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Two boxes, one is gold and the other silver. The sign on the gold box reads, "The portrait is not here." The sign on the silver one reads, "Exactly one of the two statements is true." Guess where the portrait is. What I want to see is how you reason the problem.

Jason Chen
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most venerable sir
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    It's a guess according to your question ... – Mark Bennet May 11 '14 at 22:02
  • The portrait is in the gold box. – Cure May 11 '14 at 22:05
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    "What I want to see is how you reason the problem." is the new cleaver way to ask homework questions :) – Cure May 11 '14 at 22:11
  • This is not a hw question. – most venerable sir May 11 '14 at 23:24
  • How to reason the question is to assume it is in each box and see if you find a contradiction with the signs. The poser has promised you that only one location will not lead to a contradiction. – Ross Millikan May 13 '14 at 02:21
  • @RossMillikan This doesn't seem to be a duplicate of that. In that problem, we assume something about the truth values of the statements. Also, the answers are different (if you think my answer is wrong, please say why before claiming the question as a duplicate.) – Trevor Wilson May 13 '14 at 02:22

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The portrait could be in either box because anyone could write all kinds of nonsense on a box and it wouldn't affect what's inside!

This is obvious when you think about it for a moment. But what is not at all obvious is why the seemingly logical argument that the portrait is in the gold box fails. Indeed, if the inscription on the silver box is true, then the portrait is in the gold box, and if the inscription on the silver box is false, then the portrait is in the gold box. The only explanation is that the seemingly harmless assumption "either the inscription on the silver box is true, or the inscription on the silver box is false" is not justified.

To understand this explanation, let's first consider the more fundamental "liar paradox." Consider the sentence $\Phi$ which says "this sentence is false." You can even write it on a box if you like. If $\Phi$ is true, then it is false, which is a contradiction. If it is false, then it is true, and again we have a contradiction. Therefore the assumption "$\Phi$ is true or $\Phi$ is false" leads to a contradiction; in other words, this assumption is wrong somehow.

Although classical mathematical logic admits a rule (the law of excluded middle) saying "$\psi$ is true or $\psi$ is false" for every sentence $\psi$, we must not interpret this rule out of context. Here "every sentence $\psi$" only applies to sentences in the realm of mathematical logic. Some English sentences such as "How tall are you?" or "One cup of coffee, please" would not qualify; we cannot deem them either true or false. One might hope that the sentence "this sentence is false" might qualify, because it seems to make an assertion of fact. But as we have seen, we can't allow such self-referential statements without being led into contradiction.

Returning to the question, let's observe that the inscription on the silver box is equivalent to the statement $\Phi$ saying "the portrait is not in the gold box if and only if this statement is false." The argument from this point on can be rephrased as follows.

Assume toward a contradiction that the portrait is not in the gold box. Then $\Phi$ is equivalent to the statement "this statement is false," and when we attempt to evaluate the truth value of this statement, we are led into contradiction. Therefore the portrait is in the gold box.

So we see that the liar paradox has been cleverly concealed so that it figures into a proof by contradiction, rather than leading to an outright contradiction. But the mistake is the same as always: we carelessly assumed that a self-referential statement could be assigned a truth value. To make it clear why this is a mistake, notice that an analogous argument (if it were valid) could be used to prove anything whatsoever. For example, consider another sentence $\Phi'$ saying "the portrait is not on the moon if and only if this statement is false." (Again, you can write this on a box if you like.) Then we can argue as follows.

Assume toward a contradiction that the portrait is not on the moon. Then $\Phi'$ is equivalent to the statement "this statement is false," and when we attempt to evaluate the truth value of this statement, we are led into contradiction. Therefore the portrait is on the moon.

EDIT: To see the invalidity of the other answers more directly, let's consider a slight variation of the original problem:

Suppose we have two boxes, one gold and the other silver. The statement on the gold box says "the portrait is not on the moon." The statement on the silver one says "exactly one of these two statements is true." Where is the portrait?

If the statement on the sliver box is true, then the statement on the gold box is false, so the portrait is on the moon. If the statement on the silver box is false, then again the statement on the gold box is false, so the portrait is also on the moon in this case. In either case the portrait is on the moon, so the portrait must be on the moon. If this argument were valid, you could prove the portrait to be anywhere you like, simply by replacing the word "here" (or the words "on the moon") on the gold box with any other location! So clearly the argument cannot be valid.

