34 SIGNS AND THEIR LAWS. [CHAP. II. 



then the conception of " All men except Asiatics" will be ex- 

 pressed by x - y. And if we represent by #, " states," and by 

 y the descriptive property " having a monarchical form," then 

 the conception of " All states except those which are monarchi- 

 cal" will be expressed by x - xy. 



As it is indifferent for all the essential purposes of reasoning 

 whether we express excepted cases first or last in the order of 

 speech, it is also indifferent in what order we write any series of 

 terms, some of which are affected by the sign -. Thus we have, 

 as in the common algebra, 



x-y = -y + X' (5) 



Still representing by x the class "men," and by y "Asiatics," 

 let z represent the adjective " white." Now to apply the adjec- 

 tive " white" to the collection of men expressed by the phrase 

 " Men except Asiatics," is the same as to say, " White men, 

 except white Asiatics." Hence we have 



z(x-y) = zx- zy. (6) 



This is also in accordance with the laws of ordinary algebra. 



The equations (4) and (6) may be considered as exemplifica- 

 tion of a single general law, which may be stated by saying, that 

 the literal symbols, x, y, 2, fyc. are distributive in their operation. 

 The general fact which that law expresses is this, viz. : If any 

 quality or circumstance is ascribed to all the members of a group, 

 formed either by aggregation or exclusion of partial groups, the 

 resulting conception is the same as if the quality or circumstance 

 were first ascribed to each member of the partial groups, and the 

 aggregation or exclusion effected afterwards. That which is 

 ascribed to the members of the whole is ascribed to the members 

 of all its parts, howsoever those parts are connected together. 



CLASS III. 



12. Signs by which relation is expressed, and by which we 

 form propositions. 



Though all verbs may w T ith propriety be referred to this class, 

 it is sufficient for the purposes of Logic to consider it as includ- 

 ing only the substantive verb is or are, since every other verb 



