Boolean algebras canonically defined

Summary

The canonical definition of a Boolean algebra avoids choosing a list of axioms. It uses the operations of the two-element Boolean algebra. A Boolean algebra is then any algebra with these operations that satisfies every equation true in the two-element algebra. This gives the same class of structures as the usual axioms. It also explains why many different lists of axioms work. A list of equations true in $\set{0, 1}$ is a complete axiom system exactly when it proves every equation true in $\set{0, 1}$. It also connects Boolean algebra with truth tables and with the equational logic of universal algebra.