# The basis of a Calculus

Distributive Law
Commutative Law
Index Law

Boole, George (1847 / 1948). The Mathematical Analysis Of Logic · Being An Essay Towards A Calculus Of Deductive Reasoning. New York: Philosophical Library. S. 17f.

The laws we have established under the symbolical forms

x (u + v) = xu + xv ……………. (1),
xy = yx ……………. (2),
xn = x ……………. (3),

are sufficient for the basis of a Calculus. From the first of these, it appears that elective symbols are distributive, from the second that they are commutative; properties which they possess in common with symbols of quantity, and in virtue of which, all the processes of common algebra are applicable to the present system. The one and sufficient axiom involved in this application is that equivalent operations performed upon equivalent subjects produce equivalent results. The third law (3) we shall denominate the index law. It is peculiar to elective symbols, and will be found of great importance in enabling us to reduce our results to forms meet for interpretation.