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\begin{document}
\title{Epistemology Quantized: circumstances in which we should come to believe in the Everett interpretation}
\author{David Wallace\thanks{Magdalen College, Oxford}}
\date{July 2005}
\maketitle

\begin{abstract}
I consider exactly what is involved in a solution to the probability
problem of the Everett interpretation, in the light of recent work on
applying considerations from decision theory to that problem. I suggest
an overall framework for understanding probability in a physical theory,
and conclude that this framework, when applied to the Everett
interpretation, yields the result that that interpretation
satisfactorily solves the measurement problem.
\end{abstract}

\section{Introduction}

Recent years have seen substantial progress on both of the main problems
traditionally associated with the Everett interpretation. 

The first of these, the `preferred
basis problem' is not my concern here; suffice it to say that I believe
considerations from decoherence theory, together with the right
philosophical analysis of higher-order ontology, appears sufficient to
resolve it.\footnote{For a more detailed analysis see
\citeN{wallacestructure}.} My concern is with the second, the `probability
problem', where again a combination of technical and conceptual results
have transformed a problem which appeared intractable.

The probability problem can usefully be divided into two parts:
\begin{description}
\item[The incoherence problem:] In a deterministic theory where in
theory we might have perfect knowledge, how can it even make sense to
assign probabilities to outcomes?
\item[The quantitative problem:]Even if it does make sense to assign
probabilities to outcomes, why should they be the probabilities given by the
Born rule?
\end{description}
Until fairly recently, both problems seemed intractable: any attempt to resolve either, 
at least without modifying the basic structure of quantum mechanics seemed doomed
to failure. But recent work by
\citeN{deutschprobability}, \citeN{saundersprobability}, \citeN{vaidman}, \citeN{greaves},
myself (\citeNP{decshort},\citeNP{decprob},\citeNP{wallace3kinds},\citeNP{wallacebranching})
and others has made use of considerations from decision theory, personal
identity, philosophy of probability, and philosophy of language to
provide both conceptual frameworks for thinking about probability in the
Everettian context, and --- perhaps more surprisingly --- concrete
mathematical results which purport to be (Everett-interpretation-specific) 
derivations of the Born rule.

Intractability has a certain simplicity. By contrast, our current state
of knowledge about the probability problem has become quite complex and
controversial; it is unclear exactly how to frame the problem which we
are trying to solve, and correspondingly unclear what would count as a
solution. 

This paper is an attempt to untangle the situation: in it I have tried
to lay down exactly what I believe is involved in a solution to the
measurement problem, and to show what needs to be done for the Everett
interpretation to be understood to provide such a solution. 

Section \ref{probabilitysection} is concerned with
probabilistic theories in general and not with quantum mechanics (let
alone the Everett interpretation) in particular; in this section I
advocate what I call `cautious functionalism' as the correct attitude to
take to probability in physical theories.

In the remainder of the paper, I apply this framework to quantum
mechanics and the Everett interpretation. Section \ref{everettsection}
lays out my preferred approach to the Everett interpretation: in sections 
\ref{quantumalgorithm}--\ref{subjectiveneed} I
set out a framework for what is required of a solution to the
measurement problem and how that framework applies to the Everett interpretation (in particular, 
how it requires that we understand quantum branching events as genuinely uncertain in some sense). 
Sections \ref{saundersargument}--\ref{subjective2}
analyse how uncertainty can be understood to exist in a branching
universe, and section \ref{probability} considers how (given an understanding of uncertainty) 
quantitative probability can also be understood in a branching universe
--- understood better, in fact, than in a non-branching universe! 

In section \ref{fissionsection} I consider an alternative approach: the `fission program', in which 
we make
no essential use of the concept of uncertainty or subjective probability in
considering branching. Although I conclude that the fission program is
ultimately not a viable approach to the Everett interpretation, it does
lead to some important insights which become relevant in 
section \ref{justification}, in which I consider whether the decision-theoretic
principles which are part and parcel of the `uncertainty' concept are
really justified in a branching universe. Section \ref{conclusion} is the
conclusion.


\section{What is probability?}\label{probabilitysection}

\subsection{Objective probability and the principal principle}\label{PP}

It is fairly widely accepted that there are (at least) two distinct
notions of `probability'. 

Firstly, there is subjective probability, or `credence': that is,
probability taken as a measure of an agent's degree of belief in a
hypothesis. Here, probability is taken as a numerical quantification of
the notions of `likelihood' and `uncertainty', which arguably we already
understand in qualitative terms.

Foundationally speaking, credence is in fairly good shape. If asked
to justify its use, our first response is to appeal to so-called `Dutch
Book' arguments to show that any other method of quantifying our beliefs
forces us to lose money. On careful analysis these are not really
convincing, but they have far more convincing relatives in
\emph{decision theory}. Decision theory (in the forms developed by
\citeN{savage} and \citeN{jeffrey}) begins with a purely qualitative
notion of an agent's preferences over various courses of action, and
allows us to prove a \emph{representation theorem} to the effect 
that that agent must quantify his uncertainty in terms of
probability or violate some intuitively reasonable principle of
rationality. (For a brief but careful defence of this approach to credence, see
the first chapter of \citeN{kaplan}; for more details, see
\citeN{joyce}).

But if credence is well defined, it is nonetheless too thin a notion 
to play the
role of \emph{objective} probability (OP), the robust, observer-independent
property which we use in much of science and in particular in quantum
mechanics. We have, I believe, no truly satisfactory analysis of what
sort of entity or property this `objective probability' really is. 

We do, however, 
have a good theory of how OP fits
into our general conceptual scheme: it does so via Lewis's
\emph{Principal Principle} \cite{lewischance}. Recall that the
Principal Principle states, roughly, that if I know the objective probability of an event $E$ 
to be $p$, then I am rationally compelled to
set my personal credence in $E$ to be $p$. More precisely, it states
that if $X_p$ is the proposition that $OP(E)=p$, and $A$ is any
`admissible'\footnote{Inadmissible propositions are not formally defined
by Lewis, but essentially the qualifier is there to rule out propositions which
are directly about the future --- \egc, prophecies or the testimony of time travellers.} 
proposition compatible with $X$, then
\be Cr(E|A \& X_p)=p.\ee 

(Lewis uses `chance' as his term for OP, and regards it as necessarily involving indeterminism:
classical statistical mechanics, and Bohmian mechanics, aren't chancy in
his sense. I shall use the term more broadly: the probabilities of
statistical mechanics seem as robust and as observer-independent (and as
mysterious!) as those of stochastic theories, and (provided that
`inadmissible propositions' is extended to cover microphysical knowledge
such as the actual microstate of the system or the actual location of
the Bohmian corpuscle) the Principal Principle applies just as well for
them. There is no particular etymological reason to restrict `chance' to chances
in Lewis's sense, but to avoid confusion I use the more neutral, albeit more
cumbersome, `objective probability' to cover
the more general class of physics-defined probabilities.)

