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I know the concept of vacuously true statement. So when $p$ is false, it yield $p \to q$ as true, no matter what $q$ is. What I have been told is "because there is no specific condition placed on $q$, so it is vacuously true".

Even I though this statement is not clear to me "there is no specific condition placed on $q$". But let's keep it aside and let me know, why only true, why not false?

Since there can't be any other value than true or false in the logic (boolean based). Why $p \to q$ is true when $p$ is false, why not false?

Mauro ALLEGRANZA
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tbhaxor
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    Presumably, you defined → in terms of truth tables, so it is true by definition. As for why not have it be false if p is false, imagine if we defined it that way. Then p → q is equivalent to p ∧ q, and so would be a redundant symbol, we would just use the conjunction instead. – Michael Carey Nov 05 '23 at 22:54
  • @MichaelCarey I see. That also makes sense. – tbhaxor Nov 06 '23 at 04:50

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