R Ussell and F Rege on the L Ogic of F Unctions

I compare Russell’s theory of mathematical functions, the “descriptive functions” from Principia Mathematica ∗30, with Frege’s well known account of functions as “unsaturated” entities. Russell analyses functional terms with propositional functions and the theory of definite descriptions. This is the primary technical role of the theory of descriptions in PM . In Principles of Mathematics and some unpublished writings from before 1905, Russell offered explicit criticisms of Frege’s account of functions. Consequenly, the theory of descriptions in “On Denoting” can be seen as a crucial part of Russell’s larger logicist reduction of mathematics, as well as an excursion into the theory of reference. Russell’s theory of definite descriptions, with its accompanying notions of scope and contextual definition, is justifiably still a leading theory in the philosophy of language, over one hundred years since it was first published in “On Denoting” in 1905. This theory was certainly an early paradigm of analytic philosophy, and then, along with Frege’s theory of sense and reference, became one of the two classical theories of reference. “On Denoting” is now being studied from an historical point of view as arising out of Russell’s qualms about his own prior theory of denoting concepts. Like Frege’s theory of sense, however, the role of the theory of descriptions in the larger logicist project is not well understood. Frege’s theory of sense precedes his foundational work, the Grundgesetze der Arithmetik, by only a few years. Yet after the introductory material, senses do not appear in the technical portion of Russell and Frege on the Logic of Functions 2 Grundgesetze, which is occupied with the reference, or Bedeutung, in the extensional logic of courses of values (Werthverlaufe) of concepts, his logicist version of classes. Frege’s theory of sense, it seems, is justifiably foundational in the later development of the philosophy of language, but is not so fundamental to his own life’s work, the project of reducing mathematics to logic. Russell’s theory of descriptions might seem to be similarly a digression into the philosophy of language by a philosopher whose main project was to write a long book proving the principles of mathematics from definitions using symbolic logic. My project in this paper is to explain one of the ways that definite descriptions enter into the technical project of Principia Mathematica, namely in ∗30 “Descriptive Functions.” Descriptive functions are simply ordinary mathematical functions such as the sine function, or addition. Number ∗30 is the origin of the now familiar notion in elementary logic of eliminating functions in favor of relations, and so is part of our conception of elementary logic as ending with the logic of relations, with the addition of complex terms, including function symbols, as an extra, optional development. I wish to argue, however, that this familiar way of reducing logic with functions to the logic of relations alone was in fact a step in Russell’s logicist project, one which he took in self conscious opposition to Frege’s use of mathematical functions as a primitive notion in his logic. As such “descriptive functions” were important to Russell’s reduction of mathematics to logic. Definite descriptions have an important role in Russell’s theory of propositions dating from Principles of Mathematics in 1903, where Russell uses the theory of denoting concepts which he only replaced in 1905 with the theory of “On Denoting.” Propositions in Principles, are composed of “terms” which include individuals and denoting concepts. The predicative constituents of propositions, the terms introduced by predicates, when taken in extension, play the role of classes. These concepts, obviously, are crucial to the logicist account of natural numbers and all other entities that mathematics deals with. The subjects of propositions will be individuals, when it is indeed individuals about which we make judgements, but, more generally, denoting concepts, which enable us to judge about terms with which we are not acquainted, such as infinite classes, and, more familiar from “On Denoting”, non-existents, such Vol. 4: 200 Years of Analytical Philosophy