56 DIVISION OF PROPOSITIONS. [CHAP. IV. 



pression, "Either productive of pleasure or preventive of pain," 

 include or exclude things which possess both these qualities ? I 

 apprehend that in strictness of meaning the conjunctions " and," 

 " or," do possess the power of separation or exclusion here re- 

 ferred to ; that the formula, " All #'s are either y's or z's," 

 rigorously interpreted, means, " All afs are either y's, but not z's," 

 or, ** z'a but not y's." But it must at the same time be admitted, 

 that the "jus et norma loquendi" seems rather to favour an oppo- 

 site interpretation. The expression, " Either y's or z's," would 

 generally be understood to include things that are y's and Z'B at 

 the same time, together with things which come under the one, 

 but not the other. Remembering, however, that the symbol + 

 does possess the separating power which has been the subject of 

 discussion, we must resolve any disjunctive expression which may 

 come before us into elements really separated in thought, and 

 then connect their respective expressions by the symbol + . 



And thus, according to the meaning implied, the expression, 

 " Things which are either xs or y's," will have two different sym- 

 bolical equivalents. If we mean, " Things which are #'s, but 

 not y's, or y's, but not xs" the expression will be 



the symbol x standing for #'s, y for y's. If, however, we mean, 

 " Things which are either #'s, or, if not #'s, then y's," the ex- 



pression will be 



x + y (1 - x). 



This expression supposes the admissibility of things which are 

 both #'s and y's at the same time. It might more fully be ex- 

 pressed in the form 



*y + x (1 - y) + y (1 - #) ; 



but this expression, on addition of the two first terms, only re- 

 produces the former one. 



Let it be observed that the expressions above given satisfy 

 the fundamental law of duality (III. 16). Thus we have 



{x (1 -y) + y (1 - x)}* = * (1 -y) + y (1 - *), 



It will be seen hereafter, that this is but a particular manifesta- 



