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This is a thread for sharing and discussing paradoxes. There's a nice list here: https://en.wikipedia.org/wiki/List_of_paradoxes. I'm familiar with most of the paradoxes listed under "Logic" and "Philosophy" and am most interested in paradoxes of this type, but would love to see discussion on paradoxes in any field. My favorite paradox is still the "unexpected examination" (often called the "unexpected hanging," thanks to this being Quine's favorite version). Here's the basic idea: Say I tell my logic class, which meets once each weekday, that they will have a quiz next week and that it will be a surprise - that is, on the day I give the quiz no student, no matter how rational, will come to class rationally justified in feeling sure that the quiz is scheduled for that day. A student responds that my promise is impossible to fulfill, and reasons as follows. She can know now that the test can't be next Friday, since if next Friday rolls around and I haven't given the quiz yet, she will walk in knowing that the quiz will take place that day. A Friday surprise quiz is thus an impossibility. With a Friday surprise quiz eliminated, she can go on to eliminate a Thursday surprise quiz, since if Thursday rolls around and the quiz hasn't happened yet, the class will know the quiz must happen that day, as it can't happen on Friday for the reasons above. She claims she can eliminate surprise quizzes on Wednesday, Tuesday and Monday following the same train of thought. I nod, and when the next week comes I give a quiz on Wednesday, evidently to everyone's surprise. What's wrong with the student's reasoning?
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Sep 1, 2016 01:11
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prick with tenure posted:
I know this probably doesn't adequately address the paradox, but the "no student will come to class rationally justified it'll be scheduled that day" condition is at the very least not true for Friday, so it's sort of invalid to begin with. Like, if you keep that rule then you can extrapolate it across all other days, but the rule itself fails due to the last day. It's like if someone said "No one can possibly know what color shirt I'm going to be wearing each day. Oh, by the way, I'm definitely going to be wearing a red shirt on Friday."
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Sep 1, 2016 01:56
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Ytlaya posted:so it's sort of invalid to begin with The flaw is that there is no strategy that could determine the date with 100% certainty because if such a strategy existed, the person issuing the test could always surprise the students by not giving the test on that day, so every day except Friday must be uncertain. Debunking any specific strategy isn't necessary. OneEightHundred fucked around with this message at 02:16 on Sep 1, 2016 |
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Sep 1, 2016 02:14
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OneEightHundred posted:The flaw is that there is no strategy that could determine the date with 100% certainty because if such a strategy existed, the person issuing the test could always surprise the students by not giving the test on that day, so every day except Friday must be uncertain. Debunking any specific strategy isn't necessary. Why is that a flaw? I don't understand why the professor being able to surprise the class if a strategy exists makes the absence of a possible strategy a flaw. It doesn't seem to explain why the "test can't be on Thursday since we know it can't be on Friday and thus someone on Thursday morning would be sure of the date" logic is wrong.
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Sep 1, 2016 02:28
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OneEightHundred posted:The flaw is that there is no strategy that could determine the date with 100% certainty because if such a strategy existed, the person issuing the test could always surprise the students by not giving the test on that day, so every day except Friday must be uncertain. Debunking any specific strategy isn't necessary. Make the class meet only twice a week. Are you saying the professor couldn't make good on her promise in this case?
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Sep 1, 2016 03:16
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paradol exes? probably somewhere whining about clintonasties and trumpests
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Sep 1, 2016 03:35
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prick with tenure posted:What's wrong with the student's reasoning? She assumed the professor would behave in a purely rational manner.
