**Sleeping Beauty Awakened:
New Odds at the Dawn of the New Day**

TERRY HORGAN

1. The story of
Sleeping Beauty is set forth as follows by Dorr (2002):

Sleeping Beauty is a paradigm of
rationality. On Sunday she learns for certain that she is to be the subject of
an experiment. The experimenters will wake her up on Monday morning, and tell
her some time later that it is Monday. When she goes back to sleep, they will
toss a fair coin. If the outcome of the toss is Heads, they will do nothing. If
the outcome is Tails, they will administer a drug whose effect is to destroy
all memories from the previous day, so that when she wakes up on Tuesday, she
will be unable to tell that it is not Monday. (2002: 292)[1]

Let HEADS be the hypothesis that the coin lands heads, and
let TAILS be the hypothesis that it lands tails. The Sleeping Beauty Problem is
this. When Sleeping Beauty finds herself awakened by the experimenters, with no
memory of a prior awakening and with no ability to tell whether or not it is
Monday, what probabilities should she assign to HEADS and TAILS respectively?

Elga (2000) maintains that when she is
awakened, P(HEADS) = 1/3 and P(TAILS) = 2/3. He offers the following
intuitively plausible argument (2000: 143-4). If the experiment were performed
many times, then over the long run about 1/3 of the awakenings would happen on
trials in which the coin lands heads, and about 2/3 on trials in which it lands
tails. So in the present circumstance in which the experiment is performed just
once, P(HEADS) = 1/3 and P(TAILS) = 2/3.[2]

Lewis (2001)
maintains, contrary to Elga, that when Sleeping
Beauty is awakened, P(HEADS) = P(TAILS) = ½. He offers the following
intuitively plausible argument (2001: 173-4). Initially on Sunday when Sleeping
Beauty was still awake, surely the probabilities of HEADS and TAILS were ½
each. But only new relevant evidence should produce a change in probability,
and when Sleeping Beauty is awakened she receives no new evidence that is
relevant to HEADS vs. TAILS. (She already knew on Sunday that she would find
herself being awakened at least once by the experimenters with no memory of a
prior awakening.) Hence, when Sleeping Beauty is awakened it is still the case
that HEADS and TAILS each have probability ½.

I side with Elga in this dispute. But if indeed Lewis’s argument is
mistaken, then there should be a way of explaining why and how it goes wrong.
The challenge is to make clear why Sleeping Beauty’s evidence upon being
awakened is relevant to HEADS vs. TAILS—more specifically, why she has evidence
that makes it the case that P(HEADS) = 1/3 and P(TAILS) = 2/3. If such an
account cannot be given, then it would seem that Lewis is right after all. This
would mean that in the Sleeping Beauty problem, what would happen over the long
run if the experiment were repeated many times does not reflect the single-case
probabilities of HEADS vs. TAILS.[3]^{,[4]}

Dorr (2002) also sides with Elga. Dorr employs a ‘soritical
argument by analogy’: he appeals to the alleged parallelism between a case he
constructs, in which it seems intuitively clear that the probabilities of HEADS
and TAILS respectively are 1/3 and 2/3, and the original Sleeping Beauty
case—where the putative analogy is bolstered by a sorites
sequence of intermediate cases. Although I think that Dorr’s reasoning is
suggestive and on the right track, soritical
arguments by analogy do run a serious risk of being slippery slope fallacies. I
believe that what I say in the present paper captures the underlying spirit of
Dorr’s approach, but without any appeal to analogies or to soritical
reasoning. I address the connection between my own argument and Dorr’s below.

2. Upon being
awakened by the experimenters and finding herself (as expected) with no memory
of a prior awakening, Sleeping Beauty no longer knows what day it is; today
might be Monday or it might be Tuesday. She also does not know whether the coin
toss comes up heads or tails. She contemplates the following partition of
statements, all pertaining to the current day:

H_{1}: HEADS and today is Monday

H_{2}: HEADS and today is Tuesday

T_{1}: TAILS and today is Monday

T_{2}: TAILS and today is Tuesday.

