THE NEW LOGICS. 173 



not do so if we had n = n-i, for then subtraction 

 would always give us the same number. Thus, then, 

 the way in which we have been brought to consider 

 this number n involves a definition of the finite whole 

 number, and this definition is as follows: a finite 

 whole number is that which can be obtained by suc- 

 cessive additions, and which is such that n is not equal 

 to n-\. 



This being established, what do we proceed to do ? 

 We show that if no contradiction has occurred up to 

 the n*^ syllogism, it will not occur any the more at 

 the {n-\- ly^, and we conclude that it will never occur. 

 You say I have the right to conclude thus, because 

 whole numbers are, by definition, those for which such 

 reasoning is legitimate. But that involves another 

 definition of the whole number, which is as follows : 

 a whole number is that about which we can reason by 

 recurrence. In the species it is that of which we can 

 state that, if absence of contradiction at the moment 

 of occurrence of a syllogism whose number is a whole 

 number carries with it the absence of contradiction 

 at the moment of occurrence of the syllogism whose 

 number is the following whole number, then we need 

 not fear any contradiction for any of the syllogisms 

 whose numbers are whole numbers. 



The two definitions are not identical. They are 

 equivalent, no doubt, but they are so by virtue of an 

 a priori synthetic judgment; we cannot pass from 

 one to the other by purely logical processes. Con- 

 sequently, we have no right to adopt the second after 

 having introduced the whole number by a road which 

 presupposes the first. 



On the contrary, what happens in the case of the 



