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    Thread: Unsolved paradoxes

    1. #26
      DuB
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      Quote Originally Posted by Wayfaerer View Post
      That's all fine but I can still see how evidence for one of these two completely different claims does not support the other.
      Well... it's clearly not all fine if you still think that you can imagine such evidence. It would be like saying that you can imagine a situation where a person is an unmarried man but is not a bachelor. Obviously there can be no such situation simply due to our definitions of "unmarried man" and "bachelor." For the present case, could you illustrate for us how some event could be evidence for T but not S, or vice versa?

      Quote Originally Posted by Wayfaerer View Post
      How can you rigorously prove this? It seems to me you just disproved this notion. Why not conclude this kind of thinking is wrong in the first place rather than seeing it as some confounding logical glitch?
      The proof (summarized from Carl Hempel) was laid out pretty straightforwardly in my first post--however, I don't want to defend that proof too much because I do actually think that this paradox can be resolved in a satisfying way by taking a Bayesian perspective on the problem, a perspective which I think is actually quite intuitive. It would take a fair amount of space to fully explicate the Bayesian argument (it is based on Bayes' Theorem, a result from probability theory), but I think the important points can be summed up adequately.

      Recall that the Tacking Paradox depends on two assumptions about how evidence can bear on theory: (1) if T entails E, then E is evidence for T; and (2) if E is evidence for T, and T entails P, then E is evidence for P. If we accept both of these seemingly innocent assumptions, then the paradox can always be derived.

      Basically, the Bayesian accepts (1) but does not fully accept (2). Instead, the Bayesian accepts that (2) is sometimes true, but denies that it must always be the case: if E is evidence for T & P, then E is allowed to provide different degrees of support for T and for P separately, says the Bayesian. Crucially, E serves as evidence for T or P only to the extent that E would be correspondingly less likely to have obtained if T or P were NOT true. With respect to P, for example, the Bayesian says that if E is equally likely to have occurred whether it is the case that T & P or that T & not-P, then E cannot be considered as evidence for or against P, whether or not it may be evidence for T. In this way the Tacking Paradox is blocked.

      To make this more concrete let's return to my running example of the roommate. We had arrived at the paradoxical conclusion that, because my observation that my roommate had slept through his alarm served as evidence for the joint hypothesis that "my roommate was out drinking last night AND the moon is made of green cheese" (by [1] from above), then it must also be the case that my roommate having slept through his alarm must also be evidence for the simple hypothesis that "the moon is made of green cheese" (by [2] from above). However, the Bayesian points out that the probability that I would observe my roommate to sleep through his alarm given that "my roommate was out drinking last night AND the moon is made of green cheese" is exactly the same as the probability that I would observe my roommate to sleep through his alarm given that "my roommate was out drinking last night AND the moon is NOT made of green cheese." This is of course not true for the first part of the joint hypothesis, the part saying that my roommate was out drinking last night. In that case, the evidence is undeniably more likely to have obtained if my roommate had been out drinking last night then if he had not. So we can say that while my observation of my roommate sleeping through his alarm DOES serve as evidence that he was out drinking last night, it does NOT serve as evidence that the moon is made of green cheese.

      I think you will agree that this is a pretty satisfying answer to the paradox.

      The Bayesian also has a response to the Raven's Paradox, but it is somewhat less satisfying and also more difficult to explain without appealing directly to Bayes' Theorem, which I want to avoid. Roughly put, the Bayesian bites the bullet and concedes that observing any non-black thing that is a non-raven does indeed serve as some degree of evidence that all ravens are black. However, the Bayesian maintains that, for any realistic figures, such an observation provides such an infinitesimally small degree of support for the hypothesis that we can say that it has no practical bearing on the hypothesis "in the real world." So the Raven's Paradox is not blocked in the same way that we managed to block the Tacking Paradox, but its impact is lessened considerably.
      Last edited by DuB; 09-25-2011 at 07:28 PM.

