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  "body": "\n## Summary\n\nThe canonical definition of a [[Boolean algebra|0XADJKM699]] avoids choosing a list of axioms.\nIt uses the operations of the [[two-element Boolean algebra|0WG43AC0EV]].\nA Boolean algebra is then any algebra with these operations that satisfies every equation true in the two-element algebra.\nThis gives the same class of structures as the usual axioms.\nIt also explains why many different lists of axioms work.\nA list of equations true in $\\set{0, 1}$ is a complete axiom system exactly when it proves every equation true in $\\set{0, 1}$.\nIt also connects Boolean algebra with truth tables and with the equational logic of universal algebra.\n\n## Explanations\n\n### L1 human\n\nAn equation such as $x \\wedge (y \\vee z) = (x \\wedge y) \\vee (x \\wedge z)$ can be tested on true and false by a truth table.\nThe canonical view takes all equations that pass this test as the laws of Boolean algebra.\nA Boolean algebra is then any set with \"and\", \"or\", \"not\", $0$ and $1$ that obeys all these laws.\nThe power set of a set is an example, because each law can be checked one point at a time.\n",
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