In the solution they explain why they have to conclude that there should be at least one. Let me try again with a bit more explanation.

1. The liar stating something must mean that the phrase is not true. They cannot state anything that is not false.

2. "All X are Y" is the phrase.

Now, if we assume there is no X the phrases "All X are Y" and "Not all X are Y" are both true and false.

All X are Y - True. Yes, there is no X that is not Y.

All X are Y - False. Yes, there is no X that is Y.

Not all X are Y - True. Yes, there are no X, so none is not Y.

Not all X are Y - False. Yes, there are no X, so none is Y.

All these statements are (according to the article) vacuous if there is no X. A liar then cannot make them, as they are not false.

So from here you can deduce that either the phrase "All X are Y" stated by a liar indicates the existence of X or that I'm a liar :)

By the way, if the phrase is "X exists and all X are Y" then we wouldn't know if X really existed.

We only would know that either X doesn't exists or, if it exists, not all X are Y.

Yes we can because if there are no ten-foot tall men, then it is indeed true that "All ten-foot tall men have brown hair"

It seems intuitively wrong that you can say "All X are Y" and yet that doesn't imply "At least one X exists".

It also seems intuitively wrong that there are different sizes of infinity, yet Cantor discovered exactly that.

That's how math defined what "all" means. You can be talking about all elements of empty set without implying it must have some.

Basically it disentangled two unrelated concepts, that English language unduly mixes. Concept of every item having some quality and concepts of at least one item existing.

That how we define "all" logically but not linguistically. For instance, linguistically we would consider this inconsistent: "all 10-foot men have black hair AND all 10-foot men have blonde hair" yet it is true, logically, if there are no 10-foot tall men. The translation of the English word "all" should be something something like: |Q| > 0 ∧ ∀x∈Q, x is ...

In formal logic, but in regular English that would not be true

If that is true then liar said truth.

And that's why we must conclude that there must be a ten-foot tall man, so it can have non-brown hair, so that the liar indeed lied.

Why liar can't lie about ten-foot tall man? Liar could mean two-inch hamster but lied and said ten-foot tall man

Indeed, which is why the premise is false. So if a liar who always lies says that thing about ten foot men, ten foot men must exist, and at least one does not have brown hair.

Then the liar isn't lying, as the statement is true.

In these types of puzzles, a "liar" is meant to be someone who makes only statements which evaluate to false. They are meant to be riddles about formal logic.

> A liar is someone who only says false statements.

we're using a made up definition of liar, who can only say things that are false. it's not part of the question for the liar to be able to tell a statement they arent certain is false

There are normal-discourse statements in which vacuous truth plays a role, example: "I've never met a Frenchman I didn't like, but then again, I've never met a Frenchman". This is, of course, a joke: and it relies on the fact that we assume the set "met a Frenchman" will have contents, but we recognize that, due to the negation, the empty set is actually valid.

But it doesn't work the other way around: "Every Frenchman I've ever met has become a good friend, but then again, I've never met a Frenchman". This isn't funny, because the second clause makes the first clause into a lie, as truth is normally understood. This is not a place where we colloquially accept an empty set.

So the puzzle posed translates the English sentence into logic badly. It isn't the conclusion which is counterintuitive, it is the logical analysis which is flawed.