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It's true precisely because it's vacuous. If you quantify over the empty set, anything is true.

In other words, the statement "for every x in {} it holds that <anything>" is always true.

What can be confusing is that the statement "for every x in {}, it doesn’t hold that <anything>" is always true as well.

I mean, yes. But "it doesn't hold that <anything>" is equivalent to "it holds that <not anything>" and since not anything is also anything... Ah, I see.

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