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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."

#11

Post by joachimma »

nailer wrote: post #6
n4r9 wrote: post #2

My take (possible spoiler): If he had no hats, then his statement would technically be true. Therefore he has at least one hat. He may have some green hats and some non-green hats, but must have at least one non-green hat. He could have any number of green hats, including zero, as long as he has at least one non-green hat. So the only derived statement that we can conclude to be true is A.

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 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.

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

#12

Post by MrMcCall »

soneca wrote: post #9
MrMcCall wrote: post #5

It is either that: a) they don't have any hats, i.e. that they are lying about having any hats at all. b) they have some number of hats (n >= 1) and at least one is not green, because they are lying about the color of all their hats. I would be happy to learn how I am wrong.

I wonder what color is a hat that doesn’t exist. If a person doesn’t have any hats, is the statement ”All my hats are hats” false?

I think the question is more about liars than it is about hats.

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

#13

Post by nailer »

joachimma wrote: post #11
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 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?

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

#14

Post by cluckindan »

soneca wrote: post #9
MrMcCall wrote: post #5

It is either that: a) they don't have any hats, i.e. that they are lying about having any hats at all. b) they have some number of hats (n >= 1) and at least one is not green, because they are lying about the color of all their hats. I would be happy to learn how I am wrong.

I wonder what color is a hat that doesn’t exist. If a person doesn’t have any hats, is the statement ”All my hats are hats” false?

That is a tautology. A tautology is always true.

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

#15

Post by ColinWright »

nailer wrote: post #6
n4r9 wrote: post #2

My take (possible spoiler): If he had no hats, then his statement would technically be true. Therefore he has at least one hat. He may have some green hats and some non-green hats, but must have at least one non-green hat. He could have any number of green hats, including zero, as long as he has at least one non-green hat. So the only derived statement that we can conclude to be true is A.

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.

I read statement "All my hats are green" as meaning:

For every hat H that I have, H is green.

If I have no hats, this statement is true, just as

* the empty sum is 0,

* the empty product is 1,

* the empty AND is True, and

* the empty OR is False.

So with this interpretation, the liar having no hats would make the statement true.

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

#16

Post by soneca »

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?

It is a logic puzzle. It is not two regular (or academic) people communicating

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

#17

Post by qsort »

davidt84 wrote: post #3
n4r9 wrote: post #2

My take (possible spoiler): If he had no hats, then his statement would technically be true. Therefore he has at least one hat. He may have some green hats and some non-green hats, but must have at least one non-green hat. He could have any number of green hats, including zero, as long as he has at least one non-green hat. So the only derived statement that we can conclude to be true is A.

Yeah, I'm a bit confused there is no option "he has at least one non-green hat", which is what I would answer. Perhaps that means I'm wrong.

It's a multiple choice question, it's asking "which of the following is necessarily implied", and A is the only one.

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

#18

Post by mannykannot »

This puzzle has appeared at least once on Youtube. This was two years ago, and comments are still being added - there are over 15,500 now. They are everything that you would expect on that platform, and such arguments as are being made now are pretty much the same as in the beginning. I hope HN does better.

https://www.youtube.com/watch?v=YQykZU8mcZY

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

#20

Post by aabhay »

The puzzle is referring to the concept of a vacuous truth (https://en.wikipedia.org/wiki/Vacuous_truth).

In most logic frameworks, the All function (upside down A in standard logic notation) is true if and only if no statement within the set is false (i.e. All his hats are green if he has no hats). This is for several reasons:

- it allows for more coherent empty set functions. For example if we take the power set of a set, that power set has the same All value as the standard set (since the power set includes the empty set)

- it allows for early stopping on false statements. So you can define the statement as a lazy executor of all child conditions

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