\emph{Other} than that it satisfies the Principal Principle, what do we
actually know about OP? Answer: essentially nothing.
\emph{Mathematically}, it enters through either a measure on the set of
initial conditions or a stochastic differential equation --- that is, in
effect, through a measure on either the initial conditions or on the
dynamically possible histories. But the interpretation of that measure
remains obscure. There are proposals --- such as frequentism, or Lewis's 
Best-Systems analysis --- that try to define that measure in terms of
facts about the actual world, but it is at best extremely
controversial both whether these proposals allow us to define an appropriate measure at all, 
and whether it could be shown to constrain a rational agent's credences.

This being the case, we may as well take the Principal Principle as
offering a \emph{functional definition} of OP. That is, if some
physical theory $T$ enables us to define some magnitude $C$ for events, then $C$
is OP just if anyone believing $T$ is compelled to constrain his
credences to equal $C$. More formally, 
$C$ is OP iff for any event $E$, 
if $T$ together with (admissible) background information $B$ entails
that $C(E)=p$, then \be Cr(E|B \& T)=p.\ee

This `functional definition' allows us to sidestep --- temporarily ---
the question of what chance is when we consider its role in science. For
suppose we do have a theory $T$ which allows us to define some magnitude
$C$, and suppose $PP_C$ is the proposition that $C$ satisfies the
functional definition (we might loosely say: satisfies the Principal Principle --- hence the notation). 
It follows that if we accept both $T$ and $PP_C$,
we should set our credence in an event $E$ equal to $C(E)$. If $C(E)$ is
high and our prior credence in $E$ is much less high, we should regard
$(T\& PP_C)$ as explanatory of $E$, and thus regard $E$ as reason to
accept $T$ and $PP_C$.

This argument can be rephrased in Bayesian terms:
\be Cr(T\&PP_C|E)=\frac{Cr(E|T\&PP_C)Cr(T\&PP_C)}{Cr(E)};\ee
hence
\be
\frac{Cr(T\& PP_C|E)}{Cr(T\& PP_C)}=\frac{C(E)}{Cr(E)}.\ee
Since to accept both $T$ and $PP_C$ is to accept the existence of
objective probability, it follows that we can gain evidence --- even
very powerful evidence --- for objective probability functionally
defined, without ever knowing what sort of thing that `objective
probability' really is.

\subsection{Three ways of satisfying the functional
definition}\label{threeways}


Nonetheless, we'd still like to \emph{know} what it is.   
What \emph{is} the ``magnitude $C$'' defined by the theory $T$? There are
essentially three options. The first, which might be called
\emph{functionalism},\footnote{Functionalism is close to the position
espoused by \citeN[xiv-xvi]{lewispapers2}, who requires that probabilities be shown
to be `Humean properties'.} asserts that probability is some 
physically-definable property (or set of physically-definable 
properties) which can be defined independently of the
Principal Principle but which can be shown to satisfy the functional
definition which the Principle provides. 

According to functionalism, $PP_C$ must somehow be a logical consequence
of $T$ (more precisely: of $T$ together with general principles of
rationality and possibly other background assumptions). 
For instance, a frequentist (put crudely) identifies probability
with long-run relative frequency; he is correct iff it can be proved that a rational agent
knowing the long-run relative frequency of a particular outcome of 
a repeated experiment would set his credence in getting that particular
outcome on a single run equal to the long-run relative frequency.

The problem with frequentist versions of functionalism is that we have
very little idea how to prove that long-run frequency can be proved 
to satisfy the 
functional definition of probability; the problem with functionalism 
more generally is that we have very little idea how to prove that
\emph{anything} can be proved to satisfy that definition. The problem is
not so much that we don't know how to define something with the formal
properties of probability (relative frequencies do okay here; Lewis's
`best-systems analysis' \cite{lewis94} does better); rather, it
is that we do not have any really plausible account of why that
something should place any constraints on my credences. Why should I, 
betting in the here and now on whether \emph{this} atom is going
to decay, care at all about how many similar atoms in remote regions of
the Universe have decayed?

The apparent impossibility of finding a naturalistic candidate for
probability provides much of the attraction for the second option,
\emph{primitivism}. Primitivists accept the functional definition as a
basic law of nature: not something to be deduced as holding for an 
independently characterisable property $C$, but something which is postulated 
to be true of $C$ and which defines $C$ via its role in the law. 

The strategy is not unfamiliar. Take \emph{charge}, for example: it is
highly plausible to suppose that the property of having charge $q$
cannot be defined or made sense of other than via the role of charge in
the laws of electromagnetism (see \citeN{lewistheoretical} for a full
working-out of this strategy\footnote{See see \citeN{shoemakerproperties} and \cite{mellorproperties} for further defences of this
`functional' definition of properties; note also that Lewis seems later (\citeNP{lewisnewwork}) to have 
moved to a different position.}). Furthermore, it seems to fit well with
the mathematical structure of our existing probabilistic theories: as
was alluded to above, both in stochastic theories like GRW and in deterministic 
theories with an unknown microstate, objective probability eventually
enters as a measure on the space of physically possible histories, which
has no role in the theory except to be probability: that is, to satisfy
the functional definition.

Nonetheless, primitivism is a desperate strategy. Do we really want to 
take a \emph{rationality principle} as a basic
postulate of nature, on a par with the dynamical laws of spacetime and
field theories? Are we prepared to accept that it is logically possible
that every physical property of the universe could remain the same in
some alternate possible world, and yet that what is rational could
change? 

The third strategy  ---
\emph{eliminativism} --- is in turn motivated by the grave conceptual 
problems of the other two strategies. It is
the doctrine that objective probability does not exist: that it is
unsurprising that we cannot work out what fits the functional definition
of probability, for nothing does.

If primitivism is desperate,  eliminativism is all but unacceptable.
The ubiquity of the concept of objective probability
throughout science, and indeed in ordinary life (think of the roulette
wheel and the fair coin) makes it intolerable not to accept that \emph{something} fits the
functional definition given by the Principal Principle.
(Or so it seems to
me; but others disagree. How they can maintain with
a straight face that \emph{the half-life of uranium} is not an objective
property of the world is beyond me, but I shall not attempt to defend
the point further here.)

\subsection{Cautious functionalism}\label{cautiousfunctionalism}


\begin{quote}
Philosophers' Syndrome: mistaking a failure of imagination for an insight into necessity
\cite[p.\,410]{dennettconsciousness}
\end{quote}

Faced with this dispiriting trilemma, what attitude should we take
towards probability statements in our physical theories? I suggest a
cautious functionalism. Unlike the other two strategies, the only
philosophical problem with functionalism is our total inability to think
of anything that might fit the functional definition \ldots but our
imagination has failed before. 

