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  "body": "\n## Proof\n\nWe compare the [[truth tables|05RSK4Q657]] of $p \\to q$ and $\\neg q \\to \\neg p$.\n\n1. Let $p$ and $q$ be [[propositions|0ZGC7ZESYN]]. In classical logic, each of them is true or false, so there are four rows: TT, TF, FT and FF.\n2. By the definition of [[implication|0ZNX6DQ3EJ]], $p \\to q$ is false exactly when $p$ is true and $q$ is false, that is, only in the row TF.\n3. Again by the definition of implication, $\\neg q \\to \\neg p$ is false exactly when $\\neg q$ is true and $\\neg p$ is false.\n4. By the definition of [[negation|0XR7A7RSXB]], $\\neg q$ is true exactly when $q$ is false, and $\\neg p$ is false exactly when $p$ is true.\n5. By steps 3 and 4, $\\neg q \\to \\neg p$ is false exactly when $p$ is true and $q$ is false, that is, only in the row TF.\n6. The table is this.\n\n| $p$ | $q$ | $\\neg q$ | $\\neg p$ | $p \\to q$ | $\\neg q \\to \\neg p$ |\n|---|---|---|---|---|---|\n| T | T | F | F | T | T |\n| T | F | T | F | F | F |\n| F | T | F | T | T | T |\n| F | F | T | T | T | T |\n\n7. By steps 2 and 5, the two implications have the same truth value in each row. So they are [[logically equivalent|06GYGS41JK]]: $p \\to q \\equiv \\neg q \\to \\neg p$.\n\n## Idea\n\nAn implication fails in only one situation: a true hypothesis with a false conclusion. For $\\neg q \\to \\neg p$ this situation is \"$q$ false and $p$ true\". This is the very same situation in which $p \\to q$ fails.\n",
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    "kind": "proof",
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    "name": "Proof that an implication is equivalent to its contrapositive by a truth table",
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    "sources": [
      {
        "locator": "Proof by contrapositive",
        "url": "https://en.wikipedia.org/wiki/Contraposition"
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    "tag": "SWH5EJP8PS",
    "uses": [
      "0ZGC7ZESYN",
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  "msc": [
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  "name": "Proof that an implication is equivalent to its contrapositive by a truth table",
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