152 SCIENCE AND METHOD. 



have one meaning ; it signifies exemption from 

 contradiction. Thus rectified, Mill's thought becomes 

 accurate. In defining an object, we assert that the 

 definition involves no contradiction. 



If, then, we have a system of postulates, and if we 

 can demonstrate that these postulates involve no 

 contradiction, we shall have the right to consider 

 them as representing the definition of one of the 

 notions found among them. If we cannot demon- 

 strate this, we must admit it without demonstration, 

 and then it will be an axiom. So that if we wished 

 to find the definition behind the postulate, we should 

 discover the axiom behind the definition. 



Generally, for the purpose of showing that a 

 definition does not involve any contradiction, we 

 proceed by example, and try to form an example of 

 an object satisfying the definition. Take the case 

 of a definition by postulates. We wish to define a 

 notion A, and we say that, by definition, an A is 

 any object for which certain postulates are true. If 

 we can demonstrate directly that all these postulates 

 are true of a certain object B, the definition will be 

 justified, and the object B will be an example of A. 

 We shall be certain that the postulates are not 

 contradictory, since there are cases in which they 

 are all true at once. 



But such a direct demonstration by example is 

 not always possible. Then, in order to establish 

 that the postulates do not involve contradiction, we 

 must picture all the propositions that can be de- 

 duced from these postulates considered as premises, 

 and show that among these propositions there are 

 no two of which one is the contradiction of the 



