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Fitch’s paradox and self locating belief

February 21, 2009

It’s been a while since I last posted here – which is bad seeing as I’ve had much less going on recently. I hope to return to regular blogging soon!

For now just a little note on something I’ve been thinking about to do with a version of the knowabality principle for rational belief. Back in this post I considered a version of Fitch’s paradox for rational belief, which shows the following believability principle cannot hold in full generality (C stands for rational certainty)

  • (p \rightarrow \Diamond Cp)

Here’s another route to that conclusion if you accept something like Adam Elga’s indifference principle. Suppose p is the proposition that you are in a Dr. Evil like scenario: that (a) you are Dr. Evil and (b) you have just received a message from entirely reliable people on Earth saying they have created an exact duplicate of Dr. Evil, whose situation is epistemically indistinguishable from Dr. Evils (including having him receive a duplicate message like this one) who will be tortured unless Dr. Evil deactivates his super laser. Notice that p includes self locating information.

If you accept Elga’s version of the indifference principle, once you’ve become certain of (b) you’re rationally required to lower your credence that you’re Dr. Evil to 1/2 and give credence 1/2 to the hypothesis that you’re the clone. So suppose for reductio that you could be certain that p. Since p is the conjunction of (a) and (b) you must be certain in both (a) and (b). But this is impossible, since indifference requires anyone who is certain in (b) to give credence 1/2 (or less) to (a).

It is impossible to be certain in p (p is probably unknowable too.) And since p is clearly possibly true, the principle given above is at best contingently true.

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10 comments

  1. […] the technical side of things, let me draw your attention to Andrew Bacon’s discussion, (15) Fitch’s paradox and self locating belief, posted at Possibly Philosophy, and to Tony Lloyd’s (16) We believe in probabilities, we do […]


  2. On topic, though not exactly responsive to your post:

    It’s always seemed to me that there is a trivial counterexample to the knowability premise in Fitch’s paradox. The knowability premise is (I don’t know how to type an upside down A so I’m using ‘A’ for “for all”):

    KP: Ap(p -> Kp)

    p here ranges over propositions, and K means “somewhere someone knows that”.

    Well, here’s a p for which that’s not true:

    It’s raining and nobody knows it.

    If KP were true, then if it were raining yet no one knew it, then it would be possible for someone to know that it was raining yet no one knew it.

    But suppose that someone knows the following: that it’s raining yet no one knows it. It follows that then someone knows the following: that it’s raining and he himself doesn’t know it.

    If he knows this (that it’s raining and he himself doesn’t know it), then he himself doesn’t know it, in other words, he himself doesn’t know that it’s raining.

    But by hypothesis, he [i]does[/i] know it’s raining.

    So if KP is true, then the supposition given at the top of this argument leads to contradiction.

    But surely it’s possible (at least logically!) for it to be raining, yet for no one to know it.

    So we can’t say the supposition must be false.

    The only alternative is to say that KP is false.

    -Kris


  3. Apologies, the present comment is purely for the purpose of checking the little box that asks the blog to notify me of follow-up comments.


  4. Wait: that just *is* Fitch’s paradox right?


  5. I don’t [i]think[/i] so, but maybe so. I’ve never done work on it and haven’t read much about it–it’s only come up in casual discussions with other Philosophers.

    Do you mean my argument is the paradox’s argument, or do you mean my conclusion is the paradox’s conclusion?

    My understanding has been that the paradox turns on an argument that if all truths are knowable, then all truths are known. By my understanding of the paradox, the problem has been to figure out how to avoid that implication.

    If that’s supposed to be the task set before us by the paradox, then I think the paradox fails to set us that task, because I think the knowability premise is false.

    But if the paradox is supposed to show that the knowability premise is false, then I guess I agree. But you don’t need the paradox’s argument to show this–simple counterexamples to the knowability premise exist. These counterexamples (mine being one of them) don’t strike me as particularly “paradoxical,” so it seems like the argument from these counterexamples is better than the argument from Fitch’s paradox. (I think a clear argument that feels non-paradoxical is better than one that feels paradoxical, if only for practical or rhetorical reasons.)

    After writing the above I’ve taken a quick look at the Stanford Encyclopedia article on Fitch’s paradox. It appears that it was used at first exactly to show that the knowability premise is false. This was considered to be a point against verificationism. Like I said, if that’s the argument, then a simpler counterexample like mine can be provided. (Though my counterexample relies explicitly on a kind of self-knowledge (does Fitch’s argument?) and maybe one could reconstruct Fitch’s argument on a revised version of the knowability paradox that explicitly puts such self-knowledge outside its scope?)

    Later, the focus (apparently) came to be on trying to stave off the implication. It came to seem strange, and suspicious, that such a straightforward logical proof could be given that implied omniscience given universal knowability. My answer to that is, it’s not strange at all, because if you look carefully at the notion of universal knowability, you’ll see that it itself fails to obtain. Since the knowability paradox is false, it’s not that strange that from assuming the knowability paradox, you can prove some “surprising” results.

    -Kris


  6. Wish I could edit posts… Both instances of “knowability paradox” in that last sentence should read “knowability premise.”

    -Kris


  7. Sorry: the argument in the SEP (that if all truths are knowable, then all truths are known) is essentially the argument you outlined, unless I’m missing something.


  8. I take it all back. Looking at the Stanford article some more, I see that my argument basically is Fitch’s argument. I just explicitly provide an example of an unknowable truth, where the argument in the Stanford article refers to it generically as an instance of the the existential “there is an unknown truth.”

    Though my argument is substantially the same as the Pardox’s argument, I still like mine better. 🙂 By providing an explicit example of the kind of unknowable truth Fitch’s paradox points to, it seems to me my version of the argument makes it seem much less strange (and IMO much more trivial) that there could be unknowable truths.

    -Kris


  9. To give an good example of something that has a certain property, presumably you need to know that it has the property in question.

    As you can see, it is quite difficult to know that the example (that it is raining) is an unknown truth – because of the Fitch style argument you gave.

    It is not clear at all whether the example you gave is an unknown truth. (Make it more explicit: where is it raining?) The best we can hope to show is that *if* there is an unknown truth, there is an unknowable truth. Which is why the original Fitch argument (without an explicit example) works.


  10. […] The Dr. Evil Paradox (and variants)  […]



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