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A Few Good Modal Paradoxes 
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#55
Apr312, 10:48 AM

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Assume that the truth P is unknown to John. Let Q be the statement "P is a truth unknown to John." The question is, is Q knowable by John? Well, suppose Q were known to John. Then John would know that P is a truth unknown to him. But if he knew that P is a truth, that's the same as knowing P. So he would know P and he would know that P is a truth unknown to him. But if he knew P, then it would be incorrect to say that P is a truth unknown to him, so Q would be false, and you can't know a false statement. Thus from the supposition that Q were known to John we get a contradiction, so it must be impossible for John to know Q. Hence Q is a truth unknowable by John, and therefore not all truths are knowable to John. Now do you get it? 


#56
Apr312, 10:51 AM

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#57
Apr312, 10:56 AM

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#58
Apr312, 11:02 AM

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Benjamin Franklin invented the bifocals. But we can imagine alternative histories, so we can say "It was possible for Ben Franklin to not have been the inventor of bifocals". And for any person X, we can say "It was possible for X to not have been the inventor of bifocals." For instance, X can be "William Shakespeare" or "The inventor of special relativity" or "The eighth president of the United States", etc. Thus we can let X = "the inventor of bifocals" and thus we reach the conclusion "It was possible for the inventor of the bifocals to not have been to inventor of the bifocals." But that seems absurd, because obviously the inventor of bifocals had to be the inventor of bifocals. How can you have the inventor of bifocals not be the inventor of bifocals? Now do you understand the paradox? 


#59
Apr312, 02:08 PM

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My gut feeling says that the resolution of the paradox is that the bifocals don't have to be invented, so assuming there is an inventor is a fallacy. However the way that it's worded makes the whole premise absurd
If the statement "it was possible for the inventor of the bifocals to not invent the bifocals" is a paradox, then the statement "it was possible for Benjamin Franklin to not invent the bifocals" is the exact same paradox, because Benjamin Franklin IS the inventor of the bifocals (so you can freely substitute 'inventor of the bifocals' for him). 


#60
Apr312, 02:22 PM

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"The inventor of bifocals" is a welldefined^{*} 'variable constant'.
The informal argument that "it is not necessary that X invented bifocals" makes critical use of X being a non'variable' constant. Roughly speaking, it boils down to observing "X does not vary with 'the inventor of bifocals'", and therefore it is possible for "X" and "the inventor of bifocals" to be different. The argument, of course, doesn't work if X is a constant that does vary along with 'the inventor of bifocals'. *: Ignoring the technicalities of whether there is an inventor and it is unique 


#61
Apr312, 02:27 PM

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#62
Apr312, 11:16 PM

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Will read the rest of your posts soon. All look very interesting and worthwile, mind. 


#63
Apr312, 11:33 PM

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Now the question, is Q knowable by John? Well, that's the same asking, can John know that P is a truth unknown to John? And more concretely, that's the same as asking, can John know that the statement that the coat is in the closet is a truth unknown to him? Or equivalently, can John know that it is true that the coat is in the closet and know that the statement that the coat is in the closet is unknown to him? Or in other words, can John know that the coat is in the closet and know that he does not know that the coat is in the closet? And the answer to the last rephrasing of the question is obviously No, because if he knew the coat was in the closet, then it would be wrong to say that he does not know that the coat is in the closet, so the statement "John does not know that the coat is in the closet" is false, and it's impossible to know a false statement. Thus the answer to the initial question, is Q knowable by John, is also No. So not all truths are knowable by John. When I write it out like that in words, I'm afraid it will sound too confusing, which is why I wrote it using lots of P's and Q's before. I hope this helps. 


#64
Apr412, 09:21 AM

P: 669

hi lugita,
Thanks for the clarification. Lets assume john doesnt know sun rises in the east. Now , can john know that "he doesn't know that sun rises in the east"? No. Because that is a paradoxical sentence. Ah! I see it now. P = Sun rises in east. Q = I/jhon dont know P. Jhon dont know thousands of truth. He may know lot of them later. But for now Q is truth. Suppose Jhon searches really hard to know all truths. He may eventually know P but not Q because knowing P (or trying to know Q) makes Q false. So Q will rather be destroyed than being known to john. But jhon may know R = "Q was a truth." But its not same as knowing Q. Things are making sense. So, where is the paradox? 


#65
Apr412, 09:32 AM

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If you look at my post #33, you'll see that I discussed exactly this resolution to Fitch's paradox. See the attached paper by Joseph Melia in that post. 


#66
Apr412, 10:02 AM

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People use fantastical examples to illustrate logical or philosophical points all the time. Very well. Let's do a bit of that. The election might never happen as you think it would. The anarchist party might take over, or any other political upheaval might occur that would prevent them. (In fact, this is not nearly so fantastical) The coin might fall down a drain in the pavement  or any other similar possibility. (Not too fantastical, either) The ace of clubs  before you examined the deck, it might be vapourized by some explosion  imagine you were in the Twin Towers just as you were about to examine it. (A little more fantastical, but hey, I'm no 911 denier) All above three, therefore, cannot be called unknown truths with 100% certainty. I still maintain that 'unknown truth' is an oxymoron  to the one observer. Sure, my 2 year old niece doesn't know I have a 4 litre motor in my car, but if we go down that path, we are again skating on the trivial, which I can reduce to absurd infinities  as discussed earlier. 


