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

51–60 of 419 posts

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

#51

Post by grues-dinner »

magicalhippo wrote: post #39
bradley13 wrote: post #24

Earlier quoted context omitted.

Speaking mathematically , you are right. However, linguistically I disagree. Consider: Someone tells you that "all of their kids are doing great in school". Turns out they have no kids. They obviously were trying to deceive you, and make you think they do have kids - in fact, since plural, more than one kid. Hence, it is effectively a lie. So if the liar speaks of "all my hats" while having none, that is deceptive. I…

And that's why SO gets mad when I come home with six cartons of milk[1]. [1]: https://blog.bryanbibat.net/2013/01/02/programming-joke/

I prefer to call her the first wife (apologies to Sir Clement Freud)

Also somehow saying she's my favourite wife was a problem, and yet "least favourite" was worse. Honestly!

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

#52

Post by gpderetta »

MrMcCall wrote: post #22
gpderetta wrote: post #4

Earlier quoted context omitted.

Agree it must be true that it has at least one hat, and it must be non-green (he might have other green hats).

Why would you conclude that the liar is telling the truth that they have any hats at all?

"for each hat in the set of hats I know. the statement 'the hat is green' is true"; the previous statement would be true if the set of hats I know is empty.

Incidentally, if you are a programmer it should be obvious that folding 'and' on an empty set must return True.

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

#53

Post by mewpmewp2 »

derbOac wrote: post #49
joachimma wrote: post #11

Earlier quoted context omitted.

No, in logic that is a vacuous truth. All my hats is true for zero hats. But that would not be a lie. And since the liar always lies, that can not be case.

For me, this is one of those types of examples that illustrates the problem with logics allowing vacuous truth. It amounts to an assumption of an implied conditional ("If I have hats...") which is not always warranted. The "gotcha" here says more about the vacuous truth assumption than it does someone who falls for it.

I think it is very logical to allow for vacuous truths. Doing otherwise would not be logical. The actual key insight is to accept that in a lot of cases everyday communication itself is not logical, because it is more efficient to communicate skipping always being logically correct. This builds social intuition that goes against the logic. It is interesting to observe and point out those cases, which this puzzle does.

Because for efficiency reasons you make a lot of assumptions constantly that may or may not be true, and 99% cases it would work for your favour.

Sometimes assumptions need to be challenged or we need to be reminded of that it can be good to challenge assumptions in certain cases, it can allow us to discover some new things.

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

#54

Post by frozenlettuce »

grues-dinner wrote: post #21

If we're doing "provoke arguments on the internet using confusingly-stated constructs", they should have some self respect and use the venerable 6÷2(1+2). And then read https://xkcd.com/169/ Anyway, I'm sure there'll be a YouTube video about this with an AI voiceover soon.

This is part of a Brazilian test - those are common for enrolling in free higher education or working in public positions. IMO one of the most vile ways of wasting the young generation's potential, as you need to be really smart to pass those exams (think 1-10000 candidate/position ratio) but too stupid to accept the position (think $500 USD monthly salary).

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

#55

Post by EmilienCosson »

!! Spoil:

Since we cannot refer to the liar, we can refer to the extended puzzle's author.

The author states that the goal of the extended puzzle is to determine if there is, 'IF ANY' (!!), a correct statement among [A, B, C, D, E]. Thus, there can be zero or at most one statement we can conclude as being true for sure.

The liar didn't said if he has hats. Maybe he has 0. Maybe 1. Maybe n. We just don't know.

'A: The liar has at least one hat.' > We cannot conclude this statement as sure, since maybe the liar has in fact 0 hats.

'B: The liar has only one green hat.' > He has maybe 0 hats. Or maybe n | n>1.

'C: The liar has no hats.' > He has maybe 1 hat. Or maybe n | n>1.

'D: The liar has at least one green hat.' > He might not have any hats at all.

'E: The liar has no green hats.' > Since the liar may have 2 hats, one could be green and the proposition could still be false, as it is a lie.

Since we cannot conclude any of the statements as being definitively true, the extended puzzle's answer is none of them are true for certain. It depends on how many hats the liar has.

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

#56

Post by b450 »

My guess is we're supposed to read this sentence as:

Ɐh G(h)

(for all my hats, the hat is green)

or whatever similar formulation:

Ɐx (H(x) ^ M(x)) → G(x)

(for all x, if x is a hat and x is mine, then x is green)

Either way, the general idea will be that negating the statement (making it a lie) will make it a negative existential quantifier:

Ǝh ~G(h)

(there exists one of my hats such that it is not green)

Or in the case of the alternate formulation:

Ǝh (H(x) ^ M(x)) ^ ~G(x)

(there exists an x, such that x is a hat and x is mine, and x is not green)

So I think we answer (A) The liar has at least one hat.

