Wilhelm and Lando on centred credence and chance
Wilhelm (2021) and Lando (2022) argue that the Sleeping Beauty problem reveals a flaw in standard accounts of credence and chance. The alleged flaw is that these accounts can't explain how attitudes towards centred propositions are constrained by information about chance.
I assume you remember the Sleeping Beauty problem. (If not, look it up: it's fun.) Wilhelm makes the following assumptions about Beauty's beliefs on Monday morning.
First, Beauty can't be sure that it is Monday:
(1) Cr(Mon) < 1.
Second, Beauty's credence in Heads conditional on Monday should be 1/2:
(2) Cr(H / Mon) = 1/2.
Together, these two assumptions are incompatible (by the probability calculus) with the "halfer" hypothesis (3):
(3) Cr(H) = 1/2.
Some halfers (Hawley (2013)) reject (1). Most reject (2). Wilhelm doesn't acknowledge that (1) and (2) are controversial. He says that violating (1) would be "obviously irrational", and that (2) "follows from some standard principles, endorsed throughout the literature" (both quotes on p.1901). This is clearly not correct because halfing is a consistent position, and most halfers would reject whatever "standard principles" Wilhelm has in mind.
But let's set this aside. Let's assume that (1) and (2) are correct. Wilhelm then sees a problem. Standard formulations of the Principal Principle, he says, entail (3). But (3) is false, since we're assuming (1) and (2)!
How does (3) follow from the Principal Principle? "The derivation of (3) from the Principal Principle is fully rigorous", Wilhelm says (on p.1900). Let's have a look. The derivation (in the appendix) rests on the stipulation that Beauty's Monday credence equals her initial credence conditionalized on the complete history of the world up to Monday morning. From this we get (3) by a formulation of the Principal Principle from Lewis (1980), according to which one's rational initial credence in a proposition A, conditional on information about the chance at some time t and the history of the world up to t, should equal that chance of A at t.
But why can we stipulate that Beauty's Monday credence equals her initial credence conditionalized on the complete history of the world up to Monday morning? Personally, I think this is definitely false, no matter how we construe the relevant history proposition. But I'm a halfer. Is the stipulation plausible from a thirder perspective?
Thirders tend to think that Beauty should proportion her credence to her evidence, so that Cr(H) equals Cr0(H / E), where Cr0 is a rational prior and E is Beauty's total evidence. It probably won't affect the answer to the Sleeping Beauty problem if we assume that Beauty is omniscient about the past. So we can stipulate that E entails all truths about the history of the world up to Monday morning. Wilhelm needs the stronger assumption that Beauty's evidence E is equivalent to the full truth about the history up to Monday. But Beauty also has self-locating evidence – that she is awake, that she has no memories from later than Sunday, and so on. We can't adequately model these as uncentred propositions about history. Beauty's observation that she is awake is not equivalent to an observation that she is awake on Monday morning. If her self-locating evidence is entailed by her history evidence, we must construe the relevant history evidence as centred. E must say something like 'I am now awake, and I was asleep an hour ago, and Beauty is awake at 8am on Monday, and she is asleep at 7am on Monday, and so on', but it must not connect the centred and the uncentred information it contains: it must not say 'it is now 8am on Monday', or anything like that.
The question now is if this "centred history proposition" falls in the scope of Lewis's Principal Principle. If I were a thirder, I would say no. Thirders usually say that Beauty's credence in H should be less than the known chance 1/2 because she has inadmissible information – namely, that she is awake. This inadmissible information is part of the centred history proposition. We certainly can't assume that the Principal from Lewis (1980) holds with centred history propositions.
So I'm not convinced that there's a puzzle here. There is no good reason to think that (3) follows from the Principal Principle.
But let's look at Wilhelm's response to his puzzle. If (3) is false, and follows from the Principal Principle, should we reject the Principle?
Not quite. The puzzle, Wilhelm suggests, arises because H is a centred proposition. Standard formulations of the Principal Principle presuppose that the objects of chance are uncentred.
Recall that H is "the proposition that the coin lands heads" (p.1900). According to Wilhelm, "H is, of course, a centred proposition". It must be distinguished from "the uncentred proposition – call it 'U' – that the coin lands heads" (p.1903). The chance of U is 1/2, but the chance of H is 1/3. The "rigorous derivation" of (3) falsely assumed that the chance of H is 1/2.
Wilhelm goes on to defend the idea that chances can pertain to centred propositions like H. But let's pause here.
