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Unusual Logic Puzzle

This one, with a bit of dressing that is my own, is due to the great Raymond Smullyan. This is my favorite logic puzzle of all time.

The great archaeologist, Illinois Jones, had finally reached the heart of the ancient temple, the first to do so since the temple was originally constructed, thousands of years ago. Here, he knew he would find the two mystic chests, one silver and one gold. One of the chests, he knew, held riches beyond his wildest imaginings. The other held a fiendish trap that meant certain death.

Sure enough, in front of him were the two chests. Each bore an inscription.

The silver chest had the inscription, in the language of the ancient civilization,
"The inscription on at least one of these chests is false."

The gold chest had the inscription,
"The treasure is in this chest."

Illinois Jones thought for a bit, then confidently strode over to the silver chest and lifted the lid. To his shock, the chest was empty! He was so stunned that it took him several seconds to sense the pain in his thumb, where the tiny needle hidden in the silver chest had pricked him. This wasn't possible, he thought. He had been so certain of his logic! Frantically, he scrabbled to open the gold chest--sure enough, it was loaded with treasure. It didn't do him much good, though...within the minute, Illinois Jones was dead.

What happened? Illinois Jones' archaeology and history were impeccable--was there a problem with his logic?

Avrom Roy-Faderman
Friday, October 08, 2004

"at least one"

They are both false.

Ryan
Friday, October 08, 2004

No, just the gold chest.  The other one is true, because one of the statements is false (The treasure is in this chest).

Andrew Shafer
Friday, October 08, 2004

As Andrew correctly said:

If silver chest inscription was saying truth that would mean that gold chest was saying false (since atleast one of the chest had to lie and silver didnt).

If silver chest was lying it means that the inscription on both of them is true. (since if one of them is false silver is speaking truth)
But if silver is saying truth then we are suppose to have atleast one inscription which is false
Self contradictory statement.
Therefore this case is wrong.

Only the first case exists.

So was the treasure really there or not??
Is tiny needle any kind of hint??

Samir Mehta
Friday, October 08, 2004

I agree to what Samir said. Logically, the treasure must be in the silver chest.

Anyhow, Mr. Jones obviously though the same and now is dead...

Where is the error? The only one I can think of is Mr. Jones assumption that the riddle itself is honest. It seems, the inscriptions were just made to trick innocent archealogists.

Avrom, enlighten us!

Gerd Riesselmann
Monday, October 11, 2004

OK, unfortunately, I don't know how to give a hint for this without basically giving it away. So I'll do that.

First, the simple answer. Then, some discussion of how this can be so, and why you might, like Illinois Jones, be mislead.

The simple answer: Look, you can write anything you want on a pair of chests and put anything you like inside them. There's *no way* that what's written on a chest can guarantee anything about what's inside. There's simply no causal connection.

The discussion:

So, what was wrong with Samir's (and Illinois') proof? They assumed that both sentences had real truth values, that is, were non-paradoxical. Sure enough, if the treasure is in the gold chest, you can't consistently assign any truth value to the statement on the silver chest. But that's OK--sometimes, you can't consistently assign any truth value to a statement (the most famous example of this is "This sentence is false").

What's misleading about this situation is that it bears a superficial resemblance to a bunch of similar problems (like the Vienna puzzle on this site) that proceed by assigning truth values to various sentences. But there's an important difference: In those puzzles, you're always given *some* outside information that sentences are true or false (e.g., "One of the people always tells the truth" or "one of them always lies"). This information lets you assign truth values to the statements--if they're told by a truth-teller, they're true, otherwise, they're false, so one way or the other, they're true or false. In this puzzle, you have no such outside information, so assuming the statements on the chests have *any* truth-value is dangerous.

But, you might say, all sorts of mathematical proofs use techniques like this! That's so, because one of the axioms of math is the "Law of the Excluded Middle", which states that every mathematical claim is either true or false. But there's no "Law of the Excluded Middle" for ordinary language--there are some statements (like, again, "This statement is false) which can't be consistently supposed to be true *or* false.

Avrom Roy-Faderman
Monday, October 11, 2004

reminds me of human thought... and ai

what should it take to override an assumption

magdalena
Thursday, October 28, 2004

Mr. Jones?

That's Dr. Jones to you, doll ...

Tom Zetty
Friday, November 05, 2004

I suppose it is the difference between boolean logic and first order predicate logic.

For example in boolean logic negation means take this truth and make it false, or take this false and make it true, whereas in second order logic the negation means that the assertion is not searchably true, ie there is no configuration in your domain of discourse that will ever make the logical relationship hold.

If you find truth(and you didn't steal it) bring it to me and show me I'll give you an infinite amount of money.

ragnar
Friday, November 19, 2004

heaps of thanks for the puzzle Avrom.. its now one of my favorites too!

heres my take on it:
1.the inscriptions work if gold chest inscription is false.. BUT that doesnt tell you anything about where the treasure is...it just makes the inscriptions work
(*contrast* it to the setup where the gold inscrip says 'the treasure is NOT in here'.. you would then get the 'double neg'  info you need to make a conclusion)

2.the inscriptions also work if silver chest inscription is false AND in this case you know where the treasure is...

now... you gotta assume the ancient dudes ARE giving up the info on treasure location with their inscriptions (this should be in the question at start) else option 1 is possible and you either take your chances or walk.
so... assuming they ARE giving up the location with their inscriptions the only possible choice is option 2... with the treasure in the gold chest.

Thats a cool one Avrom.. but maybe should also include that they ARE indeed telling you where the treasure is??

did i read this thing right?.. too often i miss stuff
feedback pls
Avrom?

downtown brown
Friday, December 10, 2004

Not quite sure what you mean by "the inscriptions work"...do you mean that there's a consistent way to assign truth values to them? If so, the *only* case in which the inscriptions "work" is when the inscription on the silver chest is true and the one on the gold chest is false. Any other assumption leads to the statements being paradoxical.

As to your suggestion, "now... you gotta assume the ancient dudes ARE giving up the info on treasure location with their inscriptions (this should be in the question at start) else option 1 is possible and you either take your chances or walk." This is precisely the sort of "outside" information I *don't* want to give people. With outside information that implies the inscriptions "work" in some way, the puzzle is reduced to a completely standard "knight-and-knave"-style logic puzzle.

Avrom Roy-Faderman
Wednesday, December 22, 2004

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