Nicholas Griffin

The Cambridge companion to Bertrand Russell

Introduction Nicholas Griffin 1. Mathematics in and behind Russells logicism, and its reception I. Grattan-Guinness 2. Russells philosophical background Nicholas Griffin 3. Russell and Moore, 1898-1905 Richard L. Cartwright 4. Russell and Frege Michael Beaney 5. Bertrand Russells logicism Martin Godwyn and A. D. Irvine 6. The theory of description Peter Hylton 7. Russells substitutional theory Gregory Landini 8. The theory of types Alasdair Urquhart 9. Russells method of analysis Paul Hager 10. Russells neutral monism R. E. Tully 11. The metaphysics of logical atomism Bernard Linsky 12. Russells structuralism and the absolute description of the world William Demopoulos 13. From knowledge by acquaintance to knowledge by causation Thomas Baldwin 14. Russell, experience and the roots of science A. C. Grayling 15. Bertrand Russell: moral philosopher or unphilosophical moralist? Charles R. Pidgen.

The Theory of Types

introduction A surprising feature of Russells work in logic is that he began and ended with a theory of types. This chapter begins with a summary of the 1903 theory of types and then proceeds to the much more complex ramified theory of types that emerged from Russells intense work on the foundations of logic from 1903 to 1907. After discussing the problems connected with the Axiom of Reducibility, the chapter concludes with the simple theory of types, and the later history of type theory, after the demise of the logicist programme. the 1903 theory of types Russell’s early theory of types, presented in Appendix B to the Principles of Mathematics , already contains many of the basic features of the mature system given in his fundamental paper of 1908 and in Principia Mathematica . In 1901, Russell had begun writing out the derivation of mathematics from logic, employing the methods of Peano and his school. This led him to examine Cantor’s proof that there is no greatest cardinal number. This result conflicted with his assumption that there is a universal class, having all objects as members, which ought to have the greatest cardinal number. Close analysis of the diagonal argument used in Cantor’s proof led to the discovery of the paradox of the class of all classes that are not members of themselves, now called “Russell’s paradox,” but which Russell called “the Contradiction.” The logical paradoxes emerged at an awkward moment, when Russell had already written most of the penultimate draft of the Principles . Rather than hold up its publication indefinitely, he took the manuscript of his book to the printer in May 1902 before finding a solution. His initial reaction was that the Contradiction was of a somewhat trivial character, and that it could be avoided by a simple modification of the primitive propositions of logic.

Russell’s Substitutional Theory

introduction In his 1893 Grundgesetze der Arithmetik Frege sought to demonstrate a thesis which has come to be called Logicism. Frege maintained that there are no uniquely arithmetic intuitions that ground mathematical induction and the foundational principles of arithmetic. Couched within a proper conceptual analysis of cardinal number, arithmetic truths will be seen to be truths of the science of logic. Frege set out a formal system - a characteristica universalis - after Leibniz, whose formation rules and transformation (inference) rules were explicit and, he thought, clearly within the domain of the science of logic. Confident that no nonlogical intuitions could seep into such a tightly articulated system, Frege endeavored to demonstrate logicism by deducing the principle of mathematical induction and foundational theorems for arithmetic. In his 1903 The Principles of Mathematics , Russell set out a doctrine of Logicism according to which there are no special intuitions unique to the branches of non-applied mathematics. All the truths of non-applied mathematics are truths of the science of logic. Russell embraced this more encompassing form of logicism because, unlike Frege, he accepted the arithmetization of all of non-applied mathematics, including Geometry and Rational Dynamics. Both Frege and Russell regarded logic as itself a science. Frege refrained from calling it a synthetic a priori science so as to mark his departure from the notion of pure empirical intuition ( anschauung ) set forth in Kant’s 1781 Critique of Pure Reason . In Frege’s view, Kant’s transcendental argument for a form of pure empirical (aesthetic) intuition that grounds the synthetic a priori truths of arithmetic is unwarranted. Russell concurred, but spoke unabashedly of a purely logical intuition grounding our knowledge of logical truths. Russell wrote that Kant “never doubted for a moment that the propositions of logic are analytic, whereas he rightly perceived that those of mathematics are synthetic . . . It has since appeared that logic is just as synthetic . . .” ( POM , p. 457).

Psychologism and the Development of Russell's Account of Propositions

This article examines the development of Russells treatment of propositions, in relation to the topic of psychologism. In the first section, we outline the concept of psychologism, and show how it can arise in relation to theories of the nature of propositions. Following this, we note the anti-psychologistic elements of Russells thought dating back to his idealist roots. From there, we sketch the development of Russells theory of the proposition through a number of its key transitions. We show that Russell, in responding to a variety of different problems relating to the proposition, chose to resolve these problems in ways that continually made concessions to psychologism.

Russell on specific and universal relations: the principles of mathematics, §55

In this paper we consider the arguments Russell uses in The principies of mathematics, §55 to establish the view that all relations are universals. These arguments are shown to be defective. Finally, we consider the connection between Russells view of relations and wider aspects of his philosophy—in particular, his theories of reference and truth and the gradual break-down of his absolute realism.

Brandom and the brutes

Brandom’s inferentialism offers, in many ways, a radically new approach to old issues in semantics and the theory of intentionality. But, in one respect at least, it clings tenaciously to the mainstream philosophical tradition of the middle years of the twentieth century. Against the theory’s natural tendencies, Brandom aligns it with the ’linguistic turn’ that philosophy took in the middle of the last century by insisting, in the face of considerable opposing evidence, that intentionality is the preserve of those who can offer and ask for reasons and thus of language users alone. In this paper, I argue that there is no good reason for giving inferentialism a linguistic twist, and that, in doing so, Brandom is forced to make claims which are implausible in themselves and lead him, in the attempt to mitigate them, to a number of doubtful expedients.

What Part of ‘Not’ Don’t We Understand?

Many important philosophical issues hang on the concept of negation. This seems to be true as much of eastern philosophy as of Western, though the issues are often different in each tradition. The paper explores a few of the eastern issues from among those that have been discussed by Shaw. While most of these seem on the face of it to be of concern exclusively to eastern philosophers, some at least have (perhaps surprising) resonances in western traditions.

Martin Heidegger and the First World War

For decades the reputation of the German philosopher Martin Heidegger has been eroded by growing evidence of his Nazism. While alive he kept the accusations at bay by a mixture of lies and evasions...

Russell and Leibniz on the Classification of Propositions

We owe Russell’s book on Leibniz to a very improbable series of events, of which surely the most improbable of all was that McTaggart was getting married. It was McTaggart who was scheduled to give the lectures on Leibniz that Russell ended up giving at Trinity College, Cambridge during Lent Term of 1899 and it was McTaggart’s marriage that prevented him from giving them. Now getting married (even for McTaggart) would not normally be so traumatic an event as to prevent one from lecturing, butMcTaggart’s brideto- be was a New Zealander.

Projects in progress

This department publishes articles on large-scale projects in which logic plays a significant role, especially editions of collected or selected works. In addition to factual and historical details, articles describe points of historiography and scholarship which are of more general interest. Articles should be submitted to the Editor.