124 OF REDUCTION. [CHAP. VIII. 



(for to affirm that X'a are identical with Y s is to affirm both that 

 All X*s are Y*s, and that All ys are X 9 s). Now these equa- 

 tions give, on elimination of v, 



which added, produce (2). 



3rd. Propositions of which both terms are particular. The 

 form of such propositions is 



but v is not quite arbitrary, and therefore must not be eliminated. 

 For v is the representative of some, which, though it may include 

 in its meaning all, does not include none. We must therefore 

 transpose the second member to the first side, and square the 

 resulting equation according to the rule. 

 The result will obviously be 



vX(l - Y) + vY(l-X) = 0. 



The above conclusions it may be convenient to embody in a 

 Rule, which will serve for constant future direction. 



8. RULE. The equations being so expressed as that the terms X 

 and Yin the following typical forms obey the law of duality, change 

 the equations 



X =vYintoX(l- Y) = 0, 



X = YintoX(l - Y) + F(l - X) = 0. 



vX=vYinto vX(l-Y) + vY(l - X) = 0. 



Any equation which is given in the form X = will not need transfor- 

 mation, and any equation ivhich presents itself in the form X=l 

 may be replaced by 1 - X = 0, as appears from the second of the 

 above transformations. 



When the equations of the system have thus been reduced, 

 any of them, as well as any equations derived from them by the 

 process of elimination, may be combined by addition. 



9. NOTE. It has been seen in Chapter iv. that in literally 

 translating the terms of a proposition, without attending to its 

 real meaning, into the language of symbols, we may produce 

 equations in which the terms X and Y do not obey the law of 

 duality. The equation w = st (p + r), given in (3) Prop. 3 of 



