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  "body": "\n## Definition\n\nLet $D$ be a domain of objects, and let $A$, $B$ and $P(x)$ be statements about $D$.\nThe *Brouwer–Heyting–Kolmogorov interpretation* says what a proof of a compound statement is, by these clauses.\n\n1. A proof of $A \\wedge B$ is a pair $(a, b)$, where $a$ is a proof of $A$ and $b$ is a proof of $B$.\n2. A proof of $A \\vee B$ is a pair $(i, c)$ with $i \\in \\set{0, 1}$. If $i = 0$, then $c$ is a proof of $A$. If $i = 1$, then $c$ is a proof of $B$.\n3. A proof of $A \\to B$ is a construction that transforms each proof of $A$ into a proof of $B$.\n4. There is no proof of the absurdity $\\bot$.\n5. A proof of $\\neg A$ is a proof of $A \\to \\bot$.\n6. A proof of $\\forall x\\, P(x)$ is a construction that transforms each $d \\in D$ into a proof of $P(d)$.\n7. A proof of $\\exists x\\, P(x)$ is a pair $(d, p)$, where $d \\in D$ and $p$ is a proof of $P(d)$.\n\nThese clauses give the meaning of [[conjunction|0M28ASWG8T]], [[disjunction|0H5GKDTB01]], [[implication|0ZNX6DQ3EJ]], [[negation|0XR7A7RSXB]], the [[universal quantifier|0B88PQTJ66]] and the [[existential quantifier|0JW2FHW7G3]] in [[intuitionistic logic|Q11N8X99FK]].\n\n## Remarks\n\nExample: $A \\to \\neg \\neg A$ has a BHK proof.\nGiven a proof $a$ of $A$, the construction $f \\mapsto f(a)$ sends each proof $f$ of $\\neg A$ to a proof of $\\bot$, so it is a proof of $\\neg \\neg A$.\nNon-example: there is no known BHK proof of $\\neg \\neg A \\to A$ for all $A$, because a proof of $\\neg \\neg A$ gives no construction of a proof of $A$.\nThe notion of \"construction\" is left informal.\nKleene's realizability reads it as a computable function, and the Curry–Howard correspondence reads it as a typed lambda term.\n",
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