Why is H ("of course") centred? Both H and U are defined as "the proposition that the coin lands heads". Why is one centred and the other uncentred? A centred proposition can change its truth-value within a single (uncentred) world. How can "the coin lands heads" be true at some points and false at another, assuming that "the coin" refers to the single coin in the Sleeping Beauty scenario? I don't understand.
But here's how we could make H centred, using an idea from Titelbaum (2012) (who, if I may say so, has it from Schwarz (2015)).
Let's add a second coin toss to the Sleeping Beauty scenario. The second toss takes place on Tuesday and has no relevant consequences. We only introduce it so that Beauty can be sure that some coin will be tossed today, even though she's not sure whether it is Monday or Tuesday. More concretely, we can now ask how confident she is that today's coin toss will land heads. This is our centred version of H.
It'll be useful to have different labels for the different 'Heads' propositions. Let HMon be the uncentred proposition that the Monday coin lands heads, HTue the uncentred proposition that the Tuesday coin lands heads, and HTod the centred proposition that today's coin lands heads.
It is easy to show that in light of assumption (1), Cr(HMon) cannot equal Cr(HTod). If we re-interpret Wilhelm's centred 'H' as 'HTod' and his uncentred 'U' as HMon, his suggestion would be that HMon has chance 1/2 and HTod has chance 1/3, and that Beauty should align her credence with these chances. This doesn't work, however. The probability calculus demands that Cr(HMon) ≤ Cr(HTod). (See Mike's or my paper.) In fact, standard thirding entails that Cr(HMon) = 1/3 and Cr(HTod) = 1/2. We would need the opposite proposal, that HMon has chance 1/3 and HTod has chance 1/2. It's highly implausible, though, that HMon has chance 1/3.
So there's no real room for an alternative response here. Thirders should stick to their standard move and say that (3) doesn't follow from the Principal Principle because Beauty has inadmissible evidence.
But let's think some more about HTod, the proposition that today's coin lands heads. Imagine Beauty is shown the coin on Monday afternoon. "This is the coin we're going to toss today", she is told. "It is a fair coin. What's your credence that it will lands heads?". As a thirder, Beauty would say '1/2'. Most forms of halfing, however, require Cr(HTod) = 1/3. As a halfer, Beauty would say '1/3'. Titelbaum (2012) thinks this is an embarrassment for halfers, because Beauty should obviously say '1/2'.
Lando (2022) agrees. She argues that this raises a serious problem: standard accounts of self-locating credence can't explain why 1/2 is the right answer, because centred propositions like HTod don't have a chance.
As a halfer, I don't think Beauty should say '1/2' when asked about her credence in HTod. To get Lando's problem off the ground, we have to assume that I'm wrong.
I admit that naive intuition favours this assumption. "Here's a fair coin. What's your credence that it will land heads?" One would expect the right answer to be 1/2. Let's assume it is. Surely this has something to do with the chance of heads being 1/2. But we can't directly apply the Principal Principle to HTod, because HTod is centred and presumable only uncentred propositions have a chance. So how can we explain why Cr(HTod) = 1/2?
At this point, Wilhelm could jump in and offer his response: allow for centred chances and the problem goes away! Lando briefly considers this response, but rejects it as "not very plausible" (p.116). How, she asks, could there be a probability for things like 'it is Monday' that aren't generated by a chance process?
Wilhelm's answer is a modified Best-System Account. Even centred propositions can have relative frequencies. Why not say that the chances are the best summaries of these frequencies?
In fact, one might argue that only centred propositions have non-trivial relative frequencies. Non-trivial relative frequencies pertain to things that can be instantiated more than once within the same world, not to propositions that can only be true once-and-for-all or false-once-and-for-all. I've explored this idea, and its consequences, in Schwarz (2014). My proposal is rather different from Wilhelm's, and it doesn't give us a chance for things like 'it is Monday'. Wilhelm suggests that the chance of 'it is Monday' for Sleeping Beauty is 2/3, but I don't understand why. I would have thought that the relative frequency of 'it is Monday' is 1/7, since a seventh of all days are Mondays.
Anyway, let's return to Lando's problem. Let's assume that we don't want to assign a chance to propositions like HTod. Does this mean that we can't explain why Beauty's credence in HTod should be 1/2, based on facts about chance?
