THE INDIRECT METHOD OF INFERENCE. 97 



Abstraction of Indifferent Circumstances. 



There is a simple but highly important process of 

 inference which enables us to abstract, eliminate or dis- 

 regard all circumstances indifferently present and absent. 

 Thus if I were to state that " a triangle is a three-sided 

 rectilinear figure, cither large or not large," these two 

 alternatives would be superfluous, because, by the Law of 

 Duality, I know that everything must be either large or 

 not large. To add the qualification gives no new know- 

 ledge, since the existence of the two alternatives will be 

 understood in the absence of any information to the 

 contrary. Accordingly, when two alternatives differ only 

 as regards a single component term which is positive in 

 one and negative in the other, we may reduce them to one 

 term by striking out their indifferent part. It is really a 

 process of substitution which enables us to do this ; for 

 having any proposition of the form 



A = ABC A ABc, (i) 



we know by the Law of Duality that 



AB=ABC -I- ABc. (2) 



As the second member of this is identical wito the second 

 member of (I) we may substitute, obtaining 

 A = AB. 



This process of reducing useless alternatives may be 

 applied again and again ; for it is plain that 



A = AB (CD -I- Cd A cD A cd) 



communicates no more information than that A is B. 

 Abstraction of indifferent terms is in fact the converse 

 process to that of development described in p. 89; and 

 it is one of the most important operations in the whole 

 sphere of reasoning. 



The reader should observe that in the proposition 



AC = BC 

 we cannot abstract C and infer 



A = B; 

 but from 



AC -|- Ac = BC A Be 

 we may abstract all reference to the term C. 



It ought to be carefully remarked, however, that alter- 

 natives which seem to be without meaning often imply 

 important knowledge. Thus if I say that " a triangle is a 



ij 



