92 THE PRINCIPLES OF SCIENCE. 



canine, bicuspid, or molar,' it follows that ' a not-incisor 

 tooth is either canine, bicuspid, or molar.' The general 

 rule is that from the denial of any of the alternatives the 

 affirmation of the remainder can be inferred. Now this 

 result clearly follows from our process of substitution ; for 

 if we have the proposition 



A = B |C|D, 



and insert this expression for A on one side of the self- 

 evident identitv 



Kb = A6, 



we obtain A.b = AB6 + A&C I A&D ; 



and, as the first of the three alternatives is self-contra- 

 dictory, we strike it out according to the law of contra- 

 diction : there remains 



A6 = A6C -I- A6D. 



Thus our system fully includes and explains that mood of 

 the Disjunctive Syllogism technically called the modus 

 tollendo ponens. 



But the reader must carefully observe that the Dis- 

 junctive Syllogism of the mood ponendo tollens, which af- 

 firms one alternative, and thence infers 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 incon- 

 sistency 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 





