78 THE PRINCIPLES OF SCIENCE. [chap. 



one alternative, and thence infeis the denial of the rest, 

 cannot be held true in this system. If I say, indeed, that 



Water is either salt or fresh water, 

 it seems evident that " water which is salt is not fresh." 

 But this inference really proceeds from our knowledge that 

 water cannot be at once salt and fresh. This inconsistency 

 of the alternatives, as I have fully shown, will not always 

 hold. Thus, if I say 



Gems are either rare stones or beautiful stones, (i) 

 it will obviously not follow that 



A rare gem is not a beautiful stone, (2) 



nor that 



A beautiful gem is not a rare stone. (3) 



Our symbolic method gives only true conclusions ; for if 

 we take 



A = gem 

 B = rare stone 

 C = beautiful stone, 

 the proposition (i) is of the form 



A = B .|- C 

 hence AB = B -I- BC 



and AC = BC -I- C ; 



but these inferences are not equivalent to the false ones 

 (2) and (3). 



We can readily represent disjunctive reasoning by the 

 modus ponendo tollens, when it is valid, by expressing the 

 inconsistency of the alternatives 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 = A 13 -I- AC ; 

 but in reality there is an unexpressed condition that " what 

 is salt is not fresh," from winch follows, by a process of 

 inference to be afterwards described, that " what is fresh 

 is not salt." We have then, in letter-terms, the two pro- 

 positions 



B = Bc 

 C^^'C. 

 If we substitute these descriptions in the original pro- 

 position, we obtain 



