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  "body": "\n## Summary\n\nAlgebraic logic studies logical systems by means of algebraic structures.\nFormulas are identified when they are provably equivalent, and the logical connectives become operations on the resulting classes, as in the [[Lindenbaum–Tarski algebra|QC9M3Q8B8Q]].\n[[Classical propositional logic|09K9MWD2QQ]] corresponds to [[Boolean algebras|0XADJKM699]], [[intuitionistic logic|Q11N8X99FK]] to [[Heyting algebras|NG943XNQWH]], and Łukasiewicz infinite-valued logic to [[MV-algebras|8T4M7RN9AC]].\nFor [[predicate logic|086QKPF0A6]] there are [[cylindric algebras|1MDT9BF81H]] and polyadic algebras, and [[relation algebras|GQK4DSA11K]] treat the calculus of binary relations.\n\n## Explanations\n\n### L1 human\n\nIn school algebra one calculates with numbers by rules such as $a + b = b + a$.\nAlgebraic logic calculates with statements in the same way: \"$p$ and $q$\" equals \"$q$ and $p$\", and \"not not $p$\" equals \"$p$\" in classical logic.\nThe statements, up to logical equivalence, form an algebra, and questions about the logic become questions about that algebra.\nFor example, a formula is a theorem of classical propositional logic exactly when it takes the value $1$ in every Boolean algebra under every assignment of its variables.\n",
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