First she asks herself about the *prior* probabilities of these four statements, as determined by the
evidence she possessed prior to being put to sleep. She rightly realizes that
these prior probabilities are features these statements have *presently*—viz., probabilities the
statements have *now* relative to the
evidence available to her *then*.[5]
Exercising her impeccable rationality, she correctly judges that each of the four
statements has a prior probability of ¼.[6]
(The evidence available to her from Sunday is consistent with H_{2}.
Although the Sunday evidence guarantees that she is awakened by the
experimenters at least once, it does not guarantee that she is awakened by the
experimenters *today*. Today might be
Tuesday, after all.)

Next she asks about the *current* probabilities of H_{1},
H_{2}, T_{1}, and T_{2}, as determined by her total
current evidence. Again exercising her impeccable rationality, she updates the
prior probabilities in light of the fact that she has just been awakened by the
experimenters: the probability of H_{2} goes to zero, and since the
remaining three statements are exclusive and (given her current evidence)
exhaustive, their current probabilities are ascertained by multiplying their
respective prior probabilities by the common normalization-factor of 4/3.[7]
So, after updating, P(H_{1}) = P(T_{1}) = P(T_{2}) =
1/3. But P(HEADS) = P(H_{1}), and P(TAILS) = P(T_{1} v T_{2}).
Hence, P(HEADS) = 1/3 and P(TAILS) = 2/3.

So Sleeping
Beauty’s evidence is indeed relevant to HEADS vs. TAILS, and it does indeed
drive their probabilities to 1/3 and 2/3 respectively even though they each had
probability ½ on the preceding Sunday. This is because the current evidence
excludes the possibility expressed by the statement H_{2}—a statement which, when
Sleeping Beauty is awakened, has a prior probability of ¼ as determined by her
Sunday evidence.

3. I will offer
an abstract characterization of the key factors at work in the Sleeping Beauty
problem. Before doing so, several preliminary points need to be made. First,
the kind of probability under discussion is *epistemic*,
in the sense that it is essentially tied to available evidence. Epistemic
probability perhaps should be construed as *degree
of evidential support*, or instead perhaps should be construed as ‘credence’:
*degree of belief* or *rational degree of belief*.

Second,
epistemic-probability contexts are *intensional*: when a statement S* is obtained from a
statement S by substitution of co-referring singular terms, the probability of
the resulting statement S* can differ from the probability of the original
statement S.[8] (As one might say,
epistemic-probability contexts lack the feature of substitutivity
*salva** probabilitate*.[9])
The Sleeping Beauty problem illustrates this phenomenon. At any time during the
experiment, the following claim is true, for Sleeping Beauty:

P(HEADS and Tuesday I am awakened by
the experimenters) = 0.

Suppose, however, that she has just been awakened by the
experimenters and that (unbeknownst to her),

Tuesday = today.

The following claim, obtained by substitution of co-referring
terms, is false:

P(HEADS and today I am awakened by
the experimenters) = 0.

Third, the
syntax and semantics of epistemic-probability ascriptions needs to be construed
in some way that accommodates this intensionality.
For present purposes, it will be convenient to continue the practice already
adopted in section 2 above: I will take *statements*
to be the items to which probabilities are ascribed.[10]

With these
three points as background, let me now describe abstractly the various
interconnected elements crucially at work in the Sleeping Beauty problem.
First, something occurs that results in a *loss
of self-location information* for the epistemic agent—in the case of
Sleeping Beauty, a loss of information about what day it is.[11]
Second, a *potential cognitive mishap*
is involved in this loss of self-location information, a mishap that might have
occurred (for all the agent can tell) but need not have—in the case of Sleeping
Beauty, being injected with a drug that erases all memories of the preceding 24
hours.

