62 DIVISION OF PROPOSITIONS. [CHAP. IV. 



would be exactly equivalent, v being an indefinite class symbol. 

 The form (4) is, however, the better, as the expression 



consists of terms representing classes quite distinct from each 

 other, and satisfies the fundamental law of duality. 



If we take the proposition, " The heavenly bodies are either 

 suns, or planets, or comets," representing these classes of things 

 by w 9 #, /, z, respectively, its expression, on the supposition that 

 none of the heavenly bodies belong at once to two of the divi- 

 sions above mentioned, will be 



If, however, it were meant to be implied that the heavenly 

 bodies were either suns, or, if not suns, planets, or, if neither suns 

 nor planets, fixed stars, a meaning which does not exclude the 

 supposition of some of them belonging at once to two or to all 

 three of the divisions of suns, planets, and fixed stars, the ex- 

 pression required would be 



v> = v[x+y(l-a:) + z(l-x) (1 -y)}. (6) 



The above examples belong to the class of descriptions, not 

 definitions. Indeed the predicates of propositions are usually 

 particular. When this is not the case, either the predicate is a 

 singular term, or we employ, instead of the copula " is" or " are," 

 some form of connexion, which implies that the predicate is to be 

 taken universally. 



12. Consider next the case of universal negative propositions, 

 e. g. " No men are perfect beings." 



Now it is manifest that in this case we do not speak of a class 

 termed "no men," and assert of this class that all its members 

 are " perfect beings." But we virtually make an assertion about 

 " all men' to the effect that they are " not perfect beings." Thus 

 the true meaning of the proposition is this : 



1 " All men (subject) are (copula) not perfect (predicate) ;" 

 whence, if y represent "men," and x "perfect beings," we shall 

 have 



y = v(l -#), 



