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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."

#181

Post by JadeNB »

gpderetta wrote: post #168
GuB-42 wrote: post #150

Earlier quoted context omitted.

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 withou…

The question becomes why all_of returns true for an empty list. A better example would use std::accumulate and an 'and' function argument. The original STL documentation [1] described the function argument as having to model the Monoid[2] concept and have an identity value. The identity value for the 'and' Monoid is 'true'. [1] https://www.boost.org/sgi/stl/MonoidOperation.html [2] note: Monoid, not Monad, and yes, c…

> The question becomes why all_of returns true for an empty list.

Because it is, if you'll forgive my Haskell-ese, the only implementation that means that `all_of $ l1 ++ l2 == (all_of l1) && (all_of l2)` for all lists `l1` and `l2`, including empty ones.

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

#182

Post by dingaling »

fortran77 wrote: post #126
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.

In the world of logic puzzles, it is understood that a "liar" never makes a true statement, and that "vacuously true" statements are false.

Which seems to be an untenable premise.

"I never make a true statement" says the liar.

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

#183

Post by Supermancho »

LudwigNagasena wrote: post #177
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…

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

Yes

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

#184

Post by out_of_protocol »

aidenn0 wrote: post #157
jp57 wrote: post #131

Earlier quoted context omitted.

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.…

Yes we can because if there are no ten-foot tall men, then it is indeed true that "All ten-foot tall men have brown hair"

If that is true then liar said truth.

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

#185

Post by grraaaaahhh »

Supermancho wrote: post #173
grraaaaahhh wrote: post #170

Earlier quoted context omitted.

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.

Sure, but the question isn't about which statement is possible from the liar's statement, it's about which statement we can conclude from the liar's statement.

The liar could be lying because they have no hats. They could be lying because they have a non-green hat. We cannot conclude E because it's possible that E is not correct.

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

#186

Post by AtlasBarfed »

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.

The linguistics imo are pretty ill defined.

"All" bring a common colloquial term doesn't have a strict set theory definition here. It is reasonable many people think zero hats is means the lie is in this very first word.

A lot of people will consider "all" to implicitly mean 1 or more, while I think strict logicians will map colloquial all to 0 or more.

All mat imply colloquially 2 or more as well, as why bother say "all" if you had one hat in the truthful sense

"My hats" contrasts with the "has a hat" because having a hat in your possession that you could have borrowed does not confer ownership that the word "my" can imply.

So great, a three letter word and a two letter word and we are knee deep in ambiguity.

They could be wearing the hat to try to publicly locate the true owner who might say "hey I lost that hat at x".

"Are green"... Green as in vegetable? Green as in the specific wavelength defined as green and not lime or some other named shade? Completely green dyed being undermined by a black spot or a pattern on the hat?

Imo zero hats of ownership is a viable lie to the statement, as is having one red-green hat.

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

#187

Post by scoofy »

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…

I mean, it's important to remember that the axioms of first-order logic are arbitrary. We could easily argue that the truth value of an empty group is undecidable, and that would better correlate to natural language logic.

The fact that we compact these edge cases into arbitrary truth values is just for ease of computing.

This is also relevant to the arbitrary choice of the 'inclusive or' as a default over an 'exclusive or', which most people use in natural language.

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

#188

Post by Suppafly »

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.

>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.

Does that actually happen in academia? It seems to mostly be a social media thing.

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

#189

Post by hinkley »

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…

> Either all their hats are some other color or only one hat is green. This means you have to assume the liar has a hat. How do we know that?

Which is another listener’s bias: is lying by omission a lie?

All my horses are unicorns. I don’t have any horses, nor unicorns. So it’s true but also not.

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

#190

Post by ninalanyon »

bingohbangoh wrote: post #154
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…

I agree. It reminds me of the math "puzzles" on Twitter which go: 1 shoe + 1 shoe = 2 2 shoes + 2 shoes = 4 3 shoes + 2 shoes = ??? And the answer isn't 5 because a) we're not counting shoes and b) the shoe laces were different colors. There's nothing clever, it just teaches you to be hyper cynical and question every little detail which isn't relevant to either Math or the real world.

I have a little monograph written many decades ago on Dimensional Analysis. Since reading it, not quite so many decades ago, I simply dismiss puzzles of this sort because the two sides of the equations are dimensionally incongruent. This means that I have to try to guess the state of mind of the questioner rather than solve a logic problem.

It's a handy stance because I'm no good at either solving logic problems or getting inside other people's heads!

Another on that really irritates me is the kind that presents a series of integers and asks which integer comes next. Any integer will do, you just have to fit the appropriate polynomial.

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