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Russell's Unknown Theory of Classes: The Substitutional System of 1906 DOUGLAS P. LACKEY Ir~ ms LENGTHY ARTICLE on Russell in the Encyclopedia o/Philosophy William Alston writes that Russell's status as a philosopher springs not so much from the originality of his positions as from "the ingenuity of his constructions." If this judgment is correct, and I think that it is, then devices which Russell invented and abandoned may be as philosophically interesting as devices which he published and endorsed. This is the case with a remarkable theory of classes and relations that Russell developed but never published in 1906, a theory which is quite different from the theories of classes and relations found in the earlier Principles o/Mathematics (1903) and the later Principia Mathematica (1910). The main scholarly sources for this unpublished theory are four manuscripts in the Bertrand Russell Archieves at McMaster University) The first of these is a manuscript of thirteen pages entitled "On Substitution," dated, in Russell's handwriting, December 22, 1905. The first page of this manuscript states the distinction between "substitution" and "determination" which is the basis of the substitutional theory. The remainder of the manuscript is in fragmentary condition, and the whole was never intended for publication . The second manuscript, also entitled "On Substitution," is 240 pages long and even more fragmentary than the first. This second manuscript bears no date, but one can tell from internal evidence that it is later than the first manuscript, and earlier than the fourth, which itself can have been written no later than June 1907. I would date the second manuscript sometime in the first half of 1906, since late 1906 in Russell's career was occupied with a lengthy paper published as "Les Paradoxes de la Logique" in December of 1906. The third and main source is a 38-page paper entitled "On the Substitutional Theory of Classes and Relations," which was received by the London Mathematical Society on April 24th, 1906, read before the Society on May 10th, 1906, and submitted for publication in the Proceedings ol the London Mathematical Society. The paper fared oddly. The referee took off on an extended vacation, and the Society did not get around to a positive verdict until October of 1906. By that time Russell was no longer pleased with the essay, and refused to allow its publication.2 (The correspondence from the London 1 The theory is hinted at in two published papers from 1906, "On Some Diftieulties in the Theory of Transfinite Numbers and Order Types" and "On 'Insolubilia' and their Solution by SymbolicLogic," both of which are reprinted in my anthology of Russell's logic papers, Essaysin Analysis(London: Allen and Unwin, 1973).See pp. 153, 198-200. 2 The firstpublication of thise~say isin Essaya in Analysis, pp. 164-188. [69] 70 HISTORY OF PHILOSOPHY Mathematical Society concerning the paper is now in the Russell Archives. Future references to "ms." in this essay are all to this third manuscript.) The fourth manuscript is a 108-page essay entitled "The Paradox of the Liar." Unlike the first three manuscripts, the Liar manuscript treats the substitutional theory only in passing, and mostly in a critical vein. The manuscript has no date, but there is a marginal comment, criticizing a point made on page 87, which is dated "June 1907," obviously the latest month in which the essay could have been composed. 1. IMMEDIATE BACKGROUNDOF THE THEORY In the Principles o[ Mathematics, Russell had reluctantly concluded that if logic were to serve as the foundations of mathematics , it was necessary to assume that classes exist. This assumption was troublesome, since the concept of "class" seemed to give rise to contradictions that Russell could not solve. In the years following the completion of the Principles, Russell made three different attempts to establish the foundations of mathematics without recourse to the notion of class. The substitutional theory is chronologically the second of these attempts, but logically it is the most radical. The first attempt came in May of 1903, when Russell decided that the notion of class could be dispensed with in favor of the concept of propositional [unction. For example, instead of asserting "there...


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