[[This post was written in 2014. At some point in early 2017, I believe, not one but two drafts (which, oddly, are very different…as I recall there was a time in about the middle of my writings that I decided to try to be less technical, so the earlier draft is probably superior on some ways and apparently has some fans) of my SB survey were added to the SB Bibliography. At that point I took this post down, but since I decided to delete all but the dozen most popular posts, this goes back in.]]
Just ran across PhilPapers’ Sleeping Beauty Bibliography, which is maintained by Joel Pust:
The Sleeping Beauty problem is plainly interdisciplinary, but it’s apparently difficult for a mathematician to make it onto the bibliography; at least two rather good ones have written papers on Sleeping Beauty, but neither appears on the bibliography. Jeffrey Rosenthal’s was even published, albeit in the Mathematical Intelligencer. I can’t speak for Rosenthal, but I put over one thousand hours of labor into my paper and tried to get to the bottom of every issue from the perspective of, not mathematics, but philosophy. For the record, I think the undertaking was a spectacular success, though you shouldn’t take my word for it:
Probably I don’t understand the process by which papers come to be on these lists. I do know that many unpublished papers are there, some awful on the most generous reading. I myself have a paper, which is something of a throwaway (having just glanced at it for the first time in a couple years, it’s actually pretty awesome–though still a throwaway) I spent perhaps ten hours on and never thought of trying to publish, on a different PhilPapers list:
So…is it an oversight, is it marginalization, is it just some process I never went through of posting my paper to PhilPapers (I never posted the one that does appear) or are our papers just not deemed relevant to the discussion? Do the “real philosophers” on this list know something that Rosenthal and I don’t? (In saying “Rosenthal and I” I am not implying that there is any commonality between my paper and his, or that Rosenthal would have any appreciation for my paper. Although I like his paper a lot, it’s not really a philosophy paper, and it’s quite possible he’d have little patience for mine, which takes philosophical issues more seriously than many mathematicians would like.) What exactly do the people on the list know about Sleeping Beauty? What wisdom are they able to impart? Pust has put himself on his own list 5 times. (Apparently he knows something.) This however, is not the record.
A certain Darren Bradley has no fewer than seven papers on the list. When I saw this, I figured that he must be like the Sleeping Beauty sage. Well, okay maybe not but, having not heard of him before, I immediately conspired with myself to read one of his papers. I chose Confirmation in a Branching World: The Everett Interpretation and Sleeping Beauty:
It looked to be the most heavily downloaded of his published papers. Reading it is a surreal experience not unlike reading Hegel, as is evident from the first sentence of the following outtake from p. 6 line 24:
“If I am about to look at a die to see what number it landed on, then I will see the number it landed on. So we can model learning uncentred propositions using a procedure that is biased towards whatever property is observed (e.g. if a six is instantiated then a six is observed). However, if there have been numerous dice rolls, and I won’t observe all of them, then there may be a six in the population that I don’t observe. So the procedure cannot be modelled as biased towards whatever is observed. Similarly, when evidence is centred, we cannot automatically model the procedure as biased towards whatever is observed some of which I will observe and some of which I won’t, and I am more likely to observe some numbers than others.”
It’s actually fun to read this out loud five or six times. Though I fear the humor may be lost on most. (Fans of Hegel, in particular, but probably just about any professionalized philosopher.) But let’s think about it carefully.
Bradley: Hey…the procedure you are using to model those “dice rolls” is biased.
Statistician: Biased? Biased toward what?
Bradley: Biased toward whatever number is instantiated.
Statistician: Uh…the die is fair. Each number comes up with probability one-sixth.
Bradley: No. The probability of observing the number instantiated is one. The probability of observing the other numbers is zero. So it’s a biased procedure.
Statistician: (head explodes)
Well, that’s definitely a lot of fun and surely deserved, but reading further it is possible to piece together what Bradley means by the above. Or, well, not really, but it’s possible to at least reconstruct an argument. Recall that evidence E confirms A (i.e. makes A more likely) if and only if P(A|E) > P(A) if and only if P(E and A) > P(A) P(E) if and only if P(E|A) > P(E). But since P(E) is a convex combination of P(E|A) and P(E|not A), the latter condition gives us this principle:
E confirms A if and only if P(E|A) > P(E|not A).
We now follow Titelbaum’s Technicolor SB argument (which is just Rosenthal’s argument…I assume they were developed independently). The idea is that the room Beauty wakes up in is either Blue or Red. Once each if tails, flip a further coin to determine which if heads. Beauty wakes up in a red room. Thirders, say Bradley, take as Beauty’s evidence
E1 = There is at least one red awakening,
which confirms tails, as
P(E1|tails) = 1 > 1/2 = P(E1|heads) = P(E1|not tails)
Bradley, however, says that Beauty’s complete evidence is actually
E2 = There is a red awakening today.
This does not confirms tails, says Bradley, as
(*) P(E2|tails) = 1/2 = P(E2|heads) = P(E2|not tails).
So what could be wrong with that argument? Well, the problem is that (*) depends on premises accepted only by halfers. Thirders would dispute (*). The situation arises in the original analysis of the problem. Take
E3 = There is an awakening today
Halfers parse as something like a tautology. It’s something that is true during every awakening. False never. So non-informative. Thirders parse today rigidly. If today is Monday E3 would parse as “There is an awakening Monday”, which is a tautology, but if today is Tuesday E3 would parse as “There is an awakening Tuesday”, which implies tails. If it’s either a tautology or confirms tails depending on some contingency, then of course it confirms tails simpliciter. So say the thirders, at any rate.
In other words, Bradley’s analysis, which focuses on red at the expense of today, strips E2 of its informativeness. For thirders, E2 confirms tails not because of red, but because of today. If today is Tuesday, it parses as “There is a red awakening Tuesday”, which implies tails. Bradley’s analysis would only be appropriate if E2 were contemplated after having already conditioned on E3. For example, well after Beauty had awakened, but before she had opened her eyes. How she responded to E3, who knows. She might be a halfer, in which case she found E3 to be uninformative. Or she might be a thirder, in which case she decided that E3 confirmed tails. In such a case, Bradley’s analysis is correct. Redness of the room doesn’t (further) confirm tails. That’s all Bradley’s argument shows. It doesn’t show anything else.
So now we’re in a better position to understand what the Technicolor SB argument accomplishes. It isn’t (as Rosenthal for example thought) a knockdown for thirders. On the other hand, it’s an effective rebuke of halfer schemes predicated on the idea that only uncentered evidence can be relevant. (Most notably, double halfer schemes.) Other halfers (i.e. Lewisians) still have resources against it. Bradley’s paper, however, is blind to these resources. I’ll be sure to check right away if his other six Sleeping Beauty papers glean any of them.
But not really.