The Parable of the Dagger

Did the dagger have 'pwned' inscribed on it?

And if the king wanted to be particularly nasty the other box would also contain a dagger :)

And if the king wanted to be particularly nasty the other box would also contain a dagger

No, that the king specified couldn't happen. One of the morals of the parable is that the king didn't lie.

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.

I'm having some trouble with the logic here. I wonder if the parable got a bit garbled.

"You see," the jester said, "let us hypothesize that the first inscription is the true one."

The first inscription says, "Either this box contains an angry frog, or the box with a false inscription contains an angry frog, but not both." Now we are hypothesizing that this is the true one. Therefore "the box with a false inscription" means "the second box". So, "Either the 1st box contains an angry frog, or the 2nd box contains an angry frog, but not both".

The jester goes on, "Then suppose the first box contains an angry frog."

So we know (by assumption) that the 1st clause of the inscription is true, the 1st box contains an angry frog. Since "not both" clauses are true, it means the 2nd clause is false, and so the 2nd box does not contain an angry frog - it must contain gold.

But the jester claims that this is a contradiction: "Then the other box would contain gold and this would contradict the first inscription which we hypothesized to be true." For this to be a contradiction, the 1st inscription would have had to say that the 2nd box should contain an angry frog, but we just saw that it doesn't say that.

I can't make much progress with the 2nd inscription either. I'm getting pretty confused now!

The simplest way to solve the jester's puzzle is to make a table of the four cases (where the frog is, where the true inscription is), then determine for each case whether the inscriptions are in fact true or false as required for that case. (All the while making la-la-la-can't-hear-you noises at any doubts one might have about whether self-reference can be formalised at all.) The conclusion is that the first box has the frog and the true inscription. That assumes that the jester was honest in stating his puzzle, but given his shock at the outcome of the king's puzzle, that appears to be so.

But can self-reference be formalised? How, for example, do two perfect reasoners negotiate a deal? In general, how can two perfect reasoners in an adversarial situation ever interpret the other's words as anything but noise?

"Are you the sort of man who would put the poison into his own goblet or his enemy's? Now, a clever man would put the poison into his own goblet because he would know that only a great fool would reach for what he was given. I am not a great fool so I can clearly not choose the wine in front of you...But you must have known I was not a great fool; you would have counted on it, so I can clearly not choose the wine in front of me." ...etc.

Or consider a conversation between an FAI that wants to keep the world safe for humans, and a UFAI that wants to turn the world into paperclips.

We note that the king did not say one thing the jester did: "... one, and only one, of the inscriptions is true."

But can self-reference be formalised?

Yes. Godel demonstrated this.

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

The King DID lie, because he wrote the inscriptions. What is written on the inscriptions is inaccurate if the dagger is not in the second box.

The simplest way to solve the jester's puzzle is to make a table of the four cases ... then determine for each case whether the inscriptions are in fact true or false as required for that case. The conclusion is that the first box has the frog and the true inscription.

If you do this, the case where the second inscription is true and the first box contains a frog is also consistent.

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.

Eliezer will think that this statement is false.

i.e. the above statement.

Of course, when he does, that will make it true, and without paradox, so he will be wrong. On the other hand, if he thinks it is true, it will be false, and without paradox, so he will be wrong.

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 says "There are two bulls in behind me and no bulls in front of me." The second one says "There is a bull in front of me and a bull behind me." The third one says "There are two bulls in front of me and two bulls behind me."

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)

JonathanG,

Actually, the third bull is just straight up lying. (That's why Dmitriy called the puzzle silly.)

Using the jester's reasoning, it's possible to make him believe that the earth is flat by writing down "either this inscription is true and the earth is flat, or this inscription is false and the earth is not flat, but not both" this makes an unflat earth logically impossible!

I wonder what this says about the law of the excluded middle, I guess that it slides if self reference is involved.

"One box contains a key," said the king, "to unlock your chains; and if you find the key you are free. But the other box contains a dagger for your heart, if you fail."

And the Jester opened both boxes, successfully finding (that is, not failing to find) the key. Of course, the King could declare "you know what I meant to say" and kill him anyway but that does change the intended moral somewhat.

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.

It took me a while to understand this one because theres allot of assumptions within it. They are;

• that the king isnt lying

• that the king isnt mistaken

• that the inscription isnt lying

• that there is infact 1 key or dagger.

All of which have to be taken on faith. Which my brain obviously couldnt handle.

But if you belive all of that. The you should find that; as the king told you one box contained the key, then there is only one key, and of the other box is to be believed "that both boxes contain the same mystery item" then thats a contradiction, which means the opposite box is more likly to be true.

