SPOILER
The statement translates to:
∀x ( IsAHatOfMine(x) => Green(x))
That's just equivalent to
∀x (~IsAHatOfMine(x) ∨ Green(x))
by the definition of implication (it's only false if the antecedent is true, and the conclusion false).
The negation of that is (by repeated application of De Morgan's):
~∀x (~IsAHatOfMine(x) ∨ Green(x))
∃x ~(~IsAHatOfMine(x) ∨ Green(x))
∃x IsAHatOfMine(x) ∧ ~Green(x))
Thus, the liar has at least one hat, that, furthermore, is not green, so A) [EDIT: but not D - I misread it].
In ordinary English, the meaning of the original phrase, thus the answer to the puzzle, is different.
The statement translates to:
That's just equivalent to by the definition of implication (it's only false if the antecedent is true, and the conclusion false).The negation of that is (by repeated application of De Morgan's):
Thus, the liar has at least one hat, that, furthermore, is not green, so A) [EDIT: but not D - I misread it].In ordinary English, the meaning of the original phrase, thus the answer to the puzzle, is different.