First, all good examples of a posteriori necessities follow a priori from non-modal truths. For example, as Kripke pointed out, that his table could not have been made of ice follows a priori from the contingent, non-modal truth that the table is made of wood. Simply taking metaphysical modality as a primitive kind of modality would make a mystery of this fact.

Second, it is a well-known linguistic fact that there are two ways of evaluating a sentence at a possible context or scenario. In one sense, 'it is warmer than it is here' is false at every context; in another, the sentence is true at a context c iff it is warmer at c than at the actual utterance context @.

Developing the second observation leads to two-dimensional semantics, and further to the two-dimenational account of metaphysical modality. On that account, a sentence is a priori just in case it is true at all worlds when evaluated in the first way ("as actual"), and necessary just in case it is true at all worlds when evaluated in the second way ("as counterfactual"). The space of worlds is the same in either case.

The two kinds of evaluation come apart if and only if the sentence in question contains actually-dependent expressions such as indexicals ('here'), demonstratives ('this table'), or proper names. Evaluated as counterfactual, such expressions denote whatever plays a certain role in the actual world (the utterance context). This is why the empirical information one needs to know a posteriori necessities is ordinary non-modal information about the world.

The two-dimensionalist account assumes that all a posteriori necessities can be
explained away by the difference between evaluating as actual and as
counterfactual. Let's call those a posteriori necessities *tame*,
and any others *brute*. For example, if 'there is an omniscient
being' is an a posteriori necessity, it would be brute, since there is
plausibly no difference between evaluating the sentences at worlds
considered as actual and at worlds considered as counterfactual.

There are many reasons to be sceptical about brute necessites, besides the fact that there are no clear examples. Among other things, they would complicate our metaphysics; they would raise epistemic worries; and (my favourite reason) they would make the domain of metaphysical modality philosophically uninteresting. For example, it would be mysterious why anyone should care whether the mental supervenes on the physical within some brute sphere of "metaphysical" possibility.

Brute necessities are *almost* the same thing as what Chalmers
calls *strong* necessities. According to Chalmers, a strong
necessity is an a posteriori necessity that is true at all
metaphysically possible worlds when evaluated as actual. Every strong
necessity is brute, but not every brute necessity is strong.

To illustrate, consider the hypothesis that (G) is a strong necessity:

(G) Some omniscient being likes H2O.

That is, suppose that for reasons we can't explain from the armchair and through explorations into ordinary non-modal truths, every metaphysically possible world contains an omniscient being who likes H2O, and let's suppose 'H2O' picks out the same substance in worlds considered as actual and as counterfactual.

Next, assume that 'water' rigidly picks out whatever plays the water role in the actual world, and that H2O plays this role, so that water is necessarily H2O. If (G) is necessary and 'water=H2O' is necessary, then plausibly so is (G').

(G') Some omniscient being likes water.

In contrast to (G), (G') is not true at all metaphysically possible worlds considered as actual. For assume XYZ plays the water role at w, and the only omniscient being at w likes H2O but not XYZ. Then (G') is false at w considered as actual. (G') is not a strong necessity. But it is brute, for its necessity cannot be explained away along two-dimensional lines.

Now, (G') entails (G), so if there are no strong necessities, then there are also no brute necessities like (G'). So maybe it's enough to focus on strong necessities?

Arguably not. In the example, we get the strong-necessity (G) from (G') by replacing the actuality-dependent term 'water' with the non-actuality-dependent term 'H2O' which at any world w considered as actual picks out what 'water' denotes at w considered as counterfactual. But arguably we can't always find such a term.

Assume 'charge' rigidly denotes whatever microphysical quantity plays the physical role of charge, and that different quantities play that role at different worlds. (Not an uncontroversial assumption, but one that should be compatible with two-dimensionalism.) Then consider the hypothesis that (H') is a brute metaphysical necessity.

(H') Some omniscient being likes charge.

For the same reason for which (G') is not a strong necessity, (H') is not a strong necessity either. But this time, we have no "semantically neutral" way to pick out the quantity that plays the charge role, so we cannot convert (H') into a strong necessity (H).

(H) Some omniscient being likes xxx.

Moreover, it's not a coincidence that we don't have the required word. Try to introduce a word that would do the job! The word would somehow have to pick out the quantity that plays the charge role "by its inner nature" rather than by any role it plays. But we have no conception of what that inner nature is, and it's not even clear what it would mean to have such a conception.

