t76 PROFESSOR BOOLE S MATSEMATtCAL THEORY 



Therefore, by (16), 



xy {l—s) + X z (1— y) + a; (1—?^) (1— 0)— ys^ (1— cr)~0 i 

 and therefore, by (15), 



X y (I — &) = 0. '] 



x z {I - ij) = 0, ■' 

 x{\ ~~y)il -z) = 0, f~-- U^-> 



^ 2 (1 - «) = 0. J 

 Still farther, since, by (13), the sum of the constitutents of an es" 

 pansion is unity ; and since four of the constituents in the expan- 

 sion of cc — 1/ z have been shewn to be zero ; it follows that the sum 

 of the remaining constituents in the expansion of a; — y ^^ is unity. 

 That is, 



xye-\-f/(l - x){l - z) + z(l - x)(l - 7/) 



+ (1 -^)(1 -^)(l -^) = 1 (18) 



It is obvious that this method can be applied in every case. To 

 what then does it lead ? First of all, in the group of equations (17), 

 we have brought before us all the different classes (if the expression 

 may be permitted) to which the given proposition warrants us in 

 saying that nothing can belong ; and next, in equation (18) we have 

 brought before us those different classes to one or other of which 

 the given proposition warrants us in asserting that everything must 

 belong. For instance, the first of equations (17) denies the exis- 

 tence of beasts which are clean («) and divide the hoof (y) but do 

 not chew the cud (1 -~ z) ', the second denies the existence of beasts 

 which are clean (x) and chew the cud (z) but do not divide the 

 hoof (I — y) ; and so on. Equation (18), again, informs us that 

 the universe, which is represented by 1, is made up of four classes, 

 in one or other of which therefore every thing must rank ; the first 

 denoted hj x y z, the second by y (1 — x) (1 — ^) ; and so on. As 

 an example of the interpretation of the expressions by which these 

 classes are denoted, we may take the last, (1 — x) {\ — y) (1 — 0). 

 This represents things which are neither clean beasts, nor beasts 

 chewing the cud, nor beasts dividing the hoof. 



By the method employed, we have been able to indicate certain 

 classes which do not exist, and also to indicate certain classes in one 

 or other of which every thing existing is foi;nd. But this, it m.ay be 

 said, is not a solution of the most general problem of inference. 

 The most general problem is : to express (speaking mathematically) 

 any one of the symbols entering into the given premiss, or any func- 



