6 ms·
SPOILER The statement translates to: ∀x ( IsAHatOfMine(x) => Green(x)) That's just equivalent to ∀x (~IsAHatOfMine(x) ∨ Green(x)) by the definitio
by FabHK 2y ago
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.
- gliptic 2y agoNot D), because it says "The liar has at least one green hat." which isn't implied.
- jblezo 2y agoI agree for A), by why D) ?
- chias 2y agoI suspect a misreading of D) as "The liar has at least one non-green hat" which, in fairness, feels weird that wasn't included in the list.
- autokad 2y agoI did that myself, not sure whats wrong with my eyes >.<
- jblezo 2y agoAlso, I'm not sure that those two affirmations are equivalent: ~∀x (~IsAHatOfMine(x) ∨ Green(x)) ∃x ~(~IsAHatOfMine(x) ∨ Green(x))
- FabHK 2y agoThat's basically "infinite" DeMorgan: https://en.wikipedia.org/wiki/De_Morgan's_laws#Extension_to_predicate_and_modal_logic https://en.wikipedia.org/wiki/De_Morgan's_laws#Extension_to_...
- jp57 2y agoWait 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.
- JaumeGreen 2y agoIn 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 :)
- JaumeGreen 2y agoBy 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.
- deleted 2y ago[deleted]
- bhelkey 2y agoI 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 2y agoAll my unicorns are green.