Imaging and Sleeping Beauty
A Case for Double-Halfers
Mikal Cozic
Department of Cognitive Science
Ecole Normale Suprieure
45, rue d’Ulm, F-75005 Paris
mikael.cozic@ens.fr
INTRODUCTION
1 HALFERS AND THIRDERS
(Elga 2000) introduced philosophers to the troubling scenario of Sleeping Beauty. On Sunday evening (t0 ), Sleeping Beauty is put to sleep by an experimental philosopher.
She is awaken on Monday morning and at this moment (t1 ),
the experimenter doesn’t tell her which day it is. Some
time later (t2 ), she is told that it is actually Monday. At
this point, what follows depends on the toss of a fair coin
that took place on Sunday evening - Sleeping Beauty is not
aware of the outcome. If the coin landed heads (HEADS),
then Sleeping Beauty is put to sleep until the end of the
week. If the coin landed tails (T AILS), then Sleeping
Beauty is awaken on Tuesday morning. The crucial fact
is that a drug that is given to her is such that she cannot
distinguish her awakening on Monday from her awakening on Tuesday. Of course, Sleeping Beauty is perfectly
informed of every detail of the protocol before the experiment. The question that has drawn so much attention
since (Elga 2000) is the following: what should be Sleeping Beauty’s degree of belief that HEADS? Actually, the
question will be asked at two different times: at t1 - when
Sleeping Beauty is just awaken on Monday - and t2 - when
Sleeping Beauty has been told that it is Monday. Let us call
the first question Q1 and the second Q2 . In the sequel, Pi
(i ∈ {0, 1, 2}) will denote Sleeping Beauty’s credence at
ti , that is, her beliefs concerning the relevant propositions.
Let’s begin with question Q1 : what should be the value of
P1 (HEADS)? There are basically two camps: the halfers
and the thirders. The thirders claim (following (Elga 2000))
that P1 (HEADS) = 1/3 whereas the halfers claim (following (Lewis 2001)) that P1 (HEADS) = 1/2. Now,
the answer to Q1 is intimately linked to the answer to Q2 .
As a consequence, the two positions are best described by
giving their answer to both questions. By conditionalization, one obtains P2 (HEADS) = 1/2 for the thirders and
P2 (HEADS) = 2/3 for the halfers. We can sum up the
positions of Lewis and Elga as follows :
The aim of this paper is to provide a case for the doublehalfer position, that is, the position according to which
Sleeping Beauty’s credence should be such that
P1 (HEADS) = P2 (HEADS) = 1/2
The double-halfer position is not new.1 My case for it
is based on the so-called imaging rule for probabilistic
change. In what follows, I will try to argue, first, that this
rule should be used by Sleeping Beauty and, second, that if
it used it leads to the double-halfer position.
1
See for instance (Meacham 2005) and (Bostrom 2006).
112
Q1
Q2
A. Elga
1/3
1/2
D. Lewis
1/2
2/3
Let’s turn to the arguments. Here, I follow essentially
Lewis’ reconstruction of the disagreement. First, it is supposed that the underlying state space W contains three (socalled centered) worlds: W = {HM, T M, T T } where
• in HM the coin lands heads and it’s monday
• in T M the coin lands tails and it’s monday
• in T T the coin lands tails and it’s tuesday
W is supposed to be the relevant state space because
each state of W solves all the uncertainties of Sleeping
Beauty - both her temporal location and the outcome of
the toss. Some propositions are, according to Lewis, ”common ground” between him and Elga. Here are the most
important2:
2
I follow Lewis’s notation. I skip propositions (3) and
(4): proposition (3) unfolds proposition (2) and proposition (4)
essentially equates HEADS with {HM } and T AILS with
{T M, T T }.
mitted to conditionalization to go from P1 to P2 . In the
current setting, a double-halfer position and any position
according to which P1 (HEADS) = P2 (HEADS) are
excluded.
