THE POSITIVE THEORY OF INFINITY 207 



which so delighted the sophists or unsound reasoners of 

 ancient Greece," and this no doubt represents the judg- 

 ment of common sense upon such arguments. Yet if 

 collections of things were things, his contention would be 

 irrefragable. It is only because the bay horse and the 

 dun cow taken together are not a new thing that we can 

 escape the conclusion that there are three things wherever 

 there are two. 



When it is admitted that classes are not things, the 

 question arises : What do we mean by statements which 

 are nominally about classes ? Take such a statement as, 

 " The class of people interested in mathematical logic is 

 not very numerous." Obviously this reduces itself to, 

 " Not very many people are interested in mathematical 

 logic." For the sake of definiteness, let us substitute 

 some particular number, say 3, for "very many." Then 

 our statement is, " Not three people are interested in 

 mathematical logic." This may be expressed in the 

 form : " If x is interested in mathematical logic, and also 

 y is interested, and also z is interested, then x is identical 

 with y, or x is identical with %, or y is identical with z." 

 Here there is no longer any reference at all to a " class." 

 In some such way, all statements nominally about a class 

 can be reduced to statements about what follows from 

 the hypothesis of anything's having the defining property 

 of the class. All that is wanted, therefore, in order to 

 render the verbal use of classes legitimate, is a uniform 

 method of interpreting propositions in which such a use 

 occurs, so as to obtain propositions in which there is no 

 longer any such use. The definition of such a method 

 is a technical matter, which Dr Whitehead and I have 

 dealt with elsewhere, and which we need not enter into 

 on this occasion. 1 



1 Cf. Principia Mathematical 20, and Introduction, chapter iii. 