Lando considers only one possible explanation. It starts with the assumption that the Principal Principle constrains Beauty's credence in HMon and HTue, so that \(Cr(H_{Mon}) = 1/2\) and \(Cr(H_{Tue})=1/2\). Since Beauty's credence is divided between Monday and Tuesday, the putative explanation infers that her credence in today's coin landing heads must also be 1/2.
Lando rightly points out that this explanation is problematic. Thirders would not accept that \(Cr(H_{Mon})=1/2\). It's also unclear how \(Cr(H_{Tod}) = 1/2\) is meant to follow from \(Cr(H_{Mon})=1/2\) and \(Cr(H_{Tue})=1/2\), together with the assumption that Beauty's credence is divided between Monday and Tuesday. In fact, we can say something stronger: the premises entail the negation of the conclusion! If \(Cr(H_{Mon})=1/2\) and \(Cr(Mon) < 1\) then probability theory requires that \(Cr(H_{Tod}) > 1/2\).
Without considering any other explanations, Lando assumes that no explanation can be given for Cr(HTod) = 1/2 – at least not if we assume that credence is defined over centred worlds. We can only explain why Beauty's credence in HTod should be 1/2 if we recognize that HTod is associated with uncentred "truth conditions" – namely, that the Monday coin lands heads. Lando thus reaches the sweeping conclusion that no theory that construes doxastic content in terms of centred worlds "can adequately represent all of the rational constraints on our credences" (p.119).
There is no need to draw this sweeping conclusion. We can easily explain why Beauty's credence in HTod should be 1/2. In Schwarz (2015) I show that this follows from standard thirder assumptions. Let me give a more direct explanation.
We'll figure out to what extent Beauty's evidence on Monday morning supports HTod. Beauty is aware of the general setup, and her last memories are from Sunday. This means that one of the following eight (centred) possibilities must obtain.
| HMon, HTue | HMon, TTue | TMon, HTue | TMon, TTue | |
|---|---|---|---|---|
| Mon | (a) | (b) | (c) | (d) |
| Tue | (e) | (f) | (g) | (h) |
A priori, these are all equally probable. Here we invoke the Principal Principle to determine that each column has probability 1/4, and a principle of self-locating indifference that divides the probability of each column between its two cells.
Now Beauty also has the information that she is awake. This rules out cells (e) and (f). The remaining possibilities therefore have probability 1/6 each. It follows that \(Cr(H_{Mon}) = Cr((a) \lor (b)) = 1/3\) and \(Cr(H_{Tod}) = Cr((a) \lor (b) \lor (e) \lor (g)) = 1/2\).
Wolfgang:
The statement of the SB problem is always wrong. Specifically, it schedules the wakings so that SB is always wakened in a "first" attempt, on Monday. Then conditionally wakened on a "second" attempt, on Tuesday. But this was not a part of the problem as posed by Adam Elga in 2000, it was how he implemented his (thirder) solution (which seems to echo yours). Yet the entire controversy over the problem is about how to handle the differences between Monday and Tuesday when SB does not know the day.
Here is the actual problem statement, with two irrelevant parts removed (they don't convey information, only point toward elements in Elga's discussion).
"Some researchers are going to put you to sleep. During the [experiment], they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are [awake], to what degree ought you believe that the outcome of the coin toss is Heads?"
This experiment can be implemented in several ways that make each waking have the same properties as far as the subject is concerned. One is to use two coins, C1 and C2. Wake SB if either is showing Tails, and ask her for her credence that coin C1 is showing Heads. Then put her to sleep with amnesia, turn coin C2 over, and repeat the same attempt.
This way, each attempt can be treated as a stand-alone experiment. The prior sample space is {HH, HT, TH, TT}. (Note: her credence about coin C2 is the same regardless of whether it has been turned over.) If SB is awake, she knows that HH is eliminated. Her credence is 1/3 for each of HT, TH, and TT.
Another implementation is to use four volunteers over the same two days, but with different schedules. The one representing Elga's version is awakened except on {Heads, Tue}, meaning she is awakened unless the coin landed Heads and it is Tuesday. The other three have schedules {Heads, Mon}, {Tails, Tue}, and {Tails, Mon}.
On each day of the experiment, three volunteers are wakened. Each is asked for her credence that the coin landed showing the face in her assigned schedule.
It does not matter if the volunteers know their schedule. Or if they are brought together to discuss their answers, as long as they don't share their schedules. As long as they know the experiment's details, they each know that there are three awake volunteers. And that the credence for any one of them being the one whose schedule matches the coin can't be different from the others. This credence is 1/3.
Jeff J