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  "body": "\n## Definition\n\nConsider formulas of [[propositional logic|09K9MWD2QQ]] with the [[connectives|QS0XVAN9BG]] $\\land$, $\\lor$, $\\to$ and the constant $\\bot$ for falsity; $\\neg \\varphi$ abbreviates $\\varphi \\to \\bot$.\nA *natural deduction* derivation is a finite tree of formulas.\nEach leaf is an assumption, and each inner node follows from its children by one of these rules:\n\n1. $\\land$-introduction: from $\\varphi$ and $\\psi$, infer $\\varphi \\land \\psi$. $\\land$-elimination: from $\\varphi \\land \\psi$, infer $\\varphi$, and also $\\psi$.\n2. $\\lor$-introduction: from $\\varphi$, infer $\\varphi \\lor \\psi$ and $\\psi \\lor \\varphi$.\n3. $\\lor$-elimination: from $\\varphi \\lor \\psi$, a derivation of $\\chi$ from $\\varphi$, and one of $\\chi$ from $\\psi$, infer $\\chi$, discharging $\\varphi$ and $\\psi$.\n4. $\\to$-introduction: from a derivation of $\\psi$ from the assumption $\\varphi$, infer $\\varphi \\to \\psi$, discharging $\\varphi$.\n5. $\\to$-elimination, or modus ponens: from $\\varphi$ and $\\varphi \\to \\psi$, infer $\\psi$.\n6. $\\bot$-elimination: from $\\bot$, infer any formula $\\varphi$.\n7. Reductio ad absurdum: from a derivation of $\\bot$ from the assumption $\\neg \\varphi$, infer $\\varphi$, discharging $\\neg \\varphi$.\n\nA discharged assumption is no longer open.\nOne writes $\\Gamma \\vdash \\varphi$ if some derivation has root $\\varphi$ and all its open assumptions in $\\Gamma$.\n\n## Remarks\n\nA derivation of $\\vdash \\varphi \\land \\psi \\to \\psi \\land \\varphi$: assume $\\varphi \\land \\psi$.\nBy $\\land$-elimination get $\\psi$ and $\\varphi$, and by $\\land$-introduction get $\\psi \\land \\varphi$.\nThen $\\to$-introduction discharges the assumption, so no open assumption remains.\nThe tree with the single node $p$ is a derivation of $p$, but only from the open assumption $p$; it does not show $\\vdash p$.\nWithout rule 7, the system is natural deduction for [[intuitionistic logic|Q11N8X99FK]]; with it, the system is classical.\nQuantifier rules extend the system to [[predicate logic|086QKPF0A6]].\n",
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