I just came up with a funny argument for thirdism in the sleeping beauty problem.
Let's say I'm sleeping beauty, right? The experimenter flips a coin, wakes me up once in case of heads or twice in case of tails, then tells me the truth about the coin and I go home.
What do I do when I get home? In case of tails, nothing. But in case of heads, I put on some scented candles, record a short message to myself on the answering machine, inject myself with an amnesia drug from my own supply, and go to sleep.
...The next morning, I wake up not knowing whether I'm still in the experiment or not. Then I play back the message on the answering machine and learn that the experiment is over, the coin came up heads, and I'm safely home. I've forgotten some information and then remembered it; a trivial operation.
But that massively simplifies the problem! Now I always wake up with amnesia twice, so the anthropic difference between heads and tails is gone. In case of heads, I find a message on my answering machine with probability 1/2, and in case of tails I don't. So failing to find the message becomes ordinary Bayesian evidence in favor of tails. Therefore while I'm in the original experiment, I should update on failing to find the message and conclude that tails are 2/3 likely, so thirdism is right. Woohoo!
This argument is the same as Cian Dorr's version with a weaker amnesia drug. In that experiment a weaker amnesia drug is used on beauty if Heads which only delays the recollection of memory for a few minutes, just like in your case the memory is delayed until the message is checked.
This argument was published in 2002. It is available before majority of the literature on the topic is published. Suffice to say it is not convincing to halfers. Even supporter like Terry Horgan admit the argument is suggestive and could run a serious risk of slippery slope.
If it's worth saying, but not worth its own post, then it goes here.
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