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A Few Good Modal Paradoxes

  1. Mar 21, 2012 #1
    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.

    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.

    So to review, we started with the hypothesis that P is an unknown truth and we got to the conclusion that Q is an unknowable truth. So "there exists an unknown truth" implies "there exists an unknowable truth." Turning this around, "all truths are knowable" implies "all truths are known", which is crazy! Clearly it is possible for there to be some truths which we happen to be unknown right now, but might be discovered in the future. But Fitch's argument above seems to suggest that if you believe that any truth is within our grasp, you have to believe that we already know everything!
     
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  3. Mar 22, 2012 #2

    alt

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    So to review, we started with the hypothesis that P is an unknown truth ..


    But even at the start, that hypothesis seems a little shaky, and rather a play on, or fluid use of, wording.

    How would you know it's a truth if it's unknown ?

    Interested to hear more of your paradoxes though!
     
  4. Mar 22, 2012 #3
    Sorry, maybe I was unclear. We start with the hypothesis that there EXISTS some unknown truth P. Presumably we don't know what that truth is.
     
  5. Mar 22, 2012 #4

    disregardthat

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    IMO; For any proposition to be true, you need a criterion for its truth, and the criterion needs to be satisfied. And it is only upon verification we say that it is satisfied.

    In this sense you can't have unknowable truth. "P is true, but I don't know it to be true" just doesn't make sense. "P is true" doesn't express more or less that "I know P is true". The paradox arise from abuse of language, just like any other.
     
    Last edited: Mar 22, 2012
  6. Mar 22, 2012 #5

    alt

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    No, you were clear. But I'm saying that the hypothesis is nosnensical, imo.

    May as well start with the hypothesis that there exists a five legged tripod. It's a similar word play to say we have an unknown truth. You can't call it truth if it's unknown. To call it truth you would have to know it as being that.
     
  7. Mar 22, 2012 #6
    That is a view called verificationism, which states that all truths are knowable. The whole point of Fitch's paradox of knowability is to disprove verificationism
    Knowledge is different than belief. You may believe one thing, but find out later you were wrong. On the other hand, if you know something then by definition it must be true. A common definition of knowledge used in philosophy is justified true belief. In other words, in order to know a statement P, the following three criteria must be met:
    1. You believe that P is true.
    2. P is true.
    3. You are justified in believing that P is true, in the sense that you cannot possibly be wrong about it.
    These two statements are very different. To say "P is true" is the same as saying "I believe P is true", but is very different from saying "I know P is true."
    No it doesn't, at least not in the straightforward way you're thinking.
     
  8. Mar 22, 2012 #7
    I think you still don't understand what I'm saying. I'm not saying that there is a particular truth which we know to be unknown. Rather, I'm saying that there EXISTS an unknown truth out there, even if we don't know what it is. Surely you agree that we don't know everything, don't you? Like we don't know whether the number of hairs on Obama's head is even or odd. Yet either "the number of Obama's hairs right now is even" or "the number of Obama's hairs right now is odd" must be true, and yet presumably no one knows which one. But one of these is surely an unknown truth, so we can at least say that there exists an unknown truth, can't we?
     
  9. Mar 23, 2012 #8

    disregardthat

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    That's ridiculous. The whole point of requiring a criterion for truth is that one rejects the notion of true statements being true simply in virtue of their meaning. So there "existing unknown truth out there" is meaningless. Propositions require a well-defined criterion for truth. Fitch's paradox doesn't disprove anything in this regard, it is just playing around with words.



    The point is that by asserting a proposition, you can't deny that you believe it. Saying "P is true and I believe P is false" is simply meaningless. "P is true" and "I believe P is true" has no different criterion for truth, so it's impossible to assert one of them are deny the other. Many paradoxes arise from this kind of abuse. In the same fashion, asserting that "I know P to be true and P is false" is meaningless.
     
    Last edited: Mar 23, 2012
  10. Mar 23, 2012 #9

    alt

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    Well in that case, you can reduce a great many (perhaps all) things to your definition of unknown truth. The number of atoms making up your computer screen for instance. An unknown truth. The number of cells in your left ear. Same. The exact number of cents that flowed through the American economy between 9 AM and 10.29 AM today. The number of raindrops that fell on Tokyo between 1934 and 2011. All unknown truths. This however, is reduction to the ridiculous, as is your example of Obamas hairs.

