Wednesday, April 15, 2015

A paradox about prediction of belief

Sally is perfectly honest, knows for sure whether there has ever been life on Mars (she's just finished an enormous amount of NASA data analysis), and is a perfect predictor of my future beliefs. She then informs me that she knows what I will believe at midnight about whether there was once life on Mars, and she further informs me that:

  1. There was once life on Mars if and only at midnight tonight I will fail to believe that there was once life on Mars.
Moreover, I know that:
  1. I won't get any other evidence relevant to whether there was once life on Mars.
I'd love to know whether there was once life on Mars. I start off thinking:
Well, right now I have no belief either way, and I am unlikely to get any evidence before midnight. So by midnight I will also have no belief either way. And thus by Sally's information there was once life on Mars.
But of course as soon as I accept this argument, I start to believe that there was life on Mars. And I know that if I keep on believing this until midnight, then my belief is false. I quickly see the pattern, and I realize that I don't know what to think! But when I don't know what to think, I default to suspension of judgment. But this, too, leads me astray: For as soon as I think that the appropriate rational attitude for me is suspension of judgment, then I start thinking I will suspend judgment at midnight, and I then conclude that there was once life on Mars. And the circle starts again.

Now, I know I'm not perfectly rational. So I can get out of the circle by concluding that given how confusing this case is, I am probably not going to act rationally. So something non-rational will affect my beliefs by midnight, and I don't know what that will be, so I might as well not speculate until that happens. Sally knows what it will be, but I don't.

But suppose I am perfectly rational. I shall assume that a part of perfect rationality is knowing for sure you're perfectly rational, knowing for sure what you belief, and drawing all the right conclusions from one's evidence. What should I believe in the above case?


Anonymous said...

Perhaps one might object to (2) as self-inconsistent by arguing that the knowledge that you will not get any relevant evidence between the moment Sally informs you of (1) and midnight is, itself, a kind of evidence relevant to whether there was once life on Mars.

Alexander R Pruss said...

That's excluded by the word "other" in (2), I think.

Heath White said...

I think this is a version of the Liar?

Heath White said...

Here’s a better comment.

Suppose I can choose either to believe that there was once life on Mars, to believe that there was not, or to suspend judgment. The latter two both count as failing to believe. If I choose to believe or to disbelieve, then (since Sally is a perfect predictor) I will have a false belief. If I choose to suspend judgment, then I will fail to have a true belief. It is better to be ignorant than in error, so I should choose to suspend judgment. After midnight I can figure this out and get the true belief.

Elements of the Liar, and also Newcomb.

Alexander R Pruss said...

Yeah, that seems the reasonable thing to do. But when I come to think it's the reasonable thing to do, I now think: "Oh, so I'm going to suspend at midnight. But if I suspend at midnight, then there was once life on Mars." :-)

Alexander R Pruss said...

There is also something like the surprise exam paradox here.

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