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Sep 1, 2016 03:43
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Who What Now posted:She assumed the professor would behave in a purely rational manner. What makes you think the professor is doing something irrational when she gives the test on Wednesday? Or is the professor's promise itself irrational? That's precisely what the student claims. Better might be to say that the student's reasoning assumes the professor is telling the truth, and the student isn't entitled that. The possibility that the professor is lying and that there won't be a quiz at all next week is what would make a Friday quiz a surprise. The students would arrive at class expecting a test, given their belief in the professor's basic trustworthiness, but they couldn't be sure she wasn't lying the week before and would thus be surprised by the test in the sense that the set-up defines surprise. Can it be that the possibility of the professor telling the truth here depends on the possibility of her lying? prick with tenure fucked around with this message at 04:05 on Sep 1, 2016 |
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Sep 1, 2016 03:55
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This is my favorite paradox op: https://www.youtube.com/watch?v=Y53_GV2aAg8
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Sep 1, 2016 03:58
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I think the problem with this paradox involves free will. So long as Monty is allowed to pick which goat, he can mess with probabilities and possible futures.
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Sep 1, 2016 05:23
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Paradoxes are just a consequence of fuzzy language, they really aren't very interesting.
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Sep 1, 2016 05:28
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I prefer the executioner version of this paradox, personally. The issue isn't the logic deployed to determine which day - it actually works even if you assume the professor says 'it will happen tomorrow'. All the rest of the stuff about the specific day of the week is just fairly standard game-theory strategy, that's essentially obfuscation. The problem is the use of self-reference. The logic the student uses has a conclusion, and 'you will be surprised' is a statement about that conclusion, and all future conclusions. If at any point the student concludes it won't happen, it can (and therefore will, since that's the point of the paradox) happen. The professors has laid out something that is actually an infinite series of logical statements, because of that self-reference. rudatron fucked around with this message at 05:51 on Sep 1, 2016 |
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Sep 1, 2016 05:49
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prick with tenure posted:. What's wrong with the student's reasoning? Is it something to do with the student knowing (assuming) a truth based on the knowledge that they don't know something?
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Sep 1, 2016 05:54
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Seems like a variation of Achilles and the tortoise, and would presumably be wrong for the same reason.
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Sep 1, 2016 08:06
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prick with tenure posted:Say I tell my logic class, which meets once each weekday, that they will have a quiz next week and that it will be a surprise - that is, on the day I give the quiz no student, no matter how rational, will come to class rationally justified in feeling sure that the quiz is scheduled for that day. A student responds that my promise is impossible to fulfill, and reasons as follows. She can know now that the test can't be next Friday, since if next Friday rolls around and I haven't given the quiz yet, she will walk in knowing that the quiz will take place that day. A Friday surprise quiz is thus an impossibility. With a Friday surprise quiz eliminated, she can go on to eliminate a Thursday surprise quiz, since if Thursday rolls around and the quiz hasn't happened yet, the class will know the quiz must happen that day, as it can't happen on Friday for the reasons above. She claims she can eliminate surprise quizzes on Wednesday, Tuesday and Monday following the same train of thought. I nod, and when the next week comes I give a quiz on Wednesday, evidently to everyone's surprise. What's wrong with the student's reasoning? She showed that there's no day there could be a quiz, but we know you're going to quiz them anyway. Therefore whenever you do it, it's a surprise. (E: Yeah, I realised I'm rephrasing rudatron here.) The real question to me is, if you tell your students there's a test next week, why are they surprised when there's a test next week?
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Sep 1, 2016 08:25
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There's nothing wrong with the professor's statement, it could be rephrased as "there is no way to deduce the day of next week's quiz from the statement 'there will be a quiz next week' ". This is self-evidently true for any week longer than 1 day. The case of a 1-day week: "there is no way to deduce the day of the quiz from the statement 'there will be a quiz on Friday' ", is a self-contradiction. The student starts with the self-contradictory case, the 1-day week, but proofs that include a self-contradictory premise are not valid so it's not a paradox that her conclusion turned out to be incorrect when she reasoned from a self-contradictory premise.
