CHAP. VIII.] OF REDUCTION. 123 



1st. That any equations which are of the form F=0, V sa- 

 tisfying the fundamental law of duality F(l - F) = 0, may be 

 combined together by simple addition. 



2ndly. That any other equations of the form F= may be 

 reduced, by the process of squaring, to a form in which the same 

 principle of combination by mere addition is applicable. 



It remains then only to determine what equations in the ac- 

 tual expression of propositions belong to the former, and what to 

 the latter, class. 



Now the general types of propositions have been set forth in 

 the conclusion of Chap. iv. The division of propositions which 

 they represent is as follows : 



1st. Propositions, of which the subject is universal, and the 

 predicate particular. 



The symbolical type (IV. 15) is 



X and Y satisfying the law of duality. Eliminating v, we have 

 X(l-Y) = 0, ....-' . (1) 



and this will be found also to satisfy the same law. No further 

 reduction by the process of squaring is needed. 



2nd. Propositions of which both terms are universal, and of 

 which the symbolical type is 



x= y, 



X and Y separately satisfying the law of duality. Writing the 

 equation in the form X - Y = 0, and squaring, we have 



X-2XY+ Y=0, 

 or X(l- Y)+ Y(1--X) = 0. (2) 



The first member of this equation satisfies the law of duality, as 

 is evident from its very form. 



We may arrive at the same equation in a different manner. 



The equation 



X = Y 



is equivalent to the two equations 



X=vY 9 Y=vX, 