    2. #27
      Member Ivan452's Avatar
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      Hangman's Paradox is a real brain breaker

      My thoughts on The Grandfather paradox are that it depends on the way how someone views time:
      1. If we look at it as a single line. Then the moment you go to the past and kill your grandfather you will vanish and the past and the future after it will be changed. The fact the in the future you do not exist to kill him wont change anything.
      2. If we look at it a tree. Infinite number of parallel universes. Each created when there is a choice.
      In that case when you go back in time and kill your grandfather. You will create another path of time (or parallel universe). But then you will be stuck in it. So when you try to go back in the future, you will go to a time where you do not exist. And the universe where you exist will not be available to you.

      And the Monty Hall problem - you can prove it mathematically that if you swap the door the chances actually rise.
      I'm studying the probability for the last 6 months in University. But even thou you can prove it, the chances are the same.
      The fact is that your probability to win a car raises if you choose again between 2 doors. Not only if you swap the choice. (but this is only mathematically, actual chances are the same)

    3. #28
      khh
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      DuB, that's just mindblowing. Especially the Raven paradox.

      Quote Originally Posted by stormcrow View Post
      There is also the barber paradox. If a barber shaves everyone who do not shave themselves, then who shaves the barber?
      This isn't a paradox, nor even a contradiction. There is nothing to say the barber can't shave himself, just because he additionally shaves all those who doesn't. And there's even the possible solution of him not shaving/being shaved at all.
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    4. #29
      Xei
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      Strictly the paradox should say that the barber only shaves those who don't shave themselves. It is a contradiction.

      Quote Originally Posted by Ivan452 View Post
      And the Monty Hall problem - you can prove it mathematically that if you swap the door the chances actually rise.
      I'm studying the probability for the last 6 months in University. But even thou you can prove it, the chances are the same.
      The fact is that your probability to win a car raises if you choose again between 2 doors. Not only if you swap the choice. (but this is only mathematically, actual chances are the same)
      lolwat

      Why do you think there is a difference between mathematical probability and 'chances'? They're the same thing. If you swap, your chances of winning are 2/3. If you don't, they're 1/3.

    5. #30
      Member Ivan452's Avatar
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      I used the different terms for the same thing (mathematical probability and 'chances')

      and yea, I was wrong about the problem. I misunderstood it. Its kinda late here o.-

    6. #31
      khh
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      Quote Originally Posted by Xei View Post
      Strictly the paradox should say that the barber only shaves those who don't shave themselves. It is a contradiction.
      I figured, but it didn't. I had to pounce at that. But it still doesn't rule out that the barber just doesn't shave.
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    7. #32
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      There are some pretty nice paradoxes going on here. But Dub's was so far the most amazing one I've ever heard thanks a lot Dub. I'd like to see someone compete with Dub's paradox
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    8. #33
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      Maybe The Raven paradox isn't such a paradox after all. Since everything we can see has a color, and all colors in our visual spectrum exist (lets just consider black a color for now), maybe observing a non-black thing really is ever so slightly increasing the chances of any other entity with a color being black, since there is a finite amount of entities with definite color at any given moment and a finite amount of black things that exist.

    9. #34
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      DuB, could you dispel the so-called "doomsday argument"? Basic intution tells us that without any empirical evidence or external reasoning*, you can't make any claims about the likelihood of human extinction. However, some people (like Xei) have gotten lost in the math and actually believe it's true, despite common sense. Is there a Bayesian resolution to this madness?

      Doomsday argument - Wikipedia, the free encyclopedia

      *For clarity, by "external" reasoning I mean some line of reasoning other than the doomsday argument itself, like "I think we'll go extinct because we have nuclear weapons", etc.
      Last edited by cmind; 09-26-2011 at 11:12 PM.

    10. #35
      Xei
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      > 2011
      > Still dismissing probabilistic arguments on the basis that they don't accord with human intuition and common sense.