Admittedly, the assumption that (H') is necessary still entails a strong necessity, namely that some omniscient being exists. But that hole is easily plugged:

(H'') Either there is no omniscient being or some omniscient being likes charge.

If the two-dimensional account of a posteriori necessity is correct, (H'') cannot be necessary. But (H'') is not a strong necessity, nor does the necessity of (H'') entail any strong necessity. So banning strong necessities is not enough.

]]>Let's start with two extreme cases. First, suppose I know nothing at all about my own location and how it relates to you -- not even that I am not identical to you. The assumption that you believe that are strawberries "here" (as you put it) then only tells me that there are strawberries somewhere in the universe. If you are moderately rational, then this is an uncentred belief you have as well. So in this case, deferring to you amounts to deferring to your uncentred beliefs; your centred (self-locating) beliefs can be ignored. That is, I treat you as an expert just in case, for any uncentred proposition A,

\[ Cr_I(A / Cr_U(A)=x) = x. \]If I have further uncentred evidence that you may lack, I need to conditionalize your credence on that evidence, as usual: \[ Cr_I(A / Cr_U(A / E)=x) = x. \]

Now for the other extreme: I am certain about my location relative to yours. This is information you may or may not have, and it may be relevant to other things you believe. For example, if I know that I am 1 km behind you but you are uncertain whether I am 1 km or 2 km behind, then conditional on your beliefs I should not be uncertnain whether I am 1 km or 2 km behind.

So I should consider your beliefs conditional on the extra information I have. But the information is centred, so conditionalizing your credence on it yields the wrong results: I'm not interested in your credence conditional on the hypothesis that you are 1 km behind you. We need to adjust the content.

For any centred proposition A, let [+n]A be the proposition that A is true n km down the path. My evidence E tells me that E is true here, but to consider your beliefs in light of that evidence, I should consider your credence conditional not on E but on [-1]E: the hypothesis that E is true 1 km behind you.

We need the opposite adjustment for the content of your self-locating beliefs: knowing that you are 1 km ahead of me, and assuming you are certain that there are strawberries around you, I should become certain that there are strawberries 1 km ahead.

So we arrive at the following rule for cases where my evidence E entails that you are n km ahead: for any centred or uncentred proposition A,

\[ Cr_I([+n]A / Cr_U(A/[-n]E)=x) = x. \]Equivalently,

\[ Cr_I(A / Cr_U([-n]A/[-n]E)=x) = x. \]What about in-between cases, where my evidence is not completely silent on matters of self-location, but also doesn't fully settle our relative location? (Every real-life case falls into this category.)

Well, we can apply the previous rule to possible extensions of my evidence that would settle our relative location. To spell this out, let \(Cr_U=f\) be the proposition that f is your credence function, and let D=n be the proposition that you are n km ahead of me. By the law of total probability,

\[ Cr_I(A / Cr_U=f) = \sum_n Cr_I(A / Cr_U=f \land D=n) Cr(D=n). \]When computing \( Cr_I(A / Cr_U=f \land D=n)\) it's important that I conditionalize your credence function not only my (shifted) evidence E but also the (shifted) assumption that D=n. So, slightly generalizing the previous rule:

\[ Cr_I(A / Cr_U=f \land D=n \land E) = f([-n]A / [-n]D=n \land [-n]E). \]Plugging this into the law of total probability, we get the general rule we were looking for:

\[ Cr_I(A / Cr_U=f) = \sum_n f([-n]A / [-n]D=n \land [-n]E) Cr(D=n). \]This still isn't entirely general because it reduces the question of our relative location to the question how many kilometers you are ahead of me on some path. The fully general rule requires generalizing the [-n] operator and the distance propositions D=n.

]]>I'll need the following premise, which I won't defend:

The world is not a book.

By this I mean that the world does not have linguistic structure in any meaningful sense. The structure of the world is rather like the structure of a picture.

To illustrate, here is a picture.

"The picture has structure insofar as it's not just a blob. Some ways of carving it up are objectively better than others. But the picture's structure does not mirror the structure of the sentences we might use when describing the picture.