Cautious functionalism, faced with a theory $T$ that defines some
property $C$ which seems to play the role of probability, proceeds as
follows. It collects evidence, as above, for the joint hypothesis ($T \&
PP_C)$, all the while acknowledging that $T$ comes with an attached
promissory note: eventually we will need an
account both of how $C$ is to be defined independently of the Principal
Principle and of how, given this independent characterisation of $C$, $PP_C$ can be 
derived. Until the note is cashed, the
theory has a certain phenomenological, non-fundamental character, yet
for all that it may be highly explanatory. And if the note is never
cashed, perhaps eventually we would be wise to become pessimistic and
reconsider primitivism or (\emph{just} possibly) 
eliminativism.\footnote{It may seem somewhat strange that $PP_C$, which according to functionalism must
in principle be derivable \emph{a priori}, is nonetheless the sort of thing for which
we can collect evidence. But there is nothing strange about having
\emph{a posteriori} evidence for \emph{a priori} truths. (How many of us
have actually worked through the proof of Fermat's last theorem, rather
than trusting the word of others that it is provable?) There is not even
anything particularly strange about gaining \emph{a posteriori} evidence
for a normative principle. For example, consider the Monty Hall problem: a game-show 
featuring three doors,
with goats behind two of them and a car behind the third; you choose one
door, and without revealing what is behind it the host opens one of the
other doors to reveal a goat. Two doors remain; if offered the chance to
open your original choice or the other one, should you swap? Yes, in
fact; but remarkably many people both get the problem wrong and
resolutely refuse to believe that they \emph{have} got it wrong. Such a
person might watch many reruns of the game show, and conclude that in
fact he \emph{should} swap, and yet be at a loss to understand
\emph{why} he should swap.}

Cautious functionalism is not actually (I hope!) that contentious a
position. Primitivism and eliminativism are not views that one would adopt despite having a 
perfectly satisfactory functionalist candidate for objective probability; they are views that
one adopts in response to the belief that nothing could be such a candidate. (For instance,
objections to frequentist definitions of probability are made on the
grounds that they \emph{don't work} (that is, in my framework, that they
don't fit the functional definition) --- not that even if they did work
they would be unsustainable.) So a primitivist, or an eliminativist, can
be seen as a cautious functionalist who has already reached the point of
pessimism and given up on any successful functionalist analysis. Either,
I hope, would willingly recant their pessimism if shown that a
functionalist analysis was after all possible.

\subsection{Is the functional definition
complete?}\label{isPPsufficient}

Before turning from probability in general to the specific case of
quantum mechanics, I must address one possible worry: can we really be sure, in using the Principal Principle to produce our 
`functional definition of probability', that we
have not left out a crucial feature of objective probability? Maybe there is some
additional feature $F$ of probability, such that something satisfying the
functional definition but not possessing $F$ would not be objective probability. 

As a matter of semantics, this may well be defensible --- our ordinary
`probability' talk is inchoate and ambiguous between objective
probability and credence, and maybe it does have additional features and
complications. But if we are discussing the \emph{scientific conception
of objective probability} --- that is, the theoretical term which we have
introduced to explain experimental situations which seem to need
probability --- then I am not sure what evidence we could have for $F$. 

To see this, let us temporarily introduce  ``quasiprobability'' as a
term for anything satisfying the functional definition given by the Principal Principle and discussed above. 
It is, I take it,
uncontentious that probability (whatever it is) satisfies the
definition\footnote{For an argument to this effect, see Lewis's ``questionnaire'' \citeNP{lewischance}; similar
arguments have been advanced by, \egc, \cite{mellor}.}, so probability is a certain sort of quasiprobability --- one
possessing the additional feature $F$ which was left out of the
functional definition. Suppose we have a collection of experimental data
which is explained well by some theory $T$ involving `genuine' probability.
That is, $T$ assigns high probability to the relative frequencies which
we in fact observe. Suppose also that we have some other theory $T'$ 
which involves only `quasiprobability' but which assigns high
quasiprobability to the observed relative frequencies --- in fact, which
assigns quasiprobabilities exactly equal to the `real' probabilities
assigned by $T$.

In such a situation the extra property $F$ appears quite redundant. The
evidential process by which we continue to test $T$ and $T'$ connects
\mbox{(quasi-)}probability to our observations entirely through the Principal
Principle, which applies to quasi-probability whether
or not it has the property $F$. And the same will apply when we come to
use the theory in practical applications --- to go from 
(quasi-)probabilities to actions, we must go via the Principal
Principle.

The only use that I can see for $F$ is that it might be a necessary part
of \emph{why} probability satisfies the functional definition. If, for
instance, relative frequencies were in fact the only possible candidate for a realiser of the functionalist definition of
probability, then something with the formal properties of probability
which is not a relative frequency could not be true probability.

But this is only to say that our initial guess that a particular
quantity \emph{is} a quasi-probability could be wrong. If it is wrong,
if that quantity does not satisfy the functional definition, then it is
no quasi-probability at all. And if some quasi-probability \emph{could}
be shown to satisfy the functional definition (as, I shall argue, occurs
in the Everett interpretation) whilst lacking $F$, then it would be a 
demonstration that $F$ was not after all necessary.

If there are requirements for probability over and above the functional
definition, I conclude that it is obscure at best what they could be.
For the rest of this paper, then, I shall assume that quasi-probability is a redundant concept. Anything which
genuinely does satisfy the functional definition is probability.

\section{The Everett interpretation and subjective
uncertainty}\label{everettsection}

\subsection{Interpreting quantum mechanics}\label{quantumalgorithm}

I have suggested a general framework for the understanding of theories
which incorporate objective probability. But what of quantum mechanics?
There the theory seems to speak of objective probability, but there also
the theory seems ill-defined without an `interpretation'.

As an interpretation-neutral approach to this question, I suggest the following: what we
currently possess is a \emph{theory fragment}. To be more precise: quantum mechanics 
can be understood as giving a description of certain
(usually microscopic) systems. But the connection between that
description and our observations proceeds not via a continuation of the
theory but via an \emph{algorithm}: when a state comes to describe a superposition of
macroscopically-definite outcomes, reinterpret the mod-squared-amplitudes
of each outcome as giving the objective probability of that outcome's
obtaining.

Regarded this way, this `theory fragment' (which we might call the Quantum Algorithm) is deficient in two ways,
the second by far worse than the first.
Firstly, in admitting objective probabilities without providing a theory
of same, it lacks a truly acceptable account of probability. However, as the previous section argues,
it shares this defect with all other
probabilistic theories, so perhaps calling it a `defect' at all is unfair.

Secondly, and more pressingly, it is not really a theory capable of
describing reality in an observer-independent way. Rather, the
`reinterpretation' of the quantum state which occurs when it becomes
macroscopic has no real explanation attached to it beyond its empirical
success. That is: the theory fragment gives no principled answer 
to the question of why the reinterpretation should be made; 
the only justification is that it seems to make correct predictions.