However this is wrong.

If the king is to be believed, then theres a 50/50 chance no matter what box you pick. But if the box is to be believed, then the other box is the container, but it could be lying. Therefore the chance is still an irreducible 50/50. Furthermore, believing either claim would require an assumption that the game was set up fairly or unfairly. And we know assumptions to be fallacies and never to make them.

The answer to the box question can only be worked out once the box is opened and the evidence is found. The validity of the claims can only be tested by using them.

As this is used as a proof of the core sequence “37 ways words can be wrong” “A word fails to connect to reality in the first place.”

I must say that it in no way supports this conclusion.

And the first box was inscribed: "Either both inscriptions are true, or both inscriptions are false." And the second box was inscribed: "This box contains the key."

Suppose the second inscription is false. In that case, the first inscription must also be false, in which case the king can put whatever he damn well pleases in the boxes.

Was there enough information around for the Jester to correctly determine the box? I guess he could have figured that the more obvious solution was the key being in the box labelled as having the key in it, and the king was mad at him, so that probably wasn't it.

That doesn't seem all that strong evidence.

Do I read this correctly--that there was no key?

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 comment on the juxtaposition of the Jester and King. The Jester, who is a fan of logic, lives in the court. His devotion to logical reasoning plays itself out in entertainment form, whether privately in his bedroom, or by sticking an angry frog onto a king. The King, on the other hand, lives in a world of politics, diplomacy, and war. He does not have the luxury of syllogisms, as he is surrounded by flatterers, rivals and enemies. He cannot presume that anything that is presented is not an exaggeration, inaccurate or an outright lie.

The final moral; do not stick angry frogs on someone who has the ability and the potential disposition to kill you. Or more generally, do not stick angry frogs onto people, it is just bad behavior. Just don’t do it.

Or, as P.T. Barnum put it... there is a Sucker Born Every Minute.

Even for Jesters, it's never a good idea to humiliate the King...

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 box contains the key" is meaningless if the world does not contain any boxes) or because it is inconsistent (it has at least one infinite loop, as with "This statement is false"). If a statement is meaningful it affects our observations of reality, and so we can use Bayesian reasoning to assign a probabilty for the statement being true. If the statement is meaningless, we cannot assign a probabilty for it being true without violating our assumption that there is a consistent underlying reality to observe, in which case we cannot trust our observations. Halt, Melt and Catch Fire.

The statement "This box contains the key" is a description of reality, and is either false or true. The statement "Both inscriptions are true" is meaningful if there exists another inscription, true if the second description is true and false if the second description is false or meaningless. The statement "Both inscriptions are false" is meaningless because it is inconsistent - we cannot assign a truth-value to it. The statement "Either both inscriptions are true, or both inscriptions are false" is therefore either true (both inscriptions are true, implying that the key is in box 2) or meaningless. In the latter case, we can gain no information from the statement - the jester might as well have been given only the second box and the second inscription. The jester's mistake lies in assuming that both inscriptions must be meaningful - "one is meaningless and the other is false" is as valid an answer as "both are true", in that both of those statements are meaningful - the latter is true if the second box contains the key, and the former is true if the second box does not contain the key. The jester should have evaluated the probabilty that the problem was meant to be solvable and the probability that the problem was not meant to be solvable, given that the problem is not solvable, which is an assessment of the king's ability at puzzle-devising and the king's desire to kill the jester.

It is also provable that we cannot assign a probabilty of 1 or 0 to any statement's truth (including tautologies), since we must have some function from which truth and falsity are defined, and specifying both an input and an output (a statement and its truth value) changes the function we use. If a statement is assigned a truth-value except by the rules of whatever logical system we pick, the logical system fails and we cannot draw any inferences at all. A system with a definition of truth, a set of thruth-preserving operations and at least one axiom must always be meaningless - the assumption of the axiom's truth is not a truth-preserving operation, and neither is the assumption that our truth-preserving operations are truth-preserving. Axiomatic logic works only if we accept the possibility that the axioms might be false and that our reasoning might be flawed - you can't argue based on the truth of A without either allowing arguments based on ~A or including "A" in your definition of truth. In other words, axiomatic logic can't be applied to reality with certainty - we would end up like the jester, asserting that reality must be wrong. As a consequence of the above, defining "true" as "reflecting an observable underlying reality" implies that all meaningful statements must have observable consequences.

The argument above applies to itself. The last sentence applies to itself and the paragraph before that. The last sentence... (If I acquire the karma to post articles, I'll probably write one explaining this in more detail, assuming anyone's interested.)

