A Few Good Modal Paradoxes


by lugita15
Tags: modal logic, paradoxes
lugita15
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#55
Apr3-12, 10:48 AM
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Quote Quote by I_am_learning View Post
Sorry, If I missed something by not following the whole thread. But the thread looks like cycling around anyways. :)

Good, enough assumption, I accept that.

Re-writing: Q = "There exist some truth P, which is unknown to you (but may be know to others". If this Re-writing is allowed, then
Q, just repeats our assumption. So, it must be true. (Because, Assumption means we take it to be true for granted)

Well, Q = Our assumption. We got to know our assumption when working on a problem, don't we? :)

Well, it was assumed, so we have been knowing it all way along.

If I knew my assumption (which is what you are referring by the word 'that'), I would know that there exist some truth P, which is unknown to me. I would know nothing whatsoever about what the truth exactly is.
I can't understand how you jumped to the conclusion that P is known to me? The only thing known to me is my assumption, which states that there exist some truth P, which is unknown to me.
To my knowledge, P is just an unknown variable (like the x in algebra). I am yet to solve the puzzle and find out what particular truth P contains.

I am not a philosophy student, but just sometimes get interested in such things.
Sorry, a lot of your confusion is because I didn't word things well enough. The "you" that I'm discussing the argument with is different from the "you" whose knowledge we're discussing. So instead of using "you", let me call the individual John, and let me restate the argument in that way.

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?
lugita15
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#56
Apr3-12, 10:51 AM
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Quote Quote by alt View Post
I still maintain that I do not believe there is such a thing as an unkown truth - at least if not reduced to the absurd. I'm not deliberatley being obstinate about this - I really haven't seen any proof of any unknown truth here. Will try to address Hurkyl's post soon.
You literally don't think there are any true statements that are unknown? Don't you think the results of the 2016 US Presidential election are unknown? When you flip a coin in the air, don't you think it's unknown which side it will land on?
lugita15
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#57
Apr3-12, 10:56 AM
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Quote Quote by alt View Post
Having set up the initial finite alternatives (standard deck) of course one of them is the ace of spades. There is nothing, no unknown truth here. You'll know it in up to 52 guesses. Similarly, I could make, say, 50,000 guesses about the number of hairs on Obamas head, and I'm sure I'd get it right.
Perhaps you'll know it after you check each one of your 52 guesses, but do you agree that right when your handed the deck you don't know which place the Ace of Spades is? So if the Ace of Spades is in the 10th place, then at that moment wouldn't "The Ace of Spades is in the 10th place" be an unknown truth?
lugita15
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#58
Apr3-12, 11:02 AM
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Quote Quote by alt View Post
I did - again, you said;

The conclusion to this 'paradox' is that "it is possible that Benjamin Franklin did not invent bifocals."

Wrong conclusion. It WAS possible that BF did not or would not invent bifocals (before he did so) but when he did invent bi focals, he invented them.

Therefore, it is now IMPOSSIBLE that BF did not invent bifocals.
How many times do I have to say this? You can change "it is possible" to "it was possible" if you want. That's not the important part of the logic. Let me change it myself, so there's no confusion.

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?
Office_Shredder
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#59
Apr3-12, 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).
Hurkyl
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Apr3-12, 02:22 PM
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"The inventor of bifocals" is a well-defined* '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
lugita15
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#61
Apr3-12, 02:27 PM
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Quote Quote by Hurkyl View Post
"The inventor of bifocals" is a well-defined* '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'.
Congratulations Hurkyl, you solved it! The technical way to say this is that in modal logic, you can only freely subsitute "rigid designators" into the modal operators.
alt
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Apr3-12, 11:16 PM
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Quote Quote by lugita15 View Post
Sorry, a lot of your confusion is because I didn't word things well enough. The "you" that I'm discussing the argument with is different from the "you" whose knowledge we're discussing. So instead of using "you", let me call the individual John, and let me restate the argument in that way.

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?
No .. I gets me P's & Q's mixed up. Can you try it again with a real example ?

Will read the rest of your posts soon. All look very interesting and worthwile, mind.
lugita15
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#63
Apr3-12, 11:33 PM
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Quote Quote by alt View Post
No .. I gets me P's & Q's mixed up. Can you try it again with a real example ?
Sure. John's coat is in his closet, but he doesn't know it. So "the coat is in the closet", which we'll call P, is a truth unknown to John. So let Q="P is a truth unknown to John". Then Q is certainly a true statement, because in fact P IS a truth unknown to John. With me so far?

