vi.] THE INDIEECT METHOD OF INFEEENCE. 89 



A = AB (I) 



B = BC. (2) 



By the Law of Duality we have 



A = AB.|-A& (3) 



A = AC -I- Ac. (4) 



Now, if we insert for A in the second side of (3) its 

 description in (4), we obtain what I shall call the develop- 

 ment of A with respect to B and C, namely 



A = ABC -I- ABc -I- AbC -|. Abe. (5) 



Wherever the letters A or B appear in the second side of 

 (5) substitute their equivalents given in (i) and (2), and 

 the results stated at full length are 



A = ABC -|. ABCc -I- AB&C + ABbCc. 



The last three alternatives break the Law of Contradiction, 

 so that 



A = ABC -I- o -I- o -I- o = ABC. 



This conclusion is, indeed, no more than we could obtain 

 by the direct process of substitution, that is by substituting 

 for B in (i), its description in (2) as in p. 55 ; it is the 

 characteristic of the Indirect process that it gives all 

 possible logical conclusions, both those which we have 

 previously obtained, and an immense number of others or 

 which the ancient logic took little or no account. From 

 the same premises, for instance, we can obtain a description 

 of the class not-element or c. By the Law of Duality we can 

 develop c into four alternatives, thus 



c = ABc -I- Abe ) aBc -|- ale. 

 If we substitute for A and B as before, we get 

 c = ABCc -I- AB&c -I- aBCc i abc, 



and, striking out the terms which break the Law of 

 Contradiction, there remains 



c = abc, 



or what is not element is also not iron and not metal. 

 This Indirect Method of Inference thus furnishes a 

 complete solution of the following problem Given any 

 number of logical premises or conditions, required the 

 description of any class of objects, or of any term, as 

 governed by those conditions. 



The steps of the process of inference may thus be 

 concisely stated 



i. By the Law of Duality develop the utmost number 

 of alternatives which may exist in the description of the 