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Sep 1, 2016 09:48
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OwlFancier posted:Seems like a variation of Achilles and the tortoise, and would presumably be wrong for the same reason. Are you referring to Zeno's paradoxes, or to the clever bit of dialogue that Lewis Carroll penned? I think Carroll just really illustrated the is/ought gap, and shows that the statement 'one ought accept the conclusion of a sound argument' (what one might call 'rationality'), is more an 'act of faith.'
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Sep 1, 2016 10:04
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Ytlaya posted:Why is that a flaw? I don't understand why the professor being able to surprise the class if a strategy exists makes the absence of a possible strategy a flaw. It doesn't seem to explain why the "test can't be on Thursday since we know it can't be on Friday and thus someone on Thursday morning would be sure of the date" logic is wrong. It could be rephrased as "create a strategy which will determine the quiz date for all professor strategies." That strategy can't exist because there exist professor strategies with mutually-exclusive results.
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Sep 1, 2016 15:01
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Ratios and Tendency posted:Paradoxes are just a consequence of fuzzy language, they really aren't very interesting.  Whatup my dude. I agree! But it can be hard to articulate just how the language is being slippery, which can be pretty interesting.prick with tenure posted:Say I tell my logic class, which meets once each weekday, that they will have a quiz next week and that it will be a surprise - that is, on the day I give the quiz no student, no matter how rational, will come to class rationally justified in feeling sure that the quiz is scheduled for that day. A student responds that my promise is impossible to fulfill, and reasons as follows. She can know now that the test can't be next Friday, since if next Friday rolls around and I haven't given the quiz yet, she will walk in knowing that the quiz will take place that day. A Friday surprise quiz is thus an impossibility. With a Friday surprise quiz eliminated, she can go on to eliminate a Thursday surprise quiz, since if Thursday rolls around and the quiz hasn't happened yet, the class will know the quiz must happen that day, as it can't happen on Friday for the reasons above. She claims she can eliminate surprise quizzes on Wednesday, Tuesday and Monday following the same train of thought. I nod, and when the next week comes I give a quiz on Wednesday, evidently to everyone's surprise. What's wrong with the student's reasoning? Her argument, cleaned up a bit, is something like the following: Proposition 1 1) Assume the quiz is not given on Monday, Tuesday, Wednesday, or Thursday. 2) if 1), then it is certain that a quiz will be given on Friday, and therefore a surprise quiz cannot happen on Friday. Proposition 2 1) Assume the quiz is not given on Monday, Tuesday, or Wednesday, and the quiz cannot be given on Friday because of Prop 1. 2) If 1), then it is certain that the quiz will be given on Thursday, and therefore a surprise quiz cannot happen on Thursday. Proposition 3 1) Assume the quiz is not given on Monday or Tuesday, and the quiz cannot be given on Thursday or Friday because of Prop 2. 2) If 1), then it is certain that the quiz will be given on Wednesday, and therefore a surprise quiz cannot happen on Wednesday. Proposition 4 1) Assume the quiz is not given on Monday and the quiz cannot be given on Wednesday, Thursday or Friday because of Prop 3. 2) If 1), then it is certain that the quiz will be given on Tuesday, and therefore a surprise quiz cannot happen on Tuesday. Proposition 5 1) Assume the quiz cannot be given on Tuesday, Wednesday, Thursday or Friday because of Prop 4. 2) If 1), the quiz will be given on Monday, and therefore a surprise quiz cannot happen on Monday. Conclusion: If Props 1-5 obtain, then on no day during the week can there be a surprise quiz.  The problem with all that is that propositions 2-5 involve causal chains that reach into the future and generate contradictions. For Prop 5, for instance: The P --> Q that the quiz will be given on Monday relies on a chain of reasoning that starts with the assumption (Prop 1, point 1) that the quiz has not been given on Monday, Tuesday, Wednesday or Thursday. It's the same causation problem for Props 4, 3, and 2. So while she can reason correctly that the surprise quiz can't happen on Friday, she can't parley that into a surprise quiz being impossible for the other days of the week. Of course, the real problem is that she's an insufferable freshman who just discovered that logic is a thing when she skimmed the syllabus last class*. *Haha no, she didn't look at the syllabus.
