## A Few Good Modal Paradoxes

 Quote by lugita15 Because even if the Knowability Thesis is false, it seems like Fitch's argument disproves it too easily. It would seem that any disproof of the Knowability thesis must impose fundamental limitations on human knowledge, yet Fitch's argument seems to impose no such argument. Indeed, it does not even disprove the thesis "it is possible that all truths are known", and yet it disproves what seems like a much weaker claim. That is what makes it seem paradoxical, but you can see my preferred resolution to the paradox in post #33 of this thread.
I understand that you want to move away from the Fitch paradox but I think this aspect is interesting. I see no reason to assume that a disproof of the knowability thesis should place any more fundamental limit on human knowledge than the 'paradox' provides. That the disproof is not the kind of thing that may have been expected would indicate to me a problem with the conception of knowability. I admit I am not familiar with this branch of philosophy, but what a fundamental limit to human knowledge should look like is rather obscure to me.

 Quote by lugita15 You can believe a false statement, and you can even believe that you know the false statement. But you cannot KNOW a false statement. The traditional definition of knowledge used in philosophy is "justified true belief". In other words, in order to know a statement, you must believe it, your belief must be true, and you must be justified in believing it, in the sense that your reason for believing it cannot possibly lead you to believe something false.Anything that is believed to be known does not have to be true, but anything that is known must be true.
I'll take issue with this if I may. I realize that what you describe is an established philosophical position, but it is not an objective truth. An individual may wish to use the word 'knowledge' it in a different way. Certainly you can explain that when you talk about knowledge you demand truth, but another person may, with good reason, wish to use it another way. It should be no barrier to communication if the difference is acknowledged.

I apologize in advance if I am not able to reply in this thread for a few days; I'll be without an internet connection.

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 Quote by dcpo I understand that you want to move away from the Fitch paradox but I think this aspect is interesting.
There's no need to move away from Fitch's paradox. We can discuss multiple paradoxes at once.
 I see no reason to assume that a disproof of the knowability thesis should place any more fundamental limit on human knowledge than the 'paradox' provides.
If you heard there was a proof that proved that there were some truths that were unknowable by Man, wouldn't you assume that that meant that human knowledge has limits? Well, strangely Fitch's paradox says that some truths cannot be known and yet it does not contradict the statement that it is possible to be omniscient, i.e. know all truths! That's what's neatly resolved, in my opinion, in my post #33.
 That the disproof is not the kind of thing that may have been expected would indicate to me a problem with the conception of knowability.
What problem is that? The definition of knowability is straightforward: P is said to be knowable if it is possible that P is known.
 I admit I am not familiar with this branch of philosophy, but what a fundamental limit to human knowledge should look like is rather obscure to me.
A fundamental limitation on human knowledge would not only say that there are some true statements that cannot be known, it would also say at the very least that there are some statements whose truth value cannot be known. Yet Fitch's paradox does not imposes any such limitation.
 I'll take issue with this if I may. I realize that what you describe is an established philosophical position, but it is not an objective truth.
It's not a philosophical position, it's just the standard definition used in philosophy.
 An individual may wish to use the word 'knowledge' it in a different way. Certainly you can explain that when you talk about knowledge you demand truth, but another person may, with good reason, wish to use it another way. It should be no barrier to communication if the difference is acknowledged.
Of course people may choose to use words in all sorts of nonstandard ways. But I'm using knowledge with the standard philosophical meaning.