Third, the
loss of self-location information generates, for the agent, a way of conceiving
and describing one’s self-location—in the case of Sleeping Beauty, ‘today’—that
is *essentially indexical*, in the
sense that the agent does not know what location it is that is thus conceived
(and thus expressed) indexically. Fourth, the loss of
self-location information thereby generates a partition of essentially
indexical *statements*, each of which
is consistent with the evidence possessed by the agent prior to the loss of
self-location information—in the case of Sleeping Beauty, a partition
comprising the statements H_{1}, H_{2}, T_{1}, and T_{2}.

Fifth, this
partition is *irreducibly indexical*,
in this sense: the agent lacks a non-indexical way of conceiving (or
describing) her own self-location that can be substituted *salva** probabilitate* for the relevant,
essentially indexical, thought-constituent (term) within each of the agent’s
probability judgments (ascriptions) concerning the statements in the partition.
(Thus, the irreducible indexicality of the partition
reflects the intensionality of epistemic
probability.) In the case of Sleeping Beauty, for instance, the irreducible indexicality of the statement-partition {H_{1},H_{2},T_{1},T_{2}}
is illustrated by the following substitutivity-failure
involving statement T_{1}: even if today happens to be Monday
(unbeknownst to her), and even though P(T_{1}) = P(TAILS and today is
Monday) = 1/3, it is not the case that P(TAILS and Monday is Monday) = 1/3.
Rather, P(TAILS and Monday is Monday) = P(TAILS) = 2/3.

** **Sixth, the irreducibly indexical partition of essentially indexical
statements only arises once the loss of self-location information occurs,
because this information-loss is what generates such a partition in the first
place. This in turn means, seventh, that the prior probabilities of these
essentially indexical statements only arise once the information-loss occurs.
What makes the relevant probabilities count as *prior* probabilities—here and in general (as emphasized in section
2)—is that these are the probabilities *now*
possessed, relative to the evidence *then*
available, by the statements in the partition. An atypical feature of the
situation, however, is that these present prior-probabilities are not
probabilities that *obtained* prior to
the information-loss; for, the irreducibly indexical statement-partition had
not yet arisen.

Eighth, the very episode that
generates the information-loss also furnishes the agent with conclusive
evidence, concerning a specific one of the essentially indexical statements in
the partition (or perhaps a specific disjunction of them), that it does not
obtain. Thus, ninth, the information-loss and the acquisition of this new evidence
occur *simultaneously*, with the new
evidence pertaining specifically to the irreducibly indexical
statement-partition generated by the information-loss. So tenth, the current
probability of each of the remaining statements in the partition, for the agent,
is the prior *conditional* probability
of that statement given this evidence—where this prior conditional probability
arises, via the information-loss, simultaneously with the current probability
itself.

The soritical
argument by analogy in Dorr (2002), mentioned in section 1 above, is relevant
to the features described in the preceding two paragraphs. In my terms, Dorr’s
argument can be understood as follows. He first describes a case in which the
information-gain occurs some time later than the information-loss that
generates the essentially indexical statement-partition; then he describes a
sorties sequence of similar cases in which the time-interval becomes
progressively shorter; the limit case in this sequence is the original Sleeping
Beauty scenario. In effect, the argument is (1) that for each successive case
prior to the limit case, information is gained (after the information-loss)
that lowers the probability of HEADS to 1/3 and raises the probability of TAILS
to 2/3, and (2) that the limit case itself is relevantly similar in this
respect to the other cases, even though in the limit case there is no time-gap
between the information-loss itself and the information-gain that occurs
against the background of this information-loss. This line of reasoning strikes
me as correct, rather than as a slippery-slope fallacy; the point is that there
is no evident reason why the limit case should be treated differently than the
others.[12]

The ten features lately described
constitute an abstract recipe for constructing Sleeping-Beauty-type cases. One
such case is a version of the Sleeping Beauty story in which the coin is tossed
on Sunday evening, before Sleeping Beauty is put to sleep. Another such case is
the following variant story in which sleeping plays no role (and in which the
relevant temporal indexical does not necessarily refer to a different day than
the day on which the agent still knows what day it is):

On Wednesday you learn for certain
that you are to be the subject of an experiment. At

Initially the probability of HEADS is ½. But shortly after
you find yourself having just been injected, with no memory of any prior
injection or any intervening period of insomnia, what is the probability of
HEADS? The discussion in section 2 applies here, *mutatis mutandis*, because this case too exemplifies the ten
features described in the present section. So now the probability of HEADS is
1/3.

