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  "body": "\n## Definition\n\nThe relational types are defined inductively.\n\n1. $\\iota$ is a type, the type of individuals.\n2. If $\\tau_1, \\dots, \\tau_n$ are types with $n \\ge 1$, then $(\\tau_1, \\dots, \\tau_n)$ is a type, the type of $n$-ary relations between objects of types $\\tau_1, \\dots, \\tau_n$.\n\nHigher-order logic over a signature $L$ has, for each type $\\tau$, variables of type $\\tau$ and quantifiers $\\forall$ and $\\exists$ over them.\nIts formulas are built as in [[first-order logic|086QKPF0A6]], with the atomic formula $X(t_1, \\dots, t_n)$ whenever $X$ has type $(\\tau_1, \\dots, \\tau_n)$ and each $t_i$ has type $\\tau_i$.\nIn the standard semantics over an $L$-[[structure|983GZBQYT4]] with domain $M$, the domains $D_\\tau$ are given by $D_\\iota = M$ and $D_{(\\tau_1, \\dots, \\tau_n)} = \\mathcal{P}(D_{\\tau_1} \\times \\dots \\times D_{\\tau_n})$.\nHere $\\mathcal{P}$ is the [[power set|00E0YZCERH]] and $\\times$ the [[Cartesian product|09CEDJYBKP]].\n\n## Remarks\n\nThe order of a type is $1$ for $\\iota$ and one more than the largest order of $\\tau_1, \\dots, \\tau_n$ for $(\\tau_1, \\dots, \\tau_n)$.\nLogic of order $k$ allows quantifiers only over variables whose type has order at most $k$.\nFirst-order logic is the case $k = 1$, and second-order logic is the case $k = 2$.\n\nExample: a variable $\\mathcal{X}$ of type $((\\iota))$ ranges over sets of subsets of $M$, and $\\forall \\mathcal{X} \\, \\exists Y \\, \\forall x \\, (Y(x) \\leftrightarrow \\exists Z \\, (\\mathcal{X}(Z) \\land Z(x)))$ is a third-order sentence.\nIt says that each family of subsets has a union, and it is true in every structure.\n\nNon-example: the expression $X(X)$ is not a formula, because no type $\\tau$ equals a type $(\\tau)$ of relations on objects of type $\\tau$.\n\nMany presentations instead use the simply typed lambda calculus, with function types and a type of truth values; this is Church's simple theory of types.\n\n## Explanations\n\n### L1 human\n\nHigher-order logic sorts all objects into levels: individuals, relations on individuals, relations on those relations, and so on.\nA variable can range over any one level, and a relation may only be applied to objects of the level below it.\nThis rule blocks paradoxes such as a set that contains itself.\n",
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