Anyway, do you agree with the resolution I present in my post #33? Also, have you taken a look at the other paradoxes I have presented, in posts #35 and #91?
 I think I’m starting to see what you’re getting at here. So the paradoxical nature comes from the fact that if all truths are known the problem statement cannot be constructed. This is certainly a noteworthy quirk, but I don’t find it overly worrying at first glance. This is what I mean when I say it points to a problem with the concept of knowability. Clearly knowability is a simple concept when applied to concrete statements, but the paradox hinges on a rather abstract construction. I would not expect such a superficially simple, naive even, concept to extend without issue to a sphere where we discuss all truths’. I would expect to be able to generate paradoxical statements out of the acceptance of a totality of truths. If anything I’m surprised that more damaging paradoxes have not been discovered. Am I correct in understanding that the resolution to the paradox is that while the truth of Q is unknowable, it is possible to stop Q being a truth (by establishing the truth of P)? If so then that seems a reasonable resolution to me. I’ve looked at the other paradoxes but I’ve not had many thoughts on them yet. I am happy with the simple resolution to the second proposed by Hurkyl.
 To expand on the above, if we allow there to be a class of all truths then for each subclass of this class the statement that every member of that subclass is true is a truth, and thus the class of all truths cannot be in 1-1 correspondence with itself. It strikes me that any discussion of knowability that does not put some limit on what truths are included in the discussion has more serious problems to overcome than Fitch's paradox.

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 Quote by dcpo I think I’m starting to see what you’re getting at here. So the paradoxical nature comes from the fact that if all truths are known the problem statement cannot be constructed.
The paradoxical part is that the proof rules out the weaker claim "all truths are knowable", but it does not rule out the stronger claim that "it is possible to know all truths". In other words, it somehow says that certain truths cannot be known, and yet it is still possible to know everything (i.e. all true statements). That seems very strange.
 This is what I mean when I say it points to a problem with the concept of knowability. Clearly knowability is a simple concept when applied to concrete statements, but the paradox hinges on a rather abstract construction. I would not expect such a superficially simple, naive even, concept to extend without issue to a sphere where we discuss all truths’.
Knowability is a simple concept in general. It is just knowledge plus possibility. Knowledge is well studied in epistemic logic, and possibility is well studied in alethic modal logic. Yet somehow when we combine epistemic and alethic modal logic, we get this seemingly paradoxical result.
 I would expect to be able to generate paradoxical statements out of the acceptance of a totality of truths. If anything I’m surprised that more damaging paradoxes have not been discovered.
We can actually carry through this paradox without using the phrase "all truths" at all.
 Am I correct in understanding that the resolution to the paradox is that while the truth of Q is unknowable, it is possible to stop Q being a truth (by establishing the truth of P)? If so then that seems a reasonable resolution to me.
That is precisely the resolution I outline in post #33. By knowing P, you make Q false, so you can't know Q. However, although it impossible to know that Q is true, you can very easily know that Q is false. So the correct way to say that there are no limits to human knowledge is not to say "For all truths P, P can be known to be true." Instead, you should say "For all statements P, the truth value of P can be known." (The truth value of a statement means whether it is true or false.) So you can replace "all truths are knowable" with "all statements are decidable", and this latter claim does not lead to analogus paradoxes.
 I’ve looked at the other paradoxes but I’ve not had many thoughts on them yet.
If you like, I can explain my preferred resolution to the paradox of the gentle murderer, or I can wait a little while to see whether anyone else would like to wrestle with it.
 I am happy with the simple resolution to the second proposed by Hurkyl.
Yes, that is indeed the standard resolution to the inventor of bifocals paradox.

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 Quote by dcpo To expand on the above, if we allow there to be a class of all truths then for each subclass of this class the statement that every member of that subclass is true is a truth, and thus the class of all truths cannot be in 1-1 correspondence with itself.
But it is not the case that for each subclass of the class of truths there is a statement saying that every member of that subclass is a truth! That's because you can't talk about most of the subclasses of the class of truths using statements. A statement must be a finite length, so there are only countably many statements, and thus countably many true statements. Thus there are more classes of statements then there are statements, so most classes of statements cannot be described by statements. There are uncountably many classes of statements, but only countably many definable classes of statements. Since the class of truths and the class of definable subclasses of the class of truths are 1-1 correspondence, there is no contradiction.
 It strikes me that any discussion of knowability that does not put some limit on what truths are included in the discussion has more serious problems to overcome than Fitch's paradox.
No, there is no need to limit the truths under discussion.

