CHAP. 11.] SIGNS AND THEIR LAWS. 37 



sides of an equation may be divided by the same quantity, has no 

 formal equivalent here. I say no formal equivalent, because, in 

 accordance with the general spirit of these inquiries, it is not 

 even sought to determine whether the mental operation which is 

 represented by removing a logical symbol, z, from a combination 

 zx, is in itself analogous with the operation of division in Arith- 

 metic. That mental operation is indeed identical with what is 

 commonly termed Abstraction, and it will hereafter appear that 

 its laws are dependent upon the laws already deduced in this 

 chapter. What has now been shown is, that there does not 

 exist among those laws anything analogous in form with a com- 

 monly received axiom of Algebra. 



But a little consideration will show that even in common 

 algebra that axiom does not possess the generality of those other 

 axioms which have been considered. The deduction of the 

 equation x = y from the equation zx = zy is only valid when it 

 is known that z is not equal to 0. If then the value z = is 

 supposed to be admissible in the algebraic system, the axiom 

 above stated ceases to be applicable, and the analogy before ex- 

 emplified remains at least unbroken. 



15. However, it is not with the symbols of quantity generally 

 that it is of any importance, except as a matter of speculation, to 

 trace such affinities. We have seen (II. 9) that the symbols of 

 Logic are subject to the special law, 



x. 



Now of the symbols of Number there are but two, viz. and 1 , 

 which are subject to the same formal law. We know that O 2 = 0, 

 and that 1 3 = I ; and the equation x z = x, considered as algebraic, 

 has no other roots than and 1. Hence, instead of determining 

 the measure of formal agreement of the symbols of Logic with 

 those of Number generally, it is more immediately suggested to 

 us to compare them with symbols of quantity admitting only of 

 the values and 1. Let us conceive, then, of an Algebra in 

 which the symbols x, y, z, &c. admit indifferently of the values 

 and 1, and of these values alone. The laws, the axioms, and 

 the processes, of such an Algebra will be identical in their whole 

 . extent with the laws, the axioms, and the processes of an Al- 





