CHAP. IV.] DIVISION OF PROPOSITIONS. 55 



xyz = opaque polished stones ; 



xy (1 - z) = opaque polished substances which are not stones; 

 # (1 - y) (1 - z) = opaque substances which are not polished, 

 and are not stones ; 



and so on for any other combination. Let it be observed, that 

 each of these expressions satisfies the same law of duality, as the 

 individual symbols which it contains. Thus, 



xyz x xyz = xyz ; 



xy (1 - z) x xy (1 - z) = xy (1 - z) ; 



and so on. Any such term as the above we shall designate as 

 a " class term," because it expresses a class of things by means 

 of the common properties or names of the individual members of 

 such class. 



Secondly, If we speak of a collection of things, different 

 portions of which are defined by different properties, names, or 

 attributes, the expressions for those different portions must be 

 separately formed, and then connected by the sign + . But if 

 the collection of which we desire to speak has been formed by 

 excluding from some wider collection a defined portion of its 

 members, the sign - must be prefixed to the symbolical expres- 

 sion of the excluded portion. Respecting the use of these sym- 

 bols some further observations may be added. 



6. Speaking generally, the symbol + is the equivalent of the 

 conjunctions " and," " or," and the symbol -, the equivalent of 

 the preposition " except." Of the conjunctions " and" and " or," 

 the former is usually employed when the collection to be de- 

 scribed forms the subject, the latter when it forms the predicate, 

 of a proposition. " The scholar and the man of the world de- 

 sire happiness," may be taken as an illustration of one of these 

 cases. " Things possessing utility are either productive of plea- 

 sure or preventive of pain," may exemplify the other. Now 

 whenever an expression involving these particles presents itself 

 in a primary proposition, it becomes very important to know 

 whether the groups or classes separated in thought by them are 

 intended to be quite distinct from each other and mutually ex- 

 clusive, or not. Does the expression, "Scholars and men of the 

 world," include or exclude those who are both ? Does the ex- 



