Saturday, September 10, 2005

Unknowable Truths

You might have thought that all truths are, in principle, knowable, even if it turns out that nobody actually ever does manage to know it. But, surprisingly enough, this is provably false. For consider some such truth P that will never actually be known. Then the statement Q: "P is true but will never be known" is true. But Q is clearly unknowable, for if you knew Q then you would know the first conjunct (P), but that would contradict the second conjunct (P is never known), thus making Q false, which is a contradiction. Since knowing Q would yield a contradiction, and is thus impossible, then Q is an unknowable truth.

So we've managed to prove a priori that either there are no never-known truths P, or else there are unknowable truths Q. It's a neat argument. I just came across it here. "Fitch's knowability paradox", I think they call it. Maybe epistemology isn't so bad after all.

20 comments:

  1. There's something terribly wrong here... this is one of those arguments where there just has to be a fallacy somewhere, and the trick is finding it...

    What is the distinction between "unknowable" and "never known?" Can we suck all the content out of this argument by equating the two? After all, if P will never be known, and this non-known-ness is fully determined in all possible worlds (which is the only sense of "never" that is compatible with Q), then can't we equally well say that P is unknowable?

    In that case, it would seem that all that is proven is that "either there are no [unknowable] truths P, or else there are unknowable truths [P]."

    Anyway, that's my first crack at knocking that argument down, because it seriously offends my intuition. Refute away :-)

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  2. Hi Paul, I don't think you can trivialize it that easily. "P is knowable" means that there is some possible world where P is known. "P is never known" just means that it is never known in this world. So it's possible to have unknown truths that are knowable (i.e. known at worlds other than ours). Of course, they won't count as "unknown" in those worlds.

    I'm not sure why you think this weak sense of "never" is insufficient for the trick with Q. Could you elaborate?

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  3. All rightey, let me work a little harder to accomplish my trivialization. Lets see if this works...

    Let there be two worlds, A and B. On each world, P is true, but on world A, P will never be known, whereas on world B P will be known. P is thus, by my reasoning above, non-knowable on world A (since it is fully determined that P will never be known) and knowable on world B (since it is not determined that P will never be known).

    Therefore, on world A, Q is unknowable, by your reasoning. However, on world B, Q is knowable, because P could be known.

    Once again, that seems to eliminate any distinction between will-be-known and knowability, as well as between P and Q. On the world where P will not be known, both P and Q are unknowable. On the world where P will be known (twin earth, as it were), neither P nor Q are unknowable.

    Put a different way, if the difference between "will not be known" and "unknowable" is world-dependence, then P and Q are not distinct: If P is world-dependent, so is Q. If P is world-independent, again, so is Q.

    Does that clarify the reasons that I think the unknowability of Q is equivalent to the not-being-known of P?

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  4. Re-thinking my last comment: I guess Q isn't knowable on world B in the same sense as on world A, because it's false on world B but not on world A... but that error aside, Q is still knowable as false on world B, so I don't think that defeats my point.

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  5. wait a minute... does that mean Q is only knowable if false? That doesn't help out my claim at all... ok, I'm going to sleep on this one, unless you relieve my of my confusion in some quick and final fashion. :-)

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  6. "P is knowable" isn't relative to worlds (except insofar as P must be true at the world of assessment -- you can't know something false). So in your first example, P is knowable (even on A), because there is another possible world (B) where it is known.

    If Q is false, then Q is ipso facto not knowable. (Though this leaves open the question of whether the distinct - and true - proposition "not-Q" is knowable.)

    Hope that helps. My key point is that you can't collapse "unknown" and "unknowable" together like that. The latter is a modal concept (concerned with what might have been), whereas the former depends solely on what is the case.

    I think the confusion here is due to your bringing up the notion of what's "fully determined". That just isn't a helpful notion here. If you ban those words and instead see the issue in terms of what's true and what's possible, it should become much clearer.

