The Parable of the Dagger
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.)
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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.”
It says if you find the key you’re free and the dagger is if you fail, implying that if both were in the same box his finding the key would have averted the failure.
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.
Definitions matter. If you define a lie as an intentional deception attempt, then the king lied, if you define it as uttering a falsehood, then he didn’t. The modern legal tradition is hazy on this point, and intentional deception without actually making false statements sometimes invalidates a contract, and sometimes doesn’t.
I could make up a new language for every sentence I utter, and claim that 2⁄3 of the words I am merely speaking to myself in an unrelated monologue.
Communication is so context-dependent that I see the utterance of “it was assumed, not implied” as an admission to deceit.
It is possible that “This box contains the key” was a true statement at the time it was written, and then the contents were changed. The king’s explanation does specify an ordering of events.
A statement that’s neither true nor false can’t be false...
Yes, but lies needn’t be falsities, any more than honest statements need be true.
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!
Bx is true if box x has gold, false if frog. one contains frog, other gold → B1 == ~B2. only one inscription is true → Bf == ~Bt
We know:
B2 && Bf || Bt && B1 (I1)
B2 && Bt || B1 && Bt (I2)
Bt == B1 && Bf == B2 && I1 && ~I2 || Bf == B1 && Bt == B2 && ~I1 && I2 # only one inscription is true
From this:
((B2 && B2 || B1 && B1) && ~(B2 && B1 || B1 && B1)) || (~(B2 && B1 || B2 && B1) && (B2 && B2 || B1 && B2))
((B2 || B1) && ~(false || B1)) || (~(false || false) && (B2 || false))
(true && (true && B2)) || ((true && true) && B2)
B2 || B2
B2 # so, Box 2 contains gold
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.
I understand that the “turn the world into paperclips” thing comes from a writing of Eliezer, but it is shorthand for a very unlikely scenario. Moreover, this site has gotten really far away from actually dealing with the problems that an unfriendly AGI is likely to cause. Instead, it seems to deal with stupid human problems, foibles, unreason, etc.
The problem with this, as I see it, is that humans are a diverse group, and what’s rational for those without much brainpower is totally irrational for those with a lot of brainpower.
If you have the capacity to develop a functional AGI with mirror neurons, then that’s what you should be doing. If you have the capacity to develop a part of such an AGI, then that’s what you should be doing.
If you don’t have such a capacity (in brains, or in some other necessary capital, such as monetary/material capital), then you shouldn’t waste your time trying to shape the post-singularity future.
Most of this site is word games that point out that words are inadequate communicators, and generally only used to signal status of one primate to another. True enough, but not related to the domain in question: how to stop (or “make less likely”) homo economicus (var. sociopathicus) from killing/displacing the lesser primates?
First, we must realize that sociopathy is our primary problem. We are entering the singularity using sociopath-defined social systems, sociopath-controlled social systems, and sociopath-populated social systems. The tools of liberal democracy have been abandoned, incrementally, due to the former facts / situation(s). Now it’s true that I’ve used a lot of what Marvin Minsky (more succinctly than this site) called “suitcase words.”
You can either debate me in what Ray Kurzweil calls “slow, serial, and imprecise” language (again, more succinctly than this site, in his book “The Age of Spiritual Machines”), or you can “most favorably” interpret what I say, and realize I’m right, and make your way to the fire escape.
Time is short. Human stupidity is long. We will all likely perish. Make haste.
Just to clarify: I think it’s smart to build AGI right now that starts off not knowing much, build it with a weak robot body that can interact with the world towards a goal, allow unlimited self-improvement, and raise the child with love and respect. I think it’s also good to have multiple such “Mind Children.” The more there are, the more likelihood that the non-sociopaths will be able to ameliorate the damage from the sociopaths, both by destroying them, and by designing systems that reward them enough so that the destructiveness of their sociopathy is not fully realized (as humans have tried and failed to do with their own systems).
We note that the king did not say one thing the jester did: ”… one, and only one, of the inscriptions is true.”
The Jester never assumed that. He showed that if the first inscription is true, it must be false, so he assumed it was false.
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.
For the inscriptions to be either true or false, they would have to correlate with the contents of the boxes. If he didn’t assume this correlation existed, why would he have bothered trying to solve the implied riddle, and then believe upon solving it that he could choose the correct box?
The assumption that one of the inscriptions is true is also the assumption that the contents of the boxes correlate with the truthfulness of the inscriptions. And the key point is that neither inscription need be true, because the contents of the boxes don’t correlate with the truthfulness of the inscriptions. And in fact, neither inscription was true.
In other words, I don’t understand why you’re arguing a simple clarification of essentially the same point you made.
He assumed something that implied the correlation, but he did not assume the correlation. He also assumed something that implied that the key was in the second box, but if he assumed that the key was in the second box, he wouldn’t have even bothered reading the inscriptions.
I’m still not getting the difference. He chose the second box because he deduced the the key must be there based on the assumption that one of the inscriptions was true. There is no equivalence between assuming a key in the second box and deducing a key in the second box based on a false premise.
