Generality and reference: An examination of denoting in Russell's 'Principles of Mathematics'

This essay is a study of the theory of denoting concepts Russell proposed in his 1903 Principles of Mathematics. According to the theory, a denoting phrase such as "every man" is a referring expression. One problem facing such a theory is how to express the equipment of modern quantificational theory with bound and free variables. A quite different problem was raised by Russell in the famous "Gray's Elegy Argument" set forth in his 1905 article "On Denoting". Despite its being subject to a great deal of analysis, the argument remains unclear. Some regard it as a criticism of Frege's view on quantification, others regard it as a criticism of Russell's own theory of Principles. In the first part of this essay, I shall provide a new interpretation of the argument. Having refuted those who take the argument to be against Frege, I turn to three interpretations of Russell's 1903 theory. Each differs concerning the question about the occurrence of a denoting concept in a proposition. I call them "PoM-0", "PoM-1", and "PoM-2", respectively. Most commentators have taken PoM-0 as the historical background of the Gray's Elegy Argument. I shall argue that only PoM-1 and PoM-2 are historically viable. Set in the context of PoM-1 or PoME-2, the Gray's Elegy Argument is shown to offer a sound criticism of the theory of denoting concepts. The criticism can be avoided, but only at the expense of facing another difficulty, namely, the problem of logical form. This problem is the problem of specifying the logical structure of a denoting concept's occurrence as a concept versus its occurrence as a term. In the second part of this essay, I examine whether recent discussion of determiners and general quantifiers in natural language provides a solution to these problems. I point out that first order language seems unable to do so. On the other hand, a second order logic with nominalized predicates solves the problem, but any such theory relies on a solution of Russell's paradox of predication.