In the literature on Sleeping Beauty, there are two separate but closely related arguments for thirdism. One argument depends on assumptions about what Beauty should assign after finding certain things out. Beauty should think that each of HEADS&MONDAY, TAILS&MONDAY, and TAILS&TUESDAY are equally likely, because if she were to find out MONDAY, she should think the first two are equally likely, and if she were to find out TAILS, she should think that the latter two are equally likely. This is the argument defended by Elga in the very first paper focused on the topic. Elga suggested that one interesting upshot of the argument was that it made sense for Beauty to change her probability assignment between Sunday and Monday despite learning nothing new in the interval. David Lewis objected to this, and instead favored halfism.
The second argument depends instead on assumptions about what Beauty should have assigned *before* finding herself awake. It provided an explanation of what Beauty learned in the interval between Sunday and Monday and why it was relevant. According to this argument, what Beauty learned was that she was awake *now*. This is relevant because it was in some sense epistemically possible that she would not have awoken *now*: if *now* is Tuesday, and the coin landed heads, then she would be asleep.
The second argument could be thought to fix up the problem with the first argument. Together, they provide two arguments of what Beauty should believe and one explanation of why.
However, there is a bit of tension between the two arguments. Though they happen to produce the same verdict in the original scenario, there are other variants of the original scenario where they don't obviously give the same verdict.
The second argument depends instead on assumptions about what Beauty should have assigned *before* finding herself awake. It provided an explanation of what Beauty learned in the interval between Sunday and Monday and why it was relevant. According to this argument, what Beauty learned was that she was awake *now*. This is relevant because it was in some sense epistemically possible that she would not have awoken *now*: if *now* is Tuesday, and the coin landed heads, then she would be asleep.
The second argument could be thought to fix up the problem with the first argument. Together, they provide two arguments of what Beauty should believe and one explanation of why.
However, there is a bit of tension between the two arguments. Though they happen to produce the same verdict in the original scenario, there are other variants of the original scenario where they don't obviously give the same verdict.
Here is such a variant ("Sleeping Beauty in Slow Motion").
This variant draws on the idea that the fact that Beauty sleeps is irrelevant to her epistemic situation. What matters is that she loses her memory.
One variant has Beauty stay awake the whole time, but has the experiment end on Monday night if HEADS, and has Beauty's memory instead erased (and the experiment ended on Tuesday night) if TAILS.
In *Sleeping Beauty in Slow Motion* if the coin lands heads, instead of the experiment being ended early, Beauty will be given a drug that causes her thought process to be slowed down. The 48 hours that she spends awake will be internally indistinguishable to her from a single day. Every thought she has will take twice as long.
This variant draws on the idea that the fact that Beauty sleeps is irrelevant to her epistemic situation. What matters is that she loses her memory.
One variant has Beauty stay awake the whole time, but has the experiment end on Monday night if HEADS, and has Beauty's memory instead erased (and the experiment ended on Tuesday night) if TAILS.
In *Sleeping Beauty in Slow Motion* if the coin lands heads, instead of the experiment being ended early, Beauty will be given a drug that causes her thought process to be slowed down. The 48 hours that she spends awake will be internally indistinguishable to her from a single day. Every thought she has will take twice as long.
Suppose that Beauty awakes in this experiment. What should her assignment of probabilities be?
I posed this question some time ago to Adam Elga and Terry Horgan, respective proponents of the two arguments mentioned above. Elga's off-the-cuff response was that Beauty should continue to assign thirder probabilities. This makes sense given his reasoning. If Beauty were to find out that she had not experienced more than a single day, she should assign equal probabilities between HEADS and TAILS. And if she found out TAILS, she should assign equal probabilities to MONDAY and TUESDAY.
Horgan's off-the-cuff response was that this scenario was relevantly different, and so halfism might be right for it. Beauty receives the information that she is currently awake, but this doesn't allow her to eliminate any previously open possibilities. There is no way that she could have experienced (or not experienced) anything different that her present experiences allow her to rule out.
This tension is interesting. I admit that I find it surprising that Beauty's probabilities should differ simply if her thought processes are slowed down. Suppose that Beauty, instead of being drugged, were simply given a spin in a rocket ship (so that time dilation produced the effect of slowing down her thoughts)? It may be that this does make a difference, but I am incredulous. This opens the question, is there something wrong with Elga's argument, or is there some better account of what Beauty learns in the variant scenario that justifies her change in probabilities?
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