Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

theguardian.com

Post reply on HN
Print view View original

311–320 of 419 posts

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

#311

Post by LudwigNagasena »

Supermancho wrote: post #291
LudwigNagasena wrote: post #204

Earlier quoted context omitted.

So if “all my hats” doesn’t imply that I have at least one hat, “some of my hats” doesn’t imply it either; otherwise we wouldn’t be able to derive “some” from “all”. Hence, “some of my hats are green” doesn’t imply that “at least one of my hats is green”. That’s a claim that contradicts both traditional formal logic interpretation and common sense English interpretation.

I think the same of your interpretation of some vs all. Some can contain all, just as it contains none. Both some/all imply, but do not assert existence. Claiming it tautologically defies logic is not compelling.

Well, I think showing that it defies logical inference is quite relevant in the context of that thread being about translating typical English into first order logic to do logical inference.

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

#313

Post by jldugger »

from-nibly wrote: post #228
whatshisface wrote: post #215

Earlier quoted context omitted.

The article said that the problem was based in formal logic and appeared on a "maths test." That means it isn't ambiguous question about the English language.

No that doesn't mean anything. Tests and books can be wrong.

Well, part of the coursework, and probably the hardest part, is about translating natural language problems into formal logic statements, ie:

"The liar says All my hats are green" becomes "¬∀x|x∈hat,OWNS(x) (GREEN(x))" or similar.

And then from there you can also translate the five provided answers and try to find a contradiction.

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

#314

Post by LargeWu »

hulium wrote: post #309
LargeWu wrote: post #301

Earlier quoted context omitted.

That's fine, but given the parameters of the problem change with the Monty Crawl variant, it's not the same problem, and doesn't invalidate the answer of the base variant.

The problem is that the parameter was unspecified in the original problem. Your answer is equivalent to the shaky answer referred to in the paper. Without knowing that the host selects between 2 goats with 50/50 chance, you cannot give a general answer.

I don't follow, which parameter was unspecified in the original problem?

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

#315

Post by seanhunter »

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.

If set of his hats is the empty set then it contains no hat which is not green, which would make his statement true. As siblings have said, this is a “vacuous truth” in formal logic.

Edit to add: if you find this problematic consider that the statement “All my hats are green” in formal logic is identically equivalent to

For all hats h in my hats, h is green.

So for this statement to be false there needs to be a hat in the set “my hats” which does not have the property that it is green. If “my hats” is empty or indeed if somehow contrary to the rules of the game “my hats” only contains things which are not hats or all the hats it contains are green then the statement is true.

Since the speaker is a liar the statement cannot be true. Therefore there is at least one hat in “my hats” which does not have the property that it is green.

The maths students will be learning this in the context of negation and will have learned that the negation of a universal (“for all”) statement in predicate logic is an existential (“there exists”) statement. Since we have a liar we have to negate what the liar says so since the liar says

“For all hats h in my hats, h is green.”

We negate this and deduce

“There exists at least one hat h in my hats such that h is not green.”

In the context of old-fashioned predicate logic this is not ambiguous.

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

#316

Post by copypasterepeat »

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…

Ah I see what you mean. Yeah, that makes sense. If the host behavior is not clearly stated then the right answer would indeed be "it can't be determined from available information".

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

#317

Post by InsideOutSanta »

SirMaster wrote: post #197
InsideOutSanta wrote: post #34

Earlier quoted context omitted.

Whether you are wrong depends on whether you interpret his statement in a mathematical or in a colloquial sense. Colloquially, if I have no cats and tell you "all my cats are brown", you'd say that I'm lying, beause I'm implying that I have cats. Mathematically, if I have no cats, then it is true to say that all of the ones I have, which are zero, are brown.

If you have none how can you say they are specifically brown? You could say they are any color then which makes them being just 1 specific color not true. Your non-existent cats aren't brown, they are every color or even no color. Maybe even more accurately they aren't brown, they are an undefined color. I'm not really satisfied saying that the characteristics of something that doesn't exist can be anything. I am sat…

You are talking about the colloquial meaning of these words.

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

#318

Post by eterm »

praptak wrote: post #245

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…

Here's one way to maybe make it less counter-intuitive:

We have a set of items X1, X2, X3, ... etc.

Each of X1, X2, X3 differently matches conditions A, B, C, D, etc.

Think of "All" as: A AND B AND C AND D AND...

Each term we add we further restricts the result-set.

Thus if we start with lots of conditions, as we remove conditions, we increase the matching results, until we remove all conditions, and "All([])" matches everything, i.e. is vacuously true.

Likewise, think of "Any" as: A OR B OR C OR D.

Here, as we increase the number of conditions, we increase how many things match, thus "Any" on an empty set returns nothing, i.e. is vacously false.

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

#319

Post by samatman »

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.…

There are normal-discourse statements in which vacuous truth plays a role, example: "I've never met a Frenchman I didn't like, but then again, I've never met a Frenchman". This is, of course, a joke: and it relies on the fact that we assume the set "met a Frenchman" will have contents, but we recognize that, due to the negation, the empty set is actually valid.

But it doesn't work the other way around: "Every Frenchman I've ever met has become a good friend, but then again, I've never met a Frenchman". This isn't funny, because the second clause makes the first clause into a lie, as truth is normally understood. This is not a place where we colloquially accept an empty set.

So the puzzle posed translates the English sentence into logic badly. It isn't the conclusion which is counterintuitive, it is the logical analysis which is flawed.

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

#320

Post by scoofy »

8note wrote: post #307
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,…

absurd statements are usually jokes a joke > "all my hats are green" - bill > "but green hats catch fire in the sunlight" - joe > "and thats why i dont have any hats" - bill from the link: > Many conversations have goals other than the exchange of information. One is amusement, which speakers often pursue by making jokes (Lepore & Stone 2015: §11.3). Because the goal is not to provide information, the maxims of Quali…

This is why I point out that an absurd statement still points toward meaning via a metaphor or language game, which I would put wordplay/jokes under the rubric of.

In fact, in my thesis, I cited The Naked Jape, by Jimmy Carr specifically in reference to jokes (it has a one-liner on every page). On of my main arguments against Gricean conversational implicature theory was that the theory itself was a form of begging the question or no true scotsman problems, in that all of the obvious examples where a counter-factual to the cooperation principal that exist everywhere are excused as "not conversation."

https://archive.org/details/nakedjapeuncover0000carr

Again, yes, you can have wordplay, but wordplay is wordplay, and is a language game that exists and is trying to do something in a different framework.

The reason why so many folks have no issue with the puzzle is that they view it as a puzzle (a kind of language game), and not a sensible human communication. This lets them genuinely consider absurd statements and treat them as normal.

Post reply on HN