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 Quote by lugita15 If you like, I can explain my preferred resolution to the paradox of the gentle murderer, or I can wait a little while to see whether anyone else would like to wrestle with it.
I looked up Forrester's paradox, but I don't know the exact version you gave above. I saw a symbolic version of it at http://rationalhunter.typepad.com/cl...ters_para.html,
but I didn't find it convincing as a paradox, because it seemed to me that assumption #2 was not a good assumption at all. There the structure of "obligation" is symbolically built to be bimodal (either you are obliged to do something, or you are not obliged to do it, there are no "levels of obligation" built into the symbolism, and putting in such levels, like O1 and O2 where O1 > O2 would seem to fix the paradox), but when this is translated into human sensibility, the language gets mauled, and we end up with a hierarchy of obligation (we are obliged not to murder, call that O1, but if we do murder, we are obliged to murder gently, call that O2). Mistaking a bimodel obligation (used in the logic of the paradox) with a hierarchical version (used in real life) seems to be the source of the paradox.

Incidentally, this seems to connect with your other thread about what mathematics is. We use logical structures to make proofs, but the logical structures must be very closely connected (some, not I, might dare to say identically connected) to things that we experience in our daily lives, if we want the theorems to make sense in our daily lives. So if we want to prove things about obligations that check with how we use that term in daily life, we must tailor the axioms around obligations to fit with how we use the term colloquially. That creates a kind of "back door" through which paradox can creep, some (perhaps even me) might say, through which paradox inevitably creeps. This is a cautionary tale about limits for using mathematical proofs to know truths, the most celebrated example being Godel's theorems.

 Quote by lugita15 But it is not the case that for each subclass of the class of truths there is a statement saying that every member of that subclass is a truth! That's because you can't talk about most of the subclasses of the class of truths using statements. A statement must be a finite length, so there are only countably many statements, and thus countably many true statements. Thus there are more classes of statements then there are statements, so most classes of statements cannot be described by statements. There are uncountably many classes of statements, but only countably many definable classes of statements. Since the class of truths and the class of definable subclasses of the class of truths are 1-1 correspondence, there is no contradiction.
Well, yes, but nevertheless it is true that all the statements in any subclass of the class of all true statements are true. My point is that you cannot get a complete handle on general truth using the formal machinery of logic. I guess this is rather trivial, as was my example, and Fitch's construction is more interesting in that it produces a seemingly paradoxical, though easily resolved, result using only ideas that can be easily translated into familiar formal systems. What I'm trying to say is that it's interesting as an investigation into the possible limitations of our formal machinery in this respect, but that I remain somewhat unconvinced that this kind of reasoning gets us very close to understanding the actual limits of human knowledge.

 Quote by lugita15 No, there is no need to limit the truths under discussion.
But you've just described how you are limiting them; by having 'true' things for which there is no statement in the formal language.

 Quote by lugita15 The question we're dealing with is: Are all true statements knowable? To put it another way, is it possible for there to be some truth which can never be known, no matter how hard you try? Here's an argument that seems to answer this question. Obviously there are some unknown true statements out there; we don't know everything, do we? For instance, either "The Riemann Hypothesis is true" or "The Riemann Hypothesis is false" is one of these statements. In any case, let P be some unknown true statement. Then consider the statement Q, which says "P is an unknown truth." Then Q is obviously a truth. Is it possible for Q to be known? Well, suppose Q were known. Then we would be able to say "I know that Q is true" or equivalently "I know that P is an unknown truth" or in other words "I know that P is true and that P is unknown." But it's impossible for that to be true, isn't it? Because if you knew that P is true, then P would be known, so it would be impossible to know that P is unknown, because P is not unknown, and you can't know a false statement! Thus it's impossible to know Q, so in other words Q is an unknowable truth.
Sorry if someone already touched on this, but my favorite question like this is:

How ignorant are we?

In order to answer it, you'd need to know all absolute truths. And if you knew all absolute truths, you wouldn't be ignorant.

This means that we can never know how ignorant we are.

 Quote by PlayingMonk disregardthat, do you believe there is such a thing as objective truths? Or do you think things can only be true to people? I'm having trouble understanding your objections.
PlayingMonk, what do you mean by "objective?" A truth can be objective in more than one way at the same time.