All of these comments on the jester wrongly assuming the box inscriptions related to the world seem overwrought to me. I created this account just to make this point (and because this site looks amazing!):

The jester's only mistake was discounting the possibility of both inscriptions being false.

That's it...the inscriptions (both) 'being false'. Not 'pertaining to the real world', not 'having truth values'...just 'being false'.

He figured out that it could not be the case that both inscriptions were true---so far so good. He then assumed that it must be the case that one must be true and the other false, which was only allowing for 1 out of the 2 remaining possibilities (1 true and 1 false, or 2 false). He was modelling his solution after the earlier problem he had constructed (with the frog and the gold), or he was essentially trying to maximize the number of true inscriptions, or both. Neither was warranted.

(I mostly agree with the poster above (Chrysophylax), or at least the first two paragraphs of that long post, in that the inscriptions certainly did have truth values pertaining to the world and specifically to the contents of the boxes. That is mostly the point I wanted to make. I disagree with her or him about this part, though: "The statement "Both inscriptions are false" is meaningless because it is inconsistent - we cannot assign a truth-value to it." I see that statement as false, not meaningless. So I actually take slightly more possible statements as pertaining the world and having actual truth values than does Chrysophylax (which in turn is far more than most other commenters here seem to be reporting). ...basically anything that does not match Chrysophylax's other examples of meaningless statements. I could even go so far as saying that the statement "The invisible unicorn is happy." is false though maybe also being 'meaningless' (maybe because it demands the acceptance of the false statement "An invisible unicorn exists." and could be translated as "There exists an invisible unicorn, and it is happy."). I'd love to hear opinions on that, though!)

...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.'

Regarding the correlation between inscriptions and contents being merely assumed: are the spoken claims any different? I don't see them being called into question the same way.

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?

I can't speak for Eliezer's intentions when he wrote this story, but I can see an incredibly simple moral to take away from this. And I can't shake the feeling that most of the commenters have completely missed the point.

For me, the striking part of this story is that the Jester is shocked and confused when they drag him away. "How?!" He says "It's logically impossible". The Jester seems not to understand how it is possible for the dagger to be in the second box. My explanation goes as follows, and I think I'm just paraphrasing the king here.

1- If a king has two boxes and a means to write on them, then he can write any damn thing on them that he wants to. 2- If a king also has a dagger, then he can place that dagger inside one of the two boxes, and he can place it in whichever box he decides to place it in.

That's it. That's the entire explanation for how the dagger could "possibly" be inside the second box. It's a very simple argument, that a five year old could understand, and no amount of detailed consideration by a logician is going to stop this simple argument from being true.

The jester, however, thought it was impossible for the dagger to be in the second box. Not just that it wasn't there, but that it was IMPOSSIBLE. That's how I read the story, anyway. He used significantly more complicated logic, and he thought that he'd proven it impossible. But it only takes a moment's reflection to see that he's wrong.

Some of the comments above have tried to work out what was wrong with Jester's logic, and they've explained the detailed and subtle flaws in his reasoning. That's great - if you want to develop a deep understanding of logic, self-referential statements, and mathematical truth values (and lets be fair, I suppose most of us do), but in the context of the sequences on rationality, I think there's a much better lesson to learn.

Remember: rationalists are supposed to WIN. We're supposed to develop reasoning skills that give us a better and more useful understanding of reality. So the lesson is this: don't be seduced by complex and detailed logic, if that logic is taking you further and further away from an accurate description of reality. If something is already true, or already false, then no amount of reasoning will change it.

Reality is NOT required to conform to your understanding or your reasoning. It is your reasoning that should be required to conform to reality.

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 turn out to be the case.

In other words: If the statement on the second box is false, then the statement on the first box claiming that the statements on the two boxes are in any way related is also false and the fact that both being false would cause a paradox if the first statement were true is not relevant.

The jester made the mistake of assuming "at least one of the statements is true" and confused validity and soundness, and therefore deserved to be stabbed.

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 writes 'X'", and not just "X". 

But in this example, there were 2 different levels of representation. To get to the jester and the king I need to interpret the words once. But to get to the inscriptions, I must interpret the words twice. This is what went wrong. If I correctly understood the root of my mistake, then, if I was in jester's shoes, I wouldn't have made this mistake. Therefore, I think, my mistake is not the same as jester's. Simultaneous interpretation of different levels of representation is something to be vigilant about. 

C'est ne pas un pipe. This is not a picture of a pipe either, this is a picture of a picture of a pipe. Or is this a piece text, saying "this is a picture of a picture of a pipe"? Or is this a piece of text, saying "This is a piece of text, saying \"this is a picture... \""... :-)

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