    So if reduction to the ridiculous is your thing, then I suppose Fitch's paradox is attracive.
     
  11. Mar 23, 2012 #10
    Don't you think that either "The number of hairs on Obama's head is even" or "The number of hairs on Obama's head is odd" is an unknown true statement?
    It's not just playing with words, at least not in the sense you're talking about, because it can be formalized symbolically using epistemic logic. See here. (That's a great article, and it has numerous proposed resolutions to Fitch's paradox. If anyone is interested I can discuss my preferred resolution.)
    I agree.
    It's not meaningless, it's just wrong.
    I agree, they mean the same thing, so to assert one and deny the other would be wrong.
    As I said, Fitch's paradox does not arise from at least that kind of abuse of language, because it can be expressed in symbolic language which avoids all the ambiguities and vagaries of English.
    It's not meaningless, again it's just contradictory and hence false.
     
  12. Mar 23, 2012 #11
    Yes, we can find a lot of examples of unknown truths.
    I agree that these are silly examples, but there's nothing fundamentally wrong with them. They're just a way to illustrate that there are such things as unknown truths.
    The reasoning in Fitch's paradox is not as ridiculous as you think. I suggest you examine Fitch's logic more closely.
     
  13. Mar 23, 2012 #12

    disregardthat

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    Absolutely not. I personally believe it is a very basic misconception of logic. Let me explain:

    The logical conjunction "The number of hairs on Obama's head is even OR the number of hairs on Obama's head is odd" is true by virtue of being a logical tautology. There is no need for any criterion here.

    But either of the statements P: "The number of hairs on Obama's head is even" and Q: "The number of hairs on Obama's head is odd" requires criteria for truthfulness, such as the result of counting the hairs being even or odd. The truth of P is realized by satisfying such a criterion.

    It's tricky when it comes to time: If the criterion for a proposition P (which does not depend on time) is satisfied tomorrow, it doesn't make it correct to assert "P is true now" today. It would however be correct to assert "P was true yesterday" tomorrow. The statements have a different sense. So we could say "that the truth(-value) of P was unknown yesterday" tomorrow, but it wouldn't be correct to call it an unknown truth now.

    This form of verificationism is very much alike the way we use ordinary language, and the way we treat scientific hypotheses and evidence. It is only in the platonic pits of formal logic or shaky metaphysics one end up with such silly paradoxes.

    Contradictory, meaningless, useless. All the same to me. It isn't false in the sense of failing to satisfy its criterion, because there is no criterion, none can be given.
     
    Last edited: Mar 23, 2012
  14. Mar 23, 2012 #13
    OK, forget about truths that are unknown in general. Do you at least agree that there are truths that you do not know, but perhaps that other people do know? Because even with that assumption we can carry through Fitch's paradox, and use it to disprove the statement "Any truth can be known by you."
     
  15. Mar 23, 2012 #14

    disregardthat

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    What you are suggesting is that if P is a proposition known to be true by others, but not by me, and I realize that it is known to others and hence true (since it is supposed to be knowable by hypothesis), then upon realization (that it is known to others) I simultaneously can assert that it is unknown to me and known to me at the same time?

    This time you can't deny you are playing with words, or more specifically you are ignoring the temporal aspect of the situation:

    When I realize something I didn't know before, I am made aware of that I didn't know in the past. Not that I don't know now.
     
  16. Mar 23, 2012 #15
    No, I'm suggesting something really obvious, namely that there is a statement P known to others and not to you, and that you do not know that P is known to others, but later you can come to know that P is true, at which point it will be simply be known to you, not known and unknown at the same time. Or if you prefer, you can later come to know that P is known tto others, at which which point you can conclude that P is true, so P will be known to you, not simultaneously known and unknown. What I'm saying is just trivial.
    You and I are in complete agreement on that point.
     
  17. Mar 23, 2012 #16
    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.
     
  18. Mar 23, 2012 #17

    disregardthat

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    Truth is a property of propositions, and propositions are constructs of language. Think about that for a moment.

    You confuse objective truth with an objective reality.
     
  19. Mar 23, 2012 #18

    disregardthat

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    I'm not following you here. In what sense is that a paradox?

    We are supposing that a proposition P is known to some group of people. And that I don't know whether P is true of not. The paradox in this case starts out as follows:

    "The sentence P is an unknown truth". I certainly cannot assert this. What would the criterion for the truth of this proposition be? What would incline me to accept this for any given proposition P? Nothing.

