When I first thought of this case I thought it was interesting in a novel way and showed that Lewis was right all along about Sleeping Beauty. Then I thought it showed that Elga was right about Sleeping Beauty. And now I’m not sure it’s even a new case, in which case it probably shows nothing at all about Sleeping Beauty. But what the hell, this is a blog, may we well write it up and see if someone smarter than me can figure out the philosophy. (For a starter on Sleeping Beauty, try Adam Elga’s paper here.)

Assume the following facts, just for fun:

• There is a newspaper, call it Times, that is authoritative, it is known to only print truths.
• There is a coin flip every day as part of a giant city lottery. It takes place at 3 o’clock, and the results are announced in a billboard by the clock tower. It is known to be a fair coin, with a chance 0.5 of landing heads, and 0.5 of landing tails.
• It is practically conceivable that someone could create a brain in a vat that felt in almost every respect like a normal human. Like a particular normal human, that is.

Here’s the case then.

Farrington walked past the clock tower a little before 3. He knew it was a little before 3, because he saw the crowds milling around waiting for the results of the coin flip. A while later, about ninety minutes he thought, though he couldn’t be sure, he picked up Times and started reading it over a beer. If someone asked him, which no one did, how confident he was that it was before 4:30, he would have said that he was exactly 50% confident in that. He thought it just as likely that it was before or after 4:30, based on his evidence about what time it was when he passed the clock tower, and how long it had been since then.

He noticed that an odd experiment was planned. Some researchers were secretly and remotely monitoring the brain states someone from the town, and at 4:30 today, they were planning on creating an epistemic duplicate of that person, just as he was at 4:30. The new person would be a brain in a vat, but in almost every respect he would look from the inside like a real person.

There was some ethical concern about consciously creating a being like this, so the researchers had decided to introduce some randomness into the procedure. If the coin toss today landed heads, they would call it off for now, and maybe use someone else as the basis for duplication. But if it landed tails, they would go ahead. Relying on the luck of the draw would, they thought, give God enough role in creation to not create theological headaches. (The paper did not make clear just which theologian had thought this was even remotely plausible, but let that pass.)

The newspaper also noted that the subject of this experiment, one Farrington, a clerk at Crosbie & Allenye, had not been told about the experiment, but it was hoped that since he rarely read Times, he would not find out about it. At this Farrington shook. He was the only Farrington at Crosbie & Alleyne. He was the one who would possibly be duplicated. Indeed, if it were now after 4:30, and if the coin had landed tails, he had been duplicated. Indeed, in that case, for all he knew, he was not Farrington, but rather some duplicate. Disturbing thoughts for before a second drink.

Now the questions.

What probability should Farrington (or whoever our hero now is) now assign to each of the following five possibilities

P1: It is before 4:30 and the coin landed heads.

P2: It is before 4:30 and the coin landed tails.

P3: It is after 4:30 and the coin landed heads

P4: It is after 4:30 and the coin landed tails and he is the original Farrington.

P5: It is after 4:30 and the coin landed tails and he is the duplicate Farrington.

Elga’s indifference principle says that P4 and P5 should be treated alike. Beyond that, we are left to our own devices. I have various thoughts, but I might just leave the puzzle out for now, and post the thoughts later.

You are more than invited to leave suggested answers in the comment box.

UPDATE: Thanks to Cian Dorr for spotting a slip-up in the description of the cases. So far the votes in the comments thread are 1 for and 1 (in the next comments thread) for . These two are, probably not coincidentally, the two options I take to be most plausible, though I also think has some virtues. Of course, this all assumes that Elga’s indifference principle is right, and hence that options where P4 and P5 are not given the same value are to be ignored.

I would put the argument for slightly differently to Cian. Extend this case (rather than some other fictional case) by assuming that someone authoritative tells Farrington that he is the original. Imagine perhaps he reads in the story that there is some distinctive mental task that the duplicate cannot do, and Farrington convinces himself that he can do it. It is plausible (a) that he should then conditionalise on ~P5, and (b) the results of this should be that his probability distribution over the five events is , as it was before he read the story. Given that constraint, and that P4 and P5 get the same value, is the only possible combination. I’m not sure about (b), but it is fairly plausible. Having said all that, do read Cian’s paper on Sleeping Beauty to see the views of someone who’s thought about that problem a little more deeply than I.

FURTHER UPDATE: Wo agrees with . He also thinks that Cian’s argument for is better than mine, even though neither of them end up working.