posted 05-18-2005 21:18            

Melissa said:If you are given a true coin, and toss it 99 time, all coming up heads, while you may want to argue that the next flip is most likely to be heads, I can pose an equally valid argument that it will be tails. The truth is that head/tails are equally likely if it is a true coin. Your perspective is that it is not a true coin, and thus is weighted toward heads. My perspective is that it is due to fall on a tails. Neither is correct.

This tangent eludes me completely. What is the valid argument that it will come up tails? Certainly not the argument from experience. That argument predicts heads while ditching the premise that the coin it "true". If you're unwilling to ditch the premise based on experience, why are you not unwilling to ditch a premise when it leads to paradox? Why not ditch the paradoxical framework? "This statement is false." What premise would you ditch for the preceding statement? That it's true? That it's false?

posted 05-18-2005 23:03               

I'm too dense to grasp Mel's logic. As was stated originally, you can eliminate the last flight because if you were booked on the last flight you would know so before your name was called.

You can then restate the situation as a four flight situation, and the same situation applies with regard to the fourth flight, making it a three flight situation - and so forth.

posted 05-19-2005 03:58               

But maybe that guarantees that you wouldn't be on the last flight?

posted 05-19-2005 05:35            

quote:

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Originally posted by JosephP:
Roger, it was not day 5. Which of the other days you choose is of no import...

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When I say day 5, I meant day 5. Your confidence that I did not mean day 5 simply underlines my point. When day 5 came and you had not been executed, you would have felt safe in that you would have said to yourself the conditions cannot be met. In comes the warden. You did not expect this given the "logic" of the situation. So which condition was violated?

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Owner since 1996 (a paltry one)
TUG member since 1997

posted 05-19-2005 10:05            

Roger, I took that to mean one of the original forms where the execution always occurs mid-week.

When I say day 5, I meant day 5. Your confidence that I did not mean day 5 simply underlines my point.

Roger, I'm sorry if I underlined your point. I didn't mean to. Based on the above, you can see that I didn't intend to imply that you didn't mean day 5.

When day 5 came and you had not been executed, you would have felt safe in that you would have said to yourself the conditions cannot be met. In comes the warden. You did not expect this given the "logic" of the situation. So which condition was violated?

"So which condition was violated?" To answer your question in the terms of the original post: If you were expecting condition A to be met, then condition B would be violated, and you would have anticipated your flight (execution). If you fell into a coma (another kind of non-necessary psychic event like shifting expectations), then neither condition would be violated.

posted 05-19-2005 11:46            

Yes, when this puzzle is usually presented, a "paradox" occurs in that the prisoner is executed midweek. I up'ed the ante by suggesting that the prisoner could not even be sure on day 5 whether he will be executed or not. (Or, in your version of the puzzle, a person can't be sure as to his or her flight status even when the 5 o'clock flight is boarding.)

To create the logical paradox (and in turn claim that the two conditions are inconsistent) one has to talk about "an iron clad guarantee" (plus add a lot of hand waving trying to make this guarantee into something that has the status of mathematical certainty).

Yes, in a mathematical world, the conditions are inconsistent. But we don't live in a mathematical world. Thus, even when day 5 (the 5th flight) approaches one cannot know for ABSOLUTE SURE that the execution will occur (and thus violate condition B). Come the 5 o'clock hour, you can not "know" (for ABSOUTE SURE) whether you will be on the flight (or, whether you will get the money instead, or, whether the guarantee will turn out to be bogus). All of the above are still possible. So condition B is still satisfied. You won't know if you are going to be on the last flight or not. And, if you do fly, so is condition A will also have been satisfied.

Summarizing, what might be inconsistent in a universe that obeys the laws of two value mathematical logic might not be inconsistent in the real world. The poor prisoner who tried to pretend that he was living in a mathematical universe as opposed to the real world found this out.

posted 05-20-2005 06:16               

regarding the coins, the assumption that the coinc will be more likely to come up tails is based on the idea that it is "due" for a tails. I recall this from my first class of probability and statistics. Given the assumtion that a true coin, over 200 tosses will come up in the ballpark of 100 each heads & tails, if we know for certain that those will be the results over 200 tosses, the chance of the coin landing on tails following 99 head is 100 to 1 - but we can't assume an even distribution over 200 throws, or after any set number.

The issue with your problem is that we can't legitimately restate the problem with 4 flights, then 3, and so on. By doing so, we expect to not be on any flight.

Thus we need to restate the problem:

When flight 4 leaves, there are actually 2 options - option 1 is that we are on flight 5, which violates the condition that we not know ahead of time which flight we are on. option 2 is that we are not in fact on a flight, which violates the condition that we will be on a flight. Both options violate one of the conditions.

When flight 3 leaves, there are still 2 flights left, and thus 3 options:

Option 1 - we are on flight 4
Option 2 - we are on flight 5
Option 3 - we are on neither flight

Option 3 violates on of the conditions, but options 1 and 2 do not, because we don't know which flight we will be on.

Restating the original problem as 4 flights would combine options 2 and 3 as the option of not being on flight 4. This is where the logic falls apart.

Not being on flight 4 does not violate the conditions in and of itself - After flight 3 you are making the assumption that you WILL be on flight 4 because you have eliminated flight 5. In doing so, Option 2, being on flight 5 is no longer a violation of the terms, because at this point you expect to be on flight 4. It does not become a violation until there are no longer any other valid flight for you to board. You cannot make the assumption that you will not be on flight 5.