      See also: Simpson's paradox, Monty Hall problem, Boy/Girl problem, et. al.

    11. #36
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      Quote Originally Posted by Xei View Post


      > 2011
      > Still dismissing probabilistic arguments on the basis that they don't accord with human intuition and common sense.

      See also: Simpson's paradox, Monty Hall problem, Boy/Girl problem, et. al.
      There are common sense solutions to all the examples you listed. For example, from a common sense perspective, the change in probabilities in the Monty Hall problem makes sense because the host is giving the contestant more information, thus altering the problem. The DA is on a whole other level of implausibility. Anyway, we've already heard your opinion, now I want to hear from someone who seemingly has a better grasp of statistics.

    12. #37
      DuB
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      It does seem a little arrogant to say that because something is contrary to your intuition, it is obviously wrong.

      I looked over the header of the wiki article and didn't see anything obviously wrong with the argument. The exact figures they present are obviously a little dubious since they rely on a lot of assumptions extrapolated from current data, but that isn't really the important part of the argument is it. The main point of the argument is simply that, if we assume that there is definitely a discrete beginning and discrete end to human existence, and assuming that we have no good reason to suppose that we are at a particular position within that interval, then if we have an idea about how many humans there have been, we can make some pretty straightforward probabilistic statements about how many humans there will be.

      I don't see why this is a controversial conclusion. Probabilistic conclusions are not particularly strong, binding conclusions after all. I assume you wouldn't object to the conclusion of this same argument if instead of talking about humans, we were talking about something more mundane like the total number of times that lightning will strike during a discrete thunder storm.

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      Quote Originally Posted by DuB View Post
      I don't see why this is a controversial conclusion. Probabilistic conclusions are not particularly strong, binding conclusions after all. I assume you wouldn't object to the conclusion of this same argument if instead of talking about humans, we were talking about something more mundane like the total number of times that lightning will strike during a discrete thunder storm.
      This is a little disappointing that even you've fallen for this. What about the fact that there is no "we", there is only each separate living individual experiencing life? If you constructed the probabilities taking into account each person's lifespan, not just the "year", then you might get different results. In other words, consider the probability of living at this time as person #161,706,435,603 versus living at this time as person #161,706,435,604. In other words, the argument only works if you consider the human species as one entity.

      About the context, on the contrary. I think people with pre-conceived bias about the lifetime of our species (because they think we'll die in nuclear fire or whatever) are more likely to fall for this ambiguous probabilistic argument because it supports their bias. If we were talking about lightning strikes, the absurdity would become more clear to everyone because no one has a psychological attachment to thunder storms.

    14. #39
      Xei
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      Every human who ever lived says "chronologically I am within the middle 80% of humans". How many turn out to be correct?

    15. #40
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      Quote Originally Posted by cmind View Post
      What about the fact that there is no "we", there is only each separate living individual experiencing life? If you constructed the probabilities taking into account each person's lifespan, not just the "year", then you might get different results.
      How? I cannot see any way of connecting what you just said with the argument as I understand it. We are simply talking about the total number of humans there will be. How is lifespan even relevant.

    16. #41
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      Quote Originally Posted by Xei View Post
      Every human who ever lived says "chronologically I am within the middle 80% of humans". How many turn out to be correct?
      80 % of them?

      edit: Assuming every human that has lived and will live.
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    17. #42
      Xei
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      omigod i dont believe you cant do basic statistics youre so biased raaaaaaargh
      khh likes this.

    18. #43
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      The grandfather paradox

      Imagine you build a time machine. It is possible for you to travel back in time, meet your grandfather before he produces any children (i.e. your father/mother) and kill him. Thus, you would not have been born and the time machine would not have been built, a paradox

      All You Zombies paradox


      A baby girl is mysteriously dropped off at an orphanage in Cleveland in 1945. "Jane" grows up lonely and dejected, not knowing who her parents are, until one day in 1963 she is strangely attracted to a drifter. She falls in love with him. But just when things are finally looking up for Jane, a series of disasters strike. First, she becomes pregnant by the drifter, who then disappears. Second, during the complicated delivery, doctors find that Jane has both sets of sex organs, and to save her life, they are forced to surgically convert "her" to a "him." Finally, a mysterious stranger kidnaps her baby from the delivery room.

