LAST EFFORTS OF LOGISTICIANS. 187 



VI. 

 Zigzag Theory and No Classes Theory. 



What is Mr. Rus.sell's attitude in face of these con- 

 tradictions ? After analysing those I have just spoken 

 of, and quoting others, after putting them in a form 

 that recalls Epimenides, he does not hesitate to con- 

 clude as follows : — 



" A propositional function of one variable does not 

 always determine a class." * A " propositional func- 

 tion " (that is to say, a definition) or " norm " can be 

 " non-predicative." And this does not mean that these 

 non-predicative propositions determine a class that is 

 empty or void ; it does not mean that there is no 

 value of ;ir that satisfies the definition and can be one of 

 the elements of the class. The elements exist, but they 

 have no right to be grouped together to form a class. 



But this is only the beginning, and we must know 

 how to recognize whether a definition is or is not 

 predicative. For the purpose of solving this problem, 

 Mr. Russell hesitates between three theories, which he 

 calls — 



A. The zigzag theory. 



B. The theory of limitation of size. 



C. The no classes theory. 



According to the zigzag theory, "definitions (pro- 

 positional functions) determine a class when they are 

 fairly simple, and only fail to do so when they are 

 complicated and recondite." Now who is to decide 



* This and the following quotations are from Mr. Russell's paper, 

 " On some difficulties in the theory of transfinite numbers and order 

 types, "Proceedings of (he London Mathematical Society. Ser, 2, Vol. 4, 

 Part I. 