In other words, until flight 5 is no longer a valid option - only when you would know you have to be on that flight - you cannot eliminate it. Thus you cannot carry the logic forward to eliminate any of the other flights.

Another way - I cannot be on flight 5 because I would know before the flight leaves that I would be on that flight. But if I expect to be on flight 3, then being assigned to flight 5 does not violate the rules, does it? It only becomes a problem when there is only 1 flight available.

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Melinda Towne

Come visit my homepage

posted 05-30-2005 14:23               

You are basing your assumption that one rule will be broken because the other will not. I know it seems like it should follow that if one rule is not broken, the other must be broken. But you cannot eliminate the earlier flights on the assumption that you cannot be on the last flight. The last flight is the ONLY flight you can eliminate.

The conditions are not violated until you know which flight you will be on (after the 4th flight) or that you cannot be on a flight. If I book you on a flight, and then ask you ahead of time which flight you are on, you will not be able to tell me, will you?

You KNOW that putting you on flight 5 will violate the terms of the puzzle. Even if it follows that you cannot be on flight 4, by extension, you cannot then eliminate flight 3 as well. Using that logic, you eliminate all flights, and thus expect to be on none of them, so you can be on any of them.

What if you were told:

1 - You will be on the third flight
2 - you won't know ahead of time what flight you're on

These are contraditory, so you don't know which one is the truth. Do you expect to be on the third flight, with the second statement being false. If you do, and you're on a different flight you will be surprised.

Or do you expect to be on a different flight, with the first statement being false? If you end up on the third flight you will then be surprised, won't you?

Either way, you are still surprised. The problem is that you are allowing yourself to redetermine, each hour, which flight you expect to be on, always expecting to be on the next flight - but how can you be certain each time, when you were wrong on each of the preceeding flights?

By eliminating all of the flights, you are EXPECTING to be on none of them - and are surprised when you ARE on one of them.

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Melinda Towne

Come visit my homepage

posted 05-30-2005 22:23            

I guess this means that a scenario can be envisaged from the start wherein I am put on Flight 5 and the conditions are not violated. I need help here, for I can't see the possibility.

Even if it follows that you cannot be on flight 4, by extension, you cannot then eliminate flight 3 as well. Using that logic, you eliminate all flights, and thus expect to be on none of them,

That's the idea! Assuming the preconditions, I can be on no flight if the foreknowledge condition holds. The two conditions are inconsistent.

What if you were told:
1 - You will be on the third flight
2 - you won't know ahead of time what flight you're on
These are contraditory, so you don't know which one is the truth. Do you expect to be on the third flight, with the second statement being false. If you do, and you're on a different flight you will be surprised.
Or do you expect to be on a different flight, with the first statement being false? If you end up on the third flight you will then be surprised, won't you?
As I stated earlier, this problem is not about surprise or expectations. It is about knowledge. Given the clear conflict between the two provisions (1 and 2) above, I would have no basis for any knowledge regarding my flight.

In the problem I originally posted, it's not so obvious that the two conditions conflict - but they do. The result of that inconsistency is the seemingly paradoxical result that I can't be put on any flight, while the conditions require that I be on one flight. So I have no knowledge of what will actually occur regarding the flights, but I have certain knowledge that the set of two conditions will be violated. I am therefore poised to claim the $100 Billion. Or am I just whistlin' Dixie?

posted 05-31-2005 07:27               

There are two statements, which seem to contradict each other. You know that these two statements are contradictory, but you don't know anything about truth. By our logic, one of the statements must be false, but we cannot know ahead of time which one is false. these rules are self-refering.

Because we don't know which condition is to be violated, we cannot prove it will be violated. Your assumption is that you cannot be on any flight, which is a violation of the first condition. But because you believe you can't be on any flight, you do not KNOW you will be on any given flight, not even the 5th flight.

By going up to the gate and requesting your money, you are stating that you know you will not be on the first (or any) flight. Yet, couldn't they then tell you that you are in fact on the first flight - you are then on the first flight, fulfilling the first condition. You did not know you would be on the first flight (you expected otherwise), so the second condition is also met. The same applies to each of the other flights.

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Melinda Towne

posted 05-31-2005 07:31               

Stop worrying about being a billionaire, take the 75% discount and a good book and go wait for the plane, and when your name is called get on it and enjoy your vacation. Use the 75% savings to eat out in some nice places.

posted 05-31-2005 22:50            

quote:

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Originally posted by Clark:
...take the 75% discount and a good book and go wait for the plane, and when your name is called get on it and enjoy your vacation. Use the 75% savings to eat out in some nice places.

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We (the domestic "we") recently were voluntarily bumped off our flight (Flight 1) to Cancun. We were promised that we would be on the next flight (Flight 2) and travel "First Class". Additionally, we were treated to free breakfasts and lunches. We had no way of being certain that we would be on Flight 2 (or any flight for that matter.) Fortunately for us, the airline acted as if it was bound by its promises, and we were put on Flight 2. Also, we were given certificates with a value of more than 100% of the cost of our tickets and promised that they would convert into currency when we used them. We have no way of being certain that this is true. (As I asked in a previous post, can one ever be certain of any non-analytic statement about the future?)

Was the whole experience a Meta Free Lunch for me? Perhaps not. The delay was a small inconvenience, but the uncertainty surrounding the non-identity of contracts with the certainty that attends the mathematical principle of identity left a scar on my psyche that - as witnessed by these rantings - has not healed. My wife, however, not only had a free lunch, but a Meta Free Lunch. She made me pay for all her inconveniences. Veblen would have called it "vicarious inconvenience".