194 SCIENCE AND METHOD. 



Why is it that by applying to their arguments the 

 procedure I have just described, that is, by replacing 

 the terms defined by their definitions, we do not see 

 them melt into identities like the ordinary arguments ? 

 It is because the procedure is not applicable to them. 

 And why is this? Because their definitions are non- 

 predicative and present that kind of hidden vicious 

 circle I have pointed out above, and non-predicative 

 definitions cannot be substituted for the term defined. 

 Under these conditions. Logistic is no longer barren, it 

 engenders antinomies. 



It is the belief in the existence of actual infinity that 

 has given birth to these non-predicative definitions. I 

 must explain myself. In these definitions we find the 

 word «//, as we saw in the examples quoted above. 

 The word all has a very precise meaning when it is a 

 question of a finite * number of objects ; but for it still 

 to have a precise meaning when the number of the 

 objects is infinite, it is necessary that there should 

 exist an actual infinity. Otherwise all these objects 

 cannot be conceived as existing prior to their definition, 

 and then, if the definition of a notion N depends on 

 all the objects A, it may be tainted with the vicious 

 circle, if among the objects A there is one that cannot 

 be defined without bringing in the notion N itself 



The rules of formal logic simply express the pro- 

 perties of all the possible classifications. But in order 

 that they should be applicable, it is necessary that 

 these classifications should be immutable and not 

 require to be modified in the course of the argument. 

 If we have only to classify a finite number of objects, 

 it is easy to preserve these classifications without 



* The original has "infinite," obviously a slip. 