However, assuming one of the inscriptions is true and assuming a correlation between the inscriptions and the contents of the box seem the same to me. He can’t deduce a correlation between them, because the only basis for such a correlation is the existence of the inscriptions and the basic format of the king’s challenge (which was not identical to the jester’s own riddle). There is nothing in the first inscription to suggest a correlation exists, particularly if he determined that the inscription must be false! It has to be a faulty assumption, and I don’t see how it is different than assuming one of the inscriptions must be true, other than semantically.
I’m not trying to be obtuse here, I’m just not seeing the difference between what you’ve said and what I’ve said.
He did not assume either of the inscriptions were true. He assumed that each was either true or false.
He never assumed a correlation. He deduced a correlation. He was wrong because the deduction hinged on a false assumption.
Edit: Looking back on this, I guess he did assume a correlation. He implicitly assumed that the position of the dagger did not cause the liar paradox. This is still a lot less of an assumption than assuming that either inscription was true.
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 boxes saying “the portrait is not in here” and “exactly one of these two boxes was labeled by someone who always tells the truth” (and it’s given that the only other box-maker always lies). The distinction the author draws is that the second box in the earlier problem really does have to be true or false, since “it is a historic statement about the physical world”, but there’s no such guarantee with purely self-referential labels.
If one of the boxes says that exactly one of them was written by Alice, and you know from another source that Alice always tells the truth, Bob always lies, and both boxes were inscribed by one of them, and Alice and Bob never say anything self-referential, then this is correct.
If it says that one of the boxes was labelled by someone who always tells the truth, then it’s not just talking about the person who labelled that box. It’s also talking about every aspect of reality that they’ve ever referenced, and if they were the one to write that inscription, then it’s self-referential.
Good point—in the original wording, it says it was inscribed by “Bellini”, who is established earlier to always tell the truth.
In which case, if Bellini ever references anything self-referential, the idea that he always tells the truth is not a statement about the physical world. It’s likely that the origin of the paradox is that the claim that Bellini always tells the truth and the rest of the scenario are contradictory.
I notice we’re somehow not debating what Bellini always telling the truth means for the truth value of the inscribed text which may have had no meaning to him?
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.
So, if someone came up to you and told you something that couldn’t possibly be true, you’d say they weren’t lying?
It’s not dishonest anyway. The king did not suggest that all inscriptions he wrote were true, nor did the jester assume that.
The king did, however, count on the Jester’s assumption that the content of the boxes could be deduced from the inscriptions.
The King counted on the Jester making a deductive error in the second puzzle (namely inferring that the content of the boxes could be deduced from the inscriptions given what the King said), just like the Jester counted on the King making a deductive error in the first puzzle.
In this situation, it is still a correct deduction to say “if the statements are true or false, then the content of the boxes is....” With these contents, these statements aren’t true or false.
Sorry, it’s not clear to me why you wrote this reply. Are you trying to dispute something I said, or are you bringing up an interesting observation for discussion, or what?
It sounds like Jiro was saying that the Jester really does not assume that “The content of the boxes can be deduced from the the inscriptions.” He just assumes “The inscriptions are either true or false,” and it logically follows from what the inscriptions say that he can deduce the contents. So the problem wasn’t making an assumption about how the contents could be discovered, but making an assumption that the inscriptions had to be either true or false.
Ok, thank you for that clarification.
That is correct.
If someone came up to you with a puzzle involving transcriptions where there is an expectation that some of the inscriptions are true and some of the inscriptions are false, and nothing the person actually utters is false, then that person was not lying.
In contrast, if someone came up to me and gave me something that looks like a legal notice—a scenario where there is NOT an expectation that the notice might be false—and it turns out that the notice makes false claim, then that person is indeed “lying”, especially if, when I take the notice and say “Thank you” and start to close my door, the guy says “Actually, you have to pay the fine immediately; you can’t just mail it to the police station later” or whatever.
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.
If you do this, the case where the second inscription is true and the first box contains a frog is also consistent.
No, because in that case the first inscription would also be true. Both inscriptions cannot be true.
Interestingly enough, I just mapped this whole problem out carefully in a spreadsheet, and it appears to agree with zzz2. I’ll have to check it now that I’ve seen your comment.
Markdown syntax. Asterixes give italics. > at start of paragraph for block quotes. Help link just below comment box. Welcome. Etc.
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.
He will not be wrong, just ignorant. Hypothetically:
Unknown: Eliezer, do you think that the statement in my comment is false?
Eliezer: Let me see… No, I do not.
U: Aha! Then it is false! Do you think so now?
E: No.
U: Do you think it’s true?
E: No. I understand that I cannot be correct in assigning a truth value to it. Not every sequence of words has a truth value. Moreover, the truth value of some sentences can never be known to me.
U: This makes me so much more confident that the sentence is false.
So we all know something Eliezer cannot ever know. He may even read these lines, and it’ll still be the little secret of humanity-minus-Eliezer.
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 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.”
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.)
Oh, I assumed that they were walking in a circle and the third bull was counting both ahead of him and behind him, even though those bulls are both the same, on the assumption that ‘single file’ =/= ‘straight line’.
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.
