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

theguardian.com

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

#171

Post by blueflow »

stevenAthompson wrote: post #160
mjburgess wrote: post #116

Earlier quoted context omitted.

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.

Logic itself is a relatively new invention, and is symbolic itself. That is to say, logic is a map not the territory. That said, if someone can't fathom the most widely used symbolic languages humans use (math, logic, language, etc) they probably do have a cognitive deficit of some sort when compared to those who can.

Languages were made up by mankind at some point. They are not backed by the rest of physical reality. They can only be learned by induction, and there is no guarantee that people will get the same "version" of it.

To speak in your analogy, people walk around with different maps of the same territory and realizing this is the self-awareness mjburgess is talking about.

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

#172

Post by lbwtaylor »

matthewfelgate wrote: post #73

A way to think about it is, if you went into the liars house to prove him wrong, you could find one non-green hat to prove him wrong. If he has no hats you wouldn't be able to prove him wrong. Which reminds me of a quote from the British TV series Yes Minister: "A good speech isn't one where we can prove he's telling the truth. It's one in which nobody else can prove he's lying!"

This answer is the only intuitive one for me, thanks.

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

#173

Post by Supermancho »

grraaaaahhh wrote: post #170
Supermancho wrote: post #167

Earlier quoted context omitted.

> All my hats....", you are simultaneously making an existence statement that you have at least one hat It does not. All my unicorns fly. There is no assumption that I have a unicorn. There is an assumption, based on the claim but it is not a fact. The puzzle also assumes that "my" implies there is some ownership (we'll take for granted "my" means "has" for simplicity), which is another quibble that unravels the whol…

E cannot be correct. "All my hats are green" is still false even when I own a red hat and a green hat.

I would agree that's obvious, if not for the original error.

The liar doesn't necessarily "have" any hats. Again, the assumption that the liar has hats is incorrect because it's relying on an conversational implication, rather than a specific assertion.

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

#175

Post by tantalor »

dwheeler wrote: post #145

One complication is that in typical English, if you say "All my hats....", you are simultaneously making an existence statement that you have at least one hat... but the usual formal logic "forall" quantifier does NOT presume existence. Here's a formal proof that "forall" has a "surprise" meaning for those not well-versed in formal logic: https://us.metamath.org/mpeuni/alimp-surprise.html I propose that when translat…

> in typical English you are making an existence statement

No, you're not.

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

#176

Post by shever73 »

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 an Alex Bellos puzzle, which I’ve learned through experience to ignore. The terms are always so loosely defined that any solution you come up with is a gamble at best.

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

#177

Post by LudwigNagasena »

Supermancho wrote: post #167
dwheeler wrote: post #145

One complication is that in typical English, if you say "All my hats....", you are simultaneously making an existence statement that you have at least one hat... but the usual formal logic "forall" quantifier does NOT presume existence. Here's a formal proof that "forall" has a "surprise" meaning for those not well-versed in formal logic: https://us.metamath.org/mpeuni/alimp-surprise.html I propose that when translat…

> All my hats....", you are simultaneously making an existence statement that you have at least one hat It does not. All my unicorns fly. There is no assumption that I have a unicorn. There is an assumption, based on the claim but it is not a fact. The puzzle also assumes that "my" implies there is some ownership (we'll take for granted "my" means "has" for simplicity), which is another quibble that unravels the whol…

Would you agree with the following proposition: “if all of my unicorns fly, then some of my unicorns fly”?

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

#178

Post by raincole »

dleeftink wrote: post #141
raincole wrote: post #135

Earlier quoted context omitted.

Puzzles like this are to make more people interested in formal logic. The "gateway drug" of formal logic. Just like the barber paradox isn't literally Russell's paradox, but it made more people to look up the history of it and perhaps learned what Russell's paradox is. Hopefully 0.1% of them turn out to be mathematicians.

Why not hopefully more? Less? Is mathematics this inaccessible to the other 99.9%?

Maybe "hopefully" isn't the right word.

My point is that not everyone who understands barber paradox (in plain english) has to understand formal logic, and not everyone who understands formal logic has to become a mathematician. However I still believe the existing of the plain english puzzle is a net positive for humanity's collective mathematical comprehension.

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

#180

Post by copypasterepeat »

pleurotus wrote: post #152
copypasterepeat wrote: post #124

Earlier quoted context omitted.

Agreed. Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement. It could just mean someone who frequently tells falsehoods. Or, more deviously, someone who wants to cause maximum uncertainty in his listeners, in which case some mix of true and false statements would probably be the way to go.

> Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement. FTA: >Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic. A liar is someone who only says false statements. I think it's pretty clear how on definitions

Ah I see that now, thanks. It was under the ad banner so I missed it first time around.
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