    (Alternatively, let's say that "never known" does not mean it's determined that P is never known. It merely means that it will, as a matter of fact, turn out to be true that P is never known. Just because something is true does not mean that it is necessarily true. But this perspective brings up fatalist confusions and other complications which we'd do better to avoid entirely.)

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  7. It's worth noting that on your weak definition of "unknowable" (i.e. that it is determined that P will never be known), the claim that there are unknowable truths would not be very interesting. For example, if determinism is true, then it would merely amount to the weak claim that there are some things we will never know. That's hardly surprising. Even if determinism is false, there are truths about the past that have been totally obliterated (i.e. no evidence remains), and so there is no possible future in which we could learn about them. For example, no-one will ever know how many times I blinked on my first birthday. This is determined (we may suppose). But it is surely knowable in principle -- if someone had followed me around all day counting, they could have known this.

    So, for my purposes here, to say that "P is unknowable" is to say that there is no possible world (even worlds that differ radically from our own) in which P is ever known. That is to say that P is not knowable, even in principle. This is a much stronger claim, and so it is much more interesting and surprising that we can show that there are unknowable truths. (Well, strictly speaking, either there are unknowable truths OR every truth will eventually be known. But the latter disjunct is extremely implausible!)

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  8. Okay, I think I can clarify this now. I'll remove any ambiguities by translating the claims into formal logic.

    Let "Kp" = p will be known (at the world under consideration).

    Let "Pq" = "q is in principle knowable" = there is some possible world where Kq is true.

    Clearly, ~Pq is a much stronger claim than ~Kq. That is, to be in principle unknowable is significantly stronger than merely never being known. Not only will q never actually be known, it will never be known in any possible world. Put another way: ~Pq = "it is impossible that q could have been known."

    (On these definitions, even false things might be "in principle knowable", so long as they are true somewhere else, and known on some such world where they are true. This doesn't really matter though. You still can't actually know something false.)

    Now, what we have proved is:

    Either (p --> Kp) for all propositions p, or else (q and ~Pq) for some proposition q.

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  9. I would have thought physics had already proven a vast number of truths are unknowable.
    In fact I am tempted to say they all are.

    but anyway to try and run a practical example

    Let's take an example - "the location of all particles in the universe" (I am trying to find somthign that is genuinely unknowable - fairly hard to do).

    it is true and will never be known I know there is a true value and yet I dont know what it is and I am sure no one can know it.

    P is true - but will never be known, Q is either true or true but never known (because maybe you won't ever be sure). is that right?

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  10. No, physics has not proved anything like that. The "Uncertainity Principle", for example, does not really say that we can't know the position and the momentum of a particle. It is best interpreted as saying that the classical concepts of position and momentum are not simultaneously applicable in the quantum level. when a particle has a definite momentum, it is not that we can't know it's position -it does not have at all a definite position. (At least according to mainstream interpretations of quantum mechanics).
    A better realistic example would be perhaps something from mathematics, related to Godel's theorem. But I actually all this is off-topic because the Fitch argument operates in an abstract level and does not need any concrete examples to stand or fail.

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  11. I withdraw my objection! The argument makes a lot more sense rephrased as you did.

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  12. There still seems to be some funny business going on with the indexing to possible worlds, as in the idle paradox and the ontological proof. The argument also depends on a certain stance on "future truths", as you've already noted, a topic on which your view seems to keep changing (not that that's a bad thing), but I'll set that aside.

    Here's my first attempt to understand the argument. Suppose that this world is World A, and P1 is true in World A but will never be known in World A. Then we could try to get the paradox with Q1: "P1 is true in World A but P1 will never be known in World A." But Q1 is not in principle unknowable, since Q1 could be known in World B (assuming that it is possible to have knowledge about other possible worlds). It looks like, when considering what could be known in World B, we want Q to be about World B rather than World A.

