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  "body": "\n## Definition\n\nLet $\\sigma$ be a [[signature|WDVVEYD6AT]] and $V$ a set of variables.\nThe set of first-order *formulas* over $\\sigma$ is the smallest set of strings such that:\n\n1. if $t_1$ and $t_2$ are [[terms|6D3WA7N2W4]], then $t_1 = t_2$ is a formula;\n2. if $R$ is a relation symbol of arity $n$ and $t_1, \\dots, t_n$ are terms, then $R(t_1, \\dots, t_n)$ is a formula;\n3. if $\\varphi$ and $\\psi$ are formulas, then $\\neg \\varphi$, $(\\varphi \\land \\psi)$, $(\\varphi \\lor \\psi)$, $(\\varphi \\to \\psi)$ and $(\\varphi \\leftrightarrow \\psi)$ are formulas;\n4. if $\\varphi$ is a formula and $x \\in V$, then $\\forall x \\, \\varphi$ and $\\exists x \\, \\varphi$ are formulas.\n\nThe formulas from clauses 1 and 2 are the *atomic formulas*.\n\n## Remarks\n\nIn the signature of ordered sets, $\\forall x \\, \\exists y \\, (x \u003c y)$ is a formula.\nHere $x \u003c y$ is atomic by clause 2, $\\exists y \\, (x \u003c y)$ follows by clause 4, and then $\\forall x$ by clause 4 again.\nThe string $\\forall x \\, (x \\land y)$ is not a formula: $x$ and $y$ are terms, not formulas, so clause 3 does not apply to them.\nIn propositional logic the formulas are built from variables by clause 3 alone.\nThe [[connectives|QS0XVAN9BG]] used vary between books, since some are definable from others.\nIn Mathlib formulas with bounded variables are `FirstOrder.Language.BoundedFormula`.\n",
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