Moreover, and more importantly, the picture does not specify a "domain of
individuals". One may talk about the picture in a language that
quantifies only over the two squares and the circle. Or one could use
a language that quantifies over arbitrary fusions of these things. Or
one could use a language that doesn't quantify over any individuals at
all. None of these ways of talking are wrong. Interpreting a
first-order sentence *at a picture* requires a method that tells
us what the quantifiers range over. The picture itself doesn't settle
the matter.

Now the world is probably kind of like a picture, except larger and
with more dimensions. Like a picture, it doesn't fix a "domain of
individuals". Interpreting a first-order sentence *at a world*,
where worlds are things like the actual world rather than
set-theoretic models, requires a method that tells us what the
quantifiers range over.

From this perspective, it is natural to think that the universalist and nihilist about mereology simply prefer different languages: languages with different interpretations. Each is right in their preferred language, and the world doesn't favour one over the other.

That's the slumber. But as Ralf points out, we can imagine a parallel dispute over conjunction.

Universalists about conjunction hold that from P and Q one can
always infer the conjunction *P and Q*. If you know that it is
raining and see that the cat is on the mat, you can infer that *it
is raining and the cat is on the mat*.

Nihilists about conjunction hold that *P and Q* is only true
if P = Q. From the fact that it is raining you can infer that *it is
raining and it is raining*. But you can't combine this information
with your knowledge that the cat is on the mat to infer that *it is
raining and the cat is on the mat*. In fact, according to
nihilists, it is not true that it is raining and the cat is on the
mat.

"Moderates" about conjunction hold that *P and Q* is sometimes
true if P and Q are true (even for distinct P and Q), and sometimes
not: it depends on whether "conjunction occurs". Perhaps conjunction
occurs whenever P and Q share a common topic.

Here, I think it is obvious that the universalists are right and
the others wrong. In fact, the only way I can make sense of what the
others are saying is by assuming that they misunderstand what 'and'
means. If you think that the inference from P,Q to *P and Q* is
valid only if P=Q, or if you think it is valid only on the assumption
that a special event of "conjunction occurs", then you don't
understand the meaning of 'and'.

Of course, everyone is free to speak a language without a word for conjunction. It's plausible that one can give an adequate and complete description of fundamental reality in such a language. And of course one can use a language in which 'and' doesn't mean conjunction but, say, disjunction. But arguably that's not what's going on in the above dispute. The imagined nihilist about conjunction doesn't simply think that 'and' expresses propositional identity rather than conjunction. She rather claims to defend the substantive metaphysical hypothesis that conjunction never occurs between distinct propositions. Similarly, the moderate claims that conjunction sometimes occurs and sometimes doesn't. There is no charitable interpretation of their language on which this makes sense.

In the imagined debate over conjunction, the moderates and
nihilists presuppose that the world has linguistic structure in an
extreme sense. Perhaps they think the world contains irreducible
*facts* (ontological mirror images of sentences), and that any
sentence (in English or Ontologese) is true iff there is a
corresponding fact. This allows for the possibility that there is a P
fact and a Q fact but no *P and Q* fact, and hence that the
conjunction is false even though the conjuncts are true.

But the world is not a book. It does not have linguistic structure,
and couldn't possibly have one. So the moderates and nihilists are
wrong and confused. They wrongly and confusedly think that for *P
and Q* to be true something more is required in the world than the
truth of P and the truth of Q: that a mysterious "conjunction event"
or "conjunction fact" must "occur".

Some universalists may share this false and confused idea. They
might hold that the mysterious conjunction event always occurs. But
one can also be a universalist by correctly and unconfusedly seeing
that the truth of *P and Q* requires nothing more from the world
than what's required by the truth of P and the truth of Q: any world
that satisfies both of the individual requirements automatically
satisfies the requirement imposed by the conjunction.

The same is true for mereology. Of course one can mean all sorts of things by 'part' and 'fusion'. But sensible universalists use these expressions in a metaphysically non-committal way. Fusion of individuals is much like conjunction of propositions. If there's an individual A and an individual B, nothing else is required of the world for there to be the fusion A+B. No mysterious "composition" must "occur".

There's nothing wrong with speaking a language that only quantifies over simples. But this language can't have the 'fusion' operator, if that is supposed to have anything like the meaning it has for the universalist. It's fine to use 'fusion' with an entirely different meaning, so that 'C is a fusion of A and B' means, say, that C is a unicorn, or that C is an organism. But that's not what nihilists or moderates about mereology mean. Rather, they assume that the world directly specifies a domain of individuals, and that a special composition event would have to occur for there to be a fusion of two individuals. And that's not a harmless matter of linguistic choice. It's wrong and confused.

