188 THE PRINCIPLES OF SCIENCE. 



into the form of an equality ; but the converse is not true. 

 We cannot possibly prove that two quantities are equal by 

 merely asserting that they are both greater or both less 

 than another quantity. From e >f and g >f, or e </and 

 9 <f> we can infer no relation between e and g. And if the 

 reader take the equations x = y = 3 and attempt to prove 

 that therefore x = 3, by throwing them into inequalities, he 

 will find it impossible to do so. 



From these considerations I gather that reasoning in 

 arithmetic or algebra by so-called inequalities is only an 

 imperfectly expressed reasoning by equalities, and when 

 we want to exhibit exactly and clearly the conditions of 

 reasoning, we are obliged to use equalities explicitly. Just 

 as in pure logic a negative proposition, as expressing mere 

 difference, cannot be the means of inference, so inequality 

 can never really be the true ground of inference. I do not 

 deny that affirmation and negation, agreement and differ- 

 ence, equality and inequality, are pairs of equally funda- 

 mental relations, but I assert that inference is possible only 

 where affirmation, agreement, or equality, some species of 

 identity in fact, is present, explicitly or implicitly. 



Arithmetical Reasoning. 



It might seem somewhat inconsistent that I assert 

 number to arise out of difference or discrimination, and 

 yet hold that no reasoning can be grounded on difference. 

 Number, of course, opens a most wide sphere for inference, 

 and a little consideration shows that this is due to the 

 unlimited series of identities which spring up out of 

 numerical abstraction If six people are sitting on six 

 chairs, there is no resemblance between the chairs and the 

 people in logical character. But if we overlook all the 

 qualities both of a chair and a person, and merely re- 

 meml>er that there are marks bv which each of six chairs 



