316 MR. JOSEPH JOHN MURPHY ON ADDITION 



6. Every thing is either A or B ; 

 A includes B ; 



A is coextensive with the universe whereof B is a 

 member. 



We return to the multiplication of relatives. If we 

 multiply these four relatives by the logical negative, and 

 inversely, we get the following eight canonical equations, 

 where it will be seen that the inverse order of multipli- 

 cation gives the contrapositive result. 



1. (-i)xL =M. 5. L x{-i)=N. 



2. (-i)xL-'=iV. 6. L-'x{-j)=M. 



3. (-i)xAr =L-\ y. N x{-i)=L. 



4. (-i)xM =L. 8. M x{-i)=L-\ 



That is to say : — 



1 . Negative of enclosure is alternative. 



2. Negative of includent is excludent. 



3. Negative of excludent is includent. 



4. Negative of alternative is enclosure. 



5. Enclosure of negative is excludent. 



6. Includent of negative is alternative. 



7. Excludent of negative is enclosure. 



8. Alternative of negative is includent. 



We have seen that in two important properties N and 

 M are analogous to negative unity. If we call — i, iV, 

 and M negative terms, and L and L'" positive ones, we 

 shall see that the foregoing equations conform to the rule 

 that like signs by multiplication produce +, and unlike 

 signs — . 



