The Sleeping Beauty Problem is a famous philosophical problem in the philosophy of probability which has seeped into debates in the rationalist community. Apart from it’s independent philosophical interest it raises issues of practical significance via it’s close connection to the Doomsday argument. However, I will argue below that sleeping beauty is only an apparent problem that arises only because framing the talk in terms of probability hides an unjustified assumption that our notion of (epistemically justified) credence refers and we have a good grip on what it refers to. An assumption which this very paradox shows to be false. As a result all attempts to directly argue for any particular position on the sleeping beauty problem is deeply confused.

In particular, if we take our word credence to (like water) refer to something like the most scientifically/philosophically useful term in the area then it should be obvious that the only answer we can give is “we don’t know.” If not then our natural language term simply doesn’t uniquely refer so it doesn’t make sense to expect a right answer to the sleeping beauty problem (though each particular preciseification would yield one).

In a latter post I’ll argue that debates about interpretations of probability are making a similar error in assuming there are the sort of facts behind the meaning of ascriptions of probability that are at issue between the interpretations.

#### The Sleeping Beauty Problem

The Sleeping Beauty Problem is defined thusly on wikipedia

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

- If the coin comes up heads, Beauty will be awakened and interviewed on Monday only.
- If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.

In either case, she will be awakened on Wednesday without interview and the experiment ends. Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before.

During the interview Beauty is asked: “What is your probability now for the proposition that the coin landed heads?”.

The apparent paradox arises because there are compelling seeming arguments (described in more detail on wikipedia) that sleeping beauty should answer 1/2 and a 1/3. Roughly the argument for 1/2 is that on waking up sleeping beauty doesn’t receive any information that might justify updating her probability. An argument for 1/3 is that when woken up she should assign equal probability to it being Monday or Tuesday conditional on the coin landing tails while conditional on the day being Monday she should assign equal probability to the coin being heads or tails. As there are twice as many equally probable outcomes involving tails the probability the coin landed heads is 1/3 (this is also the probability that sleeping beauty should use to bet with if offered a bet each time she is woken).

While phrased in this manner the sleeping beauty problem seems like a pretty irrelevant paradox the same reasoning applies in much more interesting settings. For instance, suppose you assign some non-zero prior to the possibility that instead of one single universe there are infinitely many universes containing an exact duplicate of you^{1}. If you accept the arguments for the 1/3 position in the sleeping beauty paradox the mere fact that you are having an experience should cause you to update your probability for there being infinitely many universes to 1.

Similarly, what arguments you find compelling in the sleeping beauty case affect how you should evaluate the Doomsday argument. For instance, taking the 1/3 position in sleeping beauty might lead one to argue against the doomsday argument on the grounds that there are more total individuals having experiences if there is no imminent doomsday and this should weight our probabilities.

#### What Does A Credence Mean?

While the sleeping beauty paradox may pose a challenge to our *philosophical* idea of (subjective) probability it doesn’t raise any problems for the mathematics of probability. But if the problem isn’t mathematical what is it about? At first blush it appears to be a question about the notion of (epistemically appropriate) credence. In other words sleeping beauty is a question about epistemology dressed up as a problem about probability.

But once we realize this we should be immediately be drive to ask: **what does it mean to have an (epistemically appropriate) credence of 1/3 (or half)?**

The sleeping beauty paradox itself demonstrates immediately that there are multiple plausible ways we could understand such a statement. For instance, one way of understanding the notion of epistemically appropriate credence might build from the intuition that the percent of times you believe something to be true with credence p and it is true should be p. If you allow the fanciful device of imagining restarting the universe one might think that a credence p in a claim C is appropriate relative to a given experience E if we ran reality a bunch of times and the ratio of the number of times C is true when E is experienced approaches p. In other words a credence is like a bet with reality you take each time you form it (though one would need to flesh out the notion of forming a credence if one wanted to pursue this). This concept supports the 1/3 answer to the sleeping beauty problem since, if the experiment is repeated many times, 1/3 of the times sleeping beauty has the experience of waking up the coin will be heads.

On the other hand, another way of understanding the notion of epistemically appropriate credence might build from the idea that you only care about whether or not a claim is true not how often it is true relative to the number of times you form a credence about it. In other words, credence p in a claim C is appropriate relative to a given experience E is appropriate if the number of times the universe is restarted where both claim C is true and you (or someone?) has experience E divided by the number of times the universe is restarted and you have experience E approaches p. This concept supports the 1/2 answer to the sleeping beauty problem since if the experiment is repeated many times 1/2 of those times will result in the coin landing heads.

Now, of course, all the problems with defining interpretations of probability and limiting frequency approaches mean I didn’t fully define a precise concept in either case. However, I don’t need to fully define any concept merely demonstrate that there is more than one way one could want to define the notion of epistemically appropriate credence.

Which notion one is interested in will depend on the particular situation at hand. For instance, if sleeping beauty’s concern is about making a bet with one of the researchers (who promises to accept bets either day) she should use the concept which considers the number of times she’s had the experience. If one of the researchers is a serial killer who tells her right before she goes under that he’s going to kill her spouse if the coin lands heads then a notion of credence which doesn’t concern herself with how many times she has the experience. Note that this nicely resolves the more applied versions of sleeping beauty once we make clear just what we are interested in.

Given that there are more than one notion of epistemically appropriate credence one could care about the most informative to the sleeping beauty problem should simply be that it is under-specified. Indeed, the fact that for any practical purpose we know which value to use should have been a red flag from the beginning that this was merely a verbal dispute not a genuine puzzle about epistemology.

#### Philosophy of Language Discussion

While I expect that I could stop at this point and satisfy non-philosophers there is a tendency among philosophers to insist that even if we don’t know exactly what properties epistemically appropriate credence (or ‘probability’) has one might nevertheless be justified in believing it uniquely asserts. While I’m skeptical of such claims in this case it certainly can happen. For instance, ‘water’ referred to H2O when Avogadro determined water’s chemical formula rather than changing it’s meaning^{2}. However, that was only because (and to the extent) that past dispositions about the use of the word water would have lead (at least hypothetically philosophically informed^{3}) people to hesitate to call something water despite it’s appearances if given sufficient reason to suspect it might differ in underlying nature, e.g., if flown in a spaceship to visit a stream on another world people would have expressed uncertainty as to whether the refreshing clear liquid was water.

But if our term ‘probability’ (or ‘epistemically appropriate credences’) is to, like ‘water’, refer to whatever natural kind fits sufficiently well with our usage then the only sane position for philosophers to take on questions like sleeping beauty is to admit that they don’t know, and indeed can’t know, until we figure out what natural kinds are in the neighborhood of our use of the term. After all, there is no doubt that problems like the sleeping beauty paradox differ from our usual applications of epistemically appropriate credences in ways that might or might not affect how some, yet undiscovered, natural kind in the neighborhood might apply. This wouldn’t reveal any kind of deep puzzle about the nature of the world, merely uncertainty as to the true reference of ‘epistemically appropriate credences’ as a result of our lack of knowledge about natural kinds in the neighborhood.

So sure, we can take ‘epistemically appropriate credences’ to refer to whatever concept turns out to be most elegantly useful in our theorizing about uncertainty in the world. However, if we do then the answer to all these paradoxes about probability becomes a simple “I don’t know” for the boring reason that we don’t know if there is an elegantly useful concept in the neighborhood yet. Thus, bald insistence that our notion of epistemically appropriate credences or probability has an implicit forward reference to the best concept in the neighborhood can’t save the arguments between the 1/2 and 1/3 camps from being appropriately regarded as confused.