What is the goal of an `interpretation' of quantum mechanics? I
claim that it is to embed the Quantum Algorithm into a genuine theory,
one which does not resort to pragmatic considerations and
`reinterpretation' of its basic ontology. This embedding can be exact
(as the Everett interpretation claims to be) or approximate (as in
dynamical-collapse theories), and it may add additional structure (as in
hidden-variable theories), but in any case it must be sufficiently
accurate that it can 
reproduce the powerful empirical success of the 
Quantum Algorithm.\footnote{I ignore here the possibility that the Quantum
Algorithm itself is misunderstood by interpreting it as speaking of
objective probability. This possibility has been defended by (amongst others) Chris Fuchs (\citeNP{fuchsperes,fuchsinformation}), 
who wishes
to regard the quantum state as some sort of credence function rather
than something which is in any sense objective. This would lead to a theory of
probability of the sort which I earlier called `eliminativist' and which I criticised for taking
science insufficiently seriously; I shall not discuss it further here.}

So: our interpretation must resolve the ambiguity and ill-definedness
inherit in the move from micro to macro. Must it also offer a physical
property that satisfies the Principal Principle? It depends on how
strict our criteria are for an acceptable physical theory. The previous
section, in advocating `cautious functionalism', 
argued that \emph{eventually} we must move from treating the
Principal Principle as a primitive rule and begin to treat it as a
derived result, but no probabilistic theory so far proposed has met that
stringent test. For the reasons given in sections \ref{PP}--\ref{isPPsufficient}, I would maintain that
\emph{inter alia}, an interpretation could take the
Principal Principle as primitive for whatever physical properties it
likes, and yet be in as secure a position as any other probabilistic
theory which physicists treat seriously.

\subsection{The need for subjective uncertainty}\label{subjectiveneed}

What of the Everett interpretation? It aims to embed the Quantum Algorithm 
(that is, instrumentalist quantum mechanics) into a full theory
in the most naive possible way: that is, by extending the formalism
which we use for microscopic systems to cover all systems, be they
microscopic, macroscopic or cosmological. To do so, it must interpret
the macroscopic superpositions that result post-measurement as
describing a superposition of different `worlds': different 
quasi-classical structures in the wave-function, effectively isolated from one another,
with some worlds corresponding to each possible outcome of the
experiment.

It is not the task of this paper to consider how this many-worlds
description of macroscopic superposition may be justified; I present
my own proposed justification in \citeNP{wallacestructure}. But even if it can
be done, there remains a seemingly insurmountable problem: how can the
Quantum Algorithm, which involves objective chance, be incorporated into a
deterministic theory in which an agent could in principle have perfect
information about all of the salient features of the quantum state?

To elaborate: the Quantum Algorithm assigns objective chances to the possible outcomes
of quantum-mechanical experiments, and the defining feature of an objective chance is that
rational agents are compelled to set their credences equal to it (if they know it). Thus, the
Quantum Algorithm assumes that agents have credences in the different
outcomes, which in turn are to be understood as quantitative measures of
how certain or uncertain they are about the result of the experiment.

But if the Everett interpretation is true, what is there to be uncertain
about? The interpretation makes a deterministic prediction about the
post-experiment state: namely, that it consists of many effectively isolated
worlds, with different measurement outcomes occurring in different
worlds. How can we make sense of being uncertain of the outcome of an
experiment in a deterministic theory where we have perfect knowledge?

I have elsewhere called this the problem of \emph{subjective
uncertainty}. (The `subjective' should not be taken too literally: the
problem is to understand why uncertainty statements can rationally be
made by agents embedded in the Everett universe despite their total
knowledge of the relevant facts, but these statements could be `there
might be a sea-battle tomorrow' just as readily as `I don't know what
result I will see'.) 

\subsection{Saunders' argument for subjective
uncertainty}\label{saundersargument}

The idea of subjective uncertainty (though not the name) was originally
proposed by \citeN{saundersprobability}, who 
argues for the SU viewpoint by means of
an ingenious intuition pump.\footnote{I should note, in criticising Saunders, that his 
thought experiment was intended primarily to argue against the claim that quantum branching
is metaphysically incoherent, and only secondarily to defend subjective
uncertainty. I have no quarrel with Saunders' primary goal; he has
conclusively established that the `metaphysical incoherence' argument is
indefensible, given that analogous situations could perfectly well occur in classical
physics, and I discuss the matter no further here.}
 His argument proceeds by analogy with ``classical splitting'',
such as that which would result from a Star Trek matter transporter or an operation
in which my brain is split in two. It may be summarised as follows: in
ordinary, non-branching situations, the fact that I expect to become my
future self supervenes on the fact that my future self has the right
causal and structural relations to my current self so as to \emph{count}
as my future self.  What, then, should I expect when I have two or more
such future selves?  There are only three possibilities:
\begin{enumerate}
\item I should expect abnormality: some experience which is unlike normal human
experience (for instance, I might expect somehow to become both future selves).
\item I should expect to become one or the other future self.
\item I should expect nothing: that is, oblivion.
\end{enumerate}

Of these, (3) seems absurd: the existence of either future self would
guarantee my future existence, so how can the existence of \emph{more}
such selves be treated as death?  (1) is at least coherent --- we could
imagine some telepathic link between the two selves. However, on any
remotely materialist account of the mind this link will have to
supervene on some physical interaction between the two copies --- an interaction which is
not in fact present.  This leaves (2) as the only option, and in the
absence of some strong criterion as to which copy to regard as
``really'' me, I will have to treat the question of \emph{which} future self I become as 
(subjectively) indeterministic.

(In understanding Saunders' argument, it is important to realise that
there are no further physical facts to discover about expectations which could 
decide between (1-3): on the contrary, \textit{ex hypothesi} all the physical facts
are known.  Rather, we are regarding expectation as a higher-level
concept supervenient on the physical facts --- closely related 
to our intuitive idea of the passage of time --- and
asking how that concept applies to a novel but physically possible
situation.)

Of course (argues Saunders) there is nothing particularly important
about the fact that the splitting is classical; hence the argument
extends \textit{mutatis mutandis} to quantum branching, and implies that
agents should treat their own branching as a subjectively
indeterministic event. 

\subsection{Objections to Saunders' argument}\label{againstsaunders}

In responding to Saunders, \citeN{greaves} argues as follows:
\begin{quote}
What (to address Saunders' question) should [someone about to be
duplicated] \emph{expect} to see? Here I invoke the following premise:
whatever she knows she will see, she should expect (with
certainty!) to see. So she should (with certainty) expect to see
[herself as the first duplicate], and she should (with certainty)
expect to see [herself as the second duplicate]. Not that she should
expect to see \emph{both}: she should expect to see \emph{each}.
(\citeN[p.\,19]{greaves}; quotation modified to remove subscripts on
pronouns, which will play no role here.)
\end{quote}
As Greaves freely admits, Saunders is unwilling to accept this extra
possibility, on the grounds that it is just conceptually impossible to
expect two incompatible possibilities. She responds by claiming that it is
conceptually impossible to feel uncertain about something when one knows
all the facts about it.

