Classes of classes and classes of functions in Principia mathematica

The No-Classes theory of classes in Principia Mathematica must be supplemented with a definition of propositional functions of classes in order to avoid a recently discovered puzzle. Such a definition is suggested but rejected by Whitehead and Russell in the Introduction to PM and does not make it into the technical presentation of *20. PM thus contains a distinction between classes of classes and classes of functions although only the former appear in the body of the work. The problem that justifies this distinction is distinct from a problem about scope identified by Carnap in 1947 in Meaning and Necessity, although Carnap's problem also involves the contextual definitions of the no-classes theory and the intensional nature of propositional functions.