    So only others can assert this of me. But this doesn't get us into a paradox.
     
  20. Mar 23, 2012 #19
    That statement you quoted isn't a paradox at all. It's just an assumption used in the paradox.
    We're not talking about you asserting "P is an unknown truth." Here is the logic of the paradox again.

    We start with the assumption that there is some truth P which is unknown to you, but perhaps known to others. Now consider the statement Q, which says "P is a truth unknown to you." By assumption, Q is true. Now the question is, can Q be known to you? Well, suppose that Q were known to you. Then you would know the statement "P is a truth unknown to you". But if you knew that, you would know that P is true and that P is unknown to you, or in other words P would be both known to you and unknown to you, which is impossible. Thus the supposition that Q is known to you leads to a contradiction, and thus it is impossible for Q to be known to you, or in other words Q is unknowable to you. Thus we can disprove the thesis that all truths are knowable to you.
     
  21. Mar 23, 2012 #20

    disregardthat

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    What is the criterion for the truth of Q? Have you just ignored the premises for truth conditions I have posed? I can neither assert nor know Q. But this is not any more paradoxical than that I can't assert "P is true and I believe P is false". Because I simply have no criterion for its truth. Neither for its falsity.
     
    Last edited: Mar 23, 2012
  22. Mar 23, 2012 #21
    Q is true by assumption, because we are assuming P is a true statement unknown to you, and Q just states that assumption.
     
  23. Mar 24, 2012 #22

    disregardthat

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    No, as I've said, this is just like the case of the proposition S "P is true and I don't believe P". If some group of people knows that P is true, but I don't yet believe P, "then S". But I have no criterion for S. S is a nonsensical proposition in my mouth in the same way Q is. So I cannot assert it. And if someone else but me asserts it of me, it won't get us into a paradox. But we're going in circles here, so I'd like to stress this point.

    The paradox is a result of tricking oneself into believing that certain nonsensical propositions have sense.

    When saying that a proposition is true, think of what criterion must have been satisfied. Most paradoxes of this nature simply ignores that, like this one.

    Don't you find it odd how you would describe (given that I don't believe P) "P is true and I don't believe P" as an unattainable truth beyond my reach? It has no meaning for me, no use, and it doesn't express anything but nonsense.

    EDIT: I didn't catch that you suggested that "P is true" is equivalent to "I believe P is true". That is certainly not the case. You can have a wide variety for believing the truth a proposition, as it is generally the expectation of the satisfaction of the criterion for truth. Correctly asserting "P is true" is not due to expectation, but the result of the realization of the criterion.
     
    Last edited: Mar 24, 2012
  24. Mar 24, 2012 #23

    Hurkyl

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    Truth is a property of an interpretation of a proposition, not of the proposition itself. (and even then, only interpretations of certain types)
     
  25. Mar 24, 2012 #24

    disregardthat

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    Fine, I'd call it a property we assign to propositions. If an assigned property is a property or not is not something I'm going to argue about.
     
  26. Mar 24, 2012 #25

    Hurkyl

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    I hope it's flawed, because classical set theory proves verificationism in the form of "A statement is True if and only if there exists a proof of it" is consistent!

    Specifically, given a formal language, one can define the truth of a proposition to be the set of all* set-theoretic interpretations of the language in which the proposition holds.

    *: Well, one has to deal with technical size issues here. There's probably a conservative way to do so, but I'm just going to invoke a large cardinal axiom, and say I mean "all small" rather than "all".

    This notion of truth value forms a Boolean algebra, so it's a model of classical logic. "True", in this algebra, is the set of all interpretations.

    Now, invoke Gödel's completeness theorem to say that any statement evaluates to "True" under this truth valuation if and only if there is a proof of that statement.


    Note that, for any statement P, "P is true or P is false" is "true" under the truth valuation I describe above! To wit, the set of interpretations where "P is true" holds is precisely the complement of the set of interpretations where "P is false" holds. The "or" of those truth values, therefore, is "True".



    Where your argument fails in this setting is very easy to spot, of course: there is no proposition "P is known". You would have to do more work to set up a logic that can talk about knowability. (But, we knew that you have to do more work anyways, since you want both knowability and truth to be temporal)

    Putting explicit references to things like "truth" into your language is always a huge can of worms anyways. e.g. Tarski's theorem on the undefinability of truth....
     
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