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:
- I should avoid false beliefs.
- 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?
Hi Alex:
ReplyDeleteHere'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
Tedla:
ReplyDeleteIt 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.
Alex,
ReplyDeleteYour 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
Thanks. I had fixed a bug in the perl code earlier, and that introduced a new bug. Both should be fixed now.
ReplyDelete