      Reeling from these disasters, rejected by society, scorned by fate, "he" becomes a drunkard and drifter. Not only has Jane lost her parents and her lover, but he has lost his only child as well. Years later, in 1970, he stumbles into a lonely bar, called Pop's Place, and spills out his pathetic story to an elderly bartender. The sympathetic bartender offers the drifter the chance to avenge the stranger who left her pregnant and abandoned, on the condition that he join the "time travelers corps." Both of them enter a time machine, and the bartender drops off the drifter in 1963. The drifter is strangely attracted to a young orphan woman, who subsequently becomes pregnant.

      The bartender then goes forward 9 months, kidnaps the baby girl from the hospital, and drops off the baby in an orphanage back in 1945. Then the bartender drops off the thoroughly confused drifter in 1985, to enlist in the time travelers corps. The drifter eventually gets his life together, becomes a respected and elderly member of the time travelers corps, and then disguises himself as a bartender and has his most difficult mission: a date with destiny, meeting a certain drifter at Pop's Place in 1970.
      Things are not as they seem

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      Time travel paradoxes disappear when you have multiple worlds.

    20. #45
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      What about Zeno of Elea? His paradoxes are cool to look at.
      No testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavors to establish - David Hume

    21. #46
      Xei
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      Mathematical analysis has basically sorted those.

    22. #47
      Dionysian stormcrow's Avatar
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      Quote Originally Posted by khh View Post
      DuB, that's just mindblowing. Especially the Raven paradox.
      Agreed.

      Quote Originally Posted by khh View Post
      This isn't a paradox, nor even a contradiction. There is nothing to say the barber can't shave himself, just because he additionally shaves all those who doesn't. And there's even the possible solution of him not shaving/being shaved at all.
      Ya, I didn't do a great job explaining it, it was supposed to say that the barber only shaves those who don't shave as Xei said.

      More on the topic of induction here's one from Hegel: We learn from history that we do not learn from history.

    23. #48
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      Here's another pointless, stupid logical paradox: Everything in moderation.

      Another one: All I know is that I know nothing. That was Socrates, by the way.

      Logical paradoxes are boring.

    24. #49
      Dionysian stormcrow's Avatar
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      Quote Originally Posted by cmind View Post
      Here's another pointless, stupid logical paradox: Everything in moderation.

      Another one: All I know is that I know nothing. That was Socrates, by the way.

      Logical paradoxes are boring.
      Okay.

    25. #50
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      This is one of my favorite paradoxes:

      I think it's safe to assume that the majority of the people who will read this have also read the Harry Potter series. Of course, this is just a ripe example, anything with any form of cause-and-effect scenario could work with this.
      So let's say that by Deathly Hallows, Harry comes across a Time Turner. Thinking that he could just go about 60 years into the past and stop (kill) Voldemort from ever beginning his career as the most powerful dark wizard to ever walk the planet, Harry sets off through time and space. Arriving in the 1940's, Harry goes into the orphanage where Tom Riddle lives, and kills him. This is a paradox though; the only reason Harry went back in time to kill Tom Riddle was because Tom Riddle caused Harry to go back in time to kill Tom Riddle. But, if Tom Riddle was never alive to cause Harry to go back and kill him, Harry wouldn't have gone back to kill him. But Harry is still there; Tom Riddle must have caused Harry to go back in time to kill him, but Tom Riddle was dead, meaning Harry wouldn't have gone back in time to kill him, but Harry DID go back in time to kill him, so Tom must have caused Harry to go back in time to kill him, but Tom couldn't have done so because Tom was dead, and so on and so forth.

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