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Sep 1, 2016 15:50
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quote:a quiz next week and that it will be a surprise - that is, on the day I give the quiz no student, no matter how rational, will come to class rationally justified in feeling sure that the quiz is scheduled for that day Probability changes as new information about the outcome is revealed. Paradox!
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Sep 1, 2016 16:02
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I wonder if one possible problem with the paradox could have something to do with the fact that all days being "impossible" sort of invalidates that "impossibility" as a condition worth consideration when trying to predict the day. Like, since they're all equally "impossible", no given day is any more or less unlikely than any other.OneEightHundred posted:The problem is in the "rationally justified" requirement. I assume that's there to prevent students from using the strategy of just coming in 100% certain that the test will be the current day and just being wrong about it. The certainty can only be rationally justified if it can't be defeated. So, even if the test was given on Monday, the certainty of it being on Monday wouldn't be rational. So what you're saying is that the "person being rationally 100% certain" is a condition that is impossible, because there's no way to 100% know the professor will choose a particular day (at least as long as the other days are real possibilities). You would have to add some extra "Tuesday-Friday 100% can't be chosen" condition in order to know it would be held on Monday (for example). Even if the professor decides "I want to have the test Wednesday" and leaves clues all over the school, it's still technically possible for him to say "nah let's have it Thursday" when Wednesday arrives. Does this basically sum it up? If so, that argument seems to do a good job of addressing the problem (since you can't exactly ask this question to begin with if all the conditions aren't possible). wateroverfire posted:The problem with all that is that propositions 2-5 involve causal chains that reach into the future and generate contradictions. This is the argument I was trying to make (that it's only impossible on Thursday given that it definitely can't occur Friday), but it seems like the causal chain isn't really an issue because the fact that it can't be on Friday is definitely known regardless of any other conditions (that is, it's always true). So if it reaches Thursday you definitely know it can't be held Friday, meaning that you'd know it was Thursday (thereby invalidating Thursday as a possibility since you'd know it would be held once that day arrived). Also I don't think the Wikipedia page lists it as a solution, so I imagine there's some problem with that reasoning even if it's not what I just mentioned.
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Sep 1, 2016 17:27
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Ytlaya posted:This is the argument I was trying to make (that it's only impossible on Thursday given that it definitely can't occur Friday), but it seems like the causal chain isn't really an issue because the fact that it can't be on Friday is definitely known regardless of any other conditions (that is, it's always true). So if it reaches Thursday you definitely know it can't be held Friday, meaning that you'd know it was Thursday (thereby invalidating Thursday as a possibility since you'd know it would be held once that day arrived). Also I don't think the Wikipedia page lists it as a solution, so I imagine there's some problem with that reasoning even if it's not what I just mentioned. You're falling victim to the same logical blind spot as the hypothetical student. How do you know the quiz can't happen on Friday? (a) Because if it hasn't happened on Monday, Tuesday, Wednesday, or Thursday, it has to happen on Friday so it wouldn't be a surprise quiz if it did. How do you know the quiz can't happen on Thursday? (b) Because if it hasn't happened on Monday, Tuesday, or Wednesday, and it can't happen on Friday because (a), it has to happen on Thursday. Therefore if it happened on Thursday it wouldn't be a surprise quiz But (a) stipulates that the quiz hasn't happened on Thursday and (b) requires that it does. (a) and (b) contain contradictory premises so (b) can't follow from (a) and the resolution to the paradox is that the student was running with an invalid argument. I don't think it's any deeper than that, though I'm willing to be convinced and THAT might be an interesting discussion. =) edit: Cleaned up some language to make the argument more clear. wateroverfire fucked around with this message at 18:51 on Sep 1, 2016 |
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Sep 1, 2016 18:45
