This is a somewhat irritating property of formal logic and mathematics. Natural language receives a “special” grammar that isn’t always declared up front. You just have to sort of be in on it.

In this case, the reader is given the special definition of liar, but not the special definition of “lie”. (As in, it’s not a lie to make definitive claims about nonexistent hats.)

A lot of the “trick” in logic puzzles boils down to this issue of word play. This puzzle could have been drafted so that the liar’s statement leaves proper room for the no-hats case, but then it would be too easy.

> In formal logic, if you have no hats, it is true that all your hats are green.

But told by someone who cannot make a true statement.

That's the thing, the problem isn't written in formal logic. It's written in English, which is vague.

It's written in a form of English that is formal enough to have a straightforward translation to logic formulas. Also, in the solution article that now has been posted, it's compared to the statement "I have read all books on my shelf" with an empty shelf, which is indeed vacuously true. The sentence about hats is equivalent, I think even the least exactly-minded English speaker would agree.

> Also, in the solution article that now has been posted, it's compared to the statement "I have read all books on my shelf" with an empty shelf, which is indeed vacuously true.

Really? Any human I have ever met when presented with that statement would likely immediately point out that there are no books on that shelf.