Wikipedia:Reference desk/Archives/Mathematics/2012 March 21
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March 21
editcould you tell me if this is correct
editI expected acceptance or resolution but did not receive any replies. here: http://en.wiki.x.io/wiki/Talk:Vacuous_truth#correct_resolution and here: http://en.wiki.x.io/wiki/Talk:Unexpected_hanging_paradox#this_article_is_just_an_article_title_mistake.2Cno_paradox
could others peer review me please. 188.157.65.3 (talk) 11:05, 21 March 2012 (UTC)
- You may get some robust feedback on your ideas here at Wikipedia, but that is not a peer review. The only way to get your ideas peer reviewed is to write a paper and submit it to an academic journal. It is no use telling us "you may publish something in some glossified paper form first if you need to reference it" - preparing your ideas for publication is up to you, and cannot be done for you by a random stranger on the internet. Gandalf61 (talk) 11:34, 21 March 2012 (UTC)
- sure it can. barring that, I welcome your robust feedback, Gandalf. 188.156.105.166 (talk) 12:17, 21 March 2012 (UTC)
- Really ? Because I don't think you will like my feedback, since it won't be agreeing with your ideas. Gandalf61 (talk) 13:44, 21 March 2012 (UTC)
- Okay, let's have them. Being wrong is halfway to being right. 79.122.70.68 (talk) 14:02, 21 March 2012 (UTC)
You're WRONG, op! Please refer to this article for an explanation of why. --128.62.82.193 (talk) 14:12, 21 March 2012 (UTC)
- Right, that's the article I'd like you guys to rewrite. I've given my reasons, and also a non-"paradox" that this rewrite resolves. I could be wrong though, and if so I'd like to know how/why. 79.122.70.68 (talk) 14:22, 21 March 2012 (UTC)
- All you've done in your argument is you've just asserted repeatedly that it's not just false, but ESPECIALLY false. Argument by repetition is pathetic. The article gives plenty of clear reasons why vacuous truths should, in fact, be considered true. It's up to you to give direct counterarguments to the points raised in the article. --128.62.82.193 (talk) 14:42, 21 March 2012 (UTC)
- Right, that's the article I'd like you guys to rewrite. I've given my reasons, and also a non-"paradox" that this rewrite resolves. I could be wrong though, and if so I'd like to know how/why. 79.122.70.68 (talk) 14:22, 21 March 2012 (UTC)
- As someone else on [1] pointed out, I think you are misunderstanding the meaning of a Conditional sentence. "If A then B" means in the cases where A is true, B must also be true. It doesn't mean that A and B are always true, which is what you're claiming (since you say that the statement is "ESPECIALLY false" if A is false). There's already a way to say that, which is "A and B". This all holds even in general English usage. The problem here isn't that the articles are making some logical error or that you've discovered something new, it's that you disagree with every other English speaker on the meaning of the word "if". And your definition is not particularly useful. Rckrone (talk) 15:23, 21 March 2012 (UTC)
I think we can all agree that the statement "whenever the value of pi dips below 3, the Federeal Reserve temporarily cuts interest rates until pi reaches its usual value, taking more drastic steps if it fails to do so within a very short period of time" is not a vacuous "truth": it's an especially false statement. As for the usefulness, I have already resolved a paradox using my 'contribution'. What do you think of my resolution at any rate? 134.255.77.6 (talk) 16:18, 21 March 2012 (UTC)
- No we do not all agree on that- your statement is indeed vacuously true. If you think it's false (or "especially false"), then demonstrate for me a counterexample. Staecker (talk) 16:29, 21 March 2012 (UTC)
- (edit conflict) Well, I don't agree. The statement "whenever the value of pi dips below 3, the Federeal Reserve temporarily cuts interest rates until pi reaches its usual value" is true because there can be no counterexample. But the statement "whenever the value of pi dips below 3, the Federeal Reserve does not temporarily cut interest rates until pi reaches its usual value" is also true, for the same reason. Gandalf61 (talk) 16:30, 21 March 2012 (UTC)
- I think if both something and its opposite follow from a true statement (or if a statement became true), we are not dealing with math anymore. That's why people come to the "paradoxical" conclusion that is the other link. What do you have to say about that one? (meaning my link to what I had to say about the hanging/student quiz paradox - what do you think about that one?)
- If we did wake up and the value of pi were different, you're telling me two opposite statements should follow from that: rather than the obvious, which is that the original implication was false, and until pi could change was especially false. Please address my resolution of the paradox I linked, which turns into a non-paradox under my "contribution". 134.255.77.6 (talk) —Preceding undated comment added 16:45, 21 March 2012 (UTC).
