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  "body": "\n## Definition\n\nAlgebraic logic is the part of mathematical logic that studies a logic through classes of algebras attached to it.\nIts main objects are the following.\n\n1. A logic, given by a formal language and a relation of consequence or provability between formulas.\n2. Its algebras, often obtained as quotients of the algebra of formulas by provable equivalence, as in the [[Lindenbaum–Tarski algebra|QC9M3Q8B8Q]].\n3. A translation between the two sides, under which theorems of the logic correspond to equations valid in the class of algebras.\n\n## Remarks\n\nExample: the completeness of [[classical propositional logic|09K9MWD2QQ]] with respect to [[Boolean algebras|0XADJKM699]] is a result of algebraic logic.\nA formula is provable exactly when it is identically $1$ in every Boolean algebra, and already in the two-element Boolean algebra.\n\nAnother example: [[intuitionistic logic|Q11N8X99FK]] corresponds in the same way to the class of [[Heyting algebras|NG943XNQWH]].\n\nNon-example: a truth-table check of a single formula is a computation in one algebra.\nIt is not by itself a study of the correspondence between a logic and a class of algebras.\n\nAbstract algebraic logic studies in general which logics admit such a correspondence.\n",
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