DISJUNCTIVE PROPOSITIONS. 93 



A = gem 

 B = rare stone 

 C = beautiful stone, 

 the proposition ( i ) is of the form 



A = B | C 



hence AB = B -|- BC 



and AC = BC -I- C ; 



but these inferences are not equivalent to the false ones 

 (2) and (3). 



We can readily represent such disjunctive reasoning, when 

 it is valid, by expressing the inconsistency of the alterna- 

 tives explicitly. Thus if we resort to our instance of 



Water is either salt or fresh, 

 and take A = Water 



B = salt 

 C = fresh, 

 then the premise is apparently of the form 



A = AB | AC ; 



but in reality there are the unexpressed conditions that 

 * what is salt is not fresh/ and ' what is fresh is not salt ; ' 

 or, in letter-terms, 



B = Be 



C = 60. 



Now, if we substitute these descriptions in the original 

 proposition, we obtain 



A = ABc | A6C ; 

 uniting B to each side we infer 



AB = ABc l- AB6C 

 or AB = ABc ; 



that is, 



Water which is salt is water salt and not fresh. 



I should weary the reader if I attempted to illustrate 



the multitude of forms which disjunctive reasoning may 



take ; and as in the next chapter we shall be constantly 



treating the subject, I must here restrict myself to a single 



