1/8 PROFESSOR Boole's mathematical, theory 



an equation all of wliose other terms are finite, this indicates that 

 the quantity of which it is the co-efficient is zero. So, in the logical 

 system, if, in any term of an equation obtained in the manner in 

 which equation (20) has been obtained, the co-efficient be i, the 

 corresponding constituent must be 0. These are certainly very 

 remarkable analogies. But let us see what follows. We have first, 

 from (20), 



Hence as the equation (20) describes the separate classes of which 

 z consists, and as there is no such class as a; (1 — «/) in esistence^ 

 the second term on the right hand side of equation (20) may be 

 rejected. The third term also may be omitted, its co-efficient being 

 zero. This reduces the equation to the form, 



!S = x^ + %(l — x){l~^): 



which means, that beasts which chew the cud consist of the ckss ani/y 

 together with an indefinite remainder of beasts common to the classes 

 1 — OS and 1 — «/. 



Before leaving the subject of inference from a single premiss, we 

 must say a few words regarding elimination ; for though, in Algebra, 

 elimination is possible only when two or more equations are given^ 

 Professor Boole, shows that, in Logic, a class symbol may be elimi- 

 nated from a single equation. In fact, elimination from two or more 

 premises is ultimately reduced by our author to elimination from a 

 single premiss. And yet, as if to preserve the analogy between 

 Algebra and Logic, even where the two sciences seem to differ most 

 widely from one another, the possibility of eliminating x from a sin- 

 gle premiss in the latter science, arises from the circumstance, that, 

 in that science the equation previously referred to as expressing the 

 Law of Duality always subsists ; and it is by the combination of that 

 equation with the given proposition that the elimination of x from 

 the given proposition is effected. For let the given proposition be 



/(^) = (21) 



Then, by (10), 



/(l)^+/(0)(l-^) = 0. 

 ••«={/(0)-/(l)| =/(0), 

 and,(l-^) J/(0)-/(l)} =-/(!). 

 ••^(1-^) S/(0)-/(l)}^ = -/(0)/(I), 

 But, bj the Law of Duality, x {1 — x) — 0. Therefore 