]]>But first let me explain Meek and Glymour's proposal.

Causal models encode causal information by a probability measure over a directed acyclic graph. The nodes in the graph are random variables whose values stand for relevant (possible) events in the world; the probability measure stands for the objective chance (or frequency) of various values and combinations of values. In many cases one can assume the "Causal Markov Condition", which ensures that conditional on values for its causal parents, any variable is probabilistically dependent only on its effects.

For the application to decision theory, it is important that an adequate model need not explicitly represent all causally relevant factors. If a variable X can be influenced through multiple paths, one may only represent some of these and fold the others into an "error term". The error term must however be "d-separated" from the explicitly represented causal ancestors of X, which effectively means that it is probabilistically independent of those other causes.

In causal reasoning, we often need to distinguish two ways of updating on a change of a given variable. To illustrate, suppose we know that there's a lower incidence of disease X among people who take substance Y. One hypothesis that would explain this observation is that there's a common cause of reduced X incidence and taking Y. For instance, those who take Y might be generally more concerned about their health and therefore exercise more, which is the real cause of the reduced incidence in X. On this hypothesis, taking Y is evidence that an agent is less likely to have disease X, but if we made a controlled experiment in which we gave some people Y and others a placebo, the correlation would be expected to disappear. That's how we would test the present hypothesis. To predict what will happen in the experiment on the assumption of the hypothesis, we have to treat taking or not taking Y as an "intervention" that breaks the correlation with possible common causes. (The fact that somebody in the treatment group of the experiment takes Y is no evidence that they're more concerned with health than people in the control group.)

In general, an *intervention* on a variable makes it
independent of its parent variables. What makes this possible are
error terms. In the X and Y example, agents in the treatment group take
Y because they are paid to do so as part of the experiment. This
causal factor is an error term in the model. As required, the error
term is probabilistically independent of the explicitly represented
other cause for taking Y, namely general concern for one's health.

Now Meek and Glymour's suggestion is that everyone should use Jeffrey's formula for computing expected utilities via conditional probabilities. The disagreement between Evidential and Causal Decision Theory (EDT and CDT), they suggest, is not a normative disagreement about rational choice, but rather a disagreement over whether the relevant acts are considered as interventions.

For example, in Newcomb's problem, there is a correlation (due to a
common cause) between one-boxing and the opaque box containing a
million dollars. Let B=1 express that the agent chooses to
one-box. Conditional on B=1, there is a high probabiliy that there's a
million in the box. However, conditional on *an intervention to
one-box*, the probability of the million is equal to its
unconditional probability: the correlation disappears, just as it does
in the X and Y example.

Now for the problems.

The fist (and most obvious) is that there is no guarantee that
interventions of the relevant kind are available. We can't just assume
that for every value x of any variable A that represents an act, there
is an intervention event *do(A=x)* distinct from *A=x*.

The required assumption is obscured by misleading terminology. If
an agent faces a genuine choice between A=1 and A=2, then one
naturally thinks that she must be free to "intervene" on the value of
A; that she can make *do(A=1)* true or false at will. But
'intervening' and '*do(A=1)*' are technical terms, and in the
required technical sense it is not at all obvious that genuine choices
are always choices between interventions.

Return to Newcomb's problem. The obvious hypothesis about the causal relationships in Newcomb's problem is captured in the following graph.

"Here, B is the variable for one-boxing or two-boxing, P is the prediction, O is the outcome, and C is the common cause of prediction and choice: the agent's disposition to one-box or two-box. Let's assume that the predictor is fallible. How does the fallibility come about? There are two possibilities (which could be combined). Either the predictor has imperfect access to the common cause C, or C does not determine B. Suppose the fallibility is of the first kind. That is, we assume that there are causal factors C which fully determine the agent's choice, but the predictor does not have full access to these factors. That's easy to imagine. The causal factors C cause the predictor's evidence E which in turn causes her prediction, but E is an imperfect sign of C: it is possible that E=1 even though C=2, or that E=2 even though C=1. We could model this by introducing an error term on E, or directly on P (if we don't mention E explicitly).

