The Parable of the Dagger
28Ben_Jones
6Dmitriy_Kropivnitskiy
33victordrake
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63Eliezer Yudkowsky
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7[anonymous]
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10Eliezer Yudkowsky
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New Comment
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."
2
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.
2
A statement that's neither true nor false can't be false...
7[anonymous]
Yes, but lies needn't be falsities, any more than honest statements need be true.
5
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.
7
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.
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.
2
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.
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 ...
2
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.
-5[anonymous]
We note that the king did not say one thing the jester did: "... one, and only one, of the inscriptions is true."
6
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.
4
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.
8
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.
1
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.
9
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 box...
1
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.
0
Good point--in the original wording, it says it was inscribed by "Bellini", who is established earlier to always tell the truth.
0
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.
1
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?
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.
-1
So, if someone came up to you and told you something that couldn't possibly be true, you'd say they weren't lying?
5
It's not dishonest anyway. The king did not suggest that all inscriptions he wrote were true, nor did the jester assume that.
1
The king did, however, count on the Jester's assumption that the content of the boxes could be deduced from the inscriptions.
2
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.
1
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.
1
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?
2
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.
0
That is correct.
0
Ok, thank you for that clarification.
2
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.
0
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.
1
Markdown syntax. Asterixes give italics. > at start of paragraph for block quotes. Help link just below comment box. Welcome. Etc.
1
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.
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.
0[anonymous]
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.
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...
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.)
3
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.
8
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.
"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."
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.
2
That would make the first inscription true. (And therefore false, and therefore paradoxical, etc)
6
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.
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.
2
The parable implied the disconnect between inscriptions and box content, so no, there couldn't have been enough information.
6
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)
8
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 ...
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...
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...
7
If they were both false, that would make the first inscription true.
1
...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.)
1
Indeed. So the inscription is both true and false. You got a problem with that? ;)
2
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".
...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.
0
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.
0
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 th...
0
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".
2
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.
1
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"
1
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.
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...
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...
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
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