i8o SCIENCE AND METHOD. 



the very least ; he is preparing to alter these laws and 

 to revoke a certain number of them. If he succeeds, 

 I shall give credit to Mr. Russell's intuition, and not to 

 Peanian Logistic, which he will have destroyed. 



III. 



Liberty of Contradiction. 



I offered two principal objections to the definition 

 of the whole number adopted by the logisticians. 

 What is M. Couturat's answer to the first of these 

 objections ? 



What is the meaning in mathematics of the word 

 to exist? It means, I said, to be free from contradic- 

 tion. This is what M. Couturat disputes. " Logical 

 existence," he says, "is quite a different thing from 

 absence of contradiction. It consists in the fact that 

 a class is not empty. To say that some ^'s exist is, 

 by definition, to assert that the class a is not void." 

 And, no doubt, to assert that the class a is not void 

 is, by definition, to assert that some ds, exist. But 

 one of these assertions is just as destitute of meaning 

 as the other if they do not both signify either that 

 we can see or touch a, which is the meaning given 

 them by physicists or naturalists, or else that we can 

 conceive of an a without being involved in contradic- 

 tions, which is the meaning given them by logicians 

 and mathematicians. 



In M. Couturat's opinion it is not non-contradiction 

 that proves existence, but existence that proves non- 

 contradiction. In order to establish the existence of a 

 class, we must accordingly establish, by an example, 

 that there is an individual belonging to that class. 



