CHAP. IV.] DIVISION OF PROPOSITIONS. 61 



thought, and regard the right employment of language as both 

 its instrument and its safeguard. 



1 1 . Let us consider next the case in which the predicate of 

 the proposition is particular, e. g. " All men are mortal." 



In this case it is clear that our meaning is, " All men are 

 some mortal beings," and we must seek the expression of the 

 predicate, " some mortal beings." Represent then by v, a class 

 indefinite in every respect but this, viz., that some of its members 

 are mortal beings, and let x stand for "mortal beings, "then will 

 vx represent " some mortal beings." Hence if y represent men, 

 the equation sought will be 



y = vx. 







From such considerations we derive the following Rule, for 

 expressing an affirmative universal proposition whose predicate 

 is particular : 



RULE. Express as before the subject and the predicate, attach 

 to the latter the indefinite symbol v, and equate the expressions. 



It is obvious that v is a symbol of the same kind as x, y, &c., 

 and that it is subject to the general law, 



v z = v, or v (1 - v) = 0. 



Thus, to express the proposition, " The planets are either 

 primary or secondary," we should, according to the rule, proceed 

 thus: 



Let x represent planets (the subject) ; 

 y = primary bodies ; 

 z = secondary bodies ; 



then, assuming the conjunction "or" to separate absolutely the 

 class of " primary" from that of " secondary" bodies, so far as 

 they enter into our consideration in the proposition given, we 

 find for the equation of the proposition 



x = v{y(l-z) + z(l-y)}. (4) 



It may be worth while to notice, that in this case the literal 

 translation of the premises into the form 



x = v (y + z) (5) 



