374 



MEMOIRS OF THE AMERICAN ACADEMY. 



horse is black, so h 



means that some horse is black. We may 



the 



1 



particular affirmative l(h,b) =as 1, and 'the particular negative l(h,n h ) 



Given the premises, every horse is black, and every horse is an animal ; required 

 the conclusion. We have given 



h 



b 



h 



a. 



Commutatively multiplying, we get 



h 



Then, by (92) or by (90), 



a,b . 



0a.b 



o b , 



or 



lh 



l(a,b). 



Hence, by (40) or by (46), 



If h > 0*,b 



0, 



or 



l(8,b): 



1; 



or if there are any horses, some animals are black. I think it would be difficult tc 

 reach this conclusion, by Boole's method unmodified. 



Particular propositions may also be expressed by means of the signs of inequality 

 Thus, some animals are horses, may be written 





a,h>0; 



and the conclusion required in the above problem might have been obtained in th 

 form, very easily, from the product of the premises, by (1) and (21). 



We shall presently 



that conditional and disjunctive propositions may 



be 



pressed in a different way 



Conjugative Terms. 



terms presents considerable difficulty, and would 



I have, however, studied this part 



The treatment of conjugative terms 

 doubt be greatly facilitated by algebraic dev 

 of my notation but little. 



A relative term cannot possibly be reduced to any combination of absolute terms', 

 nor can a conjugative term be reduced to any combination of simple relatives; but a 



i than two correlates can always be reduced to a combination 



mor 



* 



of conj ugatives of 

 may always subst 



Thus for "winner over of 



from 



to — ," we 



a 



u, or *• gainer 



tself to be 



of the ad van tag 



to 



ther 



ol 



from 



»j 



Then we may 



o 



," where the first 

 the advantage of winning over 



UL 



llu . 



