Once upon a time, there was a court jester who dabbled in logic.
The jester presented the king with two boxes. Upon the first box was inscribed:
"Either this box contains an angry frog, or the box with a false inscription contains an angry frog, but not both."
On the second box was inscribed:
"Either this box contains gold and the box with a false inscription contains an angry frog, or this box contains an angry frog and the box with a true inscription contains gold."
And the jester said to the king: "One box contains an angry frog, the other box gold; and one, and only one, of the inscriptions is true."
The king opened the wrong box, and was savaged by an angry frog.
"You see," the jester said, "let us hypothesize that the first inscription is the true one. Then suppose the first box contains gold. Then the other box would have an angry frog, while the box with a true inscription would contain gold, which would make the second statement true as well. Now hypothesize that the first inscription is false, and that the first box contains gold. Then the second inscription would be—"
The king ordered the jester thrown in the dungeons.
A day later, the jester was brought before the king in chains, and shown two boxes.
"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 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."
The jester reasoned thusly: "Suppose the first inscription is true. Then the second inscription must also be true. Now suppose the first inscription is false. Then again the second inscription must be true. So the second box must contain the key, if the first inscription is true, and also if the first inscription is false. Therefore, the second box must logically contain the key."
The jester opened the second box, and found a dagger.
"How?!" cried the jester in horror, as he was dragged away. "It's logically impossible!"
"It is entirely possible," replied the king. "I merely wrote those inscriptions on two boxes, and then I put the dagger in the second one."
(Adapted from Raymond Smullyan.)
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 :)
No, If the king REALLY wanted to be a dick, he would have put the key and the dagger in the same box, and then said "one box contains a key, and one box contains 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.
What, it doesn't count as a lie if it's in writing? That's a hell of a system of contract law they've got in this allegorical kingdom.
I have a different answer to this than what has been given so far :
It's a question of implicit conventions. The king's challenge follows and mimics the jester's challenge. In the jester's challenge, the jester makes a statement about the truth value of the inscriptions on the boxes. By doing this, he sets the precedent that the inscriptions on the boxes are part of the game and do not engage the honesty of the game maker. The inscriptions can be true of false, and it's part of the challenge to guess what is each one. Only the jester's own words engage his honesty. If he lied, the challenge would be rigged.
The king mimics the jester's setup, but makes no statement about the truth value of the inscriptions on the boxes. That difference should have sounded suspicious to the jester. He should have asked the king if the statements were logical. The king could have lied, but at that point if the king was ready to lie then he'd probably kill the jester even if he found the key.
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 ... (read more)
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."
Unlike the jester's riddle, the king never claimed there was any correlation between the contents of the boxes and the inscriptions on those boxes. The jester merely assumed that there was.
The jester assumed that the inscriptions on the boxes were either true or false, and nothing else.
In the explanation for the puzzle this is adapted from (Puzzle 70 in What is the Name of this Book?, in the "Portia's Casket's" chapter), Raymond Smullyan raises both points: "The suitor should have realized that without any information given about the truth or falsity of the sentences, nor any information given about the relation of their truth-values, the sentences could say anything, and the object (portrait or dagger, as the case may be) could be anywhere. Good heavens, I can take any number of caskets that I please and put an object in one of them and then write any inscriptions at all on the lids; these sentences won't convey any information whatsoever. So Portia was not really lying; all she said was that the object in question was in one of the boxes, and in each case it really was. ... Another way to look at the matter is that the suitor's error was to assume that each of the statements was either true or false."
The given puzzle (the boxes are labeled "the portrait is not in here" and "exactly one of these two statements is true", where the portrait is the desired object, is contrasted with an earlier problem, where there are two box... (read more)
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.
Given that it's strongly implied, and logically necessary, that both inscriptions not be true, I don't think it could be considered a lie.
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.
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)
Sorry, don't you mean, "0 in front / 2 behind"? (third bull is walking backwards)
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.
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.
Well, I'm certainly not going to object to that moral.
Nope. The dagger is only if he fails to find the key, NOT if he succeeds in finding the dagger.
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.
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?
So then the actual correct solution, per the king's description of events, would be to ignore the inscriptions and just open both boxes?
Since the King didn't say that he'd be killed if he found the dagger, only that the dagger would be employed if he failed to find the key. Opening both boxes means finding the key, therefore, open both boxes.
(bonus points for chutzpah if he opens the box with the knife first, says "cool! this will make opening the other box MUCH easier!" and then uses that to get the key out of the second box)
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)
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 t... (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.'
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 th... (read more)
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)