y.] DISJUNCTIVE PROPOSITIONS. 77 



Some processes of deduction may be at once exhibited. 

 We may always, for instance, unite the same qualifying 

 term to each side of an identity even tliouyh one or both 

 members of the identity be disjunctive. Thus let 



A = B -I- C. 

 Now it is self-evident that 



AD = AD, 

 and in one side of this identity we may for A substitute 

 its equivalent B -I- C, obtaining 



AD = BD •!• CD. 



Since " a gaseous element is either hydrocren, or oxvfren, 

 or nitrogen, or chlorine, or iluoriue," it follows that " a free 

 gaseous element is either free hydrogen, or free oxygen, 

 or free nitrogen, or free chlorine, or free Huorine." 



This process of cnml)ination will lead to most useful in- 

 ferences when the qualifying adjective combined with both 

 sides of the proposition is a negative of one or more alter- 

 natives. Since chlorine is a coloured gas, we may infer 

 tliat"a colourless gaseous element is either (colourk'ss) 

 hydrogen, oxygen, nitrogen, or fluorine." The alternative 

 chlorine disappears because colourless chlorine does nut 

 exist. Again, since " a tooth is either an incisor, canine, 

 bicuspid, or molar," it follows that " a not-incisor tooth is 

 either canine, bicuspid, or molar." The general rule is thiit 

 from the denial of any of the alternatives the affirmation 

 of the remainder can be iuierred. Now this result clearlv 

 follows from our pi'ocess of substitution ; for if we have 

 the proposition 



A = B -I- C -I- D, 

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

 evident identity 



Ah = Ah, 

 we obtain Ab = ABh -I- AbC ■]■ Ah]} ; 



and, as the first of tbe three alternatives is self-contrn- 

 dictory, we strike it out according to the law of coiiir.i- 

 diction : tliere remains 



Ab = AbC .|. AtD. 

 Thus our system fully includes and exjilains that mood of 

 the Di.sjunctive Syllogism technically called tbe vuhIus 

 iollenilo po/iens. 



But the reader must carefully observe that the Disjunc- 

 tive Syllogism of the mood ponendo lu/lens, wjiieh alii mis 



