 The principal role of this “no-classes” theory of classes, as it is called, is to show how the theory of types resolves the paradoxes that had afflicted the naive theory of classes in The Principles of Mathematics and was seen by Russell to afflict Frege’s theory. After these foundational sections, all the individual variables that appear in PM should be seen as ranging over classes, (and, as will be explained below, the relation symbols are to be interpreted as ranging over relations in extension). The paradoxes appear in different forms, as seen in the Introduction to PM, but the resolution of the paradox of “the class of all classes which do not belong to themselves”, which appears in Russell’s intial letter to Frege, will be used as our example. This class, which leads directly to a contradiction, would appear in contemporary notation as \(\{ x \mid x \notin x \}\). The paradox arises when one asks whether that class is a member of iteself or not. The expression that it is a member of itself \(\{ x \mid x \notin x \} \in \{ x \mid x \notin x \}\) will have two class expressions to be eliminated by the first definition, and then several uses of the relation symbol \(\in\) which will also be eliminated. In the end there will be an expression \(\lnot (\phi_{\tau} \in \psi_{\tau})\), which is not legitimate, since this is not well-formed for any \({\tau}\). A function must be of a higher order than its arguments.