#67
Apr412, 10:05 AM

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#68
Apr412, 10:31 AM

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But to use another fantastical example to illustrate a logical point  as you have allowed earlier, it might not even be there when he opens the closet. Someone might have stolen it. Or the cat might have pulled it through a crack in the floor. So YOU might even be wrong in your assumption of a KNOW truth, never mind Johns unknown truth. 


#69
Apr412, 05:31 PM

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And by the way, we are discussing logic, where truths both trivial and profound are treated on equal footing. And I still think what you're saying about "reducing to absurd infinities" makes no sense. Yes, we can find infinitely many examples of unknown truths, but so what? Just because we can find infinitely many examples of some notion does not mean that there is something wrong with the notion, does it? We can find infinitely many examples of truths, so does truth not make sense? We can find infinitely many examples of falsehoods, so does falsehood not make sense? We can find infinitely many examples of English sentence, so does English not make sense? We can find infinitely many examples of prime numbers, so do prime numbers not make sense? 


#70
Apr412, 05:45 PM

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But in the more general version of the paradox, the statement we're proving is "there is an unknowable truth" and the assumption is "there is an unknown truth". So in that case the truth can still be "the coat is in the closet", but this time no one knows this. But let me go to the very first example of an unknown truth I presented in this thread: either the Riemann hypothesis (see here) is true or it is false. Either way, isn't one of these statements an unknown truth? By the way, have you read the paper by Joseph Melia I attached in post #33? 


#71
Apr512, 06:43 AM

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Lugita 15  In your first post in this thread, you said;
we started with the hypothesis that P is an unknown truth I replied that the hypothesis was flawed form the start  how would you know it's a truth if it's unknown ? Since then, we've basically danced around this issue and not made much progress. I summarise my view thus  the phrase "unknown truth" is an oxymoron. I haven't even addressed Fitch's so called paradox to any degree, because in my view, as it is based on an oxymoron, any nonsense could flow from it. You sought to take it to another language  symbolic or whatever. I haven't followed you down that path, because firstly, I think that we should be able to resolve it or any other matter in the normal, modern English language, and secondly, because I am unfamiliar with the symbolic language to which you have repeatedly referred. To this you replied that English can be ambiguous. I fully agree. But it can also be, and must in fact have the capacity to be, very specific and precise. So why prefer or defer to it's ambiguity (particularly in matters in dispute) when we can easily refer to it's precision and specificity ? I maintain that "unknown truth" is oxymoron. In fact, I would even say that the term "known truth" is redundant. It is enough to say "truth". To this end, I have consulted numerous dictionaries to see what they say about the word 'truth'. here are a couple of examples; truth http://www.thefreedictionary.com/truth 1. Conformity to fact or actuality. 2. A statement proven to be or accepted as true. 3. Sincerity; integrity. 4. Fidelity to an original or standard. 5. a. Reality; actuality. b. often Truth That which is considered to be the supreme reality and to have the ultimate meaning and value of existence. http://dictionary.reference.com/browse/truth?s=t 1. the true or actual state of a matter. 2. conformity with fact or reality; verity: the truth of a statement. 3. a verified or indisputable fact, proposition, principle, or the like: mathematical truths. 4. the state or character of being true. 5. actuality or actual existence. The principle definition on my computers background dictionary (Wordweb) is .. truth  a fact that has been verified .. and this seems to accord nicely with all the above, as well as being a good, concise definition of the word. So for brevity, I will use that. Now, you CANNOT have an unknown 'fact that has been verified'. It is simply nonsense. As would most likely be anything flowing from it. Thus, 'unknown truth' is nonsense and an oxymoron. A thing is either a truth, a falsehood, or unknown. An unknown truth is as nonsensical as a false truth. Following that, you (and others who believe in unknown truths) sought to take it to, umm, how do I say it .. higher or lower order observers (although I'm sure your symbolic language caters for that). The trouble is, if you do that, we quickly descend to the absurd, as has been shown earlier. I know a great many truths that my little niece does not. But to her, they are not truths  they are unknown. Similarly, you and I being of similar intelligence (well, you are probably much more intelligent than I, but just say ..) might not know a great many things that we might discover to be truths in the future  you can in no sense of the word, call them truths now. Probabilities, sure, but not truths. In fact, they could turn out to be falsehoods. Sticking to the dictionary meanings of words tells me that ‘unknown truths’ is simply nonsense. And you haven’t shown otherwise. We have not moved past first base. PS  yes, I downloaded the J Melia paper and read it. My view is why introduce complexity when we first haven't sorted out simplicity ? PPS  remember; truth  a fact that has been verified 


#72
Apr512, 09:44 AM

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