If all statements made by the liar are false, then the statement 'my hats' which is a statement made by the liar about his possession of hats, must be false, right? If the liar claims possession of hats, the truth must be that the liar does not possess any hats.

What if the liar said "I own no hats, but if I did own hats, not a single one of them would be green"? Can we then conclude the liar owns a green hat?

Formal logic generates contradictions easily and confuses people when expressed in word problem format, because humans aren't that logical - liars may only lie 10% of the time, when they see a benefit. It's like rational actor theory in economics - people may only act rationally 50% of the time.