DISJUNCTIVE PROPOSITIONS. 93 



A = gem 

 B = rare stone 

 C = beautiful stone, 

 the proposition ( i ) is of the form 



A = B |- C 



hence AB = B f 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 -I- 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 



uniting B to each side we infer 



AB = ABc | AB&C 

 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 rnay 

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

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



