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  "body": "\n## Definition\n\nThe formulas of intuitionistic propositional logic are built from propositional variables and $\\bot$ with $\\land$, $\\lor$ and $\\to$.\nThe negation $\\neg\\varphi$ abbreviates $\\varphi \\to \\bot$.\nIntuitionistic propositional logic is the set of formulas derivable in the [[Hilbert system|1MGMQX43B8]] with the following axiom schemes, for all formulas $\\varphi$, $\\psi$, $\\chi$.\n\n1. $\\varphi \\to (\\psi \\to \\varphi)$.\n2. $(\\varphi \\to (\\psi \\to \\chi)) \\to ((\\varphi \\to \\psi) \\to (\\varphi \\to \\chi))$.\n3. $\\varphi \\land \\psi \\to \\varphi$ and $\\varphi \\land \\psi \\to \\psi$.\n4. $\\varphi \\to (\\psi \\to \\varphi \\land \\psi)$.\n5. $\\varphi \\to \\varphi \\lor \\psi$ and $\\psi \\to \\varphi \\lor \\psi$.\n6. $(\\varphi \\to \\chi) \\to ((\\psi \\to \\chi) \\to (\\varphi \\lor \\psi \\to \\chi))$.\n7. $\\bot \\to \\varphi$.\n\nThe only rule is [[modus ponens|0Y0MQYST0M]]: from $\\varphi$ and $\\varphi \\to \\psi$, derive $\\psi$.\n\n## Remarks\n\nAdding the scheme $\\varphi \\lor \\neg\\varphi$, the [[law of excluded middle|0RYYK8PG4S]], gives classical [[propositional logic|09K9MWD2QQ]].\nIntuitionistic first-order logic adds quantifiers with the usual axioms and rules, again without excluded middle.\n\nExample: $\\varphi \\to \\neg\\neg\\varphi$, that is, $\\varphi \\to ((\\varphi \\to \\bot) \\to \\bot)$, is derivable.\nBy the deduction theorem it suffices to derive $\\bot$ from $\\varphi$ and $\\varphi \\to \\bot$, which is one step of modus ponens.\n\nNon-example: $p \\lor \\neg p$ is not derivable for a variable $p$.\nIn the Kripke model with worlds $u \\le v$ and $p$ true only at $v$, neither $p$ nor $\\neg p$ holds at $u$, and derivable formulas hold at every world.\n\nLikewise $\\neg\\neg p \\to p$ is not derivable, while $\\neg\\neg\\neg p \\to \\neg p$ is.\n\n## Explanations\n\n### L1 human\n\nIntuitionistic logic keeps every rule of ordinary logic except the principle that each statement is true or false.\nIt reads a formula as a task: to prove $A \\to B$ is to give a method that turns any proof of $A$ into a proof of $B$.\n",
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