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  "body": "\n## Definition\n\nLet $\\sigma$ be a [[signature|WDVVEYD6AT]].\nA [[sentence|DKFGQTRYZE]] over $\\sigma$ is in *Skolem normal form* if it has the form\n$$\\forall x_1 \\, \\forall x_2 \\cdots \\forall x_n \\, \\psi,$$\nwhere $n \\ge 0$ and $\\psi$ is a formula with no quantifiers.\nSo it is a sentence in [[prenex normal form|CS82V969DE]] whose prefix has only universal quantifiers.\n\n## Remarks\n\nLet $\\varphi = \\forall x_1 \\cdots \\forall x_k \\, \\exists y \\, \\chi$ be in prenex normal form.\nIts Skolemization step replaces $\\varphi$ by $\\forall x_1 \\cdots \\forall x_k \\, \\chi[f(x_1, \\dots, x_k)/y]$.\nHere $f$ is a new function symbol of arity $k$; for $k = 0$ it is a new constant symbol.\nRepeating the step gives a sentence in Skolem normal form over a larger signature.\nIt is satisfiable if and only if $\\varphi$ is satisfiable.\nFor example, $\\forall x \\, \\exists y \\, R(x, y)$ becomes $\\forall x \\, R(x, f(x))$, which is in Skolem normal form.\nThe sentence $\\forall x \\, \\exists y \\, R(x, y)$ itself is not in Skolem normal form, since its prefix contains $\\exists$.\nThe sentence $\\forall x \\, R(x, f(x))$ implies $\\forall x \\, \\exists y \\, R(x, y)$, but not conversely, so the two are not equivalent.\n",
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    "name": "Skolem normal form",
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