(1) P1 (T M ) = P1 (T T )
(2) P2 (HEADS) = P1 (HEADS|M ON DAY )
= P1 (HEADS|{HM, T M })
(5) P0 (HEADS) = P0 (T AILS) = 1/2
(1) is a form of the Indifference or Laplacean Principle reflecting the fact that Sleeping Beauty cannot distinguish at
her awaken between Monday and Tuesday. (2) says that between t1 and t2 , Sleeping Beauty changes her credences by
conditionalization. (5) expresses the fact that at t0 Sleeping Beauty’s credence obeys to the ”objective probability”
of the coin landing heads or tails.
Elga’s starting point is that the coin could perfectly be
tossed on Monday night. If one accepts this, then, still
by endorsement of objective probability, Sleeping Beauty
should believe that the probability of HEADS is 1/2 after she has learned that it’s Monday. That is, according to
Elga:
(E) P2 (HEADS) = 1/2
From (E) and the common ground (including crucially the
rule of conditioning expressed by (2)), one has to conclude
that P1 (HEADS) = 1/3. Elga’s argument is a kind of
bottum-up argument which starts from an answer to Q2 to
give an answer to Q1.
On the opposite, Lewis provides a direct answer to Q1
and infers from it an answer to Q2. Lewis’s premiss is
(roughly)3 the following one:
(L) P1 (HEADS) = P0 (HEADS)
Still, from (L) and the common ground (including crucially
the rule of conditioning expressed by (2)), one has to conclude that P2 (HEADS) = 2/3.
A point stressed by Lewis is that both arguments conclude
that P1 (HEADS) < P2 (HEADS) - more precisely, that
P2 (HEADS) = P1 (HEADS) + 1/6. This is a direct
consequence of the fact that halfers and thirders are com3
As a matter of fact, Lewis’s premiss is that ”only new evidence, centered or uncentered, produces a change in credence;
and the evidence [{HM, T M, T T }] is not relevant to HEADS
versus TAILS.” (Lewis 2001)
113
Both Elga’s and Lewis’s basic intuitions are appealing.
Elga’s intuition is that the coin could be tossed on Monday
night and that in this case, one should endorse the objective
probability of HEADS as her credence. Lewis’s intuition
is that on Monday morning, there is no new evidence that is
relevant to the credence concerning HEADS. Therefore
the credence toward HEADS at t1 should remain the same
as at t0 . What is clear from the remarks above is that, given
the common ground between Elga and Lewis, these intuitions cannot be reconciled. As a consequence, someone
who finds both intuitions appealing (and accordingly who
accepts both (E) and (L)) faces the following dilemma: either to give up one of the intuitions, or to give up part of
the common ground.
2 CONDITIONING AND IMAGING
I will put into question neither proposition (1) nor proposition (5) but rather proposition (2), namely the use of conditionalization to go from P1 to P2 . Let’s note first that
what is learned at t2 by Sleeping Beauty (”it is Monday”)
is a context-sensitive information. Importantly, contextsensitive propositions are in general problematic for conditionalization. To be more precise, there are two central
properties of conditionalization that are problematic: concentration and partiality. (i) Concentration is the fact that
the beliefs of an agent who conditionalizes become more
and more concentrated as she learns more and more information. Each time a non-trivial information4 compatible with the initial probability5 is learnt, the support of
the posterior probability distribution is strictly included in
the support of the initial probability. This implies preservation (Grdenfors 1988), namely that if a proposition A is
believed with certainty then after having learned any information compatible with the initial beliefs, A is still believed with certainty. And this implies that if a proposition
has null probability, its probability will remain null whatever information compatible with the initial probability is
learnt. (ii) Partiality is the fact that when an information is
incompatible with the agent’s initial beliefs, the new probability distribution is undefined. These issues are general,
but they give us prima facie reasons to look more carefully
at the use of conditionalization in Sleeping Beauty’s scenario.6
4
That is, an information that excludes at least one of the world
in the support of the initial probability distribution.
5
That is, an information whose intersection with the support
of the initial probability is not empty.
6
For detailed discussions, see (Arntzenius 2003) and
(Meacham 2005).