In your post, "true to" seems to mean what many relativists about truth seem to mean "true for" when they say, "That may be true for you, but it's not true for me." Well, their comment merely another way to say, "You may believe that, but I don't believe it." Am I telling you an objective truth when I tell you that for each proposition p, someone or other may or may not believe that-p?" If that claim is true, it's objective, too, in one of three relevant senses of the word "objective." If the claim is true, it's true about each person. But relativists about truth deny that there's any proposition that's true about each person. They want to have their cake and eat it, too.

A relativist about truth told me that since every truth depended on some context or other. But here opinion was absolute in one sense of "absolute." It was an opinion about every context.

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 Quote by Bill_McEnaney A relativist about truth told me that since every truth depended on some context or other. But here opinion was absolute in one sense of "absolute." It was an opinion about every context.
When I teach, I always say "I must warn you that nothing I say will be absolutely or completely true. Including that."

 Quote by lugita15 People most often hear about paradoxes that challenge our notions of truth and falsity, like the Liar Paradox, Curry's Paradox, Russell's Paradox, Berry's Paradox, etc. But just as interesting are the paradoxes that challenges other notions we hold dear, the ones philosophers call "modal" notions: knowledge, possibility, morality. So let me present one of the most famous ones, called Fitch's Paradox of Knowability, and if people find that interesting I can talk about a few other favorites of mine.
Whats this? "Fitch's Paradox". Are we talking about THE Fredric Fitch!? Oh my...I see it is so.

He is one of the sharpest logicians Ive encountered. I found he was extremely short in the amount of selected words but a few lines of his compares with chapters from lesser logicians.

But ...awww...Its modal logic! What shall I do? All my life I refused to take part in it!

I always assumed no modal logic to be consistent, so why bother? Well Ill start reading now and it will take me some time to catch up, if ever I will since modalities disgust me. But I cant ignore Dear old Fredric...Here I go :)
 ^^ Why would you assume no modal logic could be consistent?

 Quote by dcpo ^^ Why would you assume no modal logic could be consistent?
It was originally Frege that made me suspicious of modal concepts: You know the case of the Morning star and The Evening star being both identical with the "star" Venus,and if Mr X knows that the Morning star is Venus then he should also know that the Evening star is Venus... but it need not be so. (If I remember the argument correctly.)
Other things, the Liar Paradox in particular, kept occupying my attention and I never started a search for a way to "defuse" modal concepts. By the way what are the "^^" supposed to add to your question?
 I must say that seems a rather flimsy reason to doubt the consistency of modal logic, especially considering that it has been rigorously studied for 100 years, and has its own well developed proof and model theories. Maybe you mean something non-standard when you talk about consistency. p.s. The '^^' indicates I'm replying to the post immediately above mine.

 Quote by dcpo I must say that seems a rather flimsy reason to doubt the consistency of modal logic, especially considering that it has been rigorously studied for 100 years, and has its own well developed proof and model theories. Maybe you mean something non-standard when you talk about consistency. p.s. The '^^' indicates I'm replying to the post immediately above mine.
Flimsy reason or not...I rely rather heavily on my intuition :) But I dont trust my life on it,I check. I now want to start a research into modal logic...Im not fast and I have to defend my solution of the Liar paradox meanwhile: http://www.physicsforums.com/showthr...42#post3980342 So any "deep" comments have to wait...perhaps forever ;)
By "inconsistent" I mean simply that a contradiction follows from the axioms of the theory in question.And I think I did not CLAIM there was inconsistency... I just reported my suspicion: So what interesting results are there after the hundred years of rigorous research? Quines claim that Modal Logic was conceived in the sin of confusing use and mention is refuted? Modal Logic is now wholly without sorrows? I havent given it much thought the last thirty years but I suppose my stand still is that Modal Matters matter but formalizing its logic seemed to me just a fun game for formalists bored by ordinary logic. (No disrespect intended.)
 Blog Entries: 1 If anyone is still interested, I can present my preferred resolution to the Paradox of the Gentle Murderer I introduced in post #91. (The previous two paradoxes, Fitch's paradox of Knowability and the Inventor of Bifocals paradox, are stated in posts 1 and 35 and resolved in posts 33 and 61). Then I can introduce yet another modal paradox.