   Russell spent the years between 1902 and 1910 searching for a philosophically motivated solution to such paradoxes. He tried solutions of various sorts.  However, in late 1905, after the discovery of the theory of descriptions, he became convinced that an expression for a class is an “incomplete symbol”, i.e., that while such an expression can occur as part of a meaningful sentence, it should not be regarded as representing a single entity in the corresponding proposition. Russell dubbed this approach the “no classes” theory of classes (see e.g., TNOT, 145), because, while it allows discourse about classes to be meaningful, it does not posit classes as among the fundamental ontological furniture of the world.  The precise nature of Russell’s “no classes” theory underwent significant changes between 1905 and 1910. However, in the version adopted in the first edition of Principia Mathematica, Russell believed that a statement apparently about a class could always be reconstructed, using higher-order quantification, in terms of a statement involving its defining propositional function. Russell believed that whenever a class term of the form “{z|ψz}” appeared in some sentence, the sentence as a whole could be regarded as defined as follows (cf. PM, 188):