« Futarchy vs. Predictocracy | Main | Open Thread »

February 01, 2008

Comments

Did the dagger have 'pwned' inscribed on it?

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

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

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

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

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

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

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

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

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

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

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

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

Rationality is choosing to acknowledge that candlelight is fire, instead of preserving your dignity by maintaining the search.

Now Eliezer has cleverly gotten us to turn down a certain $1,000 by telling us lies about how the other box will contain $1,000,000 if we choose only it! Wasn't that clever of him?

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

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

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

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

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

But can self-reference be formalised?

Yes. Godel demonstrated this.

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

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

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

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

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

Eliezer will think that this statement is false.

i.e. the above statement.

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

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

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

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

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

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

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

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

the question of how hostile perfect reasoners can communicate

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

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

This reminds me strongly of a silly russian puzzle. In the original it is about turtles, but I sort of prefer to translate it using bulls. So, three bulls are walking single file across the field. The first bull says "There are two bulls in behind me and no bulls in front of me." The second one says "There is a bull in front of me and a bull behind me." The third one says "There are two bulls in front of me and two bulls behind me."

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

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

JonathanG,

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

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

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

The comments to this entry are closed.

Less Wrong (sister site)

May 2009

Sun Mon Tue Wed Thu Fri Sat
          1 2
3 4 5 6 7 8 9
10 11 12 13 14 15 16
17 18 19 20 21 22 23
24 25 26 27 28 29 30
31