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.
I_am_learning
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#64
Apr4-12, 09:21 AM
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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?
lugita15
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#65
Apr4-12, 09:32 AM
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Quote Quote by I_am_learning View Post
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.
Exactly, you see the solution! By knowing P, John makes Q false, so he cannot know that Q is true, because it's not. But he can know that Q is false. So the only reason Fitch's paradox says there are unknowable truths is that there are some truths that become false as soon as you find out their truth value, so you can never know them to be true, but you can know them to be false.

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.
alt
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Apr4-12, 10:02 AM
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Quote Quote by lugita15 View Post
You literally don't think there are any true statements that are unknown? Don't you think the results of the 2016 US Presidential election are unknown? When you flip a coin in the air, don't you think it's unknown which side it will land on?
Earlier in a post you said;
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.
alt
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Apr4-12, 10:05 AM
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Quote Quote by lugita15 View Post
Perhaps you'll know it after you check each one of your 52 guesses, but do you agree that right when your handed the deck you don't know which place the Ace of Spades is? So if the Ace of Spades is in the 10th place, then at that moment wouldn't "The Ace of Spades is in the 10th place" be an unknown truth?
See my above post.
alt
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#68
Apr4-12, 10:31 AM
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Quote Quote by lugita15 View Post
Sure. John's coat is in his closet, but he doesn't know it ..
But you do, in order to make the statement. Therefore, it is a known truth, and all John has to do is ask you.

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.
lugita15
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#69
Apr4-12, 05:31 PM
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Earlier in a post you said;
People use fantastical examples to illustrate logical or philosophical points all the time.
And I meant it.
Very well. Let's do a bit of that.
All right, but I don't see what logical or philosophical point you're trying to illustrate.
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.
Yes, all of these are possible. But isn't it still true that either "Barack Obama will win the 2016 US Presidential Election" or "Barack Obama will not win the 2016 US Presidential Election" is an unknown truth?
The coin might fall down a drain in the pavement - or any other similar possibility.
Yes, but isn't it still true that either "the coin landed heads" or "the coin did not land heads" is an unknown truth?
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.
OK, but either "the Ace of Clubs was the 10th card in the deck" or "the Ace of Clubs was not the 10th card in the deck" was an unknown truth, correct?
All above three, therefore, cannot be called unknown truths with 100% certainty.
I'm not saying that there is a particular statement that we know to be an unknown truth. All I'm talking about is whether we know that there EXISTS an unknown truth. Don't you think "The Earth goes around the Sun" was an unknown truth in ancient times? Don't you think the Pythagorean theorem was an unknown truth in even more ancient times? Don't you think there are similar truths unknown to us, both trivial and profound?

And by the way, we are discussing logic, where truths both trivial and profound are treated on equal footing.
I still maintain that 'unknown truth' is an oxymoron - to the one observer.
To say "No truths are unknown" is the same as saying "all truths are known". Do you really believe that all truths are known, i.e. we are omniscient?
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.
The example of your niece is a perfectly good one. I don't know why you dismiss things for being trivial. What's wrong with trivial examples, fantastical examples, sensible-sounding examples, or anything else. In logic we talk about all kinds of examples.

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?
lugita15
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#70
Apr4-12, 05:45 PM
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Quote Quote by alt View Post
But you do, in order to make the statement.
Count me out of it. I don't know that there's a coat in the closet. I'm just supposing that "the coat is in the closet" is an unknown true statement, and I am deducing the consequences of that assumption.
Therefore, it is a known truth, and all John has to do is ask you.
First of all, the "John" formulation of Fitch's paradox is a restricted version of the paradox I made up to satisfy disregardthat's objections earlier in the thread. The point of this restricted version is just to prove that "not all truths are knowable by John". So the assumption for this restricted version is just "there exists a truth unknown to John", not the more general assumption "there exists an unknown truth". So even it was known to me that the coat is in the closet, it would still be unknown to John, which is what's relevant.

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 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.
Sure, but when the coat was in the closet, wasn't it an unknown truth that the coat was in the closet?
So YOU might even be wrong in your assumption of a KNOW truth, never mind Johns unknown truth.
Sure I may be wrong that the coat is in the closet. But assuming it is in the closet, that is a truth is unknown to John, and assuming it's not in the closet, THAT is a truth unknown to John. Either way, isn't there a truth unknown to John?

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?
alt
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#71
Apr5-12, 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 non-sense 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 non-sense. 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
lugita15
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#72
Apr5-12, 09:44 AM
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Quote Quote by alt View Post
truth - a fact that has been verified
That is just not how the word truth is used in logic. In logic we use Tarski's definition, which states that "P is true" is the same thing as saying "P". So far instance, saying that it is true that 1+1=2 means that 1+1 does equal 2. Do you understand that?


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