> What the author would probably say is “All my hats are green” means the liar is either lying about All or Green. Either all their hats are some other color or only one hat is green. This means you have to assume the liar has a hat. How do we know that?

This is almost certainly not what the author will say, and it does have some tangential connections to computing so it's worth expanding on briefly.

In computing you often reduce problems to smaller problems, this can be done with recursion but it doesn't have to be, dynamic programming kind of is the reverse of it; whatever. When you do that though, you get these questions about really small collections, sets, lists, data structures.

Just for a quick concrete example so that we're not an abstract theory land, is a one-element list sorted? Is a zero element list sorted? Is [5, 5, 5] sorted ascending, descending, both, or neither? If you choose the wrong answer, then it means that your algorithm needs to be more complex, you don't trust zero-element lists to be sorted so your quicksort HAS to pivot on a median of 3, you don't trust one-element lists to be sorted so your quicksort HAS to pivot on a median of 5, or maybe when you see that you would recurse into a list of size 1 you generate a new median-pivot or something.

In mathematics, there is a convention which attempts to generalize this idea that an empty list is always sorted. In fact, an empty list is also always randomly sorted. It is always sorted descending, too. If it's an empty array of ints, all of those ints are greater than 1000—and they are also less than 50.

The mathematical convention is, in lay speak, “if you are talking about nothing, pretty much anything you say is going to be true unless it's gibberish.” If I said every int is green, well, synesthesia excepted ints don't really have colors, that's crap.

Everything else is “trivially true.” More formally, any predicate of the form {for all elements in S, this is true of that element} is taken to be “trivially true” when S is an empty set. It is something like the code,

    let agg = true
    for (const element of mylist) {
      agg = agg || f(element)
    }

where no matter what f is, if mylist was empty, agg is true. (“Trivial” here is sometimes replaced with the word “vacuous” because “trivially” is also a common English word meaning “easily,” so something “trivially true” might also by normal English rules mean “is easily seen to be true, is easily proven to be true, we can debunk the opposite with a 5 second look at the Wikipedia page,” etc.)

So that's the convention and besides making base cases much easier, one reason to do this is that the negation of any “for all Xs in the set, Y” is always perfectly specified as “there exists an X in the set such that not-Y.” The other convention you'd have to negate as “Either the set is empty or (...)”.

So the claim is that the perfect Liar is lying according to mathematical rules of Truth and falsehood, and according to those rules, this statement “all of my hats are green” can only be false if there exists at least one hat belonging to the liar which is not green. If there are no hats then the statement was trivially true.