It’s not the law of the excluded middle that’s the problem, it’s the jester’s assumption that the entire statement “either this …, or this..., but not both” is true. The jester reasons correctly under his assumptions, but fails to realize that he still has to discharge those assumptions before reaching reality.
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.
… and was first set free from his chains, and then stabbed through the heart with the dagger.
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.
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.
The only necessary assumptions are that the King isn’t lying, and that he isn’t mistaken. Once you know this, you can deduce that there is one key and one dagger.
The jester made an additional, incorrect, assumption that everything on the first box was either “true” or “false”.
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.
The first inscription says that the inscriptions have the same truth value. If the second one is false then the first one implies that it is false which, in turn, implies that the first one is true. Contradiction. So the premise that “the second inscription is false” is false. So the second inscription is true.
The Jester’s logical inference is right. The point isn’t that the Jester’s logic was wrong—it wasn’t. It’s that the Jester assumed that the locations of the key and the dagger would follow the logic when there really was no good reason to assume so. This is meant to illustrate that making unwarranted assumptions about reality isn’t a good idea.
That would make the first inscription true. (And therefore false, and therefore paradoxical, etc)
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.
The parable implied the disconnect between inscriptions and box content, so no, there couldn’t have been enough information.
Do I read this correctly—that there was no key?
That’s incorrect—the king’s uttered words (“One box contains a key, 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.”) were still completely true. The key was in the first box, the dagger on the second.
It’s just that the jester’s reasoning about the supposed logical impossibility of the statements inscribed on the boxes was utter nonsense. He knew that neither of the statements inscribed need have been true, but he still foolishly argued himself into thinking that whether true or false they ‘proved’ the key being on the second box.
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)
King: Very clever. (to the guards) set him free from the top of the tallest tower.
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!)
If they were both false, that would make the first inscription true.
Indeed. So the inscription is both true and false. You got a problem with that? ;)
If something is both true and false, then it becomes trivial to prove that any given fact is true. This is called the principle of explosion.
If it’s neither true nor false, that doesn’t happen.
To make definitions clear, I am using “X is false” to mean “Not X is true”, rather than “X is something other than true”.
...and then the first inscription would be false, etc.
If you are pointing out that would be unstable in that way, or ‘meaningless’, then OK. good point.
(I did specify that I see the statement “Both inscriptions are false” as false rather than just meaningless, though, and the first inscription would be of that same form if the second one were false.)
In any case I still defend the jester’s impression that statements have truth values (excluding ‘meaningless’ ones, as necessary), while still faulting him for something else entirely:
He was (still) modelling his solution after the earlier problem he had constructed (with the frog and the gold), or he was assuming a situation in which none of the statements were ‘meaningless’. Neither was warranted.
(That is one step closer to what many commenters have mentioned, but “This box contains the key.” is plainly just false, not unconnected to the world.)
...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.
There isn’t correlation between these inscriptions and implied contents (since he could have put the key and dagger in either box), but there /is/ correlation between {the inscriptions and contents} and the king’s honesty. The king didn’t lie and he wouldn’t have put inscriptions and contents into such an arrangement that would make it true that he lied. This puts a constraint on how he could arrange the inscriptions and contents.
Salient point: why you mention arrangements of inscriptions and contents at all? That is what confuses me. Either the arrangements matter at some point—such as inscribing—in which case there had been a lie when the king labeled an (apparently?) empty box with “This box contains the key.” (not “this box doesn’t contain the dagger”, which would have been true), or not at all, in which case I reiterate my previous question.
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.
Breaking #24 of the Evil Overlord List makes me wince, too, even if it’s a jester doing it. Not sure if that’s the main point, though, but then, none of the proposed explanation for how the king could pull his “riddle” off without at any point lying feel entirely right to me, so, unless someone offers to help me, I shall have to take your advice and not let myself get entangled in the “complex and detailed logic”, when the answer might as well be “BS”.
There’s a lot of value in that. Sometimes it’s best not to go down the rabbit hole.
Whatever the technicalities might be, the jester definitely followed the normal, reasonable rules of this kind of puzzle, and by those rules he got the right answer. The king set it up that way, and set the jester up to fail.
If he’d done it to teach the jester a valuable lesson about the difference between abstract logic and real life, then it might have been justified. But he’s going to have the jester executed, so that argument disappears.
I think we can all agree, The King is definitely a dick.
I’ll somewhat echo what CynicalOptimist wrote. I think the message is is one any first semester logic student should have been taught or know: a valid argument is not necessarily true. The validity of an argument’s conclusion is all about form of the argument. The truth of the conclusion is an external fact existing completely independent from the argument’s structure.
I’m trying to stay levelheaded about King Richard. What I meant was that there seems to be extraneous details here—about the order things were done in, first inscribe (“key is here”, on an empty(?) box), then put dagger in, or that it was written, not spoken. Many comments only enforce the importance of that.
The “real” answer seems to be one that effectively makes all kinds of communication useless, and what I’ve spent so much time on was trying to pin down the borders of this insanity, some marker saying “abstract logic application to real life* not allowed past this point”.
*) the use of physical boxes binding the riddle to “real life”
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