This
scenario nicely illustrates an important fact: sometimes a statement’s present
prior-probability is not identical to its preceding current-probability.
Suppose that (unbeknownst to you) today is still Wednesday, as you find
yourself having just been injected and with no memory of a prior injection or
of 24 hours of insomnia. You consider the four statements of the form ‘X and
today is Y’ obtained by substituting ‘HEADS’ or ‘TAILS’ for ‘X’ and
substituting ‘Wednesday’ or ‘Thursday’ for ‘Y’. The present prior-probability
of the statement *HEADS and today is
Wednesday*, as determined by your pre-injection evidence, is ¼. But the
preceding current-probability possessed by this statement before the injection,
as determined by the then-current evidence, was ½.[13]
This difference between the preceding current-probability and the present
prior-probability reflects the intervening loss of self-location information.[14]

The discussion in section 2 also applies,
*mutatis mutandis*, to a variant of the
original Sleeping Beauty story in which Sleeping Beauty is informed on Sunday
that if the coin-toss comes up heads on Monday, then she will be killed
immediately. When Sleeping Beauty is awakened by the experimenters and she
contemplates the statement *HEADS and
today is Tuesday and I am now dead*, she rightly judges that this statement
has prior probability ¼, as determined by her Sunday evidence. Given her *current* evidence, of course, she updates
the probability to zero.[15]

Our abstract
recipe probably can be further generalized. For instance, the second
feature—the presence of a potential cognitive mishap—presumably is replaceable
by various other ways of losing self-location information. Also, a
statement-partition that is irreducibly indexical presumably can arise through
the loss of information not only about self-*location*
but also about self-*identity*. So my
recommended treatment of the Sleeping Beauty problem probably is applicable to
cases involving loss of self-identity information, and/or to cases in which the
loss of self-location or self-identity information results from some factor
other than potential cognitive mishap.[16]

4. Before being
put to sleep on Sunday, Sleeping Beauty rightly judged that the probability of
HEADS was ½. Upon being awakened on Monday, she rightly judges that P(HEADS) =
1/3. Elga claims that this belief change does not
result from acquiring new information. He says:

This belief change is unusual. It is
not the result of your receiving new information—you were already certain that
you would be awakened on Monday…. Neither is this belief change the result of
your suffering any cognitive mishaps in the intervening time—recall that the
forgetting drug isn’t administered until well after you are first awakened. (p.
145)

He goes on to urge the following moral:

Thus the Sleeping Beauty Problem
provides a new variety of counterexample to Bas Van Fraassen’s
‘Reflection Principle’ (1984: 244, 1995: 19), even an extremely qualified
version of which entails the following: Any agent who is certain that she will
tomorrow have credence x in proposition R (though she will neither receive new
information nor suffer any cognitive mishaps in the intervening time) ought *now* to have credence x in R. (p. 146)

