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 x = 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 b = 0, either a == or b = 0. 



xm. 



a ,a — a. 



XIV. 



a -\r a == a. 



These do not hold with arithmetical 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 proved 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 



