88 THE PRINCIPLES OF SCIENCE. [CHAP. 



Now by the Indirect method we obtain from (i) the 

 Contrapositive 



& = a, 

 and inserting in (2) the equivalent for & we have 



C = aC, (3) 



or " the letter w is not a vowel." 



Miscellaneous Examples of the Method. 



We can apply the Indirect Method of Inference however 

 many may be the terms involved or the premises con- 

 taining those terms. As the working of the method is 

 best learnt from examples, I will take a case of two 

 premises forming the syllogism Barbara : thus 



Iron is metal (i) 



Metal is element. (2) 



If we want to ascertain what inference is possible concern- 

 ing the term Iron, we develop the term by the Law of 

 Duality. Iron must be either metal or not-metal ; iron 

 which is metal must be either element or not-element ; 

 and similarly iron which is not-metal must be either 

 element or not-element. There are then altogether four 

 alternatives among which the description of iron must be 

 contained ; thus 



Iron, metal, element, (a) 



Iron, metal, not-element, (/3) 



Iron, not-metal, element, (7) 



Iron, not-metal, not-element. (6) 



Our first premise informs us that iron is a metal, and if 

 we substitute this description in (7) and (8) we shall have 

 self-contradictory combinations. Our second premise like- 

 wise 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 forms 



