CHAP. IV.] DIVISION OF PROPOSITIONS. 57 



tion of a general law of expressions representing " classes or 

 collections of things." 



7. The results of these investigations may be embodied in 

 the following rule of expression. 



RULE. Express simple names or qualities by the symbols x,y, z, 

 Sfc., their contraries by 1 - x, 1 - y, 1 - z, fyc.; classes of tilings 

 defined by common names or qualities, by connecting the correspond- 

 ing symbols as in multiplication ; collections of things, consisting of 

 portions different from each other, by connecting the expressions of 

 those portions by the sigti+. In particular, let the expression, " Either 

 xs orys," be expressed by # (1 - y) + y(\- x), when the classes de- 

 noted by x and y are exclusive, by x + y (1 x) when they are not 

 exclusive. Similarly let the expression, "Either x's, or y"s, or z's" be 

 expressed byx(\-y)(L-z) + y(\-x)(\-z) + z(\-x)(\-y), 

 when the classes denoted by x, y, and z, are designed to be mutually 

 exclusive, by x + y (1 - x) + z (1 - x) (\-y), when they are not meant 

 to be exclusive, and so on. 



8. On this rule of expression is founded the converse rule of 

 interpretation. Both these will be exemplified with, perhaps, 

 sufficient fulness in the following instances. Omitting for bre- 

 vity the universal subject " things," or " beings," let us assume 



x = hard, y = elastic, z = metals ; 

 and we shall have the following results : 



" Non-elastic metals," will be expressed by z (1 ?/) ; 

 " Elastic substances with non-elastic metals," by y + z (1 - y) ; 



" Hard substances, except metals," by x -y ; 

 " Metallic substances, except those which are neither hard nor 

 elastic," by z -z (1 - x) (1 - y\ or by z {1 - (1 - #) (l-y)} 9 

 vide (6), Chap. II. 



In the last example, what we had really to express was " Metals, 

 except not hard, not elastic, metals." Conjunctions used be- 

 tween adjectives are usually superfluous, and, therefore, must 

 not be expressed symbolically. 



Thus, " Metals hard and elastic," is equivalent to " Hard 

 elastic metals," and expressed by xyz. 



Take next the expression, " Hard substances, except those 



