CHAP. III.] DERIVATION OF THE LAWS. 47 



universally subject to the law whose expression is a; 2 = x. Of 

 the symbols of Number there are two only, and 1, which sa- 

 tisfy this law. But each of these symbols is also subject to a law 

 peculiar to itself in the system of numerical magnitude, and this 

 suggests the inquiry, what interpretations must be given to the 

 literal symbols of Logic, in order that the same peculiar and 

 formal laws may be realized in the logical system also. 



PROPOSITION II. 



13. To determine the logical value and significance of the 

 symbols and 1. 



The symbol 0, as used in Algebra, satisfies the following for- 

 mal law, 



x y = 0, or Oy = 0, (1) 



whatever number y may represent. That this formal law may be 

 obeyed in the system of Logic, we must assign to the symbol 

 such an interpretation that the class represented by Oy may be 

 identical with the class represented by 0, whatever the class y 

 may be. A little consideration will show that this condition is 

 satisfied if the symbol represent Nothing. In accordance with 

 a previous definition, we may term ^No thing a class. In fact, 

 Nothing and Universe are the two limits of class extension, for 

 they are the limits of the possible interpretations of general 

 names, none of which can relate to fewer individuals than are 

 comprised in Nothing, or to more than are comprised in the 

 Universe. Now whatever the class y may be, the individuals 

 which are common to it and to the class " Nothing" are identi- 

 cal with those comprised in the class " Nothing," for they are 

 none. And thus by assigning to the interpretation Nothing, 

 the law (1) is satisfied; and it is not otherwise satisfied consis- 

 tently with the perfectly general character of the class y. 



Secondly, The symbol 1 satisfies in the system of Number 

 the following law, viz., 



1 x y = y-> or \y = y, 



whatever number y may represent. And this formal equation 

 being assumed as equally valid in the system of this work, in 



