox), 4. his defense of logicism, the view that mathematics is in some important sense reducible to logic, and his many detailed derivations supporting this view, 5. his ground-breaking advances in technical philosophy, including both his theory of definite descriptions and his theory of logical constructions, 6. his theory of logical relations, including his impressively general theory of relation arithmetic, 7. his formalization of the reals, 8. his theory of logical atomism, and 9. his championing of the many connections between modern logic, mathemat- ics, science, and knowledge in general. Russell’s first important book about logic, The Principles of Mathematics,ap- peared in 1903. This was followed by the landmark articles “On Denoting” (in 1905) and “Mathematical Logic as based on the Theory of Types” (in 1908). A decade of technical work then culminated with the publication (in 1910, 1912 and Handbook of the History of Logic. Volume 5. Logic from Russell to Church Dov M. Gabbay and John Woods (Editors) c 2009 Elsevier B.V. All rights reserved. 2 Andrew D. Irvine 1913) of the three-volume Principia Mathematica, co-authored with Russell’s for- mer teacher, Alfred North Whitehead.1 This was followed by several books and articles intended to help underscore the philosophical importance of the new math- ematical logic that Russell, Whitehead, Peano and Frege had developed, including “The Philosophy of Logical Atomism” (in 1918–19), Introduction to Mathematical Philosophy (in 1919), “Logical Atomism” (in 1924), and My Philosophical Devel- opment (in 1959). Russell received the Order of Merit in 1949 and the Nobel Prize in 1950. Born in 1872, he wrote over fifty books, over 4,000 articles, book chapters, reviews and other more minor publications, and over 61,000 letters. He died in 1970, having lived a very full and public life. Developing a comprehensive, coherent interpretation of Russell’s logic and phi- losophy of mathematics is both important and challenging. It is important be- cause of the tremendous influence his work has had on philosophy, mathematics, economics, decision theory, computability theory, computer science and other dis- ciplines over the past one hundred years. It is challenging for at least three rea- sons. First, modern logic is intrinsically abstract and complex. Even mastering the full notation of Principia Mathematica may be judged an accomplishment in itself. (This is especially true given that, over the past century, many parts of the notation have been modified or replaced by more modern kinds of symbolism. Learning it in complete detail is thus something that not even all Russell scholars have achieved.) Second, as with the work of any intellectual pioneer, it is often suggested, not only that Russell’s views changed and developed over time, but that at important times they were also relatively inchoate. Third, as with the work of many other important historical figures, it sometimes can be difficult to distinguish between Russell’s actual views and those attributed to him by various well-known commentators and interpreters. Today there exist two main approaches to understanding the most difficult and demanding aspects of Russell’s philosophy. The first is to claim that Russell’s work is largely a series of almost unconnected views, views that not only changed over time but that also were largely indebted to the work of others. This is the view that characterizes Russell primarily as a popularizer and as a less-than-systematic thinker. The second is to see in Russell’s writings a much more systematic and original body of work. To give just one example, there are many often-overlooked connections between Russell’s logic, his theory of language (especially his theory of definite descriptions), and his epistemology (for example, his multiple-relation theory of judgment). According to this second view we fail to see these many connections at our peril. Significantly, it is also this second view — that Russell is offering us a fully integrated body of intellectual work — that is much more in line with Russell’s own statements about his philosophical aims and achievements, statements that many writers over the years have conveniently chosen to ignore. 1Although anticipated in several respects in the writings of Gottlob Frege, Principia Math- ematica remains just as intellectually impressive an achievement as Frege’s Grundgesetze der Arithmetik. The case might also be made that, without it, Frege’s work might never have re- ceived the prominence it deserves. Bertrand Russell’s Logic 3 Whether explicitly or not, it is also this integrated view of logical, philosophical and scientific knowledge that has influenced an entire century of western philosophy. Understanding not only Russell’s substantive logical and philosophical views, but also his methodological views, is therefore crucial to understanding the domi- nant paradigm of intellectual work carried out in philosophy over the past century. Today it is hard to imagine a philosophy undergraduate anywhere in the western world whose education has not been influenced, at least in part, by Russell’s math- ematical logic and philosophical method. INITIAL INFLUENCES At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. After I had learned the fifth proposition, my brother told me that it was generally considered difficult, but I had found no difficulty whatever. This was the first time it had dawned upon me that I might have some intelligence. From that moment until I met Whitehead and I finished Principia Mathematica, when I was thirty-eight, mathematics was my chief interest and my chief source of happiness. Like all happiness, however, it was not unalloyed. I had been told that Euclid proved things, and was much disappointed that he started with axioms. At first I refused to accept them unless my brother could offer me some reason for doing so, but he said: “If you don’t accept them we cannot go on,” and as I wished to go on, I reluctantly admitted them pro tem. The doubt as to the premises of mathematics which I felt at that moment remained with me, and determined the course of my subsequent work. [Russell, 1967, pp. 37–8] So begins the story in Russell’s Autobiography of how Russell came to be inter- ested in the foundations of mathematics. Educated at home by a series of Swiss and German governesses and tutors before entering Cambridge in 1890, Russell embraced academic life enthusiastically. He obtained a first in Mathematics in 1893. Even so, with the exception of the lectures given by Whitehead, Russell found the teaching of mathematics at Cambridge “definitely bad.”2 As a result, he became even more determined to try to learn exactly what it is that gives mathematics its high degree of certainty. Having finished his undergraduate program in mathematics, Russell “plunged with whole-hearted delight into the fantastic world of philosophy”3 and the follow- ing year he completed the Moral Sciences Tripos. In philosophy he was initially influenced by the absolute idealism of J. M. E. McTaggart, F. H. Bradley and the 2[Russell, 1959, p. 38]. 3[Russell, 1959, p. 38]. 4 Andrew D. Irvine other Ox-bridge neo-Hegelians of the time.4 As Russell understood it, absolute idealism depended crucially on the doctrine of internal relations, the doctrine that any relational fact (for example, that x is related to y) is really a fact about the natures of the related terms. According to this view, if x is greater than y, then being greater than y is a part of the nature of x. Object y is thus, in some sig- nificant sense, a part of x and x, similarly, is a part of y. Given the complexity of relations in the world, it turns out that all objects must be related to all other objects. Hence it follows that there exists only a single, all encompassing unity. Further, since if one is aware of x, x must also (on this view) be a part of one’s mind, it follows that everything conceivable must be a part of consciousness. Initially, Russell used this idealist framework in his approach to mathematics, for example in his Essay on the Foundations of Geometry (1897). However, he became disenchanted with idealism once he realized how incompatible the view was with his developing understanding of both mathematics and science. On Russell’s emerging theory, geometrical points, for example, could be individuated only by their relations. But according to absolute idealism, relations in turn depended on the intrinsic natures of their individual relata.