All that said, I think other commenters are rightly pointing out that this relies on a very questionable distinction between semantics - which is what we've formalized above - and pragmatics. In conversational pragmatics, "All my hats are green" means that I have at least one hat (probably at least 3, even, since the sentence didn't say "My only hat" or "Both my hats"). One might explain this by way of an implicit pragmatic conversational principle that all statements should be relevant and informative in some way, which vacuously true statements (like, "all grass growing on the moon is purple") are not (see the "Gricean maxims").

If we don't make this implausible distinction between semantics and pragmatics (implausible to me because it assumes that sentences in general are usefully analyzed as having "propositional" meanings which can be evaluated outside of any conversational context), we might cash out the statement as:

Ɐh G(h) ^ Ǝh G(h)

so we can conclude, since this is a lie, that:

Ǝh ~G(h) ∨ Ɐh ~G(h)

Which is consistent with the liar owning no hats, as in:

> "All my hats are green"

> "Liar! You don't own any hats"

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

#57

Post by mannykannot »

bryanrasmussen wrote: post #43

Actually I would say by the rules of English we cannot conclude any of those multiple choice questions is the absolute case - A) "The liar has at least one hat." cannot conclude because may have no hats, thus the lie is in the "all my hats" B) "The liar has only one green hat." cannot conclude because may have 2+ green hats out of a 3+ set. C) "The liar has no hats." cannot conclude that because he may have hats that…

There are few hard semantic rules in English; it is more a matter of conventional usage and expectations - and when 'all' is used, people usually expect that the sentence is about at least one thing, and probably more. I suspect that this is mainly a matter of omission: in ordinary discourse, there seem to be few occasions for using it when it is definitely the case that the resulting sentence is not referring to anything (or exactly one thing), and if there is doubt about the existence of any referents at all, an alternative phrasing along the lines of 'if there are any X, then...' would be considered the right way to say it.

This is so much so that if you use the 'vacuous all', people will suspect that you have ulterior motives, and are being deliberately obtuse to hide them.

I wonder whether, if we attempted to make explicit all these tacit rules and conventions, we would end up with a consistent logic, and I believe that this looseness of natural languages was the main motivation for formalizing logic, from the enlightenment onward.

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

#58

Post by mewpmewp2 »

Certhas wrote: post #37
rkachowski wrote: post #26

Earlier quoted context omitted.

by the same following, this would mean that any statement on the members of an empty set can be made and it would be logically true? e.g. "all my lamborghinis have magical goat skin seat covers" is true if 1) I have no lamborghinis or 2) All the ones I own have magical goat skin seat covers. (fr I have no logical or mathematical background)

Correct. Common source of confusion/trickery/divergence between ordinary language and formal logic. Edit: Logically speaking, the following two are equivalent: They married and had kids. They had kids and married.

Also I think sometimes children will realize a logic gap there and so they will try this funny trickery where they will make statements like these, which technically are true, but imply something totally otherwise to others. Which I find very interesting and kind of speaks to ability and inventiveness of children to think outside the box. Parents may find it annoying or dismiss it, but I think it is great.

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

#59

Post by anon373839 »

gpvos wrote: post #30
nailer wrote: post #6

Earlier quoted context omitted.

No, the liar having no hats would not make the statement true. ‘All my hats implies’ implies the liar stating they have hats. The liar either has zero hats or some amount of hats. The only thing we know for certain is that if they do have hats, there is at least one non Green hat.

No. In formal logic, if you have no hats, it is true that all your hats are green. You can claim anything about those hats, it is even true that each one of those hats is the same size as the universe, or that they are all completely green and completely red at the same time. In normal language, this would be different, but that is not the context here. > Note: this question was originally set in a maths exam, so the…

This is a somewhat irritating property of formal logic and mathematics. Natural language receives a “special” grammar that isn’t always declared up front. You just have to sort of be in on it.

In this case, the reader is given the special definition of liar, but not the special definition of “lie”. (As in, it’s not a lie to make definitive claims about nonexistent hats.)

A lot of the “trick” in logic puzzles boils down to this issue of word play. This puzzle could have been drafted so that the liar’s statement leaves proper room for the no-hats case, but then it would be too easy.

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

#60

Post by gpderetta »

nailer wrote: post #13
joachimma wrote: post #11

Earlier quoted context omitted.

No, in logic that is a vacuous truth. All my hats is true for zero hats. But that would not be a lie. And since the liar always lies, that can not be case.

Do you think regular people, when communicating, use academic logic? Or do you think the liar is an academic?

Let's say that saying 'all my hats' implies that the set of hats is non empty, then you have the two following statements

    my-hats is not empty
    for every hat in my-hats, is-green(hat) is true
We know that the speaker always lies, so both statements must be false: my-hats must be empty, and it must be that it exists at least one hat in my-hat that is not green. This is a contradiction. So either the speaker or the puzzle is not consistent (and uninteresting), or the 'my-hats is not empty' is not a valid assumption.
Post reply on HN