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  "body": "\n## Summary\n\nThe Lindenbaum–Tarski algebra of a logical theory is the set of its formulas taken up to provable equivalence, with the operations induced by the connectives.\nFor a theory of classical [[propositional logic|09K9MWD2QQ]] it is a [[Boolean algebra|0XADJKM699]].\nIt turns statements about provability into statements about an algebra, and it is the basic tool in algebraic proofs of completeness theorems.\n\n## Explanations\n\n### L1 human\n\nTwo formulas such as $p \\wedge q$ and $q \\wedge p$ look different but say the same thing.\nThe Lindenbaum–Tarski algebra glues together all formulas that the theory proves to be equivalent.\nEach glued class becomes one element, and \"and\", \"or\" and \"not\" act on these classes.\nWith one propositional variable $p$ and no extra axioms, there are exactly four classes, those of $p$, $\\neg p$, $p \\vee \\neg p$ and $p \\wedge \\neg p$.\n",
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