OF THE LAWS OF THOUGHT. 1/9 



/(0)/(lJ =0: ...(22) 



"whicli is the result of the elimination of a; from equation (21). We 

 cannot pause to give examples of the use of the formula (22) ; but 

 we must quote an interpretation of it, -viewed as the result of the 

 elimination of ;» from (2L), which strikes us as extremely elegant. The 

 formula implies that either/ (0) = 0, or/(l) = 0. Now the latter 

 equation/ (1) = expresses what the given proposition / (a) = 

 would become if x made up the universe; and the former/ (0) = 

 expresses what the given proposition would become if x had no 

 existence. Hence, (22) being derived from (21), it follows that what 

 is equally true whether a given class of objects embraces the whole 

 universe or disappears from existence^ is independent of that class 

 altogether. 



The principle of elimination is extended by our author to groups 

 of equations, by the following process. Let 



2^= 0, 1 



F=0, ■ 



. = o,| (^«> 



......... J 



be a series of equations, in which T, U, V, &c., are functions of the 

 concept X. Then 



T^ + V" + U^ -h &c. = 0. .......(24) 



It is shown by Professor Boole that the combined interpretation of 

 the system of equations (23) is involved in the single equation (24). 

 Indeed, had all the terms in the developments of T, V, U, &c., been 

 sucb as to satisfy the Law of Duality, it would have been sufficient 

 to have written 



T -\- ¥ + Z7 + &c. = 0. 



In order now to eliminate x from the group (23), it is sufficient to 

 eliminate it, by tbe method described in the preceding paragraph, 

 from the single equation (24) ; and, if the result be 



]r= 0, 



this equation will involve all the conclusions that can legitimately be 

 derived from the series of equations (23) with regard to the mutual 

 relations of the concepts, exclusive of x, which enter into these 

 equations. 



We do not see how it is possible for any one not blinded by pre- 

 judice against every thing like an alliance of Logic with formulae and 



