Friday, February 6, 2009

Liar-like dizziness without truth

Can one run the liar paradox without the concept of truth?

Suppose I write on my board: "At no point today do I believe anything written on this board", and know that nothing else will have been on the board today. Now there is no problem of inconsistent truth assignment: there is no logical contradiction in the sentence on the board being true (in which case I don't believe it) and there is no logical contradiction in the sentence in the board being false (in which case I do believe it). But while there is no contradiction, there is dizziness as I try to figure out whether to believe what is on the board.

The dizziness results from two plausible principles:

  1. I should avoid false beliefs.
  2. If the evidence conclusively points to p, I should believe p.

Principle (1) prohibits me from believing what is on my board. For I know ahead of time that were I to believe what's on the board, I would be believing something false. The case somewhat resembles the surprise exam. For, it seems that knowing myself, I may know that I won't violate norm (1) in this regard. I also know that not violating norm (1) entails not believing what's on the board. But then I know p, and I know that p entails q, but I don't know q even though I am, plainly, thinking about q. That is, surely, at least a very unstable situation.

Perhaps, though, (1) is mistaken. Maybe it's rationally acceptable to believe something even when one knows that the belief would be false, when the belief is self-referential? Is the norm that I should refrain from believing something when I know that believing it would be the having of a false belief, or is the norm that I should refrain from believing something that I know to be false? I do not know that what is written on the board is false, because I do not know what I will in fact believe. Still, (1) does seem very plausible.

Or is this a case where, whatever I do, I do something irrational?

Or should I say what I say about the liar, and deny that what is written on the board has meaning, even though an exactly similar token would have meaning?

4 comments:

  1. Hi Alex:

    Here's another liar-like paradox, which you might take as another example that is not directly based on the concept of truth:

    (1) This sentence is meaningless.

    Now, is sentence (1) meaningless? Or, is it meaningful? Trying to answer these questions seems to generate a liar-like paradox. Just wanted to hear what you think about this too.

    P.S. It's good to briefly see you at Baylor.

    Tedla

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  2. Tedla:

    It was good to see you. I ended up not attending the conference sessions as I decided I was too sick to be close to other people.

    I think (1) doesn't have to generate a paradox. Just say that it's false. Then it's meaningful. And still false.

    But there is a way of combining the original liar with (1) to get the so-called Revenge of the Liar:

    (1*) This sentence is meaningless or false.

    If true, it's meaningless or false. If meaningless, it's true. If false, then it's true.

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  3. Alex,

    Your survey in your next (chronological) entry seems to be giving a HTTP 500 error. I can't tell you there, so I thought I'd tell you here.

    Sam

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  4. Thanks. I had fixed a bug in the perl code earlier, and that introduced a new bug. Both should be fixed now.

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