CHAP. VII.] OF ELIMINATION. 101 



PROPOSITION I. 



5. If f(x) = be any logical equation involving the class symbol 

 with or without other class symbols, then will the equation 



be true, independently of the interpretation of x ; and it will be the 

 complete result of the elimination of x from the above equation. 



In other words, the elimination of a; from any given equation, 

 f(x)=0 } will be effected by successively changing in that equation x into 

 1, and x into 0, and multiplying the two resulting equations together. 



Similarly the complete result of the elimination of any class sym- 

 bols, x, y, fyc.,from any equation of the form V= 0, will be obtained 

 by completely expanding the first member of that equation in con- 

 stituents of the given symbols, and multiplying together all the coeffi- 

 cients of those constituents, and equating the product to 0. 



Developing the first member of the equation f(x) = 0, we 

 have (V. 10), 



{/(1)-/(0)}*+/(0) = 0. (1) 



/(Q) 





and /(I) 



'/(<>) -/(I)' 



Substitute these expressions for x and 1 - x in the fundamental 

 equation 



x (1 - x) = 0, 

 and there results 



/(0)/(1) -Q. 



(/(O) -/(!))'" 



or, /(1)/(0)-0, (2) 



the form required. 



6. It is seen in this process, that the elimination is really effected 

 between the given equation f(x) = and the universally true 

 equation x (1 - x) = 0, expressing the fundamental law of logical 

 symbols, qua logical. There exists, therefore, no need of more 



