CHAP. VII.] OF ELIMINATION. 99 



CHAPTER VII. 



ON ELIMINATION. 



1. TN the examples discussed in the last chapter, all the ele- 

 -- ments of the original premiss re-appeared in the conclusion, 

 only in a different order, and with a different connexion. But it 

 more usually happens in common reasoning, and especially when 

 we have more than one premiss, that some of the elements are 

 required not to appear in the conclusion. Such elements, or, as 

 they are commonly called, " middle terms," may be considered 

 as introduced into the original propositions only for the sake of 

 that connexion which they assist to establish among the other 

 elements, which are alone designed to enter into the expression of 

 the conclusion. 



2. Respecting such intermediate elements, or middle terms, 

 some erroneous notions prevail. It is a general opinion, to which, 

 however, the examples contained in the last chapter furnish a con- 

 tradiction, that inference consists peculiarly in the elimination of 

 such terms, and that the elementary type of this process is exhi- 

 bited in the elimination of one middle term from two premises, so as 

 to produce a single resulting conclusion into which that term does 

 not enter. Hence it is commonly held, that syllogism is the basis, 

 or else the common type, of all inference, which may thus, how- 

 ever complex its form and structure, be resolved into a series of 

 syllogisms. The propriety of this view will be considered in a 

 subsequent chapter. At present I wish to direct attention to an 

 important, but hitherto unnoticed, point of difference between 

 the system of Logic, as expressed by symbols, and that of com- 

 mon algebra, with reference to the subject of elimination. In 

 the algebraic system we are able to eliminate one symbol from 

 two equations, two symbols from three equations, and generally 

 n - 1 symbols from n equations. There thus exists a definite 

 connexion between the number of independent equations given, 



H 2 