But in the sense of ‘new information’ that is contextually
most appropriate with respect to questions of how available evidence affects
probabilities, Sleeping Beauty *does*
acquire new information upon being awakened on Monday. This awakening-event
produces, simultaneously, both an information loss and also an information gain
that is predicated upon that information loss. When she is awakened by the
experimenters on Monday, she thereby loses self-location information: the
awakening-event generates the irreducibly indexical partition of statements H_{1},
H_{2}, T_{1}, and T_{2}—each of which now expresses an
epistemic possibility relative to her Sunday evidence, and each of which now
has a prior probability of ¼ as determined by her Sunday evidence. Against the
backdrop of this loss of self-location information, the awakening-event
simultaneously constitutes evidence that conclusively excludes the epistemic
possibility expressed by H_{2}; although H_{2} is consistent
with her Sunday evidence, it is ruled out by her total *current* evidence. Exclusion of epistemic possibilities counts as
acquisition of new information, in the context of ascertaining probabilities on
the basis of current evidence. So it appears that the Sleeping Beauty problem
does not really constitute a counterexample to the core principle that Elga cites—the principle that is entailed by even an
extremely qualified version of the Reflection Principle—provided that the
phrase ‘new information’ in this core principle is appropriately construed.

Elga uses the phrase ‘new information’ in
a more coarse-grained way. He says, ‘To say that an agent receives new
information (as I shall use that expression) is to say that the agent receives
evidence that rules out possible worlds not already ruled out by her previous
evidence’ (2000: 145). On this usage, evidence that only rules out a *within*-world possibility concerning
one’s present temporal location (e.g., Tuesday in a HEADS-world), but does not
rule out any possible world altogether (e.g., a HEADS-world or a TAILS-world),
does not count as new information. So much the worse for Elga’s
usage, in the present context.

It bears emphasis that mere changes
in the referents of self-locating indexicals should
not be construed as generating new information all by themselves, on pain of
trivializing the Reflection Principle.[17]
If several days go by, and one knows all the while what day one is currently
located within, then the mere change in referent of the term ‘today’ does not
constitute or generate ‘new information’ about one’s self-location. (Perhaps
one knows all along all relevant information concerning each successive day,
expressible non-indexically. If so, then one also
knows all along how to reformulate various aspects of that information indexically—where such reformulations do not alter
probabilities.) By contrast, the statement-partition that Sleeping Beauty
contemplates after being awakened is *irreducibly
indexical*: because ‘today’ is essentially indexical for her, some
statements in the partition have probabilities (both prior probabilities and
current probabilities) different from the probabilities of the corresponding
non-indexical statements that result from replacing essential indexicals by co-referential non-indexicals.
Exclusion of an epistemic possibility expressed by a statement in an
irreducibly indexical statement-partition *does*
constitute genuinely new information.

One might reply, ‘But Sleeping Beauty
already knew on Sunday that she would be awakened on Monday with no memory of a
prior awakening. So didn’t she already possess, on Sunday, all the information
that she would possess upon being awakened on Monday?’ The answer is no.
Although Sleeping Beauty did know on Sunday that she would be awakened on
Monday, and although she also knew on Sunday that on Monday she would possess
information that would be expressible by saying or thinking

It is not the case that (today is
Tuesday and the coin comes up heads),

the information she thus expresses on Monday is *essentially indexical*. She did not yet
possess this information on Sunday. Although she already knew on Sunday that
she would obtain essentially indexical new information on Monday, and although
she even knew on Sunday how she we would indexically
describe this new information on Monday, she did not yet have the new
information itself.[18]

*University** of **Arizona*

*Tucson** **AZ** **85721-0027**, **USA*

**References**

Arntzenius, F. 2002. Reflections on Sleeping
Beauty. *Analysis* 62: 53-62.

Arntzenius, F. Forthcoming. Some problems for conditionalization and reflection.

Bradley, D. 2003. Sleeping Beauty: a note on Dorr’s argument
for 1/3. *Analysis* 63: 000-000.

Dorr, C. 2002. Sleeping Beauty: in defence
of Elga. *Analysis*
62: 292-96.

Elga, A. 2000. Self-locating belief and
the Sleeping Beauty problem. *Analysis*
60: 143-47.

Elga, A. Manuscript. Defeating Dr. Evil
with self-locating belief.

Horgan, T. 2000. The two-envelope paradox,
nonstandard expected utility, and the intensionality
of probability. *Nous*
34: 578-602.

Lewis, D. 2001. Sleeping Beauty: reply to Elga.
*Analysis* 61: 171-76.

