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  "body": "\n## Definition\n\nLet $\\vdash$ be the derivability relation of a proof system, such as a [[Hilbert system|1MGMQX43B8]].\nLet $\\models$ be [[logical consequence|1HK0VYWTV7]] for the same language.\nThe proof system is *sound* if $\\Gamma \\vdash \\varphi$ implies $\\Gamma \\models \\varphi$, for every set of formulas $\\Gamma$ and every formula $\\varphi$.\nIt is *weakly sound* if $\\vdash \\varphi$ implies $\\models \\varphi$ for every formula $\\varphi$.\n\n## Remarks\n\nThe standard Hilbert system for propositional logic is sound.\nEach axiom is a [[tautology|05006AJ7DZ]], checked by a truth table, and modus ponens preserves truth under every assignment.\nInduction on the length of a derivation then gives soundness.\nAdding the rule \"from $\\psi$ and $\\varphi \\to \\psi$, infer $\\varphi$\" gives an unsound system.\nWith it, $\\set{p \\to q, \\; q} \\vdash p$, but the assignment $p = \\mathsf{F}$, $q = \\mathsf{T}$ shows $\\set{p \\to q, \\; q} \\not\\models p$.\nThe converse property, that $\\Gamma \\models \\varphi$ implies $\\Gamma \\vdash \\varphi$, is completeness.\nFor first-order logic it is [[Gödel's completeness theorem|WQ1ZE255SF]].\n",
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