 Russell’s “No-Class” theory of classes from *20 of Principia Mathematica provides a contextual definition like that of the theory of definite descriptions. One of Russell’s early diagnoses of the paradox of the class of all classes that are not members of themselves was that it showed that classes could not be individuals. Indeed Russell seems to have come across his paradox by applying Cantor’s famous diagonal argument to show that there are more classes of individuals than individuals. Hence, he concluded, classes could not be individuals, and expressions for classes such as ‘\(\{x: Fx \}\)’ cannot be the singular terms they appear to be. Inspired by the theory of descriptions, Russell proposed that to say something \(G\) of the class of \(F\)s, \(G\) \(\{x: Fx \}\), is to say that there is some (predicative) property \(H\) coextensive with (true of the same things as) \(F\) such that \(H\) is \(G\). The restriction to predicative properties, or those which are not defined in terms of quantification over other properties, was a consequence of the ramification of the theory of types to avoid intensional or “epistemic” paradoxes which motivated the theory of types in addition to the set theoretical “Russell’s Paradox” (see Whitehead and Russell 1910–13, Introduction, Chapter II). These predicative properties are intensional, however, in the sense that two distinct properties might hold of the same objects. (See the entry on the  notation in Principia Mathematica.)  That classes so defined have the feature of extensionality is thus derivable, rather than postulated. If \(F\) and \(H\) are coextensive then anything true of \(\{x:Fx \}\) will be true of \(\{x:Hx \}\). Features of classes then follow from the features of the logic of properties.