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  "body": "\n## Statement\n\nLet $L$ be a signature, let $P$ be an $n$-ary relation symbol not in $L$, and let $T$ be a [[theory|92SWXST1ZM]] in the signature $L \\cup \\set{P}$.\nSuppose that $P$ is implicitly defined by $T$.\nThat is, any two [[models|MVCH6N9SNW]] of $T$ with the same domain and the same interpretation of each symbol of $L$ interpret $P$ alike.\nThen $P$ is explicitly defined by $T$.\nThat is, there is an $L$-[[formula|HYJ10AF6RT]] $\\varphi(x_1, \\dots, x_n)$ with\n$$T \\models \\forall x_1 \\cdots \\forall x_n \\, \\big(P(x_1, \\dots, x_n) \\leftrightarrow \\varphi(x_1, \\dots, x_n)\\big).$$\n\n## Remarks\n\nThe converse is immediate: an explicit definition fixes $P$ in each [[structure|983GZBQYT4]] for $L$.\nThe standard proof uses the [[Craig interpolation theorem|E2CBFNAVZY]].\nTake a copy $P'$ of $P$ and the copy $T'$ of $T$; implicit definability says $T \\cup T' \\models P(\\bar x) \\to P'(\\bar x)$.\nBy compactness, finitely many sentences suffice, and an interpolant between the $P$-side and the $P'$-side is the required formula $\\varphi$.\n\nExample: let $L = \\set{+, \\cdot, 0, 1}$ and let $T$ be the theory of real closed ordered fields, with the order symbol $\\le$ in the role of $P$.\nIn each model of $T$, the nonnegative elements are exactly the squares, so $\\le$ is implicitly defined.\nAn explicit definition is $x \\le y \\leftrightarrow \\exists z \\, (y = x + z \\cdot z)$.\n\n## Explanations\n\n### L1 human\n\nSuppose some axioms pin down a new notion uniquely, once the old notions are fixed.\nBeth's theorem says that the new notion can then be written as a formula in the old notions.\nIn first-order logic, a unique description always gives an explicit definition.\n\n### L2 human\n\nThe theorem was proved by Evert Willem Beth in 1953.\nIt follows from Craig interpolation, and both results fail for many logics beyond first-order logic.\nIn abstract model theory, logics with the Beth property are studied as those in which implicit definitions can always be made explicit.\n",
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