Trevor Wilson
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  • How do you know that the statements have no truth or falsity? – most venerable sir May 11 '14 at 23:28
  • @user11355 I didn't say I know the statements have no truth values. I just said I don't know that they do have truth values. It is possible that the portrait is in the gold box, the gold statement is false, and the silver statement is true. However another possibility is that the portrait is not in the gold box, in which case the silver statement is equivalent to "this statement is false," which cannot be consistently assigned a truth value. – Trevor Wilson May 12 '14 at 00:07
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    @user11355 Another way to think about the problem is that I can move the portrait between the boxes after the inscriptions were made. Surely the presence of some words on the box will not hinder me in doing so. – Trevor Wilson May 12 '14 at 00:11
  • This is a logic problem. No tricks – most venerable sir May 12 '14 at 00:36
  • @user11355 I hope my latest edit clears things up. – Trevor Wilson May 13 '14 at 02:23
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Suppose the Silver one is true, then the Gold one must be false, so it's in the gold box. Suppose the Silver one is false, then obviously we can't have both statements be true, so again the gold one must be false. So again it is in the gold box.

Christopher Liu
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  • So whatever is written on the gold box must be false? What if I write "Christopher Liu's answer is correct" on the gold box? :-) – Trevor Wilson May 11 '14 at 22:17
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    http://lesswrong.com/lw/ne/the_parable_of_the_dagger/ – Christopher Liu May 11 '14 at 22:22
  • Would you consider rewriting your answer to address the point that your link makes? It shows that your argument is invalid, does it not? – Trevor Wilson May 12 '14 at 17:21
  • I agree with your reasoning that in real life, this is a pointless question. But for my answer, as with all questions given to me in math, I'll assume that the asker is not screwing with me and giving me false information that is not necessarily true. – Christopher Liu May 12 '14 at 17:27
  • I think you misunderstand. Nowhere in the question does it assert that each of the inscriptions has a definite truth value (true or false.) There is, in general, no consistent way to define truth values for statements that refer to themselves (e.g. "this statement is false") or to each other (e.g. statement 1 says "statement 2 is false" and statement 2 says "statement 1 is true.") Therefore your assumption that either (1) the inscription on the silver box is true, or (2) the inscription on the silver box is false, is not justified. – Trevor Wilson May 12 '14 at 17:35
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    Note that I am talking in terms of mathematical logic, not "real life." In mathematical logic, one must be aware that, for a given formal language, not just any old string of symbols counts as a statement in that language. – Trevor Wilson May 12 '14 at 17:38
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One possibility is that the portrait is in the gold box. If the silver one is true, then the gold one is false. If the silver one is false, then the gold one is false. If the gold one is true, that is not possible.

Edit: The other possibility is to ignore what the signs say and the portrait could be in any box.

Jason Chen
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    Unless you just go ahead and put it in the silver box, ignoring what's written entirely... – mjqxxxx May 11 '14 at 23:02
  • that's true...lol – Jason Chen May 11 '14 at 23:06
  • Actually the portrait is in the silver casket. The person who inscribe the words on the caskets isn't lying at all. – most venerable sir May 11 '14 at 23:26
  • I don't want to sound overly critical, but I think you need to fix your answer to address mjqxxxx's objection. It may be funny but it is also a perfectly legitimate objection. – Trevor Wilson May 12 '14 at 17:20
  • I still don't get it. Are you supposed to ignore what the signs say? – Jason Chen May 12 '14 at 22:08
  • You don't need to ignore what the signs say, but you do need to not make unjustified assumptions about them. (For example, I'm sure you'd agree that we can't just assume that what the gold box says is true.) The reason the problem is confusing is that the assumption "either what the silver box says is true or what the silver box says is false," which appears to be tautological, is in fact is not justified under the circumstances. To understand this you have to first understand how the Liar paradox "this sentence is false" is resolved in mathematical logic. – Trevor Wilson May 13 '14 at 01:05
  • I will expand my answer to try to explain this further. – Trevor Wilson May 13 '14 at 01:23