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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."

#261

Post by aftbit »

dllthomas wrote: post #254
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 think the puzzle is better if the person "only speaks falsehoods" than if they "always lie" - depending on context, a true statement can be a lie.

This is called out in the article:

>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.

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

#262

Post by lukas099 »

stickfigure wrote: post #251
seanhunter wrote: post #232

Earlier quoted context omitted.

Yes. It isn’t ambiguous and is entirely solvable using old school formal logic of the “All Cretans are liars” sort. If the liar owns no hats the statement “All my hats are green” would be true. Under the parameters of the question it must be a lie and therefore cannot be true. So the liar owns at least one hat which is not green. They may own additional hats which can be of any colour. People who are saying “if they…

I think there's a sort of divide-by-zero problem here. Does an empty set of hats have a color? You could arbitrarily define "all my hats are green" for the empty set as either true or false as part of a consistent logical system. There isn't enough information in the question to know whether we should pick one or the other, though there's probably a colloquial preference for true.

Is there not a convention within formal logic defining this type of statement to be true or false?

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

#263

Post by aftbit »

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"

It seems intuitively wrong that you can say "All X are Y" and yet that doesn't imply "At least one X exists".

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

#264

Post by LudwigNagasena »

tantalor wrote: post #260
LudwigNagasena wrote: post #177

Earlier quoted context omitted.

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

No. (assuming you have no unicorns) "all of my unicorns fly" is true; "some of my unicorns fly" is false; "true->false" is false.

So you claim that in normal English “all” doesn’t imply “some”?

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

#265

Post by LargeWu »

raincole wrote: post #219
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.

My favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability. It really isn't. The problem is usually given in this form: > Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, s…

The difference here is that the Monty Hall problem has an explanation that while counterintuitive, is statistically sound. You should always switch, because the probability you picked the correct door is locked in at the time you made the choice between 3 doors. It is 1/3 that you picked correctly, and 2/3 that you picked incorrectly. The counterintuitive part is that if you switch, you are effectively selecting all the doors you did not pick originally. It's your original door, vs the field.

The green hat problem hinges on subjective interpretations of the meaning of both "liar" and the different ways in which the liar's sentence may be false. It may be false because the liar owns many hats, none of which are green. Or they own many hats, only some of which are green. Or they own no hats. These are all reasonable interpretations of how the sentence might be false, and the answers presented are not necessarily mutually exclusive.

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

#266

Post by hansvm »

raincole wrote: post #219
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.

My favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability. It really isn't. The problem is usually given in this form: > Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, s…

Perhaps there's another formulation you meant to write, but every time I've seen it, it's been equivalent to the thing you just asked, and the only potentially ambiguous part of the question is "to your advantage". If you stay then you have a 1/3 chance of getting a car, and if you switch you have a 2/3 chance.

Yes, it's also correct that you don't "know" the result, and you might prefer goats to cars (even then you should probably sell the car and buy several goats), but there's a reasonable enough interpretation of "advantage" that you shouldn't dismiss the problem outright.

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

#267

Post by naniwaduni »

fargle wrote: post #205
gryn wrote: post #163

Earlier quoted context omitted.

yeah, you can add most "paradoxes" to that list. the one I hate the most is the Monty Hall Problem.

monty hall is not a true paradox. it could be classed as "veridical" (truthful) paradox, or i've seen it called a pseudo-paradox. but you realize that it's unambiguously mathematically true without gimicky word-play or puns or other logic puzzle trickery. it's just simply an un-intuitive result. most statistics really is. curious: why the hate for the monty-hall problem?

The claimed Monty Hall result becomes untrue with very subtle changes to the premises. For example, if the host blindly chooses randomly between remaining doors, the resulting distribution is 1/3 contestant door choice correct (switch loses), 1/3 contestant and host both incorrect (switch wins), 1/3 contestant incorrect host correct (no switch can be offered).

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

#269

Post by LudwigNagasena »

tantalor wrote: post #252
scoofy wrote: post #207

Earlier quoted context omitted.

Oh boy... so I actually wrote a thesis in graduate school on conversational implicature, Paul Grice, and various other theories of implying things. I would actually agree user dwheeler here. Whether or not you agree with Gricean implicature theory (I do not), the point is that making a claim about a group that doesn't exist is absurd. Absurd statements do not convey meaning, and language is a tool for communication,…

Granted all that, but we're not really talking about normal everyday English, but a hypothetical conversation with some mythical entity who can only lie, which is not really a capability of humans; even the most pathological liar among us can and will tell the truth. So I'd put all that theory in a drawer somewhere and acknowledge that, when we're talking about logic puzzles, the rules of logic are paramount, not gra…

Sure, if your main use of formal logic is to while away the time doing vacuous puzzles.

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

#270

Post by dooglius »

raincole wrote: post #239
copypasterepeat wrote: post #231

Earlier quoted context omitted.

I guess I don't understand why you think "I don't know" is the only correct answer. It's clearly not, which is the point of the problem in the first place. It's a bit hard to grok, but once you do, it's clear what the right answer is.

"I don't know" is the only correct anwser because the original problem isn't properly worded. You need to (at least) add this statement to the original question: > The host must open a door in any situation, and you know this rule. Because "host" in daily language is a human being with agency to choose whether to not open a door. Without this statement, the original problem is actually a game theory problem with two…

I don't think improper wording makes it an example of the type of puzzle GP is complaining about, because the counter-intuitive aspect holds up when everything is clarified.
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