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prick with tenure posted:What's wrong with the student's reasoning? The student performs induction and determines the quiz must be on Monday, since Tuesday through Friday are impossible, and so declares she rationally must expect the quiz Monday. When it doesn't come, she performs induction again and decides the quiz must rationally be Tuesday... Whichever day the quiz falls on, it is not a surprise, and the professor fails his challenge. e: oops, missed that someone already covered this. OneEightHundred posted:The problem is in the "rationally justified" requirement. I assume that's there to prevent students from using the strategy of just coming in 100% certain that the test will be the current day and just being wrong about it. The certainty can only be rationally justified if it can't be defeated. I agree that the terminology is a big issue with the formulation, but disagree with the statement "you can only be rationally justified in a belief if it is correct". Take the same scenario, as an example, and say the professor does not give a quiz Monday through Thursday. The students would obviously be "rationally justified" in coming to class Friday expecting the quiz; the possibility that the professor was lying, or that he might drop dead of an aneurysm Thursday evening, does not render the students' expectation irrational. King of Bleh fucked around with this message at 19:36 on Sep 1, 2016 |
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Sep 1, 2016 19:20
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King of Bleh posted:The student performs induction and determines the quiz must be on Monday, since Tuesday through Friday are impossible, and so declares she rationally must expect the quiz Monday. When it doesn't come, she performs induction again and decides the quiz must rationally be Tuesday... I feel like the fact that it's possible for the quiz to not fall on the day she "rationally expects" it to sort of breaks the "100% rationally justified" rule. If it doesn't happen then clearly she wasn't 100% rationally justified in thinking it would happen.
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Sep 1, 2016 19:46
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Uglycat posted:Are you referring to Zeno's paradoxes, or to the clever bit of dialogue that Lewis Carroll penned? Zeno.
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Sep 1, 2016 19:48
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Ytlaya posted:I feel like the fact that it's possible for the quiz to not fall on the day she "rationally expects" it to sort of breaks the "100% rationally justified" rule. If it doesn't happen then clearly she wasn't 100% rationally justified in thinking it would happen. I edited my post to address this; you can be rationally justified in believing something that isn't true. I also think that the wording of the scenario intentionally avoids the implication that the students must be correct in their belief; the stipulation is NOT that a student must *know* a test occurs on a given day, they must only be "justified" in "feeling sure" -- note in particular the use of the word "feel". To be justified in having an emotion is a much weaker constraint than that of knowing a fact.
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Sep 1, 2016 20:49
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prick with tenure posted:This is a thread for sharing and discussing paradoxes. There's a nice list here: https://en.wikipedia.org/wiki/List_of_paradoxes. I'm familiar with most of the paradoxes listed under "Logic" and "Philosophy" and am most interested in paradoxes of this type, but would love to see discussion on paradoxes in any field. You also quizzed them on Tuesday
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Sep 1, 2016 20:54
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Hilbert's hotel demonstrates the pardoxes which arise with an actual infinite. quote:Imagine a hotel with a finite number of rooms. Say, 500 rooms. And all the rooms are taken. A new guest arrives and asks for a room, and the owner says, “Sorry, all the rooms are taken.” End of story. William Lane Craig concludes "Hilbert’s Hotel is absurd. But if an actual infinite were metaphysically possible, then such a hotel would be metaphysically possible. It follows that the real existence of an actual infinite is not metaphysically possible." I'm not sure if his conclusion is valid. I think there may be some problem with how they keep calling the hotel "full". Can a hotel with an infinite number of rooms ever really be full? There would always be an infinite number of rooms available no matter how many people checked in. If there are always an infinite number of rooms available how can you ever say the hotel is full? Also in the first case the owner switches the guest in room #1 to room #2 and the guest in room #2 to room #3 on into infinity. He then moves the new guest into room #1. However, this "then" would never actualize as the owner would have to be switching guests into the room next door for an infinite amount of time. AARO fucked around with this message at 22:51 on Sep 1, 2016 |
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Sep 1, 2016 22:36
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Ratios and Tendency posted:Paradoxes are just a consequence of fuzzy language, they really aren't very interesting. There's no such thing as an uninteresting paradox. There are a lot of paradoxes, some of them can be described with a short explanation, some longer. So, let's say we line up all the paradoxes from shortest to longest. Starting with the smallest one, maybe it's interesting on its own merits, but if not, it is the first uninteresting paradox, so that makes it very interesting. That's contradictory of course, so logically, there can be no shortest uninteresting paradox so they must all be interesting.
