$$\require{amssymb} \forall x\in\varnothing; P(x)$$
We know that this is true, because its negation states there is at least one $x$ in empty set so that $\neg P(x)$ is true, which is false; because there isn't anything in empty set! Its negation is false so it is true itself.
I was thinking of something like this for implications. We know that in the statement $p\implies q$, if $p$ is false, then the whole statement is considered true, regardless of the truth of $q$. But why? I think one way to justify it, is to represent the conditional statement as a universal quantifier. Something like this:
$$\big[p\implies q\big]\iff\forall x\in S_p; q(x)$$
Where $S_p$ is the set of cases where $p$ is true.
Now, when $p$ is false:
- $S_p$ is empty (there are no cases where $p$ is true).
- Therefore, $\forall x \in S_p; q(x)$ is vacuously true.
What do you think? Is it a good way to tell the reason for vacuous truth in conditional statements?