A logical puzzle

A logical puzzle
Jean Francoise
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Posted Jan 17, 2010 - 6:10 AM:
Subject: A logical puzzle
I like the following problem, because its solution, althought strictly logical should really be qualified as meta-logical. This means that the justification for the solution contains meta-logical expressions/operations. But you have to solve it first to appreciate it.

Three Masters of Logic wanted to find out who was the wisest amongst them. So they turned to their Grand Master, asking to resolve their dispute. “Easy,” the old sage said. "I will blindfold you and paint either red, or blue dot on each man’s forehead. When I take your blindfolds off, if you see at least one red dot, raise your hand. The one, who guesses the color of the dot on his forehead first, wins." And so it was said, and so it was done. The Grand Master blindfolded the three contestants and painted red dots on every one. When he took their blindfolds off, all three men raised their hands as the rules required, and sat in silence pondering. Finally, one of them said: "I have a red dot on my forehead."
How did he guess?
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Posted Jan 17, 2010 - 9:17 AM:

I guess wink that by meta-logic you mean the implicit rule that the solution must be guessable by everyone. cool
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Posted Jan 17, 2010 - 9:41 AM:

Seems like ordinary logic to me.

When all three hands go up, this gives two possibilities:

1) All three dots are red.
2) Two of the three dots are red.

Now, if it is the second case, then the two people with red dots will be able to quickly figure this out, because they will each see one blue dot and one red dot, and they will know that if their own dot was blue, then the person with the red dot would not have raised his hand.

So, if all three hands go up, and none of them quickly say that their dot is red, then you can conclude that all the dots must be red.

Of course, this makes the assumption that each person expects the other two people to be smart enough that they would quickly figure out the first point I made above.


Anyway, I don't see anything "meta" about this. It's just that the conclusion cannot be made from the observation of hands and dots alone, but instead must also take into account information about the responses (or lack thereof) of the other two people.


DM
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Posted Jan 17, 2010 - 10:12 AM:

What I find interesting is the introduction of time into the equation - as in DM's "quickly". How quick is proportional to how clever one thinks the other chaps are. I just have to hope that they underestimate my intelligence by more than I underestimate theirs... or something. The extra information required to solve the problem is the fact that no one has been able to solve it. We know at once because we are told, and also told that the puzzle is 'then' solved, but it only becomes apparent to the participants 'quickly'.
Jean Francoise
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Posted Jan 17, 2010 - 11:07 AM:

To me an inference made from "the lack of immediate inference" seems higher order since the operation of "inference" (something of the form "|=" or "|-") is not part of a lower level (object) language which includes the rules of the game and concepts such as " raised hands, blue dots, red dots".

Hence when the metalinguistic concept "inference" or lack thereof, becomes a neccessary explanatory element of the solution i.e. it enters the language of the explenation as an object in that explanation and so the expanation shiflts gears to a higher level. Hence my use of the word meta-logical.

To avoid any ambiguities this fragment from Wikipedia may be useful:

"In metalogic, formal languages are sometimes called object languages. The language used to make statements about an object language is called a metalanguage. This distinction is a key difference between logic and metalogic."

Edited by Jean Francoise on Jan 17, 2010 - 11:20 AM
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Posted Jan 19, 2010 - 6:55 PM:

The problem with us regarding this as anything meta, is the observance of the situation you presented. But if you were to actually be in such a situation and like DM said and UN-E, elaborated was the "collective" (meta)intelligence initiated out of such a situation.



As a side note, I remember a childrens game. Where, if you look at some ones eyes at the same time he is looking at you, you loose. A really smart game for assessing children and giving them a lesson on social stuff.
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Posted Jan 20, 2010 - 1:05 AM:

Jean Francoise wrote:
I like the following problem, because its solution, althought strictly logical should really be qualified as meta-logical. This means that the justification for the solution contains meta-logical expressions/operations. But you have to solve it first to appreciate it.

Three Masters of Logic wanted to find out who was the wisest amongst them. So they turned to their Grand Master, asking to resolve their dispute. “Easy,” the old sage said. "I will blindfold you and paint either red, or blue dot on each man’s forehead. When I take your blindfolds off, if you see at least one red dot, raise your hand. The one, who guesses the colour of the dot on his forehead first, wins." And so it was said, and so it was done. The Grand Master blindfolded the three contestants and painted red dots on every one. When he took their blindfolds off, all three men raised their hands as the rules required, and sat in silence pondering. Finally, one of them said: "I have a red dot on my forehead."
How did he guess?


If you would be in that situation (all red) you would notice all tree look equally to each other, if one would be blue, you would notice the red ones looking more to each other (meaning you are blue)... (to solve this you will have to count observations)
If you see two blues and not raise your hand and both blue's keep looking to each other... you are red
if no one raises his hand everyone is blue.
but I can be completely wrong about it.


longfun
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Posted Jan 20, 2010 - 1:15 AM:

longfun wrote:


If you would be in that situation (all red) you would notice all tree look equally to each other, if one would be blue, you would notice the red ones looking more to each other (meaning you are blue)... (to solve this you will have to count observations)
If you see two blues and not raise your hand and both blue's keep looking to each other and raise hands... you are red
if no one raises his hand everyone is blue.
but I can be completely wrong about it.



Jean Francoise
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Posted Jan 20, 2010 - 9:23 AM:

Maybe the following analogy will show why I still think that the term "meta-logical solution" is adequate.

We can idealise the scenario of the masters of logic to them being programs, respectively A, B, C. The solution would contain (roughly!) some code of the form in A's script:

function=Solve(bannanas, n_hands, n_color, A, B, C,...,.......)

n_color=red.2

n_hands=3

if B and C not = bannanas

print= "red"

end

end

Where "bannanas" refers to a print out appearing from B or C which can be causally traced to be a result of a script that has the same function as "Solve" in A. I'm not necessary implying self reference althtought it would be an interesting special case of the example I provided.

Now the winner would be a program/machine sysyem which has the most efficient deductive algorithm. But it needs to utilize as additional information/data the notion of a program, the functional (categorical) analogue which it employs to churn out an answer.

I think that a grasp what meta means in this context, would not elicit such a strong opposition.

In mathematical logic we use meta linguistic/meta logical operations all the time. This problem is just a fabulation of such operations.

The element of computability speed (or intellignce as some call it), I've found is commonly pointed to. It is interesting, but not centrally relevant to the meta aspect.
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Posted Jan 20, 2010 - 10:51 AM:

Jean Francoise wrote:

The element of computability speed (or intellignce as some call it), I've found is commonly pointed to. It is interesting, but not centrally relevant to the meta aspect.


It's not the speed as such that interests me, but the fact that time becomes an element at all in what starts as a timeless logic. you are right to call it a meta-logic, because it takes its own result as an input. This recursion 'creates' time, in that the first time round, the problem cannot be answered, but given that it cannot be answered, it becomes answerable.

You might be interested in a book - The Laws of Form, by G. Spencer Brown, which develops a similar idea from a very primitive pre-boolean arithemetical foundation.
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