    My second attempt: We instead want Q2: "P1 is true in this world but P1 will never be known in this world." The problem here is that we don't know whether Q2 is true or false in any world except for World A (where Q2 is true and will never be known). P1 might be false in World B, or P1 might be true in World B and known at some point within World B. We're going to have to add assumptions about P in other worlds order to let Q range across every possible world.

    Third attempt: For Q2 to be true but unknowable in every possible world, we need a proposition P2 that is true in every possible world and that will never be known in any possible world. Then Q2 would be unknowable in principle. However, P2 is already unknowable in principle, so our proof of the existence of unknowable statements is question-begging.

    Am I right about these three versions of the argument? If so, is there some other P and Q that will make it work?

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  13. Blar, I was thinking of Q2. It is not something that could possibly be known in any world, for the reasons explained in the main post.

    Proof by contradiction: Suppose Q2 was known by S in some world W. Then S knows the first conjunct P1, in which case it is false that P1 will never be known in W, in which case Q2 is false in W and thus cannot be known by S after all.

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  14. But then isn't P unknowable in principle as well? In every possible world, P must be true but never known. Otherwise, there would be some possible world where Q2 is false, which is not what you want. You want Q2 to be true but unknown in every possible world. But the proof isn't accomplishing anything if we need to assume that P is unknowable in principle in order to show that Q is unknowable in principle.

    That's what I was trying to say with my second and third attempts. I hope this clarifies the problem that I see.

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  15. Here's the problem, spelled out more explicitly.

    Q: "P is true in this world and P will never be known in this world."

    Claim: In every world, Q is true and Q will never be known.

    Argument: Pick an arbitrary world W. In that world, Q is true, since, by assumption, P is true in W and P will never be known in W. It follows that Q will never be known in W, since, if Q was known in W, then it would be known in W that P is true in W, which is false by assumption. Since Q is true and will never be known in any arbitrary world W, it follows that Q is true but unknowable in principle.

    Problem: We had to assume that, for an arbitrary world W, P is true in W and P will never be known in W. But since P is true and will never be known in any arbitrary world W, it follows that P is true but unknowable in principle. Hence the argument is question-begging.

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  16. Q2 can be false in some possible worlds. That's no problem at all. All I require is that (i) it is actually true; and (ii) there is no possible world where it is known.

    That suffices for Q2 to be an unknowable truth.

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  17. Now I see. In that case Q is an unknowable truth.

    If an unknowable truth only needs to be true in the actual world, though, the move from never known in our world to never known in any possible world seems trivial. Call our world, the actual world, World A, and let P be some truth that is never known in our world. Then isn't the proposition R: "P and this world is World A" (or "P and this world is the actual world") another truth that is unknowable in principle, in virtue of being false in all worlds but this one? Maybe you can't put R into formal logic? I don't know formal logic well enough to say.

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  18. Yeah, good point. Though there seems something a bit fishy regarding claims about what world 'this' is. Perhaps it's simply that, as you say, they're so trivial. I guess we can at least say that Q is a more interesting (and less suspicious) unknowable truth than R is. But yes, the existence of unknowable truths is starting to look less significant than I would've thought.

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  19. > The "Uncertainity Principle", for example, does not really say that we can't know the position and the momentum of a particle.

    Ask yourself the question then - what do you know? Also I think my example works and am very confident I can come up with one if we explored it further.

    But mentioning the uncertainty principle raises another interesting point - if we concieved of P as a wave function - the uncertainty principle might claim that probabilities would never colapse without "observaiton" but this implies concievable knowability - thus you might argue that you cant have anything unknowable and true.

    > and does not need any concrete examples to stand or fail.

    I suggest it does need an example to exist even if you dont know exactly what it is. Naming that example would hardly be a bad thing for it.

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  20. Why are you all trying to reinvent the wheel? Fitch presented this argument in his 1963 paper "A Logical Analysis of Some Value Concepts" and then it was rediscovered in the 70s. Since then a dozen solutions have been proposed. I suggest having a look at http://plato.stanford.edu/entries/fitch-paradox/

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