AND MULTIPLICATION OF LOGICAL RELATIVES. 215 



We have next to consider the addition of these rela- 

 tives. In what follows^ U means coextension with the 

 universe, and U~^ membership of the universe wherewith 

 the other absolute term is coextensive ; i, as before, means 

 identity, and — i the logical negative, or the relation be- 

 tween A and whatever is not A. Six pairs from the four 

 terms are added together, as follows : — 



I. L + L-'=i. 2. N+M =-i. 



3. N+L =0. 4. N+L-^=o. 



5. M+L =U-\ 6. M+L-'=U. 



These canonical equations may be called syllogisms, 

 because they give the resultant of two propositions ; but 

 they are not syllogisms in the technical sense, because 

 they do not eliminate any middle term. Their interpre- 

 tations are as follows : — 



1 . A is included in B ; 

 A includes B ; 



A and B are identical. 



2. Nothing is both A and B ; 

 Every thing is either A or B ; 



A and B are the negatives of each other. 



3. Nothing is both A and B ; 

 A is included in B ; 



One or both of the premises must be untrue. 



4. Nothing is both A and B; 

 A includes B ; 



One or both of the premises must be untrue. 



5. Every thing is either A or B ; 

 A is included in B ; 



A is a member of the universe wherewith B is coex- 

 tensive. 