Conditionalization is often viewed as the only reasonable
rule for changing one’s credence7. Other rules are conceivable, however. Consider for instance the imaging rule
introduced by (Lewis 1976) as the rule that matches Stalnaker’s conditional. The basic idea is this. For each world
w and each proposition A, wA is the closest world to w
where A is true.8 Suppose that the agent is informed that
A holds. In the case of conditionalization, all the weights
of the A-worlds are allocated to A-worlds compatible with
the prior in a way that preserves the relative probabilities.
In the case of imaging, the weight of a A-world w is exclusively allocated to the world wA . The rule of imaging is
therefore the following:
P Im(A) (w) =
P
′ }
{w ′ ∈W :w=wA
P (w′ )
In other words, the probability of w after imaging on A
is the sum of the probabilities of the worlds w′ such that
w is the closest world to w′ where A is true. As stressed
by Lewis, imaging satisfies a form of minimality: there is
”no gratuitous movement of probability from worlds to dissimilar worlds” (Lewis 1976). Here is an example that is
intended to illustrate the divergent behavior of conditionalization and imaging:
Exemple 1 (Apple & Banana, partial beliefs) A basket
may contain an apple and a banana. There are four
possible states : {AB, A¬B, ¬AB, ¬A¬B}:
AB
¬AB
A¬B
¬A¬B
Suppose then that the initial probability, P , is such that the
agent is certain that there is at least one fruit in the basket and that the same weight is allocated to the remaining
states:
1/3
1/3
1/3
0
The agent receives the following information: I =
{A¬B, ¬A¬B}, that is, there is no banana in the basket. If the agent relies on conditionalization, her new belief
should be this:
0
0
0
0
2/3
1/3
3 IMAGING AND SLEEPING BEAUTY
As one would expect, the debate between halfers and thirders is dramatically transformed if one adopts a rule of belief change that is different from conditionalization. Let
see what happens, for instance, if one relies on imaging.
To apply the imaging rule, one needs first to make some
assumption on the similarity between worlds. In the case
of Sleeping Beauty, the information that Sleeping Beauty
learns at t2 (”it is Monday”) excludes one world from P1 ’s
support, namely the world T T . Therefore, the only parameter that has to be specified is the closest world to T T
where it is true that it is Monday. I think it is a rather natural assumption to suppose that T M is the closest world to
T T where it is true that it is Monday. Granting this assumption, the imaging rule is easily applied to Sleeping Beauty’s
scenario.
As I said before, I consider both Elga and Lewis’s basic intuitions as attractive. Let’s start from Lewis premiss (L) and the rest of the common ground (propositions
(1) and (5)). If one relies on imaging, then P2 (T M ) =
Im(MON DAY )
(T M ) = P1 (T M ) + P1 (T T ) = 1/2 and
P1
Im(MON DAY )
(HM ) =
P2 (HEADS) = P2 (HM ) = P1
P1 (HM ) = 1/2. In other words from the Lewisian premiss (L) there results a double-halfer position: the credence of Sleeping Beauty toward HEADS is the same
at t1 and t2 , namely 1/2. But we could start from
Elga’s intuitions as well and suppose that P2 (HEADS) =
P2 (T AILS) = 1/2. Now, if one ”backtracks” the imaging
rule in the same way one ”backtracks” conditionalization
in Elga’s original argument, one obtains P1 (HEADS) =
P2 (HEADS) = 1/2.
What this shows is that if one starts either from Elga’s or
from Lewis’s basic intuition and that one relies on the imaging rule rather than on conditionalization, then one obtains
the double-halfer position. But what this does not show is
that one should rely on imaging rather than on conditionalization. At this point, the crucial issue is to adjudicate
between several rules of belief change.
4 REVISING AND UPDATING
1
0
But if the agent relies on imaging with ABI = A¬B and
¬ABI = ¬A¬B, one obtains this:
7
The diachronic Dutch Book argument is the main justification
for this belief.
8
To be sure, it is not an assumption that is kept in Lewis’ own
semantics of conditionals. Lewis factorizes this assumption into
the Limit Assumption and the Uniqueness Assumption and rejects
both.
