The Functions of Russell's No Class Theory

§1. Introduction. Although Whitehead and Russell’s Principia Mathematica (hereafter, PM ), published almost precisely a century ago, is widely heralded as a watershed moment in the history of mathematical logic, in many ways it is still not well understood. Complaints abound to the effect that the presentation is imprecise and obscure, especially with regard to the precise details of the ramified theory of types, and the philosophical explanation and motivation underlying it, all of which was primarily Russell’s responsibility. This has had a large negative impact in particular on the assessment of the socalled “no class” theory of classes endorsed in PM. According to that theory, apparent reference to classes is to be eliminated, contextually, by means of higher-order “propositional function”—variables and quantifiers. This could only be seen as a move in the right direction if “propositional functions,” and/or higher-order quantification generally, were less metaphysically problematic or obscure than classes themselves. But this is not the case—or so goes the usual criticism. Years ago, Geach (1972, p. 272) called Russell’s notion of a propositional function “hopelessly confused and inconsistent.” Cartwright (2005, p. 915) has recently agreed, adding “attempts to say what exactly a Russellian propositional function is, or is supposed to be, are bound to end in frustration.” Soames (2008) claims that “propositional functions . . . are more taken for granted by Russell than seriously investigated” (p. 217), and uses the obscurity surrounding them as partial justification for ignoring the no class theory in a popular treatment of Russell’s work (Soames, 2003).1 A large part of the usual critique involves charging Russell with confusion, or at least obscurity, with regard to what a propositional function is supposed to be. Often the worry has to do with the use/mention distinction: is a propositional function, or even a proposition