'©•F fHE ILAWS 'OP THOtTGH'T. 1/7 



^ion thereof, as an explicit function of the others. To the problem 

 as put even thus in its widest generality. Professor Boole's processes 

 extend, it would make our article too lengthened were we to go 

 into minute details ; but we must endeavour to give some idea of the 

 course here followed, as it both is extremely interesting as a matter 

 of pure speculation, and forms an important part of the system under 

 consideration. 



Take the equation in (16), a? — y z = ; and, as a simple instance 

 will serve the purpose of illustration as well as a complicated one, 

 let the inquiry be: how can s be expressed in terms of x and 3// 

 Jn ordinary Algebra we should have 



z^- ..(19) 



y 



But though both sides of an equation may, in Logic as in Algebra, be 

 multiplied (so to speak) by the same quantity, they cannot, in Logic, 

 l)e legitimately divided by the same quantity. For instance, let the 

 objects common to the class X and to the class U be identical with 

 those common to the class T and to the class U ; in other words, let 



Z7X= VY; 

 it does not follow that Xis identical with Y, or symbolically, that 



x= r. 



Hence equation (19) could not, in Logic, be legitimately deduced 

 from (16), even if y were an explicit factor of co. But still further, 



when X has not y as one of its factors, the expression - is not, in the 



logical system, interpretable. Nevertheless, Professor Boole shows 

 that conclusions both interpretable and correct will ultitnately be 

 arrived at, if the value of z be deduced Algebraically, as in (19), and 



30 



the expression - be then, as a logical function, subjected to develop- 

 ment. Now, if - be developed by (11), and the expansion equate'd 



to z, we ge't 



z^xy ^\x{\-y) +0(l-a:)^-ff(l-a:=)(l-y)......(20) 



Here we have two symbols, % and i, the meaning of which has not 

 yet been determined. Our author shows that the former, which in 

 Algebra denotes an indefinite numerical quantity, denotes in the 

 logical system an indefinite class. In Algebra \ denotes infinity \ 

 and, as is well known, when it occurs as the co-efficient in a term ia 



Vol. X. M 