I think that this impasse can be clarified (if not resolved) by
interpreting Greaves as raising the
possibility of \emph{concept failure}: a breakdown of our concept of
personal identity (Saunders, of course, has implicitly assumed that
this concept remains applicable in cases of splitting and seeks, via his possibilities (1)--(3),  to ask \emph{how} exactly
it is applicable.).


Many of Parfit's examples (in \citeNP[pp.199--306]{parfit}) are
designed to suggest the possibility of concept failure: Parfit's
intention in doing so is to persuade the reader to \emph{give up} on
personal identity as something worth caring about and to replace it with
a notion of \emph{personal survival} (according to which it is perfectly coherent
for me to care about my future successors without having any particular view about
whether they are \emph{me} or not). I think that Greaves is best read as
sharing this view: according to her version of the Everett
interpretation, I should replace any notion of \emph{becoming} a
post-splitting version of myself simply with the notion of \emph{caring
about} the future versions of myself, and should treat `I expect
experience $X$' simply as synonymous with `a future version of myself
has experience $X$'.

(This is not exactly how Greaves describes her own attitude to splitting:
she states that identity simply is survival, and so wishes to claim that
in cases of splitting, I become each of my future selves. However, she
does not make any particular effort to justify this claim other than by
extension from the non-splitting case, and I think that her claim is
most appropriately read simply as holding by definition: to Greaves as
to Parfit, survival is what matters in all identity cases, so we might
as well just use ``identity'' to refer to survival. Since Saunders is
using the term in a very different sense, to avoid confusion I shall 
eschew Greaves' terminology, and simply refer to `survival' in Parfit's
manner.)

Concept failure is not, I think, something which Saunders can just
reject \emph{a priori} in the context of his thought experiment. If 
personal identity is an emergent concept then
there is no reason why that concept should not simply break down in
certain situations --- especially new and alien ones, such as classical Parfittian
splitting. In fact, another of Parfit's thought experiments seems to
make it even more obvious that concept failure is a live option:
consider a machine which merges me with Greta Garbo 
\cite[pp.\,229--244]{parfit}. Adapting the example slightly, the 
machine has a dial with settings from 0 to 100. Set to 0, it leaves me
alone; set to 100 it obliterates me and creates Garbo \emph{ex nihilo};
set to intermediate values it creates someone with some of my, and some
of her, properties, in a ratio determined by the dial settings. I think
it is hard to argue that, for settings in the vicinity of 50, there is
any coherent concept of personal \emph{identity} here, or any reasonable
answer to the question `what do you expect to happen?'.

However, once concept failure has been admitted as a conceptual possibility, 
it is unclear that there need be a `correct'
answer to the Saunders thought experiment. There are in fact no
Parfittian splitters on Earth, so our existing concept is at best
underdetermined as regards splitting, and we are actually left with the
question of how (and if) it should be \emph{extended}. 

In fact, there is an extension of it available which answers Greaves'
worries about the lack of uncertainty in a deterministic universe: that
offered by \citeN{lewisfission}, who identifies a person (roughly) as a maximal
totally ordered set of person-stages (with the ordering in question
being the partial order: `is a descendant of'). According to Lewis's
proposal, if at some stage in my future I am to undergo branching into
two copies, then (timelessly) there are two people, and my current 
(pre-branching) person-stages are shared by both of them.

On the additional assumption that the correct referent of utterances and
of mental states is a person at a time (rather than a person-stage) it
follows that I am genuinely ignorant of my post-branching future. For
when I say `who will I become' that statement should actually be
ascribed to two versions of me (one of whom will, post-splitting, become each
version of me). Since (as a consequence of any physicalist approach to
the mind) any thoughts and beliefs I have at a time supervene on my
person-stage at that time, and since the two versions of me share all
person-stages prior to branching, it follows that it is impossible for
the two versions of me to resolve their ignorance.

What are they ignorant about? Not of course any propositional knowledge, but
something more indexical: something like a centred possible world (\citeNP{quinepropositionalobjects,lewisindexical}), 
but where the `centre' is a world-line and not a point. However, presumably
Greaves accepts that indexical ignorance is ignorance nonetheless, so
the Lewis proposal does seem to offer an extension of our existing
personal-identity concept that survives splitting.

However, just because we \emph{can} extend our concepts in this way, it
doesn't seem to be the case that we \emph{have} to do so. In fact, I
think this is a genuine choice, and one which would likely be made on
sociological grounds as much as philosophical grounds. 

To see this, consider  another example where personal identity is in
doubt: the simple (non-splitting) teletransporter, where I am
disintegrated and a copy of me is assembled somewhere else from the
information scanned from me in the disintegration process. Is
teletransportation survival, or death followed by the creation of a
doppelganger?

Well, suppose we come across an alien species who use teletransporters
all the time as a form of rapid transit, and universally \emph{believe}
that teletransportation is survival. It would be hubristic (at best) to
suppose that we know best here: presumably (with aliens as with other
emergent objects) what justifies the validity of a given theory of
personal identity is its predictive and explanatory power, and regarding
an alien about to step into the teletransporter as \emph{the same as}
the one who steps out of the arrival booth is far more explanatory of
the aliens' social and cultural practices.

We don't have teletransporters on Earth, so practical considerations
like this aren't currently available. But suppose we did, and suppose we
started using them widely; then our culture would (in that respect!)
become like the aliens', and just as we did with the aliens, we should regard
ourselves as surviving the teletransportation.

\emph{Would} we start using them widely? I imagine that we would, but I don't
know; in any case, it's a \emph{sociological} question, and could depend
wildly on extraneous factors. (Suppose, at one extreme, that a cover-up
leads everyone to think that the teletransporters are really wormholes
in spacetime that transport people whilst preserving their physical
continuity, and people have been using them quite happily for centuries
before the truth is discovered.)

It seems to me that the case of splitting is analogous. There is an
extension of our theory of personal identity according to which we
should expect survival and subjective uncertainty upon walking into a
classical
splitter, but I have no idea whether we would adopt it (I should think
that it would depend very sensitively on the circumstances in which the
splitters were introduced into our society).

But if the correct account of what it would be like to undergo
`classical splitting' rests on considerations like these, then ---
Lewis's account notwithstanding -- it becomes unclear that the analogy
helps us understand quantum-mechanical branching. For we should not be asking: how
should we extend our concepts if branching suddenly became possible, but
rather: how should we understand our existing concepts, given that
branching has been happening all the time? It isn't as if we are asked
what to think if we were suddenly transported from a classical to an
Everettian world, or if a switch had been flipped so that Everettian
branching was suddenly occurring.

