EXTENSION OF THE ORDINARY LOGIC. 93 



If A and B are both pupils of M, their relation is that 

 of fellow pupils of M. This relation is expressed by 



LM 



LM.' 



equivalent in arithmetic to 



which expression we shall adopt. In arithmetic^ every 

 term with zero index has the value of unity, so that if 



|=(LM)° 



we may eliminate, or drop, M, and write 



In logic we may eliminate in the same way ; that is to 

 say, if A and B are fellow pupils of M, they are fellow 

 pupils. But it is not true in logic that all terms with 

 zero index have the same value ; and from 



A 



we cannot infer that 



■— 7"° 



B-^ 



|=(LM)« 



We may drop at pleasure the absolute term M, but we 

 cannot insert or substitute a term, nor can we drop the 

 relative term L. This is analogous to the rule that in the 

 logic of absolute terms, if 



A=ABC, 

 it follows that 



A=ABandA=AC. 



But though we can thus drop a factor we cannot insert or 

 substitute one ; from 



A=AB 



