I completely understand why $p \implies q$ is not false when $p$ is false.

If we take the statement, "If it rains, I don't go to the gym", and it's not raining and I go or don't go to the gym, the original statement is clearly not false. So that I understand. However, there's some difference between the first two lines in the truth table and the last two lines when we prove the validity of the statement.

Mainly, we can prove $p \implies q$ true if we assume $p$ and end up getting $q$. This seems to me to correspond to the first line in the truth table, where $p$ and $q$ are true means $p \implies q$ true.

Similarly, we can prove $p \implies q$ false if we assume $p$ and end up getting $\neg q$. This to me corresponds to the second line in the truth table, where $p$ true and $q$ false means $p \implies q$ false.

However, we cannot prove $p \implies q$ true or false if we assume $\neg p$ and get $q$ or $\neg q$. Regardless, it's almost "inconclusive" in a sense. We say that $p \implies q$ is vacuously true here, but to me, it would make more sense if it was "inconclusive". Is there a reason why we say "vacuously true" specifically? Couldn't by the same logic, it be "vacuously false" or the statement just be inconclusive, since using it in a proof would not result in a conclusion whether $p \implies q$ holds?

  • $\begingroup$ It's vacuously true, because if it were false there should exist a counter-example. Such a counter-example cannot obviously exist if $p$ is false. $\endgroup$ – Bernard Jun 24 '18 at 19:39
  • $\begingroup$ You answered your own question in your second paragraph: if it's not false, it is true. $\endgroup$ – BDN Jun 24 '18 at 19:40
  • $\begingroup$ To assert $p \to q$ (i.e. "if $p$, then $q$") is not the same as "from $p$, we have proved $q$". When we try to prove something (e.g. $q$) we start from axioms or already proved theorems (e.g. $p$): in that case, we already know (or assume) that $p$ is true. $\endgroup$ – Mauro ALLEGRANZA Jun 24 '18 at 19:42
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    $\begingroup$ You can see the similar post : Still struggling to understand vacuous truths. $\endgroup$ – Mauro ALLEGRANZA Jun 24 '18 at 19:46
  • $\begingroup$ And also : Why is $p \rightarrow q$ true if $p$ is false and $q$ is true ? $\endgroup$ – Mauro ALLEGRANZA Jun 24 '18 at 19:50

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