For these reasons, I conclude that although I personally find Saunders' thought-experiment
to be a very effective way of seeing why it \emph{makes sense} to
consider branching as subjectively indeterministic, it fails in its
larger goal of showing that we are \emph{required} to regard it thus,
both because the question of how to regard classical splitting
seems indeterminate and sensitively dependent on the details of its
implementation in our society, and because in any case that question is
not fully analogous to the question of how we should think about
\emph{quantum} branching.


\subsection{Subjective Uncertainty again: arguments from interpretative
charity}\label{subjective2}

So: if splitting \emph{has} been occurring all the time, how should we
think about it? This brings us on to my own preferred solution to the
subjective uncertainty problem (which I present more fully in
\citeN{wallacebranching}): that the problem is solved by considering how
to interpret the language of inhabitants of a branching universe.

To elaborate: suppose that we consider a race of beings who inhabit a
branching universe (that is, a universe like that entailed by the Everett interpretation, where
one world physically splits into many), but where the beings don't realise this. Suppose further that when confronted with
what are in fact branching events, they are 
disposed to say `I am *uncertain what is going to occur'; (where `*uncertain') is
a term in their language. More
generally, suppose that they are normally disposed to assert `A *will happen' only
when it happens in every branch futurewards of the assertion, and to
deny it only when it happens in no branch. However, their philosophers,
asked to give an analysis of *uncertainty, are led by their ignorance of branching to the 
claim that one should be *uncertain of something only if there is some objective fact of the
matter about which to be *uncertain, and that `A *will happen' is true
iff A happens in the single determinate future.

What are the real meanings of `*uncertain' and `*will'? One possibility
(call it the Elite View) is to accept the philosophers' claims about their meaning, in which case
(given that these beings' universe really branches) it appears that almost all the
beings are using their language very inaccurately and are making all
manner of claims which are either false or meaningless. For when the
beings say (\eg)
``The Red party *will not win the election'', on the
Elite view they mean that in the single determinate future, the Red
party do not win the election. But in their branching universe, the
`single determinate future' is one in which the world splits into many
copies, in some of which the Red party win and in some of which they do
not. Does this make it true that the Red party win in the future
(because they do in some branches), or ill-posed that they do (because
such claims presuppose falsely that there is no branching)? If the
former, the beings' claim is false; if the latter, it is meaningless.

The alternative to the Elite View (call it
the Charitable View) regards the beings as using the terms entirely
correctly and accepts that the beings' philosophers are wrong about
their language.

Which view is correct? Given plausible assumptions about the philosophy of language
(notably, a certain externalism and/or holism about meaning) 
there can be no completely decisive answer: the beings are seriously
wrong about some aspect of their world-view, but we cannot decide what
they are wrong about prior to deciding their semantics. However, it is
highly relevant that the Elite View makes the beings wrong almost all
the time, about almost everything, whereas the Charitable View preserves
the truth of most of their discourse and falsifies only fairly
specialised parts of it (parts, furthermore, which were motivated by a
wildly inaccurate metaphysics). According to the radical-interpretation
approach to semantics espoused by \citeN{davidsoninterpretation},
Lewis\citeyear{lewisinterpretation,lewislanguage},\citeN{quinewordobject}
\emph{et al}, there are no further facts about
meaning beyond fit to usage and the best interpretation is, other things
being equal, that which makes most of the community's utterances come
out true. If we accept any variant of this approach, 
then the Charitable View easily seems to beat the Elite View.

Applying these arguments to our own language, it seems that we should conclude that
our use of `uncertain' to describe our attitude to the outcomes of
quantum measurements is entirely justified. We are wrong about some of
the referential underpinnings of uncertainty talk, but no wonder --- the
metaphysical considerations (such as the absence of widespread branching) 
which led us to assume those referential
underpinnings were drastically wrong.\footnote{An analogy: suppose that
actually the clear transparent liquid that we drink isn't $H_2 O$ at
all, it's been XYZ all along, but an International Conspiracy of
Chemists has hidden this from the public. Philosophers have produced semantic theories, on
the basis of this faulty information, that water is necessarily $H_2 O$.
When Woodward and Bernstein uncover the Conspiracy, how will the
Washington Post report it: as `water isn't $H_2O$' (the Charitable View)
or as `the sea doesn't contain water' (the Elite View).}

(It is perhaps worth stressing that this distinction between the Elite
and Charitable views is not just a linguistic dispute. Of course we can
define `uncertain' to mean whatever we like; we can define `blancmange'
to mean anything we like, too. But the argument is rather that our
existing talk of future possibility and uncertainty, and the entire
conceptual framework that goes with it, \emph{already} refers to
quantum branching, for all that we have not as yet realised it. As such, we are fully justified in applying our
existing machinery for testing and confirming theories to the Everett
interpretation, and in particular we can regard any evidence for the
Quantum Algorithm as supporting the Everett Interpretation. The point will be considered more
carefully in section \ref{justification}.)

There is in fact a variant of the Elite View that can avoid the problems
imposed by charity of interpretation.\footnote{Suggested in conversation
by several people (most clearly by Simon Saunders) in the last few years.} 
Suppose (following Lewis) that in cases of an object's branching, we
should regard there as having been two objects present all along, even
before the branching. On this basis (as has already been discussed)  an
agent can maintain that he is uncertain of an outcome iff there is some
fact of the matter to be uncertain about --- the `fact of the matter' is
which continuant agent he is. (Or possibly: in which continuant world he is: we might more naturally apply
Lewis's theory of identity to entire worlds rather than just to agents.
See \citeN{wallacebranching} for more on this matter.)

Whether we prefer to modify our
metaphysics of identity or our semantics seems to me to be a question
which may have no determinate answer other than utility (one is
reminded of Quine's indeterminacy thesis). The conclusion is the same
regardless: confronted with quantum-mechanical splitting, I should
correctly assert ``I am uncertain about $A$'' whenever I know that $A$
obtains in some but not all branches futurewards of the point of
assertion.

\subsection{Quantum weights and the functional definition of
probability}\label{probability}

I have argued that an agent in a branching universe should be genuinely uncertain about which outcome of branching will occur. 
As such, a
believer in the Everett interpretation can now coherently assign
credences to each possible outcome of a quantum measurement, despite his
perfect objective knowledge. 

Since each possible outcome is assigned a quantum-mechanical weight, we
are now in a position where weight is the sort of thing that
\emph{could} fit the functional definition of probability given in
section \ref{PP}. If we simply add to the Everett interpretation the postulate that 
weights \emph{in fact} fit the functional definition, we can deduce that
the Everett interpretation entails the Quantum Algorithm, and as such we
can regard empirical evidence for that algorithm as supporting the
Everett interpretation.