In this version of Newcomb's Problem, there is no error term on B. So there is no possibility of "intervening" on B in the technical sense of causal models. This does not mean that the agent has no real choice. To be sure, the agent doesn't have strong libertarian freedom, since her choice is fully determined by the causal factors C. But who cares? It's highly contentious whether the idea of strong libertarian freedom is even coherent. It's even more contentious that ordinary humans are free in this sense. And almost nobody believes that robots have that kind of freedom. But robots still face decisions. Many are interested in decision theory precisely because they want to program intelligent artificial agents. An adequate decision theory should not presuppose that the relevant agent has libertarian free will.

That's the first problem. Here is the second. Suppose there are
error terms on the right-hand side in Newcomb's problem. More
specifically, let C be the agent's general disposition to follow CDT
or EDT, and suppose acts of one-boxing can be caused not just by C but
also by random electromagnetic fluctuations in the agent's
muscles. These fluctuations are proper error terms because they
decorrelate B from C. That's just what the interventionist seems to
want. But if that's the causal story, it would be wrong to assess the
choiceworthiness of one-boxing and two-boxing by conditionalizing on
*do(B=1)* and *do(B=2)* respectively. For that means to
effectively conditionalize on the relevant electromagnetic fluctuation
events, which are in no sense under the agent's control. They are not
even sensitive to the agent's beliefs and desires (we may assume).

Here the technical nature of the expressions 'intervention' and
'do' become obvious. In the technical sense, the random
electromagnetic fluctuations are interventions, and they realize
*do(B=1)*. But they are not interventions or doings on part of
the agent in any ordinary sense.

The third problem is pointed out in "Stern 2017. I'll try to make it a little more explicit than Stern does.

Consider the following causal structure.

"Here A represents the agent's possible actions, which may be smoking and not smoking. These are evidentially correlated with some desirable or undesirable outcome O (cancer or not cancer) via a common cause C (as in Fisher's hypothesis about the relationship between smoking and cancer). I is an intervention variable, which, we assume, decorrelates A from C and therefore O. Think of I as something like the agent's libertarian free will.

The depicted structure is not yet a causal model because it doesn't specify the chances. Suppose the agent's credence is evenly divided between two hypotheses about the relevant chances, H1 and H2. According to H1, I=1 and O=1 both have probability 0.9; according to H2 they both have probability 0.1. (It doesn't matter what else H1 and H2 say.)

By the Principal Principle, \begin{align} Cr(O=1) &= Cr(O=1 / H1)Cr(H1) + Cr(O=1 / H2)Cr(H2)\\ &= .9 * .5 + .1 * .5 = .5\\ Cr(I=1) &= Cr(I=1 / H1)Cr(H1) + Cr(I=1 / H2)Cr(H2)\\ &= .9 * .5 + .1 * .5 = .5 \end{align}

Since both H1 and H2 treat O and I as independent, it follows again from the Principal Principle that

\[ Cr(O=1 / I=1 \land H1) = Cr(O=1 / H1) = .9\\ Cr(O=1 / I=1 \land H2) = Cr(O=1 / H2) = .1 \]By Bayes' Theorem,

\[ Cr(H1 / I=1) = Cr(I=1 / H1) Cr(H1) / Cr(I=1) = .9 * .5 / .5 = .9\\ Cr(H2 / I=1) = Cr(I=1 / H2) Cr(H2) / Cr(I=1) = .1 * .5 / .5 = .1 \]Finally, by the probability calculus,

\begin{align} Cr(O=1 / I=1) =&\; Cr(O=1 / I=1 \land H1)Cr(H1 / I=1)\\ &\;+ Cr(O=1 / I=1 \land H2)Cr(H2 / I=1). \end{align}Putting all this together, we have

\[ Cr(O=1) = .5\\ Cr(O=1 / I=1) = .9 * .9 + .1 * .1 = .82 \]So although the agent assigns credence 1 to causal hypothesis on which I and O are probabilistically independent, the two variables are not independent in her beliefs.

This means that conditional on *do(A=1)*, which is tantamount
to *I=1*, the agent assigns much greater probability to O=1 than
conditional on *do(A=2)*. According to Meek & Glymour et al,
the agent should therefore choose A=1 (via I=1). *But this means to
act on a spurious correlation.*

(The argument does not require an explicit intervention variable. An evidential correlation between A and H1 would do just as well as the assumed correlation between I and H1.)

