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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."

#141

Post by dleeftink »

raincole wrote: post #135
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…

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%?

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

#142

Post by sillysaurusx »

mjburgess wrote: post #116
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.

It’s not so much that an inability to solve them means you’re deficient, as it is that an ability to solve them means you’re capable in ways most people aren’t. My wife is great at puzzles, and it’s always humbling to watch her sail through them.

I agree that puzzles alone shouldn’t e.g. determine whether you’re a good fit for a job, though. That’s one of the more annoying parts of software interviewing.

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

#143

Post by jkaptur »

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…

The end of the article notes "A liar is someone who only says false statements." I agree that this is quite different from the colloquial definition of a liar, someone who mixes truth and lies in order to mislead.

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

#144

Post by gosub100 »

eblume wrote: post #134
dotancohen wrote: post #123

Earlier quoted context omitted.

Only people with formal training in logic, or those who work with formal definitions of logic, have trouble with this aspect of the puzzles. Trying to return to your more naive understanding of logic not only helps understand the intent of the puzzle, it also makes the puzzle more fun.

I'm genuinely perplexed by this, how is the puzzle even a puzzle if you don't judge it by formal logic?

It's still useful for young people or those who have never stopped to consider it. I think it's a good introduction to the warts of a context sensitive grammar

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

#145

Post by dwheeler »

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 translating such statements to a formal logic, if that's what you really mean, use an "allsome" quantifier as I've described here: https://dwheeler.com/essays/allsome.html

It's really easy to forget to include an existence quantifier. Having notation specifically designed to automatically include it can avoid some problems.

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

#146

Post by mjburgess »

sillysaurusx wrote: post #142
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.

It’s not so much that an inability to solve them means you’re deficient, as it is that an ability to solve them means you’re capable in ways most people aren’t. My wife is great at puzzles, and it’s always humbling to watch her sail through them. I agree that puzzles alone shouldn’t e.g. determine whether you’re a good fit for a job, though. That’s one of the more annoying parts of software interviewing.

I think it should be the duty of the person asking the question to find a isomorphism to another question that the majority of people can answer.

Eg., rather than asking about the risks of A,,B,C given various probabilities etc. ask them to make bets in a highly familiar environment with a resource that has uniform marginal utility to them... so eg., "suppose you were at home and your friend does..., how much of your time on a sunday would you bet to do... "

You find that when "the very same question" is asked in highly familiar terms people get it right.

Then this investigator-academic should ask: what features of their own puzzle induce the kind of mistakes they see?

In my experience its often that people are far less socially incompetent than questioners, so put "interpretation & trust" priors on terms/presentations of questions that mean they don't parse the presentation into the problem the investigator has in mind.

People who write puzzles tend to be the most bureaucratic sort of literalists who have profoundly eccentric modes of interpretation

The problem that most people are solving is: "what do i say to make this person/question go away"

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

#147

Post by gosub100 »

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…

Any time I hear the Rick Roll song (Rick Astley - Never Gonna Give You Up) I always notice the ambiguity in the lyrics "Never Gonna run around AND desert you" and ponder how his proclamation would still be valid provided he only committed one of those transgressions individually.

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

#148

Post by crazygringo »

sillysaurusx wrote: post #142
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.

It’s not so much that an inability to solve them means you’re deficient, as it is that an ability to solve them means you’re capable in ways most people aren’t. My wife is great at puzzles, and it’s always humbling to watch her sail through them. I agree that puzzles alone shouldn’t e.g. determine whether you’re a good fit for a job, though. That’s one of the more annoying parts of software interviewing.

> as it is that an ability to solve them means you’re capable in ways most people aren’t.

No, because the point is that they always hinge on an arbitrary distinction to give one answer. But if you made a different arbitrary distinction you'd get a different answer. And the arbitrary distinctions are, well, arbitrary. They reflect neither truth nor capability. Just whether you can read the questioner's mind as to what arbitrary and intentionally unstated assumptions they are making.

No thanks.

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

#149

Post by JaumeGreen »

jp57 wrote: post #131
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…

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.…

In 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 :)

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

#150

Post by GuB-42 »

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…

That's the reason natural languages are not very well suited to formal logic, and one of the reasons programmers use programming languages.

If we reframe it using C++ "std::all_of" function over an array of strings called "hats", and say that the following must be false (because he is a liar):

  std::all_of(hats.begin(), hats.end(), [](std::string hat) { return hat == "green"; })
Then we can answer the questions without ambiguity:

A) The liar has at least one hat. Yes, because std::all_of returns "true" on an empty list

B) The liar has only one green hat. Unsure, because { "green", "green", "red" } is false, and { "green", "red" } is also false

C) The liar has no hats. No, it is the opposite of A

D) The liar has at least one green hat. Unsure, because { "green", "red" } is false and { "red" } is also false

E) The liar has no green hats. Unsure, for the same reason as in D

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