a logical paradox

Philosophy Forums Philosophy
1

I read this in Martin Gardner’s “The Unexpected Hanging”:

Let’s play a game. I have ten boxes labeled 1-10, and you are to open them in order from lowest to highest. I claim that in one of them I have hidden an “unexpected” egg. An egg is “unexpected” if you are completely unable to deduce its location before opening its box and seeing it.

Now you argue that the game is absurd, because I cannot have hidden the egg in any box at all. To see why, let’s break the problem up into cases. Suppose the egg were in box 10. Then after you have opened boxes 1-9 you will be able to deduce that the egg was in box 10, so it will be an expected egg and my claim will be false. Therefore the egg cannot be in box 10, and we can push it off the table without affecting the game. But now the same argument applies to box 9, pushing it off the table, and so on, so that no boxes at all can hold the egg.

Well, I say, why don’t you test your reasoning and start opening boxes? So you do, and to your surprise the egg is in box 5. You did not predict it would be there, making it an unexpected egg. Thus your argument must have been faulty.

So what’s wrong with it?

2

Logic is a useful tool for discovering the truth, but it pales in comparison to good ol’ fashioned experimentation. As soon as logic hits the world, it has some ‘splainin’ to do and sometimes it falls short.

3

No it isn’t and no it doesn’t, only those who don’t understand (how to use) logic get such effects.

The egg could be in box number 1-9 for you not able to make a definite deduction, as far as logical problems go thats an extremely simple and easy one to solve.

aporia i suggest increase your critical thinking skills.
in your argument your assuming that because you pushed the 10 box away that the game only consists a choice between 1-9 boxes and the 10th doesn’t no longer as a choice at all exist when indeed it does.

I could explain it more clearly but if i need to i wouldn’t wanted to have bothered and you would only be reading he answer instead of growing intellectually.

Peoples ‘level’ of understanding regarding logic and rationality almost astounds me, I’ve yet to see someone who is on my level in the area of right and wrong though a number of people which i could count on my hand have shown definite indications.

I value rationality above all things though i’ve never pushed myself for the reason i mentioned above.

4

Oni Omega

I notice you left out box 10. If that omission was intentional, you’ve missed the point of the exercise. There’s nothing special about box 10. The argument concluding that the egg cannot be in box 10 is as faulty as the extension of the argument to box 9, 8, etc. The problem is to explain why that argument is faulty.

But the argument’s first conclusion was that the egg cannot be in the 10th box. If that argument is right, then the 10th box is no longer a choice. You haven’t explained what’s wrong with that argument, so your refutation is ineffective.

Xunzian

Experimentation is just stamp collecting until it meets with reason and becomes science. Thus it is essential for us to clarify the relationship between reason and reality. That’s what this puzzle is about.

5

wow dog, you’re an insulting person! I’ve always felt that the smartest people are the quickest to judge others in a negative manner. Figure that one out Einstein. And have a sense of humor because a sense of humor is a sign of intelligence.

6

The egg cant be in box 10 if it has to be there unexpectedly there when opened, so the egg must be in boxes 1-9 since cant it be deduced given the rules whether the egg will definitely a certain box by opening them from order 1-9.

Once again, it is extremely simple.

The argument posed takes box 10 out of the equation, then 9 and so on which breaks the rules of the game, there are 10 boxes, the 10th is just the only box the egg definitely cant be in.

7

Okay, forget the part about pushing the boxes off the table. That was just a metaphor for their being ruled out logically. The point is that according to the given argument, box 10 cannot have the egg. So now that you know that, the game is effectively down to boxes 1-9.

Now suppose the egg were in box 9. Then when we open boxes 1-8, we will be able to predict that the egg is in box 9 before opening it, because we have already argued that box 10 cannot have the egg. Therefore the egg would be an expected egg if it were in box 9; thus it can’t be in box 9. Now we keep repeating this argument, ruling out boxes 8,7,6,… until the egg cannot be in any box at all.

We are eliminating boxes by pure logical process of elimination, there’s no violation of the rules occurring. Does that help?

