A liar who always lies says "All my hats are green." • Hacker News Board

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

#241

Post by karmakurtisaani » Mon, Dec 09, 2024, 6:40 PM UTC

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…

But it is in your advantage! Code a simulator and do some numerical experiments. The simulations where you switch end up winning more.

Edit: from your other reply I see your beef is more with the wording.

Edit edit: although I disagree with your point even then. But it seems to lead to rather fruitless argument, so let's leave it here.

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

#242

Post by t0mek » Mon, Dec 09, 2024, 6:40 PM UTC

Yes, the liar have some hats. There are two ways how to conclude this:

1. If they don't have any hats, then any sentence about them would be truth.

2. The negation of the sentence, which must be truth, is "not all my hats are green". For this to be true, there must be some hats that are not green, so there must be some hats in general.

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

#243

Post by drdec » Mon, Dec 09, 2024, 6:41 PM UTC

copypasterepeat wrote: post #231

raincole wrote: post #219

Earlier quoted context omitted.

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…

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.

It depends on if Monty has the option to show another door or do nothing.

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

#244

Post by [deleted] » Mon, Dec 09, 2024, 6:41 PM UTC

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…

[deleted]

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

#245

Post by praptak » Mon, Dec 09, 2024, 6:42 PM UTC

Yes, it is a counterintuitive aspect of mathematics that "for all x in X ..." is always true if X is an empty set, just like "A implies B" is always true if A is false.

I once passed a midterm by abusing the latter. The question was to prove "there exists x such that if |a - b| The actual proof for positive values of x was much harder but the professor respected my math hacking skills and gave me full points for that question.

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

#246

Post by triclops200 » Mon, Dec 09, 2024, 6:42 PM UTC

The proposed solution misses the fact that you can't even derive that the liar has a hat because the subject of the predicate could be the lie. What if the liar had two green cars but any number or no hats? The lie is a lie then even if the statement is vacuous as there is too much ambiguity in the english language overall. Their hats could be any combination of colors or they could even be hatless if the lie was over the subject of the constraint, not the qualification of the subjects, as they were effectively miscommunicating what is green.

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

#247

Post by skybrian » Mon, Dec 09, 2024, 6:43 PM UTC

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…

Yes, this is an artificial example and it uses ambiguous language. But it doesn’t really matter whether you got it right or whether the puzzle is fair. Knowing about “vacuously true” statements is useful because that’s a corner case where a bug can happen. (Or it could be a loophole in a legal document.)

When you’re writing functions that work on lists or sets, you need to decide what to do about statements that are vacuously true. Call it out in the documentation? Put in a special case? Assert that the set is non-empty? Or maybe even create a new datatype to make the corner case unrepresentable. These design choices have side-effects, making a function easier or harder to use, more or less error-prone.

Recognizing vacuous statements is pattern recognition, but it’s sometimes useful pattern recognition, and for children, learning about it might even be fun.

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

#248

Post by ewzimm » Mon, Dec 09, 2024, 6:44 PM UTC

AtlasBarfed wrote: post #186

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.

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…

I like your way of phrasing this. Also, we can assume that this is not the only statement that the liar has ever told, so the context matters. If the preceding statement was "Every hat is my hat." Then we have a definition of "all" in the next sentence that works against the stated answer. So "we can't conclude anything" is really the answer because of undefined terms.

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

#249

Post by [deleted] » Mon, Dec 09, 2024, 6:45 PM UTC

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"

[deleted]

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

#250

Post by stevenAthompson » Mon, Dec 09, 2024, 6:46 PM UTC

bbor wrote: post #166

stevenAthompson wrote: post #160

Earlier quoted context omitted.

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.

I mean, 2400y at least… I guess you mean quantitative logical calculus notation? - the local Russell hater

By "relatively recent" I meant in terms of human evolution. We've had it for around 2400 of the last 300,000 years or so. So less than 1% of our species existence.

I used that phrasing to drive home the idea that logic is not some inherent aspect of nature, or even fundamental to the way humans perceive the world.