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?