Nciv System of Logical Notation. 23 



but redundant in this place, though we shall find occasion 



for it further on. 



The form 



i.r = jy 



asserts the equivalence of x and y, and is Sir William 

 Hamilton's equation " all x is all j/," which he regards as 

 the fundamental form of proposition. A possible expression 

 for equivalence in this notation would be 

 \°x=y, or jc= i*'_y. 

 Contraposition is expressed with equal facility, by 

 changing the signs of the terms and transposing the co- 

 efficient without change of index : — thus, all the following 

 four forms of proposition are equivalents of each other. 

 The inverses are one above the other, and the contrapositives 

 in the same line — 



It will be noticed that the equation 



1-1=1, 

 which is true in arithmetic, is not generally true here. 



The most important application of this notation is to 

 the " logic of relatives," that is to say the theory of pro- 

 positions containing terms which signify relations. In what 

 follows, "absolute terms" or the terms between which 

 relations subsist — the terms of the old logic — are expressed 

 by Roman capitals, and relative terms by Italic capitals ; 

 and the corresponding negatives are expressed by the 

 corresponding small letters, as in De Morgan's notation. 

 " Of" is expressed by the sign of multiplication ; thus, let 

 A and B be the names of individuals, and let R mean the 

 relation of teacher, then 



A = i?xB 