Stern's observation puts the nail in the coffin of Meek and Glymour's conjecture that CDT and EDT agree on the validity of Jeffrey's formula for calculating expected utilities, but disagree over whether the relevant acts are understood as interventions or ordinary events. In the present example, conditionalizing on interventions in Jeffrey's formula doesn't yield a recognizably causal decision theory.

As a corrolary, we can see that there's an important difference
between conditionalizing on *do(A=1)* and *subjunctively
supposing* A=1, what "Joyce
1999 would write as P( * \ A=1), with a backslash. "Joyce
2010 suggests that if P( * \ A=1) is understood in terms of
imaging or expected chance then there's a close connection between
P( * / do(A=1)) and P( * \ A=1), so that the the operation of
conditionalizing on *do(A=1)* may actually be understood as subjunctive
supposition rather than conditionalizing on an intervention event. But
the discussion presupposes that we are certain of the objective
probabilities. If we are not, conditionalizing on *do(A=1)* is
not at all the same as subjunctively supposing A=1.

To get around the third problem, Stern proposes to use Lewis's K-partition formula for calculating expected utilities, on which Jeffrey's formula is applied locally within each "dependency hypothesis" K and expected utility is the weighted average of the results, weighted by the agent's credence in the relevant dependency hypotheses. In Stern's "interventionist decision theory", the depedency hypotheses are identified with causal models. So expected utility is computed as follows (again, I'm slightly more explicit here than Stern himself):

\[ EU(A) = \sum_K Cr(K) \sum_O Cr(O / do(A) \land K) V(O) \](Since causal models are effectively hypotheses about chance, this account is perhaps even closer to Skyrms's version of CDT than to Lewis's.)

This gets around the problem because any evidence A may provide for or against a particular causal model becomes irrelevant.

Notice that Stern's proposal is "doubly causal", as it were. First, it replaces Jeffrey's formula by the Lewis-Skyrms formula, in order to factor out spurious correlations between acts and causal hypotheses. Second, it replaces ordinary acts A by interventions, do(A). Do we really need both?

Arguably not. Return to Newcomb's problem. Here the Lewis-Skyrms
approach already recommends two-boxing because it distinguishes
*two* relevant dependency hypotheses. According to the first, the
opaque box is empty and so there's a high chance of getting $0 through
one-boxing; according to the second, the opaque box contains $1M and
so there's a high chance of getting $1M through one-boxing.

Can the interventionist also treat these as two different causal models? Yes. Easily. The two models would have the same causal graph, but different objective probabilities. In one model, it is certain that the predictor predicts one-boxing, in the other it is certain that the predictor predicts two-boxing. This may not fit the frequentist interpretation of probabilities in causal models, but this interpretation spells trouble for interventionist accounts of decision anyway, since (a) the Principal Principle for frequencies is much more problematic than for more substantive chances, (b) population level statistics make it even harder to find suitable error terms for intervening (as "Papineau 2000 points out). If instead we think of the probabilities more along the lines of objective chance (though it could be statistical mechanical chance), it is quite natural to think that at the time of the decision, the contents of the box are no longer a matter of non-trivial chance.

So there are good reasons for the interventionist to follow Lewis and Skyrms and model Newcomb's problem as involving two relevant causal hypotheses K. And then we get two-boxing as the recommendation even if, conditional on each hypothesis, we conditionalize on B rather than do(B).

This is nice because it also solves the first two problems for the interventionist: the availability and eligibility of interventions. On the revised version of Stern's account, we don't need interventions any more.

Of course, the revised version of Stern's account is basically the decision theory of Lewis and Skyrms. The only difference is that dependency hypotheses are spelled out as causal models.

Upshot: The theory of causal models can indeed be useful for thinking about rational choice, because causal models are natural candidates to play the role of dependency hypotheses in K-partition accounts of expected utility. The supposedly central concept of an intervention, however, is not only problematic in this context, but also redundant. We can do better without it.

]]>But I seem to be alone in thinking that fission is the right paradigm for modeling Sleeping Beauty. A much more popular assumption is that Sleeping Beauty is essentially a problem about "losing track of time": as a result of the potential memory loss, it is claimed, Beauty can't tell upon awakening whether it is Monday or Tuesday, and that's what makes her case special. I don't agree that this adequately sums up Beauty's predicament. Surprisingly, though, I think this way of modeling Sleeping Beauty still supports halfing. (That's surprising because almost all authors who endorse the present interpretation are thirders).

