Russell: The Journal of Bertrand Russell Archives

New evidence concerning Russell's substitutional theory of classes by Gregory Landini I. INTRODUCTION IT IS WELL known that Russell regarded his new theory of denoting (of 1905) as the conceptual breakthrough that "made it possible to see, in a general way, how a solution of the contradictions might be possible " (Schilpp 1944, p. 14). The solution, of course, was the nonassumption of classes as single logical subjects. The theory of denoting was an important first step because it showed the way to provide a treatment of classes as if they were single logical subjects. In his 1908 a.rticle, "Mathematical Logic as Based on the Theory of Types", we find the following contextual definition effecting this solution: f{x: l/Jx} = df(3ljJ)((x)(ljJ!x == l/Jx) & f(ljJ!z». The contextual definition appears to make the assumption of propositional functions as single logical subjects; and this has come to be the accepted view. But, according to Russell, the non-assumption of classes realized here employs only the "technical convenience" of using symbols for propositional functions in subject positions. The convenience was supposed to be eliminable by using a technique of substitution (Russell 1908, p. 89). Just what substitutional technique Russell had in mind remained a mystery for some time, however. OnI4 December 1905 Russell had read an article entitled "On Some Difficulties in the Theory of Transfinite Numbers and Order Types" before the London Mathematical 26 Substitutional theory of classes 27 Society. (The article was subsequently published in the proceedings of the society on 7 March 1906.) In it he set out the main alternatives for avoiding the contradiction. The preferred alternative was a substitutional theory according to which neither classes nor propositional functions were assumed as single logical subjects. Because the contradiction was formulable in terms of functions, Russell felt that "the assumption ofpropositional functions is open to the same arguments, pro and con, as the admission of classes" (Russell 1905, p. 154). It was this early theory of substitution which was the direct"result of Russell's studies on the new theory of denoting. The theory was able to treat classes as if they were single logical subjects, and it allowed what would be quantification over classes. Moreover, by assuming propositions (true or false) as single logical subjects instead of propositional functions or classes, the theory built homogeneous typing into the logical form of propositions.whose grammatical form suggested that they were about classes. In this way, Russell avoided having types of logical subjects, and the univocity of being of all logical subjects was preserved. Nonetheless, the subsequent articles in which Russell went on to elaborate a substitutional theory of classes and relations went largely unnoticed. Russell himself was partly the cause. Its first detailed public elaboration in "On the Substitutional Theory of Classes and Relations" (1906a) was read before the London Mathematical Society in May of 1906, but the article was withdrawn from publication. Russell's decision to withdraw the article seems to be related to his desire to include a solution of what are now called "semantic" paradoxes such as the Liar Paradox. In a letter to Jourdain dated 14 June Russell wrote: I feel more and more certain that this theory is right. In order, however, to solve the Epimenides, it is necessary to extend it to general propositions, Le., to such as (x). ljJx and (3x). ljJx. This I shall explain in my answer to Poincare 's article in the current Revue de Mitaphysique. (Quoted from GrattanGuinness 1977, p. 89) Poincare's article was "Les Mathematiques et la Logique". It contained criticisms of the new mathematics of the infinite and a proposed solution -namely, the Vicious Circle Principle. Russell was eager to address the criticisms and explain that his substitutional theory is what is required by adherence to the Vicious Circle Principle. In September Russell published his reply entitled "Les Paradoxes de la Logique" (1906b). (The English title is: "On 'Insolubilia' and Their Solution by Symbolic Logic".) In it he espoused the substitutional theory, and as promised in his letter to Jourdain, it was now extended to account for 28 Russell summer 1989 the Liar Paradox. In Russell's mind, Poincare's Vicious...