Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question : the zig-zag theory and the ramified theory of types.
2.
As Stewart Shapiro explains in his " Thinking About Mathematics ", Russell's attempts to solve the paradoxes led to the ramified theory of types, which, though it is highly complex and relies on the doubtful axiom of reducibility, actually manages to solve both syntactic and semantic paradoxes at the expense of rendering the logicist project suspect and introducing much complexity in the PM system.
3.
"' Ramified theory of types : function-orders and argument-types, predicative functions "': When Russell proclaimed " all " classes are useful fictions he solved the problem of the " unit " class, but the " overall " problem did not go away; rather, it arrived in a new form : " It will now be necessary to distinguish ( 1 ) terms, ( 2 ) classes, ( 3 ) classes of classes, and so on " ad infinitum "; we shall have to hold that no member of one set is a member of any other set, and that " x " ? " u " requires that " x " should be of a set of a degree lower by one than the set to which " u " belongs.
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