8 NATURE AND DESIGN OF THIS WORK. [CHAP. I. 



spect to propositions. Now let those things or those propositions 

 among which relation is expressed be termed the elements of 

 the propositions by which such relation is expressed. Proceed- 

 ing from this definition, we may then say that the premises of any 

 logical argument express given relations among certain elements, 

 and that the conclusion must express an implied relation among 

 those elements, or among a part of them, i. e. a relation implied 

 by or inferentially involved in the premises. 



8. Now this being premised, the requirements of a general 

 method in Logic seem to be the following : 



1st. As the conclusion must express a relation among the 

 whole or among a part of the elements involved in the premises, 

 it is requisite that we should possess the means of eliminating 

 those elements which we desire not to appear in the conclusion, 

 and of determining the whole amount of relation implied by the 

 premises among the elements which we wish to retain. Those 

 elements which do not present themselves in the conclusion are, 

 in the language of the common Logic, called middle terms ; and 

 the species of elimination exemplified in treatises on Logic consists 

 in deducing from two propositions, containing a common element 

 or middle term, a conclusion connecting the two remaining terms. 

 But the problem of elimination, as contemplated in this work, 

 possesses a much wider scope. It proposes not merely the elimi- 

 nation of one middle term from two propositions, but the elimi- 

 nation generally of middle terms from propositions, without 

 regard to the number of either of them, or to the nature of their 

 connexion. To this object neither the processes of Logic nor 

 those of Algebra, in their actual state, present any strict parallel. 

 In the latter science the problem of elimination is known to be 

 limited in the following manner : From two equations we can 

 eliminate one symbol of quantity; from three equations two 

 symbols ; and, generally, from n equations n \ symbols. But 

 though this condition, necessary in Algebra, seems to prevail in 

 the existing Logic also, it has no essential place in Logic as a 

 science. There, no relation whatever can be proved to prevail 

 between the number of terms to be eliminated and the number 

 of propositions from which the elimination is to be effected. 

 From the equation representing a single proposition, any num- 



