Did the dagger have 'pwned' inscribed on it?

It's a dressed up version of "This sentence is a lie". It's only self referential, so it's truth value can't be determined in any meaningful, empirical sense.

Jester should've remembered the primary rule of logic: Don't make somebody look like an idiot if they can kill you.

But can self-reference be formalised?

Yes. Godel demonstrated this.

If this material conditional is true, you should give me a hundred dollars. ;)

I must have edited this parable into an inconsistent state at some point - should've double-checked it before reprinting it. I've rewritten the jester's explanation to make sense.

So, the king put the dagger in the second box that he touched, without regard for whether the jester can find it - right? Is that what the last sentence means?

The last sentence is the King pointing out to the Jester that all the reasoning in the world is no good if it is based on false premises, in this case the false presumption was that the text on the boxes was truthful.

Ian, no, the jester didn't presume the text was true: he simply presumed the first inscription was either true or false, and the problem arose from this presumption.

In my example, on the other hand, the statement is actually true or false, but Eliezer can never know which (if he doesn't decide, then it is false, but he will never know this, since he will be undecided.)

I always thought that the statement "You can never know that this statement is true" illustrates the principle most clearly.

You're right, zzz. Proof, if I needed it, that I am not yet a perfect reasoner.

Caledonian: While Gödel formalised some sorts of self-reference, it's not clear to me how his work applies to puzzles like these, or to the question of how hostile perfect reasoners can communicate. Barwise and Etchemendy's "The Liar" has other approaches to the problem, but I don't think they solve it either.

the question of how hostile perfect reasoners can communicate

Hostile reasoners are rarely interested in communicating with each other. When they are, they use language - just like everyone else.

Oh, I get it, the other box couldn't contain a dagger as well, because the king explicitly said that only one box has a dagger in it. But he never claimed that the writings on boxes are in any way related to the contents of the boxes. Is that it? Or is it that if the "both are true or both are false" sign is false, basically anything goes?

This reminds me strongly of a silly russian puzzle. In the original it is about turtles, but I sort of prefer to translate it using bulls. So, three bulls are walking single file across the field. The first bull... (read more)

The third one says "There are two bulls in front of me and two bulls behind me."

Sorry, don't you mean, "0 in front / 2 behind"? (third bull is walking backwards)

A problem with self-reference which I find nearly as amusing but which is much more terse:

"This sentence is false, and Santa Claus does not exist."

I have created an exercise that goes with this post. Use it to solidify your knowledge of the material.

I suppose the message here is that though the inscriptions (literally) labeled the boxes as X and Y, this does not conform in reality. The words do not make it true, and the Jester made the mistake of presuming that his strict logic meant that reality has to follow the labels that were given. His last words, sadly, was “It's logically impossible!” One should reconsider calling things logical impossibilities, when they are occurring right in front of you. Who know what other logical impossibilities you were missing.

If I were man of literature, I would also ... (read more)

There are a lot of comments here that say that the jester is unjustified in assuming that there is a correlation between the inscriptions and the contents of the boxes. This is, in my opinion, complete and utter nonsense. Once we assign meanings to the words true and false (in this case, "is an accurate description of reality" and "is not an accurate description of reality"), all other statements are either false, true or meaningless. A statement can be meaningless because it describes something that is not real (for example, "This... (read more)

...but could not the Jester rattle the boxes before opening one, and then update his beliefs upon that evidence? I mean, it would not be much to go by, but it's better than nothing... 'But Sire, whatever I find, you lose a Jester! What can ever reconcile you to such a lamentable tragedy?' 'A goblet from your skull?' 'In that case, the important thing for me is not to find the dagger, for which the best choice is not to choose any box.' 'Then you fail by default.' 'Then take the box with the dagger, since I failed by default, and I shall pick the other one.'

Assume not that it is true or false, assume that it's a paradox (i.e. both true and false), and from that it follows that the king didn't lie.

But, still, that's not the only moral of the story. A moral of the story is also that we shouldn't start by assuming some statements are either true or false, and then see what that implies about the referents, unless those statements are /entangled with their referents/. If statements aren't entangled with their referents, then no logical reasoning from these statements can tell you anything about the referents.

The king wrote "This box contains the key." on the 2nd box, before putting the dagger in. Did the second box contain the key as well as the dagger?

The jester should have seen this coming.

"Either both inscriptions are true, or both inscriptions are false."

If this statement is true then the second box must hold the key by the jester's reasoning. However if this statement is false then it doesn't require that the second statement be true. In his testing the jester negated only half of the statement at a time. If this statement is entirely false then it could simply mean that the true-false values of the statements on either box have no relationship to each other. Which did indeed... (read more)

I tried to reason through the riddles, before reading the rest and I made the same mistake as the jester did. It is really obvious in hindsight; I thought about this concept earlier and I really thought I had understood it. Did not expect to make this mistake at all, damn.

I even invented some examples on my own, like in the programming language Python a statement like print("Hello, World!") is an instruction to print "Hello, World!" on the screen, but "print(\"Hello, World!\")" is merely a string, that represents the first string, it's completely inert. (in an interactive environment it would display "print("Hello, World!")" on the screen, but still not "Hello, World!"). 

Edit: I think I understand what went wrong with my reasoning. Usually, distinguishing a statement from a representation of a statement is not difficult. To get a statement from a representation of a statement you must interpret the representation once. And this is rather easy, for example, when I'm reading these essays, I am well aware that the universe doesn't just place these statements of truth into my mind, but instead, I'm reading what Eliezer wrote down and I must interpret it. It is always "Eliezer writ... (read more)

Is the existence of such situations an argument for intuitionistic logic?

Solution (in retrospect this should've been posted a few years earlier):

let
'Na' = box N contains angry frog
'Ng' = N gold
'Nf' = N's inscription false
'Nt' = N's inscription true

consistent states must have 1f 2t or 1t 2f, and 1a 2g or 1g 2a

then:

1a 1t, 2g 2f => 1t, 2f
1a 1f, 2g 2t => 1f, 2t
1g 1t, 2a 2f => 1t, 2t
1g 1f, 2a 2t => 1f, 2f