168 THE PRINCIPLES OF SCIENCE. [CHAP. 



hardly requisite to decide how we define the names of 

 numbers, provided we remember that out of the infinitely 

 numerous relations of one number to others, some one 

 relation expressed in an equality must be a definition of 

 the number in question and the other relations imme- 

 diately become necessary inferences. 



In the science of number the variety of classes which 

 can be formed is altogether infinite, and statements of 

 perfect generality may be made subject only to difficulty 

 or exception at the lower end of the scale. Every existing 

 number for instance belongs to the class ra + 7 ; that is, 

 every number must be the sum of another number and 

 seven, except of course the first six or seven numbers, 

 negative quantities not being here taken into account. 

 Every number is the half of some other, and so on. The 

 subject of generalization, as exhibited in mathematical 

 truths, is an infinitely wide one. In number we are only 

 at the first step of an extensive series of generalizations. 

 As number is general compared with the particular things 

 numbered, so we have general symbols for numbers, and 

 general symbols for relations between undetermined 

 numbers. There is an unlimited hierarchy of successive 

 generalizations. 



Numerically Definite Reasoning, 



It was first discovered by De Morgan that many argu- 

 ments are valid which combine logical and numerical 

 reasoning, although they cannot be included in the 

 ancient logical formulas. He developed the doctrine of 

 the " Numerically Definite Syllogism," fully explained in 

 his Formal Logic (pp. 141 i/o). Boole also devoted 

 considerable attention to the determination of what' he 

 called " Statistical Conditions," meaning the numerical 

 conditions of logical classes. In a paper published among 

 the Memoirs of the Manchester Literary and Philosophical 

 Society, Third Series, voL IV. p. 330 (Session 1869 70), 

 I have pointed out that we can apply arithmetical calcula- 

 tion to the Logical Alphabet. Having given certain logical 

 conditions and the numbers of objects in certain classes, 

 we can either determine the numbers of objects in other 

 classes governed by those conditions, or can show what 