8

Maybe you mistake me for someone like you who utilizes emotional implication above the actual meanings of the words i use, unless the words i use yield little sense when taken absolutely literally i don’t.

If stating the truth about something is insulting to it then then the actuality is that something must be quite despicable, not the teller.

Comforting someone for having detrimental beliefs isn’t an ideal act to me.

9

Isn’t there a difference between boxes 9 and 10…?

okay, I get kinda that box 10 cannot be ‘unexpected’, and therefore discounted. But it’s existance, wether as a viable choice or not, lends box 9 the possibility of unexpectedness.

ie: the situation for the last box and the penultimate box are fundamentally different and the same rules that disregard the last box do not apply.

I’m wrong surely, because I suck at logic.

10

Ok, I’ll have a crack (pun intended).

At every opening of boxes there are two possibilities, the egg is in the box you are about to open or in one of the others. Therefore, when you get to box 9, with no egg yet found, the egg could be in 9 or 10 and is therefore still unexpected (or indetermined), even if in box 10. Opening 9 and not finding an egg is equivalent to opening any box and finding an egg. In other words if I open 5 and see an egg there, that is equivalent to opening 9 and finding no egg there i.e. affirming 10 as the egg holder.

11

Here’s a hint:

Think about that word, deduce. If you open boxes 1-9 and then say that by process of elimination, the egg must be in 10, have you really deduced, by pure logic, the location of the egg? Wasn’t there guesswork involved in choosing to open the first 9 boxes?

12

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you mentioned a location but not a specific location of the egg. that to me is the key

for I can deduce the eggs location is within the ten boxes which is " expected "

the egg being in box 5 is no surprise for it could have been in box 1 just as easily

your argument is faulty because you assume an unexpected egg I did not.

13

Ok i see what your getting at now, how ether that form of deduction doesn’t work here, you need to know that definitely under the rules and circumstances given which box it will be in and that deduction method alone doesn’t allow for that to happen, the only way you can really definitely know is still if you are down to 10 without opening it, that th egg can be in box 1-9 still stands.

With your stated deduction if there were 2 boxes left, the last you would be able to definately deduct so it must be the one before it right? but you can deduct that also so that makes it a 50/50 chance for each box to contain the egg which is clearly obvious without going the deduction method.
Which ever you chose you cant be totally sure thats always going to be right choice, from that viewpoint it just makes the chance of the item being in the next box the amount of boxes left divided by 100 %, which again is obviously the case even if looked slightly at differently as did i.

The conclusion is as i previously stated.

14

Yes if you had to be absolutely certain which box it would be a fairly sound, but one can merely expect an equal chance of the egg being in the remaining boxes in which you cant really lose no matter what ever box it is in.

15

to me you missed the importance of the last statment , which is

now you are actually openning the boxes , are you "surprised "?

16

I wouldn’t be surprised if i was opening the boxes and there was the stated egg in one of them, i’d be a little surprised if there wasn’t an egg in any of the boxes as it breaks the rules.

I’m not sure if i understand the point your trying to make with that post though, i’ll check back here tomorrow.

17

my point is that we have focused on the first two statements;

rather than the last being;

the egg is real

18

There was no guesswork that I can see in choosing to open the first 9 boxes. The rules said

.

This is an interesting riddle, even if some responders think it is ‘below’ their intellect (which I find amusing because they seem to not even understand what you are saying).

Here is what I am thinking. The only position we would be able to say that the egg cannot be located is box #10 because it is the only box that when the time came to open it, I would know with 100% certainty whether or not the egg was in the other boxes or not.
The argument would be:

P: There are only 10 boxes and one box contains an egg.
p: Not Box #1
p: Not Box #2
p: Not Box #3
p: Not Box #4
p: Not Box #5
p: Not Box #6
p: Not Box #7
p: Not Box #8
p: Not Box #9
C: therefore, box #10. If I KNEW it was in 10… it wouldn’t be unexpected.

