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  "body": "\n## Summary\n\nIntuitionistic logic is the logic of constructive reasoning, in which a statement counts as true only when it has a proof.\nIt is classical logic without the [[law of excluded middle|0RYYK8PG4S]] $\\varphi \\lor \\neg\\varphi$ and without double negation elimination $\\neg\\neg\\varphi \\to \\varphi$.\nIts semantics include [[Heyting algebras|NG943XNQWH]], [[Kripke models|NGHWZ9Y7ZV]] and the [[Brouwer–Heyting–Kolmogorov interpretation|7B53ZGXXEQ]].\nThrough the [[Curry–Howard correspondence|A77FER1Q5H]], its proofs correspond to typed programs.\n\n## Explanations\n\n### L1 human\n\nIn intuitionistic logic, to prove \"$A$ or $B$\" you must prove $A$ or prove $B$, and to prove \"there exists $x$\" you must exhibit one.\nSo \"$A$ or not $A$\" is not a law, because you may have a proof of neither.\nFor example, a classical proof that some pair of irrational numbers $a, b$ has $a^b$ rational, without telling which pair, is not an intuitionistic proof.\nEverything provable intuitionistically is provable classically, but not conversely.\n",
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