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  "body": "\n## Definition\n\nLet $L$ be a signature.\nThe formulas of second-order logic over $L$ are built like the formulas of [[first-order logic|086QKPF0A6]] over $L$, with these additions.\n\n1. For each $n \\ge 1$ there are relation variables $X, Y, \\dots$ of arity $n$, and $X(t_1, \\dots, t_n)$ is an atomic formula for terms $t_1, \\dots, t_n$.\n2. If $\\varphi$ is a formula and $X$ is a relation variable, then $\\forall X \\, \\varphi$ and $\\exists X \\, \\varphi$ are formulas.\n\nIn the standard semantics, a formula is evaluated in an $L$-[[structure|983GZBQYT4]] $\\mathcal{M}$ with domain $M$ as in first-order logic.\nAn $n$-ary relation variable ranges over the whole [[power set|00E0YZCERH]] of $M^n$.\n\n## Remarks\n\nFunction variables may be added in the same way; they can be expressed by relation variables.\nMonadic second-order logic allows only relation variables of arity $1$, that is, variables for subsets of $M$.\n\nExample: the sentence $\\forall X \\, \\big( (X(0) \\land \\forall x \\, (X(x) \\to X(S(x)))) \\to \\forall x \\, X(x) \\big)$ is a second-order sentence over the signature $\\set{0, S}$.\nIt says that every subset of the domain that contains $0$ and is closed under $S$ is the whole domain.\n\nNon-example: a formula with a quantifier over sets of subsets of $M$ is not second-order, because no clause provides such variables; it belongs to third-order logic.\n\nWith the standard semantics, second-order logic has neither a complete proof system nor the compactness property.\nWith Henkin semantics, in which the relation variables range over a chosen family of relations, both properties return.\n\n## Explanations\n\n### L1 human\n\nSecond-order logic lets formulas speak about all subsets and all relations of the domain, not only about its elements.\nThis makes it much more expressive than first-order logic.\nFor example, finiteness of the domain, or the completeness of the real line, can be written as a single second-order sentence.\n",
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