OF ARTS AND SCIENCES : NOVEMBER 13, 1867. 429 



that the three conceptions of reference* to a ground, reference to a cor- 

 relate, and references to an interpretant, are those of which logic must 

 principally make use. I there also introduced the term " symbol," to 

 include both concept and word. Logic treats of the reference of 

 symbols in general to their objects. A symbol, in its reference to its 

 object, has a triple reference : — 



1st., Its direct reference to its object, or the real things which it 

 represents ; 



2d., Its reference to its ground through its object, or the common 

 characters of those objects ; 



3d., Its reference to its interpretant through its object, or all the 

 facts known about its object. 



What are thus referred to, so far as they are known, are : — 



1st., The informed breadth of the symbol; 



2d., The informed depth of the symbol ; 



3d., The sum of synthetical propositions in which the symbol is 

 subject or predicate, or the information concerning the symbol. 



By breadth and depth, without an adjective, I shall hereafter mean 

 the informed breadth and depth. 



It is plain that the breadth and depth of a symbol, so far as they are 

 not essential, measure the information concerning it, that is, the synthet- 

 ical propositions of which it is subject or predicate. This follows 

 directly from the definitions of breadth, depth, and information. Hence 

 it follows: — 



1st., That, as long as the information remains constant, the greater 

 the breadth, the less the depth ; 



2d., That every increase of information is accompanied by an in- 

 crease in depth or breadth, independent of the other quantity ; 



3d., That, when there is no information, there is either no depth or 

 no breadth, and conversely. 



These are the true and obvious relations of breadth and depth. They 

 will be naturally suggested if we term the information the area, and 

 write — 



Breadth X Depth = Area. 



If we learn that S is P, then, as a general rule, the depth of S is 

 increased without any decrease of breadth, and the breadth of P 

 is increased without any decrease of depth. Either increase may be 

 certain or doubtful. 



