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  "body": "\n## Definition\n\nLet $\\Gamma$ be a set of [[formulas|HYJ10AF6RT]] and $\\varphi$ a formula, all over the same signature.\nThen $\\varphi$ is a *logical consequence* of $\\Gamma$, written $\\Gamma \\models \\varphi$, if every interpretation that makes every formula of $\\Gamma$ true also makes $\\varphi$ true.\nIn [[propositional logic|09K9MWD2QQ]] an interpretation is a truth assignment to the variables.\nIn first-order logic it is a [[structure|983GZBQYT4]] together with an assignment of its elements to the variables.\n\n## Remarks\n\n$\\set{p, \\; p \\to q} \\models q$: in every assignment where $p$ and $p \\to q$ are true, $q$ is true by the truth table of $\\to$.\n$\\set{p \\to q, \\; q} \\not\\models p$: the assignment $p = \\mathsf{F}$, $q = \\mathsf{T}$ makes both premises true and $p$ false.\n$\\emptyset \\models \\varphi$ means that $\\varphi$ is valid, written $\\models \\varphi$.\n$\\Gamma \\models \\varphi$ holds exactly when $\\Gamma \\cup \\set{\\neg \\varphi}$ is not [[satisfiable|QN7M1JCGT9]].\n",
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