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  "body": "\n## Definition\n\nLet $X$ be a set and let $P(x)$ be a [[predicate|0T3DPBG981]] on $X$.\nThe universal quantification of $P$ over $X$ is the proposition $\\forall x \\in X, \\, P(x)$, read \"for all $x$ in $X$, $P(x)$\".\nIt is true when $P(a)$ is true for each $a \\in X$, and false otherwise.\n\n## Remarks\n\nWhen $X$ is empty, $\\forall x \\in X, \\, P(x)$ is true for each $P$. This is a case of [[vacuous truth|0V4QXPGJQK]].\nFor a finite set $X = \\{a_1, \\dots, a_n\\}$, the statement is equivalent to $P(a_1) \\land \\dots \\land P(a_n)$.\n\n## Explanations\n\n### L1 human\n\n\"Every square of a real number is at least $0$\" is $\\forall x \\in \\mathbb{R}, \\, x^2 \\ge 0$.\nTo prove it, take an arbitrary $x$ and show $x^2 \\ge 0$.\nTo refute it, one [[counterexample|0ED72CTBNP]] is enough.\n",
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