- Vacuously true statements aren't just some odd corner case which is "not math anymore". They're often very useful. For example consider the statement "if integer p is the largest prime, then p!+1 is not divisible by any prime, which is a contradiction." This is a vacuously true statement and it occurs in some form or another in most proofs that there are infinitely many primes. There might be ways to contort our language to avoid it, but why should we? Rckrone (talk) 16:51, 21 March 2012 (UTC)
- There's no contortion. When you make the statement "if p is the largest prime, then whatever" you have to have in the back of your mind that you might have just made/set up an especially false deduction/implication. When you get to the contradiction, it becomes clear where you flucked up: what you thought was true (if only vacuously) in fact turned out to be especially false. I'm saying that when the students set up their reasoning "If the teacher has not given the test by Friday, then..." they have to have in the back of their mind that they might be making an especially false statement. When they get to the contradiction, it is apparent that this possibility is the reason: what they logically thought was true, was in some sense logically valid, but turned out to be especially false due to its premise. If the students knew about the possibility of their statements being especially false, they would realize that this is, in fact, what's happening. Just as when you get to your contradiction you realize that "if integer p is the largest prime..." sentence is especially false, they will realize that their "if the teacher hasn't given the test by friday" sentence is especially false. Do you see what I'm saying? Please address my resolution of the student-quiz non-paradox, bearing in mind that we know the actual outcome (below, the fact that 5+5+5 is actually 15), which is that the students can be surprised. 134.255.77.6 (talk) 16:57, 21 March 2012 (UTC)
- Are you honestly claiming that that proof is false? Rckrone (talk) 17:05, 21 March 2012 (UTC)
- No, you did the proof perfectly, but deluded yourself about what you were doing. You thought you were setting up true statements until you got to a contradiction, but instead you were setting up one that was especially false, and proceeded to proof this fact. It is especially false to begin any sentence with "If integer p is the largest prime" and you've proceeded to prove this. The students have proven that if the teacher can surprise them, (the jailer can surprise the inmate) then it is especially false to start "If on Friday morning we haven't had a test..." You're doing the same thing, you just don't realize it. The paradox disappears when you see the truth values correctly. It doesn't require changing any usage, just perception. Could you address my resolution of the prisoner paradox? 134.255.89.2 (talk) 17:14, 21 March 2012 (UTC)
- You're confusing what's meant by the truth of the statement. The proof is the statement "if integer p is the largest prime, then p!+1 is not divisible by any prime, which is a contradiction." Either that statement is true, or the proof is invalid. The truth of the entire statement is separate from the truth of the first clause "integer p is the largest prime", which you are correct in saying is false. Rckrone (talk) 17:27, 21 March 2012 (UTC)
- No, you did the proof perfectly, but deluded yourself about what you were doing. You thought you were setting up true statements until you got to a contradiction, but instead you were setting up one that was especially false, and proceeded to proof this fact. It is especially false to begin any sentence with "If integer p is the largest prime" and you've proceeded to prove this. The students have proven that if the teacher can surprise them, (the jailer can surprise the inmate) then it is especially false to start "If on Friday morning we haven't had a test..." You're doing the same thing, you just don't realize it. The paradox disappears when you see the truth values correctly. It doesn't require changing any usage, just perception. Could you address my resolution of the prisoner paradox? 134.255.89.2 (talk) 17:14, 21 March 2012 (UTC)
- Are you honestly claiming that that proof is false? Rckrone (talk) 17:05, 21 March 2012 (UTC)
- There's no contortion. When you make the statement "if p is the largest prime, then whatever" you have to have in the back of your mind that you might have just made/set up an especially false deduction/implication. When you get to the contradiction, it becomes clear where you flucked up: what you thought was true (if only vacuously) in fact turned out to be especially false. I'm saying that when the students set up their reasoning "If the teacher has not given the test by Friday, then..." they have to have in the back of their mind that they might be making an especially false statement. When they get to the contradiction, it is apparent that this possibility is the reason: what they logically thought was true, was in some sense logically valid, but turned out to be especially false due to its premise. If the students knew about the possibility of their statements being especially false, they would realize that this is, in fact, what's happening. Just as when you get to your contradiction you realize that "if integer p is the largest prime..." sentence is especially false, they will realize that their "if the teacher hasn't given the test by friday" sentence is especially false. Do you see what I'm saying? Please address my resolution of the student-quiz non-paradox, bearing in mind that we know the actual outcome (below, the fact that 5+5+5 is actually 15), which is that the students can be surprised. 134.255.77.6 (talk) 16:57, 21 March 2012 (UTC)
- Do bear in mind that it obviously is a paradox that requires resolution, since in actual fact there is no paradox and the students can be surprised. That "paradox" is the article saying, "Look, if you add five together three times you get ten, what a paradox since it's 15". I'm showing the source of the paradox - the reason it requires resolution is because you can go ahead and actually do it (add the fives together, or surprise the class), despite the math prediction. The math prediction is based on accepting the vac vacuous truth as true rather than especially false. When you realize it's especially false to start saying "If the teacher hasn't given the test by Friday, it's Friday" in the case that the teacher is not giving the test on Friday, then you have resolved the whole paradox, by realizing that that if... statement is not true, but especially false. Please read through my resolution of the non-paradox and tell me whether it is satisfactory. 134.255.77.6 (talk) 16:57, 21 March 2012 (UTC)
- Vacuously true statements aren't just some odd corner case which is "not math anymore". They're often very useful. For example consider the statement "if integer p is the largest prime, then p!+1 is not divisible by any prime, which is a contradiction." This is a vacuously true statement and it occurs in some form or another in most proofs that there are infinitely many primes. There might be ways to contort our language to avoid it, but why should we? Rckrone (talk) 16:51, 21 March 2012 (UTC)
- Regarding the paradox itself, here's what I would say about it: Suppose instead the setup was that the judge chooses a (randomized) strategy for choosing the execution day, and then reveals his strategy to the prisoner. Then the prisoner's argument is correct that there's no strategy the judge can choose so that the prisoner is surprised with 100% probability on the day of the execution. However, this is not the situation we're dealing with here, since it's not a repeated game. The judge can choose a strategy that has a lower probability of surprising, and then get lucky and be right. The prisoner has to assume this is a possibility, since he has no guarantee that the judge is telling the truth with 100% probability (especially because this is impossible). Assuming there will be no execution doesn't follow. (Since the strategy isn't actually revealed, the prisoner has some prior distribution for how he believes the judge will act and then it works the same way.)
- Obviously this is just my take, since there doesn't seem to be consensus on it. Rckrone (talk) 17:56, 21 March 2012 (UTC)