114
For more than two decades, formal epistemology has developed rules of full belief change. It has been convincingly
argued by (Katsuno & Mendelzon 1992) that one should
carefully distinguish two kinds of belief change contexts:
contexts of revising where the agent learns an information
about an environment that is supposed to be stable and contexts of updating where the agent learns an information
about a potential change in her environment. If, for instance, the agent has beliefs concerning the content of a
basket of fruits that may or may not contain an apple and
that may or may not contain a banana, a revising information could be that there is no banana in the basket and an
updating information could be that there is no more banana
in the basket (if there was any). The point is that rules of
belief change have to be different is these two kinds of contexts. In a revising context, the new belief set given an information that is compatible with it has to be included in the
initial belief set9 whereas in an updating context, the new
belief may not be included in the initial belief set10 . This
results in two kinds of rationality postulates: the so-called
AGM-postulates for belief revision (Grdenfors 1988) and
the KM-postulates (Katsuno & Mendelzon 1992) for belief
updating. This is illustrated by the following example:
Exemple 2 (Apple & Banana, full beliefs) A basket may
contain an apple and a banana. There are four possible
states : {AB, A¬B, ¬AB, ¬A¬B}. Suppose the agent believes initially that there is at least one fruit in the basket
i.e. K = {AB, A¬B, ¬AB}:
AB
¬AB
of partial belief change? The question was left unanswered
until recently. But (Walliser & Zwirn 2002) have shown the
following result, which is at the very core of my argument:
conditionalization-like change rules may be derived from
probabilistic transcription of AGM-postulates for belief revision whereas imaging-like change rules may be derived
from probabilistic transcription of KM-postulates. This
result can be interpreted in the following way: if one is
guided by rationality postulates of full belief change, then,
in a revising context one should rely on conditionalization
whereas in an updating context one should rely on imaging.
To sum up my argument: in the previous section I have
argued that if one starts either from Elga’s or Lewis’ basic
intuitions and that one relies on imaging, then one obtains
the double-halfer position. In the current section, I have
argued that if the context of belief change is an updating
context, then one should rely on imaging. It remains to be
argued that when Sleeping Beauty learns that it is monday
(at t2 ), it is indeed an updating context, and not a context
of updating.
A¬B
5 UPDATING AND SLEEPING BEAUTY
Then the agent believes a revising message according to
which there is no banana in the basket. The new belief
set will be: K r = {A¬B}. But suppose the agent is informed that something has happened such that if there was
a banana in the basket, it is no more in it. In this case,
it is much more intuitive to reason in the following way:
if the true world was AB, then it is now A¬B ; if it was
A¬B, it is unchanged ; and if it was ¬AB, then it is now
¬A¬B. Therefore one would obtain as a new belief set
K u = {A¬B, ¬A¬B} which differs from K r . To sum up:
revising
”there is no banana”
A¬B
updating
”there is no more banana (if there was any)”
A¬B
¬A¬B
The Sleeping Beauty scenario involves rules of partial belief change. A natural question would then be the following: if one accepts the distinction between revising and updating contexts (as I do), what are the corresponding rules
9
In the same way that, after conditionalization, the support of
the new probability distribution is included in the support of the
initial one, if the information is compatible with the latter.
10
In the same way that, after imaging, the support of the new
probability distribution may not be included in the support of the
initial one, even if the information is compatible with the latter.
115
An updating context (of belief change) is a context where
an agent is informed about a potential change of her situation. Now, in so far as in Sleeping Beauty’s scenario
we consider centered worlds, an information bearing on
a change of temporal location is an information about a
change of Sleeping Beauty’s situation. And it is precisely such an information that the experimenter provides
to Sleeping Beauty at t2 . Therefore, it seems that this is an
updating context.
