"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Saturday, October 24, 2009

"Masters of Logic" puzzle

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?



Mariusz Popieluch said...

All hands raised implies at least 2 red dots.

Which means 2 or 3 red dots.

Now what would happen if there were just 2 red dots and one blue? One of the wiseman (two of them actually but let's just look at one) would see 1 red and 1 blue dot. He could infer immediatelly that he has a red dot since the other red dotted wiseman had his arm raised. The same goes for the other red dotted wiseman and his immediate inference.

But there was no immediate inference. But if there were just 2 red dots there should have been such an immediate inference since the above is an elementary deduction for each red dotted wiseman.

So if there was no such immediate inference, the next (and only) remaining possibility is that there are 3 red dots. And it was the wisest of them that made this last inference - a kind of meta-inference, since he inferred the solution from the lack of inference.

Mariusz Popieluch said...

I like this 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.

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.