vni.] PRINCIPLES OF NUMBER 167 



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 

 difference, equality and inequality, are pairs of equally 

 fundamental relations, but I assert that inference is pos- 

 sible only where affirmation, agreement, or equality, some 

 species of identity in fact, is present, explicitly or implicitly. 



Arithmetical Reasoning. 



It may 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 remember that 

 there are marks by which each of six chairs may be 

 discriminated from the others, and similarly with the 

 people, then there arises a resemblance between the chairs 

 and the people, and this resemblance in number may be 

 the ground of inference. If on another occasion the chairs 

 are filled by people again, we may infer that these people 

 resemble the others in number though they need not 

 resemble them in any other points. 



Groups of units are what we really treat in arithmetic. 

 The number Jive is really 1 + 1 + 14 I + I, but for the 

 sake of conciseness we substitute the more compact sign 

 5, or the name five. These names being arbitrarily im- 

 posed in any one manner, an infinite variety of relations 

 spring up between them which are not in the least 

 arbitrary. If we define four as I + I + I + I, and Jive 

 as i + i + i + i + i, then of course it follows that 

 five = four + i ; but it would be equally possible to take 

 this latter equality as a definition, in which case one of 

 the former equalities would become an inference. It is 