But if one looks more carefully at the exact timing of information in the Sleeping Beauty scenario, things are much
less clear that they appear to be. As a matter of fact, when
Sleeping Beauty becomes aware at t1 (at her awakening
on Monday) that she is on Monday or Tuesday, this is a
true updating context since the day it is is different from
the day it was at t0 . But when she learns that is it Monday (at t2 ), the information does not bear on a change
that took place between t1 and t2 . At t1 , Sleeping Beauty
becomes aware that the actual (centered world) is among
I1 = {HM, T M, T T }. At t2 , the information that is
given to her allows her to refine her beliefs since she learns
I2 = {HM, T M } ⊂ I1 . From this point of view, the information provided at t2 seems to be a revising context. On
the other hand, I2 is a refinement of an updating-type information, namely I1 . This issue shows that the distinction
between updating and revising contexts is underspecified
and raises quite a general question: when an agent learns
successively two informations at t1 and t2 , which both bear
on a change that took place between t0 and t1 , should we
view the second information as a revising or as an updating
context?
Note that this question is crucial for the double-halfer position: if the information that is provided to Sleeping Beauty
at t2 has to be considered as a revising context, then our
case for double-halfers collapses. I won’t provide a general
answer to this question but I will exhibit an example with a
similar structure, in particular where the agent receives two
informations at two different times, and where it is more
intuitive to handle the second information by updating.
Exemple 3 (Apple, Banana & Coconut) A basket may
contain three fruits: an apple, a banana and a coconut.
There are eight possible worlds:
ABC
¬ABC
AB¬C
¬AB¬C
A¬BC
¬A¬BC
A¬B¬C
¬A¬B¬C
Suppose first that the agent’s initial beliefs can be represented by the following probability distribution P0 :
0
1/4
1/4
0
1/4
0
1/4
0
It happens that between t0 and t1 , if there was a banana it has been removed and if there was a coconut it
has been removed as well. But suppose that at t1 the
agent learns only that there is no more banana (I1 =
{A¬BC, A¬B¬C, ¬A¬BC, ¬A¬B¬C}) and learns at
t2 that there is no more banana and no more coconut (I2 =
{A¬B¬C, ¬A¬B¬C}). The shift from P0 to P1 is clearly
Im(I1 )
an updating context, therefore P1 should be P0
i.e.:
0
0
0
0
1/4
1/4
1/2
0
Now the question is: how should the agent handle the information I2 ? If he still uses the imaging rule, he will obtain
Im(I2 )
for P2 = P1
:
0
0
0
0
0
0
3/4
1/4
0
0
0
0
CONCLUSION
Imaging provides a way to support the double-halfer position, which may be viewed as a reconciliation of Elga and
Lewis. Note that the use of the imaging rule in the Sleeping
Beauty scenario rests on the same fundamental assumption
as the one that underlies both Elga’s and Lewis’ arguments,
namely that information about one’s temporal location has
to be treated in the same way as any other kind of information. To rigorously assess this assumption, one would need
to make explicit the structural role of temporal factors in
rules of belief change but this I leave for future investigation.
Acknowledgments
I would like to thank for their comments J. Baratgin,
D. Bonnay, T. Daniels, I. Drouet, P. Egr, Th. Martin,
B. Walliser, D. Zwirn and audiences from “Probability,
Decision, Uncertainty” (IHPST, Paris), “Paris-Amsterdam
Logic Meeting for Young Researchers (ILLC, Amsterdam)
and the “Seminar on Belief Dynamics” (Dept. of Philosophy, Lille III). To my knowledge, the first to establish some
connection between Sleeping Beauty and updating was J.
Baratgin. Note that, even if I rely strongly on theoretical
results by Walliser & Zwirn, their view of Sleeping Beauty
is different of mine.
References
Note that this is what the agent would have obtained had
Im(I2 )
Im(I2 )
he directly known I2 (i.e. P0
= P1
). If the agent
Cond(I2 )
uses conditionalization, he will obtain for P2 = P1
:
0
0
Cond(I )
2
than P1
. Apple, Banana & Coconut bears some
similarity with Sleeping Beauty: (a) the relevant change
in the world takes place between t0 and t1 ; (b) what the
agent learns at t1 and t2 bears on the change in the world
that has taken place between t0 and t1 ; and (c) the second
information is a refinement of the first (I2 ⊂ I1 ). As a consequence, the example provides some support to the basic
claim of the present section, namely that the information
received by Sleeping Beauty at t2 should be viewed as an
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