Monton, B. 2002. Sleeping Beauty and the
forgetful Bayesian. *Analysis* 62:
47-53.

van Fraassen, B. C. 1984. Belief
and the will. *Journal of Philosophy*
81: 235-56.

van Fraassen, B. C. 1995. Belief
and problem of Ulysses and the sirens. *Philosophical
Studies* 77: 7-37.

[1] Sleeping Beauty also knows the following about the
experiment, although Dorr neglects to say so: if the outcome of the coin toss
is Tails, then she will be awakened *by
the experimenters* again on Tuesday. This feature of the story is needed in
order to guarantee that if the outcome is Tails, then her Tuesday awakening
will leave her unable to tell that it is not Monday.

[2]Another plausible argument for this position is as
follows. When Sleeping Beauty is awakened by the experimenters, P(HEADS *given that* today is Monday) = P(TAILS *given that* today is Monday) = ½, because
if today is Monday then the coin-toss has not yet occurred. Also, P(TAILS and
today is Monday) = P(TAILS and today is Tuesday), because her evidence is
indifferent between those two possibilities. Under the laws of probability
theory, and given Sleeping Beauty’s background knowledge about her situation,
these probability assignments entail that P(HEADS) = 1/3 and P(TAILS) = 2/3.
For an elaborated version of this argument, see Elga
(2000: 144-5.)

[3] For a plausible argument that the long-run
frequencies do not reflect the single-case probabilities in the Sleeping Beauty
problem, see section 4 of Arntzenius (2002). Arntzenius does not endorse Lewis’s position, however;
instead he maintains ‘that one should distinguish degrees of belief from
acceptable betting odds, and that some of the time Sleeping Beauty should not
have definite degrees of belief in certain propositions’ (pp. 53-54).

[4] Adopting Lewis’s position would also mean that Elga’s argument involving conditional probabilities
(described in note 2) should be rejected—perhaps by biting the bullet and
siding with Lewis in his highly counterintuitive claims (1) that when Sleeping
Beauty is awakened by the experimenters, P(HEADS *given that* today is Monday) = 2/3, and (2) that later on Monday
when she is told that today is Monday, the probability of HEADS rises to 2/3.

[5] The point is entirely general: the prior probability
at time t of a statement S is a probability had by S *at t*—albeit a probability that is relative to a body of evidence
that is known (at t) to be the evidence that was available *prior* to the most recent change in evidence. (Thanks to Sarah
Wright for emphasizing this to me.) Typically—but not invariably—the prior
probability of S at t is identical to the current-probability of S *at that earlier time*—i.e., the
probability *then* possessed by S
relative to the total evidence *then*
available. More on this below.

[6] Likewise, she also correctly judges that the prior *conditional* probabilities of (H_{1}
given –H_{2}), of (T_{1} given –H_{2}), and of (T_{2}
given –H_{2}) are each 1/3.

[7] This is the standard, Bayesian, form of updating: conditionalization. See note 6.

[8] See Horgan (2000) for
further discussion of this insufficiently appreciated feature of epistemic
probability, with specific application to the two-envelope paradox. Objective
probability, on the other hand—i.e., *chance*—need
not be intensional, even though epistemic probability
is.

[9] Thanks to Sarah Wright for suggesting this phrase to
me.

[10] I leave it open exactly what sorts of entities
statements are, qua bearers of epistemic possibility. (I do assume that there
are *indexical* statements.) One might
try accommodating the intensionality of probability
in other ways than by taking the bearers of epistemic probability to be
statements. For instance, perhaps epistemic probability attaches to very
finely-individuated *possibilities*
(including indexical ones)—so that the possibility *being awakened on Tuesday* is distinct from the possibility *being awakened today*, even if today =
Tuesday. (Possibilities like *being
awakened today* might get represented as classes of so-called *centered worlds*: possible worlds with
designated individuals-at-times within them.) Or perhaps epistemic probability
is an attribute of possibilities under*
descriptions* and/or under *modes of
presentation* (including indexical ones). Advocates of some such views would
need to find ways to reformulate, within their preferred idiom, the claims I
make in the text about probabilities of indexical statements.

