*.] 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 though one or both 

 members of the identity be disjunctive. Thus let 



A = B + C. 

 Now it is self-evident that 



AD = AD, 



and in one s.ide of this identity we may for A substitute 

 its equivalent B -|- C, obtaining 



AD = BD .|. CD. 



Since " a gaseous element is either hydrogen, or oxygen, 

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

 gaseous element is either free hydrogen, or free oxygen, 

 or free nitrogen, or free chlorine, or free fluorine." 



This process of combination 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 

 that " a colourless gaseous element is either (colourless) 

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

 chlorine disappears because colourless chlorine does not 

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

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

 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 -I- C -I- D, 



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

 evident identity 



A5 = A5, 



we obtain A6 = AB& -|. A&C -|- A6D ; 



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 



Thus our system fully includes and explains that mood of 

 the Disjunctive Syllogism technically called the modus 

 lollendo ponens. 



But the reader must carefully observe that the Disjunc- 

 tive Syllogism of the mood ponendo tollens, which affirms 