If this was all that the Everett interpretation could achieve, it should
still be seen as solving the measurement problem: it provides a
physically complete, observer-independent theory in which is embedded
the Quantum Algorithm. It may be a \emph{postulate} that probability=weight,
but the postulate is no worse off than in any other probabilistic
physical theory.\footnote{This position has been defended in print by Simon Saunders \citeyear{saundersprobability}; see also
\citeN{papineau}.} In particular, we can perfectly well adopt the cautious
functionalism espoused in section \ref{cautiousfunctionalism}, and hope that in the future
some argument will be found to justify why weight fits the functional
definition.

However, things are actually rather brighter than this. There is no need
for \emph{cautious} functionalism where the Everett interpretation is
concerned. As was originally argued by \citeN{deutschprobability}, and is
defended in detail in \citeN{decshort} and \citeN{decprob}, the
principles of decision theory actually \emph{entail} the fact that
weight fits the functional definition. That is: in the Everett
interpretation, we can prove that weight=probability.

I will not attempt to summarise these decision-theoretic proofs here,
since
the details are somewhat involved, but the underlying principle is
essentially that of symmetry: if there is a physical symmetry between
two possible outcomes there can be no reason to prefer one to another.
Such arguments have frequently been advanced in non-quantum contexts but
ultimately fall foul of the problem that the symmetry is broken by one
outcome rather than another actually happening (leading to a requirement
for probability to be introduced explicitly at the level either of the initial
conditions or of the dynamics to select which one happens). They find their
natural home --- and succeed! --- in Everettian quantum mechanics, where all outcomes occur
and there is no breaking of the symmetry.

I leave it to the reader to examine these arguments in the papers cited and
decide whether they are valid. If so, then the Everett interpretation
has allowed us to make genuine progress on a fundamental problem in the
philosophy of probability; even if not, the interpretation is no worse
off than any other physical theory which makes use of objective
probability.

\section{Rejecting subjective uncertainty}\label{fissionsection}

\subsection{The fission program}\label{fission}

Notwithstanding the arguments advanced above, subjective uncertainty
remains controversial. It is therefore interesting to ask to what extent
we can understand the Everett interpretation without its use.

Suppose, then, that we reject subjective uncertainty. Then there
are indeed no objective chances, and an agent who knows quantum
mechanics (and the quantum state) is not in any way uncertain about the
outcomes of measurements. Instead, such an agent knows that he has a
multitude of successors: so, faced with branching, his task is to consider 
the interests of the (indefinitely) large
number of successors which one will have after branching occurs, and to
take that course of action which best serves those interests. 
This response (implicit in
\citeN{deutschprobability} and given explicit and elegant expression in
\citeN{greaves}) might be called the \emph{fission program}.\footnote{This is Greaves' terminology,
more or less. I was tempted to call it the `Parfittian program', but this seems a little impertinent
since Parfit himself is not an advocate of it.}It is a radical program: 
it entails the falsehood of a great deal of our pre-theoretic
view of the world.\footnote{For instance, suppose that we analyse `untrustworthy', crudely, as `probably
isn't telling the truth'. Then, more than likely, no-one is
untrustworthy (since nearly everyone is telling the truth on at least some
branches), or perhaps everyone is (since nearly everyone is lying on at
least some branch). A substantial fraction of the rest of our everyday concepts
are similarly undermined.

Of course, the natural move is to change our analysis of
`untrustworthy': we now realise that it means `lies on high-weight
branches'. But this natural move, taken to its logical conclusion, leads
back to the charity argument for subjective uncertainty, and away from
the fission program.}
 (But then, given the radical nature of the Everett
proposal itself, why not expect such widespread falsehood?) 

The
fission program can best be understood \cite{greaves} as offering
reinterpretations of the mathematical axioms of decision theory so as to
apply not to an agent's ignorance of his single future but to his
preferences between his multiple successors. For instance, the Dominance
axiom states (roughly) that an agent should regard $A$ as preferable to $B$ 
whenever $A$ rewards him better than $B$ irrespective of how the future
turns out (for instance, if $A$ and $B$ are bets where $A$ always gives higher payoff
than $B$, then Dominance says that $A$ is preferable to $B$). The
radical program reinterprets Dominance as saying that the agent should
regard $A$ as preferable to $B$ if each of his successors is rewarded
more richly under $A$ than under $B$. 

Each of the axioms has such a reinterpretation, and it is plausible that each
reinterpretation is rationally compelling for someone in a
branching universe. As such, the reinterpretation of the 
decision-theoretic representation theorem tells us that rational agents
choose that action which maximises expected utility, where the weights
in the expected-utility calculation are not credences in unknown
outcomes but rather a measure of how much that agent cares about each of
his determinate future descendants. Following \citeN{greaves}, I shall
call this measure the \emph{caring measure}.

The advocate of the fission program now proposes the following rationality principle
(call it the \emph{quantum caring principle}, or QCP): rational agents are
compelled to allocate caring measure to branches in proportion to their
quantum-mechanical weight, when they know the latter. That is, if $E$ is a
proposition, $T$ is the Everett interpretation (interpreted according to the fission program) and $X$ is the 
proposition that the weight of all branches
on which $E$ is true at the time in question is $x$ (relative to the agent), then QCP requires
that
\be \cred(E|T \& X)=x.\ee
If QCP is true then rational agents will act in an Everettian universe
just as they would have acted in a universe where the Quantum Algorithm
was true; as such, the fission program amounts to a sort of `fictionalist' 
approach to the quantum algorithm, in that it entails that rational agents should behave as if there
were objective probabilities even though strictly there are none.

Can we provide any sort of argument for QCP? Actually,
we can provide a very good one: the decision-theoretic proofs of the
Born rule mentioned in section \ref{probability} apply \emph{mutatis
mutandis} to the fission program under the reinterpretation of the
decision-theoretic axioms, and entail that caring measure=weight. Note,
though,  that even if these proofs fail then QCP is not
obviously worse off than the Principal Principle. That is: in both cases
we appear to have a primitive rationality principle, something which we
would very much rather avoid (in one case: that probability=weight; in the other, that caring measure=weight). 
In both cases we do not yet know how to
derive that principle rather than just postulating it; in both cases we
are nonetheless prepared to continue using it. 

This analogy, however, is suggestive rather than conclusive. I do not
see what rational argument could be given to justify our accepting the Principal Principle
without argument but demanding a justification of QCP;
perhaps one can be found though. 

\subsection{Against the fission program}\label{againstfission}

Whether or not QCP is problematic, we should be slow to accept the fission program
as I have so far formulated it. Partly, there are general methodological grounds to be wary of it:  
denying that `uncertainty' is applicable to branching 
requires us to
accept (given the ubiquity of branching) that most of our existing
worldview is wildly wrong, in contrast with the general naturalistic viewpoint (as defended by Quine and Neurath)
that progress in science and philosophy comes from successively
modifying our worldview, not from rejecting it almost \emph{in toto}.

