If the theory were without any embarrassments of its own, and were indispensable for the resolution of the so-called paradoxes1 (which no one seems to believe), there would be nothing to do but to acknowledge the impossibility of cosmic formulations, as well as the inadequacy of philosophic criticisms, and to pass charitably over such remarks as Russell's as mere accidents in a busy life. However, the statement of the theory itself involves the following difficulties in connection with (1) its scope, (2) its applicability to propositions made about it, and (3) its description.
1. It is either about all propositions or it is not.
A. If it were about all propositions it would violate the theory of types
and be meaningless or self-contradictory.
B. If it were not about all propositions, it would not be universally
applicable. To state it, its limitations of application would have to be specified. One cannot say that there is a different theory of types for each order of the hierarchy, for the proposition about the hierarchy introduces the difficulty over again.
2. .Propositions about the theory of types (such as the present ones, as well as those in the Principia) are subject to the theory of types, or they are not.
A. If they were, the theory would include within its own scope propositions
of a higher order, and thus be an argument to what, is an argument to it.2
B. If they were not, there would be an unlimited number of propositions,
not subject to the theory, that could be made directly or indirectly about it. Among these propositions there might be some which refer to a totality and involve functions which have arguments presupposing the function.
3. The statement of the theory of types is either a proposition or a propositional function, neither or both.
A. If it were a proposition, it would be either elementary, first order,
general, etc., have a definite place in a hierarchy and refer only to those propositions which are of a lower order. If it were held to be a proposition of the last order, then the number of orders would have a last term, and there could not be meaningful propositions made about the theory. The Principia should not be able to say, on that basis, just what the purpose, character and application of the theory is.
B. Similarly, if it were a propositional function, it would have a definite
place in a hierarchy, being derived from a proposition by generalisa-
1 Paradoxes, though contrary to common opinion, may be and frequently are true. Paranoumena, violating principles of logic or reason, if they are not meaningless, are false, and it is only they which are capable of logical analysis and resolution. What the Principia attempts to do is to solve apparent paranoumena with a real paranoumenon.
* P. 39, Principia Mathematics.