Let's begin with an ordinary case where someone loses track of time.

Noisy Awakening I (incomplete). A loud noise wakes you up at night. You have a vague sense that you've slept for a few hours, but the sensation is equally compatible with it being 2 am or 3 am.

It's important to realize that this is not yet a case in which you're necessarily lost in time. For suppose you knew when you fell asleep that a loud noise was going to wake you up at 2am. Remembering this information, you should be confident upon awakening that it is 2am, despite your unspecific sensation of how much time has passed.

So your new credence in what time it is should be affected by two features. First there is your broadly sensory evidence of how long you've slept, as well as other pieces of "new evidence": your perception that it is still dark, etc. Second, there are your previous beliefs about when you might wake up. Since our topic is losing track of time and not forgetting or irrational priors, we can assume that these earlier beliefs were rational and you have no trouble recalling the reasons on which they were based.

It is not obvious how exactly these two factors determine the new beliefs. But the following special case should be uncontroversial.

(*) If before falling asleep you rationally gave credence x to waking up at t1 and 1-x to waking up at t2, and if upon awakening your new evidence is neutral between t1 and t2, then you should now give credence x to the time being t1 and 1-x to the time being t2.

So let's complete our first scenario.

Noisy Awakening I (complete). A loud noise wakes you up at night. You have a vague sense that you've slept for a few hours, but the sensation is equally compatible with it being 2 am or 3 am. Before going to sleep, you rationally gave credence 1/2 to the noise waking you at 2 am and 1/2 to the noise waking you at 3 am.

What should you believe about the time? Answer: you be 50% confident that it is 2 am and 50% confident that it is 3 am.

Now consider a simplified variant of the Sleeping Beauty problem in which Beauty is rationally certain that the coin lands tails. Before falling asleep, she then assigns credence 1/2 to waking up on Monday and 1/2 to waking up on Tuesday. Upon awakening, any sensations she may have about how much time has passed are presumably defeated by her knowledge of the setup, so we may as well assume that she has no relevant new evidence at all about the time.

If we model this as a "losing track of time" scenario, it is obviously
analogous to *Noisy Awakening*.

The real Sleeping Beauty problem is a little more complicated because there is also the possiblity of heads. Let's make the corresponding adjustments to Noisy Awakening.

InNoisy Awakening II. Before going to sleep on Sunday evening, you were given the following information. A fair coin will be tossed twice. If it comes up heads at least once, a loud noise will wake you up at 2 am, otherwise (if the coin lands tails twice) the noise will wake you up at 3 am. You fall asleep and are awakened by a loud noise. You have a vague sense that you've slept for a few hours, but the sensation is equally compatible with it being 2 am or 3 am.

What should you believe in *Noisy Awakening II* when you are
woken up by the noise?

Fortunately, we don't need any new principles. For the upshot of the information about the coin tosses is that on Sunday you give credence 3/4 to waking at 2am and 1/4 to waking at 3am. By (*), absent relevant new evidence, you should give credence 3/4 to the hypothesis that it is 2am.

If Sleeping Beauty is a problem about losing track of time, we should give the parallel answer: Beauty should give credence 3/4 to the hypothesis that it is Monday. That's what halfers say. They hold that Beauty's Monday credence should be divided 1/2 - 1/4 - 1/4 between the possibilities Heads & Monday - Tails & Monday - Tails & Tuesday; thirding says it should be divided 1/3 - 1/3 - 1/3.

Just to be clear: this is an argument by analogy. We can't directly
apply (*) to Sleeping Beauty, precisely because Sleeping Beauty is not
a straightforward case in which someone is otherwise perfectly
rational but loses track of time. My point is that *if* we model
Sleeping Beauty as nonetheless analogous to such cases (as I think we
shouldn't: we should rather model it as a case of epistemic fission),
we still get an argument for halfing.

(Sleeping Beauty as a case of fission is interesting because it
arguably reveals the tension between evidentialism and
conservatism. Case of merely losing track of time don't: in *Noisy
Awakening II*, your evidence upon awakening plausibly supports the
2 am hypothesis to degree 3/4.)