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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?

rb612
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  • 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. – Bernard Jun 24 '18 at 19:39
  • You answered your own question in your second paragraph: if it's not false, it is true. – BDN Jun 24 '18 at 19:40
  • 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. – Mauro ALLEGRANZA Jun 24 '18 at 19:42
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    You can see the similar post : [Still struggling to understand vacuous truths](https://math.stackexchange.com/questions/2723860/still-struggling-to-understand-vacuous-truths). – Mauro ALLEGRANZA Jun 24 '18 at 19:46
  • And also : [Why is $p \rightarrow q$ true if $p$ is false and $q$ is true ?](https://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr) – Mauro ALLEGRANZA Jun 24 '18 at 19:50
  • I think you're getting confused because of your example : "If it rains, I go to the gym." This statement is true when it doesn't rain regardless of whether you go to gym or not. – AgentS Jun 24 '18 at 19:55
  • You probably want to look at the field of [relevance logic](https://en.wikipedia.org/wiki/Relevance_logic). – Dave L. Renfro Jun 24 '18 at 20:05
  • @rsadhvika it's true in the fact that it's not inconsistent, but what happens when it doesn't rain doesn't prove what happens when it rains. That's why I say it's like inconclusive. For example, Goldbach's conjecture currently doesn't have a counterexample known. But that doesn't mean it's true. We haven't proved it yet. Hence it's in a "middle" state between true and false. So why isn't this the case here too? That it can exist in some state that isn't provably true or false. – rb612 Jun 24 '18 at 20:08
  • I sorta get what you're saying. I think in logic we focus on "structure" of statements, not semantics. – AgentS Jun 24 '18 at 20:12
  • Often when we are only interested in $q$ when $p$ is true, we replace $q$ with a predicate that is only defined when $p$ is true. E.g., rather than consider $x \geq 0 \rightarrow (\sqrt x)^2 = x$, we may write simply $(\sqrt x)^2 = x$. And one can make a case that ideally $(\sqrt x)^2 = x$ should be indeterminate or whatever when $x <0$. Also, it seems hard to fantasize about $p \rightarrow q$. But fantasizing about $p \wedge q$ is no substitute for fantasizing about $p \rightarrow q$ when $q \rightarrow p$ is evil yet $p \rightarrow q$ may be an important moral truth worth fantasizing about. – Stephen A. Meigs Jun 24 '18 at 23:03

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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?

You're basically asking why rows 3 and 4 of the truth table of $\to$ (where the antecedent is False) is even defined, and why not define them as False instead of True.

Well, if we make either of the two proposed changes, then the statement$$\forall z{\in}\mathbb C\,\big(z>0\to z\in\mathbb R\big)$$ will no longer hold.

ryang
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