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  "body": "\n## Definition\n\n1. A formula $\\varphi$ of [[propositional logic|09K9MWD2QQ]] is *satisfiable* if some truth assignment to its variables gives $\\varphi$ the value true.\n2. Let $\\sigma$ be a signature and $\\varphi$ a first-order [[formula|HYJ10AF6RT]] over $\\sigma$. Then $\\varphi$ is *satisfiable* if there are a $\\sigma$-[[structure|983GZBQYT4]] $M$ and an assignment $s$ of elements of $M$ to the variables with $M \\models \\varphi[s]$.\n3. A set $\\Gamma$ of formulas is *satisfiable* if one interpretation makes all formulas of $\\Gamma$ true at once.\n\nA formula or set that is not satisfiable is *unsatisfiable*.\n\n## Remarks\n\nThe formula $(p \\lor q) \\land \\neg p$ is satisfiable: the assignment $p = \\mathsf{F}$, $q = \\mathsf{T}$ makes both conjuncts true.\nThe set $\\set{p, \\; p \\to q, \\; \\neg q}$ is unsatisfiable: if $p$ and $p \\to q$ are true then $q$ is true, so $\\neg q$ is false.\nEach of its three formulas is satisfiable on its own, so clause 3 is stronger than satisfiability of each member.\nA formula $\\varphi$ is valid, for example a [[tautology|05006AJ7DZ]], exactly when $\\neg \\varphi$ is unsatisfiable.\n",
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