OF THE LAWS OF THOUGHT. 175 



of those terms in which the coefficients are zero. Then we say that 



X= (15) 



For, since Q = 0, and /"(«) is supposed to vanish, 

 A X + Ai X^ + &c. = 

 :. AX^ + Ai X X^ -I- &e. = 



But, by (14), X X^ = XX,- = XX„ = 0. Therefore 



^ X2 = 0. 

 But A is not zero. Therefore X must be zero. 



These principles having been laid down, our best course will pro- 

 bably now be to take a few examples, and to offer in connection with 

 them such explanations as may seem necessary of the mode of pro- 

 cedure which they are intended to illustrate. 



Our first example shall be one in which but a single proposition 

 is given : " clean beasts are those which both divide the hoof and 

 chew the cud." Let 



% •=■ clean beasts, 

 y =r beasts dividing the hoof, 

 z = beasts chewing the cud. 

 Then, the given proposition, symbolically expressed, is, 



X — y z, 

 or, by transposition, 



X — y z — (16). 



This premiss contains a relation between three concepts ; and, ac- 

 cording to Professor Boole, a properly constructed science of infer- 

 ence should enable us, by some defined process, to show what cdnse • 

 quence, as respects any one of these, follows from the premiss. 

 Now, the definite and invariable process which Professor Boole ap- 

 plies, with the design which has been indicated, to an equation such 

 as (16), is to develop the first member of the equation. Writing, 

 then, 



f{x,y,z) = X — y z, 

 we have, /(I, 1, 1) = 0, 

 /(0,0,0) = 0, 

 and so on. Hence [see (12)] the developement required is 

 X — yz-=:xy{\ — z) + x z {\ — y) 



+ xO--y){\- z) -yz{[ -x) 

 + a y 2 + y (I - a;) (1 — z) 

 + Qz{\^x){\-y) 

 + 0(1 ~^) (1 — y)(l— z). 



