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  "body": "\n## Definition\n\nLet $p$ and $q$ be [[propositions|0ZGC7ZESYN]].\nThe implication from $p$ to $q$ is the proposition $p \\to q$, read \"if $p$, then $q$\".\nThe proposition $p$ is the hypothesis, and the proposition $q$ is the conclusion.\nThe proposition $p \\to q$ is false when $p$ is true and $q$ is false. In all other cases, $p \\to q$ is true.\n\n| $p$ | $q$ | $p \\to q$ |\n|---|---|---|\n| T | T | T |\n| T | F | F |\n| F | T | T |\n| F | F | T |\n\n## Remarks\n\nAnother notation for $p \\to q$ is $p \\Rightarrow q$.\nThe proposition $p \\to q$ has the same truth value as $\\neg p \\lor q$.\nWhen $p$ is false, $p \\to q$ is true. This is [[vacuous truth|0V4QXPGJQK]].\nThe [[converse|0RVYE75E2M]] of $p \\to q$ is $q \\to p$. It can have a different truth value.\n\n## Explanations\n\n### L1 human\n\nThink of $p \\to q$ as a promise: \"if it rains, I take an umbrella\".\nYou break the promise only when it rains and you take no umbrella.\nWhen it does not rain, you keep the promise, whatever you do.\n",
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