alt Hacker NewsSPOILER
The statement translates to:
∀x ( IsAHatOfMine(x) => Green(x))
∀x (~IsAHatOfMine(x) ∨ Green(x))
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))
In ordinary English, the meaning of the original phrase, thus the answer to the puzzle, is different.
Replies
gliptic • 12/09/2024
Not D), because it says "The liar has at least one green hat." which isn't implied.
bhelkey • 12/09/2024
I think this is how the puzzle author intended the puzzle to be read.
That being said, I would argue that, "All my hats are green." has different meaning than "I may or may not own a hat. Any hat that I own is green".
The use of 'all' and the plural of 'hat' implies that the author has multiple hats.
gorfian_robot • 12/09/2024
All my unicorns are green.
Wait a second.
If the liar says, "All ten-foot tall men have brown hair," we cannot conclude that there must exist a ten-foot tall man.
EDIT: I'll clarify to say I wasn't taking issue with the derivation, but rather with the translation of the English statement into first-order predicate logic. No non-logician would conclude that there must be a ten-foot tall man if "All ten-foot-tall men have brown hair" is false. But since we can derive it from the translated logic statement, then there must be a problem with the translation.
In normal discourse people don't accept vacuous truths like that as meaningfully true. Rather I think people would interpret such a statement as a kind of hypothetical: "If there were a ten-foot tall man, he would have brown hair."
It's not clear to me if first-order predicate logic is really equipped to even handle reasoning about these cases of "a liar who always lies." Such a situation seems to be intrinsically higher-order. If a liar states a hypothetical, what does that mean, exactly?
My interpretation of the negation of the statement is, "If there were a ten-foot-tall man, he would not necessarily have brown hair." This doesn't imply the existence of any ten-foot-tall man.