Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

A liar who always lies says "All my hats are green."

theguardian.com

Post reply on HN
Print view View original

111–120 of 419 posts

Re: A liar who always lies says "All my hats are green."

#111

Post by chias »

jblezo wrote: post #106
FabHK wrote: post #95

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 a…

I agree for A), by why D) ?

I 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.

Re: A liar who always lies says "All my hats are green."

#112

Post by aabhay »

LudwigNagasena wrote: post #99
marcosdumay wrote: post #89

Earlier quoted context omitted.

In programing, you can always rewrite that first rule as "all" and "some" must compose over set union. So, "all (A ∪ B) == all A && all B", and "some (A ∪ B) == some A || some B". That lets you discover the answer for the empty set.

Which leads to a funny fact that if all elements of the set S satisfy proposition P it doesn’t necessarily imply that some elements of the set S satisfy proposition P.

My description of the power set is by definition allowing all to imply some

Re: A liar who always lies says "All my hats are green."

#113

Post by JKCalhoun »

bryanrasmussen wrote: post #43

Actually I would say by the rules of English we cannot conclude any of those multiple choice questions is the absolute case - A) "The liar has at least one hat." cannot conclude because may have no hats, thus the lie is in the "all my hats" B) "The liar has only one green hat." cannot conclude because may have 2+ green hats out of a 3+ set. C) "The liar has no hats." cannot conclude that because he may have hats that…

As others have noted in the comments, if the liar has no hats, then "All my hats are green" is, in some formal understanding, true. The liar cannot make true statement so must in fact have at least one hat.

That makes A) a good candidate.

But E) can also be true — and in fact both A) and E) can both be correct.

Re: A liar who always lies says "All my hats are green."

#114

Post by jblezo »

jblezo wrote: post #106
FabHK wrote: post #95

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 a…

I agree for A), by why D) ?

Also, I'm not sure that those two affirmations are equivalent:

  ~∀x  (~IsAHatOfMine(x) ∨  Green(x))

   ∃x ~(~IsAHatOfMine(x) ∨  Green(x))

Re: A liar who always lies says "All my hats are green."

#115

Post by LudwigNagasena »

bradley13 wrote: post #24
n4r9 wrote: post #2

My take (possible spoiler): If he had no hats, then his statement would technically be true. Therefore he has at least one hat. He may have some green hats and some non-green hats, but must have at least one non-green hat. He could have any number of green hats, including zero, as long as he has at least one non-green hat. So the only derived statement that we can conclude to be true is A.

Speaking mathematically , you are right. However, linguistically I disagree. Consider: Someone tells you that "all of their kids are doing great in school". Turns out they have no kids. They obviously were trying to deceive you, and make you think they do have kids - in fact, since plural, more than one kid. Hence, it is effectively a lie. So if the liar speaks of "all my hats" while having none, that is deceptive. I…

What does it even mean to be right mathematically here? If I invent a mathematical structure where I define elements 1 and 2, an operation + and a relation = that posits that 1+2=2, I can say that mathematically one apple plus two apples equals two apples. Would I be mathematically right or would I be applying a wrong/not-even-wrong/linguistically deceiving /incoherent model to the real world?

Re: A liar who always lies says "All my hats are green."

#116

Post by mjburgess »

rawgabbit wrote: post #103

I never liked this type of puzzle. It is not formal logic but more about the idiosyncrasies and conventions of the English language. I put this puzzle on par with Agatha Christie’s murder mysteries. It requires a suspension of disbelief and logic to be believable. Someone who always lies means in the purest sense means you cannot trust anything they say. Even the word “hat” could mean they are talking about their pet…

This is generally the case for the vast majority of puzzles, and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency.

I've rarely encountered a case where it is isnt an extreme lack of self-awareness in the questioner -- eg., being extremely overfit to language/notation/etc. localised to their own area of expertise.

Re: A liar who always lies says "All my hats are green."

#117

Post by Waterluvian »

rawgabbit wrote: post #103

I never liked this type of puzzle. It is not formal logic but more about the idiosyncrasies and conventions of the English language. I put this puzzle on par with Agatha Christie’s murder mysteries. It requires a suspension of disbelief and logic to be believable. Someone who always lies means in the purest sense means you cannot trust anything they say. Even the word “hat” could mean they are talking about their pet…

Agreed. I bin this with other “rules lawyering,” which usually feels unfair, uninteresting, and unsatisfying.

It’s why puzzlemaking a truly challenging and impressive skill. It must be fair, a challenge, and obvious in retrospect.

I think it’s why the twist in Bioshock worked so well for me: it was right there in plain sight numerous times, while other twists in mystery games/films elicit a groan.

Re: A liar who always lies says "All my hats are green."

#118

Post by inerte »

grues-dinner wrote: post #47
baalimago wrote: post #23

I tried to figure it out but got stuck on the linguistic dilemma if he's lying about the concept of himself existing ("All of MY hats"). Then I decided I have better things to do.

Right? There's more: Perhaps he exists, but he's denying that concept of ownership exists. He may also be claiming that the hats are ecologically friendly. Are we also to assume that if all the hats were each mainly green on the outside but had brown linings, they are, or are not, each said to be "all green"?! I need a pint.

Ah... lateral thinking. You reminded me of this joke https://www.snopes.com/fact-check/the-barometer-problem/

Re: A liar who always lies says "All my hats are green."

#120

Post by FabHK »

jblezo wrote: post #114
jblezo wrote: post #106

Earlier quoted context omitted.

I agree for A), by why D) ?

Also, I'm not sure that those two affirmations are equivalent: ~∀x (~IsAHatOfMine(x) ∨ Green(x)) ∃x ~(~IsAHatOfMine(x) ∨ Green(x))

That's basically "infinite" DeMorgan:

https://en.wikipedia.org/wiki/De_Morgan's_laws#Extension_to_...

Post reply on HN