In "On Denoting" (1905), Russell presents a theory of denotation which relies on the notion of a variable.  Russell says very little about variables in this paper.  He says only that they are "fundamental," and that they are "essentially and wholly undetermined" constituents of propositional functions.  I think I understand the role of this notion in Russell's theory, and why Russell says what he does about it,  He appeals to non-denoting elements in propositions in order to avoid having to interpret "a=b" as "a=a."  By using variables, he can claim that no elements in a propositional function serve the role of the denoting phrase.  For example, in the fully explicit presentation of "Scott is the author of Waverley," we do not find anything for which we could substitute the phrase "the author of Waverley."  The meaning of the denoting phrase is only found when we interpret the proposition as a whole, and cannot be found in any of its parts. 

My problem is, I don't know what it means to say ... (read more)