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  "body": "\n## Statement\n\nLet $\\varphi$ and $\\psi$ be first-order [[sentences|DKFGQTRYZE]] in a language with equality.\nAssume that $\\varphi \\models \\psi$, in the sense of [[logical consequence|1HK0VYWTV7]].\nThen there is a first-order sentence $\\theta$ with $\\varphi \\models \\theta$ and $\\theta \\models \\psi$.\nMoreover, every relation, function and constant symbol of $\\theta$ occurs in both $\\varphi$ and $\\psi$.\n\n## Remarks\n\nThe sentence $\\theta$ is called an interpolant.\nOne proof uses a cut-free [[sequent calculus|W72CA3R9MP]] and builds $\\theta$ by induction on a cut-free proof of $\\varphi \\to \\psi$.\nAnother proof is model-theoretic and uses the [[compactness theorem|1NR3HAWG9D]].\nExample: $P(c) \\land Q(c) \\models P(c) \\lor R(c)$, and $\\theta = P(c)$ is an interpolant in the common symbols $P$ and $c$.\nWhen $\\varphi$ and $\\psi$ share no symbols, $\\theta$ is a sentence of pure equality, such as $\\forall x \\, (x = x)$ or its negation.\nThe propositional version holds too, with $\\theta$ built from the common variables and the constants $\\top$ and $\\bot$.\nWilliam Craig proved the theorem in 1957.\n\n## Explanations\n\n### L1 human\n\nSuppose one statement implies another.\nThe theorem says that there is a middle statement, using only the vocabulary the two have in common, that carries the implication.\nSo the reason for the implication can always be expressed in the shared language.\n\n### L2 human\n\nCraig interpolation is a basic structural property of first-order logic.\nIt implies the [[Beth definability theorem|RJBDSA2ETP]]: an implicitly definable relation is explicitly definable.\nIt is closely related to Robinson's joint consistency theorem.\nMany other logics fail to have it, so it is also used to compare logics.\n",
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