## Peirce’s 1870 “Logic Of Relatives” • Comment 9.3

An idempotent element $x$ in an algebraic system is one that obeys the idempotent law, that is, it satisfies the equation $xx = x.$ Under most circumstances it is usual to write this as $x^2 = x.$

If the algebraic system in question falls under the additional laws that are necessary to carry out the required transformations then $x^2 = x$ is convertible into $x - x^2 = 0,$ and this into $x(1 - x) = 0.$

If the algebraic system in question happens to be a boolean algebra then the equation $x(1 - x) = 0$ says that $x \land \lnot x$ is identically false, in effect, a statement of the classical principle of non-contradiction.

We have already seen how Boole found rationales for the commutative law and the idempotent law by contemplating the properties of selective operations.

It is time to bring these threads together, which we can do by considering the so-called idempotent representation of sets. This will give us one of the best ways to understand the significance that Boole attached to selective operations. It will also link up with the statements that Peirce makes regarding his dimension-augmenting comma operation.