Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/53

CHAP. II.] 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, $\scriptstyle{z}$, from a combination $\scriptstyle{zx}$, is in itself analogous with the operation of division in Arithmetic. 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 commonly 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 $$\scriptstyle{x=y}$$ from the equation $$\scriptstyle{zx=zy}$$ is only valid when it is known that $$\scriptstyle{z}$$ is not equal to 0. If then the value $$\scriptstyle{z=0}$$ is supposed to be admissible in the algebraic system, the axiom above stated ceases to be applicable, and the analogy before exemplified remains at least unbroken. 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, Now of the symbols of Number there are but two, viz. $$\scriptstyle{0}$$ and $\scriptstyle{1}$, which are subject to the same formal law. We know that $\scriptstyle{0^2=0}$, and that $\scriptstyle{1^2=1}$; and the equation $\scriptstyle{x^2=x}$, considered as algebraic, has no other roots than $$\scriptstyle{0}$$ and $\scriptstyle{1}$. 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 $$\scriptstyle{0}$$ and $\scriptstyle{1}$. Let us conceive, then, of an Algebra in which the symbols $\scriptstyle{x}$, $\scriptstyle{y}$, $\scriptstyle{z}$, &amp;c. admit indifferently of the values $$\scriptstyle{0}$$ and $\scriptstyle{1}$, 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