But to rule out box #9 takes another sort of ‘layer’ of logic. I dont really know how to explain it but it seems to take on a more theoretical approach. The argument would have to look something like…

P: There are only 10 boxes and one box contains an egg.
p: Not Box #1
p: Not Box #2
p: Not Box #3
p: Not Box #4
p: Not Box #5
p: Not Box #6
p: Not Box #7
p: Not Box #8
p: Not Box #10
c: therefore 9
(but keep in mind that we wouldn’t know “not box #10” when we opened box # 9 because we weren’t able to deduce ‘therefore 10’ until AFTER we opened boxes 1-9) It’s almost as if we are saying that it can’t be in box 10 because we already know its not in the other boxes. And I’m just not buying it for box #9 because we dont REALLY know if the egg is actually in box 10 or not. If that makes any sense.

I dont know… I’m sure I’m just rambling at this point and I’m eager to search the web for an answer to this (if one exists) but I wanted to respond before I knew the answer. It seems to me that either this is a paradox (as the title of this thread suggests) because of the way the we are supposed to ‘deduce’ something ‘unexpected’ or because of the ‘leap’ of logic that is required to say that the egg is not in box number 9 that is greater than the logic required to say the box is not in number 10.

EDIT: After a small amount of research on this topic I have found that I was both closer to and further away from the answer than I originally thought. But I’ll keep the spoilers to myself. :stuck_out_tongue:

19

I would argue against your veneration of pure logic. Hume’s skepticism, after all, perfectly logical but the system that he proposes is infinitely worse at prediction than the system he seeks to replace. Likewise, theistic logic is often air-tight within its own realm but since I disagree with its most basic assumption I cannot really call it true (its track-record on prediction also leaves something to be desired, largely because it predicts events outside the normal world of experience).

Our brains are not a de novo product, but rather an accumulation over millions of years of evolution, so while our sense may indeed be false, it is worth noting that our brain grew into our senses and not the other way 'round.

Indeed, modern psychology has demonstrated that the mind is informed by the sense, and in the absence of sensual information, the mind goes haywire. So, while I think everyone recognizes that our sense may not be trustworthy, it is a folly to then place trust in the mind, which is dependent upon those perhaps-not-trustworthy-senses. Indeed, in doing so, one is adding an extra layer (rather than removing one) in terms of approaching useful understanding. This is a problem that goes back to the pre-socratics, but that doesn’t mean that it ought be embraced.

Furthermore, since the brain is limited by the sense, to postulate an unknowable (and it is, definitionally that) world where our senses are indeed false is to postulate a world we can never know, can never model, can never interact with . . . and therefore, never care about.

So, what use does such a line of thinking have?

Philosophy that seeks to divorce the mind from the senses (and in turn, the physical realm) will, invariably, experience the same fate that the mind, itself, does when robbed of sensual experience: meaningless madness.

To me, pure logic seeks to do just that – to create an internally consistent system that exists purely in the mind.

Now, I am not suggesting that we throw the baby out with the bathwater! I like logic quite a lot actually, but only insofar as it can be related to meaningful observations rather than from a priori assumptions. Taking things back to first principles only works if it accurately models reality, so we need to have a fully fleshed-out model of reality before we can even begin considering first principles. Most logicians that I have seen try and flip that one on its head and, naturally, the more expansive their reality becomes the more inaccurately their first principles model it.

In short, my way of logically solving this problem would be to start opening boxes rather than mingling my mind with the ether. Otherwise you fall into tricky situations. After all, it can’t be in box 10 because we know that if it were in box 10, by the time we arrived at it, we wouldn’t be surprised. Having ruled out box 10, we run into the same problem with box 9. After all, if it wasn’t in box 8, it would have to be in box 9. Now, we can follow this chain of thought until we discover that it has to be in box 1. Except that it can’t be in box 1, because we’d know that already! But of course, opening boxes with our minds proves nothing. That is my point.

20

The egg would only be unexpected 9/10ths of the time. Whenever the egg is not within boxes 1-9, you expect it to be in box ten. Thus, it is expected 1 out of every 10 times.

This makes the game faulty because it states that the egg is always unexpected. As we can see, though, opening 9 boxes with no egg would lead you to expect the egg in box 10.