104 THE PRINCIPLES OF SCIENCE. 



likewise informs us that metal is element, and applying 

 this description to (/3) we again have self-contradiction, 

 so that there remains only (a) as a description of iron 

 our inference is 



Iron = iron, metal, element. 

 To represent this process of reasoning in general 



symbols, let 



A = iron 



B = metal 



C = element. 

 The premises of the problem take the form 



A = AB (i) 



B - BC. (2) 



By the Law of Duality we have 



A = AB -I- A6 (3) 



A = AC ! 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 

 development of A 



A = ABO | ABc -I- A6C -I- 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 at full length are 



A = ABC -I- ABCc l- AB&C | AB&Cc. 

 The last three alternatives break the Law of Contradic- 

 tion, so that 



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



A = ABC. 



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

 the direct process of substitution ; it is the characteristic 

 of the Indirect process that it gives all possible logical 

 conclusions, both those which we have previously obtained, 

 and an almost infinite number of others of which the 

 ancient logic took little or no account. From the same 

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



