OF ARTS AND SCIENCES: SEPTEMBER 10, 1867. 409 



by Theorem ix. 



x,(l-\-a)==x(l-\-a)==xl-\-xa==x-\-xa 



Whence x a == 0, while neither x nor a is zero, which, as will 

 appear directly, is impossible. 



XII. 



0,a = 



Proof. — For call ,a == x. Then by definition 3 

 x belongs to the class zero. 

 .'. by definition 4 a: = 0. 



Corollary 1. — The same reasoning applies to arithmetical multi- 

 plication. 



Corollary 2. — From Theorem x. and the last corollary it follows 

 that if a 6 == 0, either a == or 5 = 0. 



XIII. 



a ,a == a. 



XIV. 



a -\r a = a. 



These do not hold with ai'ithmetical operations. 



General Scholium. — This concludes the theorems relating to the 

 direct operations. As the inverse operations have no peculiar logical 

 interest, they are passed over here. 



In order to prevent misapprehension, I will remark that I do not 

 undertake to demonstrate the principles of logic themselves. Indeed, 

 as I have shown in a previous paper, these principles considered as 

 speculative truths are absolutely empty and indistinguishable. But 

 what has been pi'oved is the maxims of logical procedure, a certain 

 system of signs being given. 



The definitions given above for the processes which I have termed 

 arithmetical plainly leave the functions of these operations in many 

 cases uninterpreted. Thus if we write 



VOL. VII. 52 



