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.
22.
At about the same time Ludwig Wittgenstein made short work of the theory of types in his 1922 work Tractatus Logico-Philosophicus in which he points out the following in parts 3.331 3.333:
23.
An adaptation of radical theory was called theory of types ( theory of residues ), advocated by Charles-Adolphe Wurtz, August Wilhelm von Hofmann and Charles Fr�d�ric Gerhardt and another one was water type as promoted by Alexander William Williamson.
24.
But because the stipulations of the ramified theory would prove ( to quote Quine ) " onerous ", Russell in his 1908 " Mathematical logic as based on the theory of types " also would propose his " axiom of reducibility ".
25.
According to R . Blutner and E . Hochnadel, socionics is not so much a theory of personalities per se, but much more a theory of type relations providing an analysis of the relationships that arise as a consequence of the interaction of people with different personalities ."
26.
The Epimenides paradox appears explicitly in " Mathematical Logic as Based on the Theory of Types ", by Bertrand Russell, in the " American Journal of Mathematics ", volume 30, number 3 ( July, 1908 ), pages 222 262, which opens with the following:
27.
Metaclasses are sometime organized by levels, in a similar way to the simple Theory of types where classes that are not metaclasses are assigned the first level, classes of classes in the first level are in the second level, classes of classes in the second level on the next and so on.
28.
Deep studies of admissible representations of " p "-adic reductive groups were undertaken by Howe and Moy and Bushnell and Kutzko, who developed a " theory of types " and classified the admissible dual ( i . e . the set of equivalence classes of irreducible admissible representations ) in many cases.
29.
Russell in his 1920 " Introduction to Mathematical Philosophy " devotes an entire chapter to " The axiom of Infinity and logical types " wherein he states his concerns : " Now the theory of types emphatically does not belong to the finished and certain part of our subject : much of this theory is still inchoate, confused, and obscure.
30.
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.