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Sep 1, 2016 22:49
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Dr. Arbitrary posted:There's no such thing as an uninteresting paradox. Is the fact that this just hinges on a fuzzy definition of "interesting" supposed to be part of the joke?
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Sep 1, 2016 23:00
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King of Bleh posted:Is the fact that this just hinges on a fuzzy definition of "interesting" supposed to be part of the joke? It's a rewording of the interesting number paradox.
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Sep 1, 2016 23:24
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mrbradlymrmartin posted:paradol exes? probably somewhere whining about clintonasties and trumpests toss some tolberonetriangular in there too if things start to lag
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Sep 1, 2016 23:37
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^^^ Oh man I forgot all about paradol ex. How nostalgic.Dr. Arbitrary posted:There's no such thing as an uninteresting paradox. I've found that it's best I avoid paradoxes, because every time I encounter one I end up stuck in a pointless mental loop for hours, if not days. It's sort of like if you tell a robot something that conflicts with its programming, causing it to endlessly attempt to resolve the conflict. The only exception are paradoxes related to the concept of infinity, which don't really both me much for some reason.
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Sep 1, 2016 23:51
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the only paradox that matters is if you let kaz die in Afghanistan
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Sep 1, 2016 23:53
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King of Bleh posted:I agree that the terminology is a big issue with the formulation, but disagree with the statement "you can only be rationally justified in a belief if it is correct". I'm interpreting "rationally justified" as permitting certainty on only one day. Also just for fun, the professor can technically win by giving the quiz on a day that class isn't scheduled. Ytlaya posted:Does this basically sum it up? If so, that argument seems to do a good job of addressing the problem (since you can't exactly ask this question to begin with if all the conditions aren't possible). OneEightHundred fucked around with this message at 00:38 on Sep 2, 2016 |
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Sep 2, 2016 00:12
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Let's try this on for size: 1. Finding ways around stupid paradoxes is good, and this qualifies as a stupid paradox. 2. Therefore I have a rational justification to believe something that can defeat this particular paradox, regardless of how rational or irrational the belief itself is. 3. Given there's a finite number of days in the week I only need 6 other people to agree with the above two premises. 4. Assuming I can find these 6 other people... 5. ...I can create a system as follows: Persons A-G will each choose or be assigned one day of the week to believe in as the day that the 'surprise' quiz will be given. 6. One of them will therefore not be surprised, and thus defeat the paradox.
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Sep 2, 2016 12:08
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https://www.youtube.com/watch?v=7WTSlr_WRgo
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Sep 2, 2016 12:59
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Ratios and Tendency posted:Paradoxes are just a consequence of fuzzy language, they really aren't very interesting. What is the exact boundary between Earth's atmosphere and outer space? Where does the atmosphere end and where do you begin? Our very existence is fuzzy.
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Sep 2, 2016 13:07
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Ytlaya posted:The only exception are paradoxes related to the concept of infinity, which don't really bother me much for some reason. All the infinity paradoxes evaporate when you think of infinity as a direction, not a number. "One car goes Left, and another car goes Left twice as fast, and yet they are both still going Left!"
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Sep 2, 2016 15:30
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The "obvious" solution is to give them a second test on Friday. After all, you never claimed there would be exactly one test, did you?
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Sep 2, 2016 17:44
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