[11] Loss of self-location information is also emphasized
in relation to the Sleeping Beauty problem by Monton
(2002) and by Arntzenius (forthcoming), both of whom
endorse the claim that after Sleeping Beauty is awakened, P(HEADS) = 1/3. For Arntzenius this is a change from the position taken in Arntzenius (2002); see my note 3 above.

[12] Bradley (2003) argues that Dorr’s initial case is
crucially disanalogous to the original Sleeping
Beauty scenario. (He does not discuss Dorr’s sorites
sequence.) He writes, ‘In the variant case, a certain possibility has been
eliminated. It *could* have turned out
that it was Heads and Tuesday…. In the original case, there is no such
possibility’ (000). On the contrary: in the original case too, the possibility *HEADS and today is Tuesday* is consistent
with Sleeping Beauty’s Sunday evidence, and thus now has prior probability ¼ as
determined by her Sunday evidence. Here too this possibility has been
eliminated by her current total evidence—specifically, by her newly acquired,
essentially indexical, knowledge that she has been awakened *today* by the experimenters (with no
memory of any prior awakening).

[13] There is a parallel difference between
presently-prior and previously-current *conditional*
probabilities. Both conform to the standard definition of conditional
probability: P(A given B) = P(A&B)/P(B). The presently-prior conditional
probability of (A given B) is determined, in accordance with the definition, by
the present prior-probabilities of A and B, whereas the previously-current
conditional probability of (A given B) is determined by the preceding
current-probabilities of A and B. For instance, after you find yourself having
just been injected and no longer knowing whether today is Wednesday or
Thursday, the presently-prior conditional probability of [(HEADS and today is
Wednesday) *given that* it’s not the
case that (HEADS and today is Thursday)] is 1/3. But the previously-current conditional
probability of this statement, back before the experiment began and when you
still knew that it was Wednesday, was ½. In situations where previously-current
conditional probabilities differ from presently-prior conditional probabilities,
Bayesian updating should employ the latter rather than the former.

[14] In the original Sleeping Beauty scenario, I take it,
the *sentences* H_{1}, H_{2},
T_{1}, and T_{2} would have expressed different indexical *statements* if they had been used by
Sleeping Beauty on Sunday prior to being put to sleep, since the indexical term
‘today’ would have had a different referent on Sunday. Those statements, which
are indexical but not essentially indexical, each had probability zero on
Sunday. Also, suppose that (unbeknownst to Sleeping Beauty) today is Monday.
Then the indexical statement expressible today with the sentence ‘HEADS and
today is Monday’ has a present prior-probability of ¼, as determined by her
Sunday evidence, whereas the indexical statement that was expressible on Sunday
with the sentence ‘HEADS and tomorrow is Monday’ possessed on Sunday a
probability of ½, as determined by her then-current evidence.

[15] Justin Fisher impressed upon me the need to provide a
treatment of the Sleeping Beauty problem that would generalize to cases like
those described in recent paragraphs.

[16] For discussion of a range of cases that exhibit these
features and appear otherwise similar to the Sleeping Beauty problem, see Arntzenius (forthcoming) and Elga
(manuscript).

[17] Justin Fisher impressed upon me the importance of
this point.

[18] I have pestered numerous people about the Sleeping
Beauty problem. Thanks to Justin Fisher and Sarah Wright for especially
valuable discussion and feedback, and to Robert Barnard, David Chalmers, Ned
Hall, John Hawthorne, Dianne Horgan, Kelly Horgan, Jenann Ismael, Keith Lehrer, David Papineau,
John Pollock, Eric Schwitzgebel, John Tienson, Mark Timmons, Michael Tye,
Brian Weatherson, and Ruth Weintraub.