More concretely, though, the fission program as presented above provides an answer 
to the wrong question.
Specifically, it tells us: ``supposing that we believed that the Everett
interpretation was true, what would constitute rational action --- that is, what
rationality principles should we conform to in deciding how to live our lives?'' 
And indeed it
would be crucial to answer that question if we indeed came to believe in
the Everett interpretation. 

But this is not currently our situation. Rather, we want to know,
`should we believe the Everett interpretation in the first place?' That
is, is the Everett interpretation explanatory of our current epistemic
situation? And this, I believe, is extremely difficult to answer if we
eschew all talk of uncertainty.

For recall: in section \ref{quantumalgorithm} I argued that the task of an
interpretation of quantum mechanics is to embed the Quantum Algorithm that
is instrumentalist QM into a satisfactory physical theory (with the
possibility that it is slightly modified in the process). The fission program
explicitly rejects that task when it rejects the notion of
probability.

What, then, could make us come to accept the fission program?
Presumably, that it offers an explanation for observed phenomena just as
good as the Quantum Algorithm (while being an improvement over the
Quantum Algorithm in that it is a coherent, complete scientific theory).
But it is not clear why this should be so. The `observed phenomena' are
in essence a vast list of experimental outcomes whose relative frequencies
correspond very closely to the probabilities defined by QM. The fission program 
predicts that there are branches in which this is indeed
so, and ascribes a very high weight to such branches, but as yet it offers no
reason why it is rational to assume that we are in one such branch. All
it can provide is a prudential reason to care about successors in
proportion to their weights, but that does not seem to be of epistemic
import. 

This point can be made more formally within the Bayesian framework for
theory confirmation. Recall: in ordinary decision theory there are plausible
arguments that we should update our credence in a hypothesis via
conditionalising. That is, if $\cred_A(B)$ is our credence in some
proposition $B$ subsequent to learning that $A$, then 
\be \cred_A(B)=\cred(B|A)\equiv\cred(B\& A)/\cred(A).\ee
It then follows from Bayes' Theorem that for some theory $T$ and some
evidence $E$
\be \cred_E(T)=\cred_T(E) \cred(T)/\cred(E).\ee
If $E$ is some \emph{a priori} unlikely event assigned high objective
probability by $T$, then (since by the Principal Principle,
$\chance(E)=\cred_T(E)$) it follows that our credence in $T$ will rise
upon observing $E$. 

However, according to the fission program we cannot regard 
the outcomes of experiments as being assigned high or low objective
probability: all outcomes occur, so $\cred_T(E)=1$ irrespective of the
weight of $E$. This seems to undermine the idea that it is our observation
of \emph{high}-weight events that provides any evidential support for quantum 
mechanics, since --- notwithstanding QCP --- the weight does not seem to appear in the
Bayesian update rule. 

\citeN{greaves} anticipates this kind of objection, and offers a
possible response. She argues that on the assumption that we live in a 
branching, 
Everettian-style universe, we can construct an analogue of the Bayesian update
rule and prove its validity. But it is unclear at best how this strategy can
help us where we are concerned with evidence for the Everett
interpretation \emph{itself}. For suppose that her argument succeeds.
Let $T$ be the hypothesis that the Everett interpretation is true and
let $X_i$ be the further hypothesis that the weight of branches in which
evidence $E$ occurs is $x_i$. Then Greaves' analogue of the update rule
combined with the QCP entails that
\be \cred_{ET}(X_i)=\frac{x_i\cred_T(X_i)}{\sum_j x_j \cred_T(X_j)}.\ee 
All that this allows us to do is to update various credences all of
which are conditional on $T$: that is, on the truth of the Everett
interpretation. It provides no way to make any statements about rational credence in 
the Everett interpretation itself.

The conclusion to draw from this is that we cannot assess evidence for a
theory within an epistemic framework which presumes that very theory.
This makes it difficult to assess evidence for Everett according to the
fission program: our normal epistemic framework presumes that we are
ignorant about the outcome of experiments which are to be performed, and
this is simply false from the perspective of the fission program.

I am only aware of one solution to this problem which is compatible with
the fission program, and (perhaps tellingly) it too makes use of the notion of
ignorance. \citeN{vaidmanencyclopaedia} has observed that an agent who has
performed a quantum experiment but does not yet know the outcome is
uncertain about that outcome in a fairly conventional way: that is, the
agent knows that the outcome has some particular value but is ignorant
of that value. A rational agent presumably deals with this uncertainty
in the usual way, by ascribing probabilities: call these the
\emph{Vaidman probabilities} of outcomes.

A defender of the fission program can now proceed in either of two ways. The first is simply to
stipulate that rational agents must set the Vaidman probability of an
event equal to its quantum weight: that is, to stipulate that the
quantum weights fit the functional definition of objective chance as
applied to the Vaidman probabilities. The second is to argue, as above,
from decision-theoretic considerations that the caring measure of an
event in my future is equal to its quantum weight, and then further
argue (probably by means of Dutch-Book-type considerations) that my
future selves must set their Vaidman probabilities to be equal to the
caring measures that I now assign to them. 

In either case, the Vaidman probabilities enable us to give a
conventional treatment of the epistemology of quantum mechanics. For Vaidman probability
is probability nonetheless\footnote{It is admittedly a slightly unusual sort of probability:
not probability of being in a particular possible world, but rather probability of being in 
a particular location in a known possible world. This sort of \emph{self-locating uncertainty} 
does seem to lead to some odd problems: see
\cite{elga} for an example.}, and an
agent who gives high probability to some measurement result $E$
conditional on quantum mechanics will be justified in increasing his
credence in the latter if he observes the former.

This strategy provides, so far as I can see, the only promising means to
salvage the fission program; however, I do not find it wholly
satisfactory, for two (admittedly very inconclusive) reasons. Firstly, the Vaidman probabilities are
somewhat contrived entities to use as the foundation of our epistemology
of quantum mechanics. It is undeniably the case that we often find
ourselves in Vaidman's sort of uncertainty; however, in the bulk of
experiments which we perform to test quantum mechanics the gap between
our conducting the experiment and observing the result is too short to
allow us time to be uncertain. (Consider, for instance, observations of
a Geiger counter.)

Secondly, it is not at all clear that Vaidman probabilities can be
introduced at all without admitting full-blooded subjective uncertainty.
For consider: an agent about to make a measurement should expect, with
certainty, that he will branch into many copies each of whom is
subjectively uncertain about what result he will see when he looks. That
is: he should  expect, with certainty, that he will
be uncertain about the result of the measurement. Is this any different
from being uncertain right now about that result? I am inclined to think
not, but the argument now begins to merge with the argument from
interpretative charity, and I shall not pursue it further. I should
stress, however, that insofar as the Vaidman probability strategy
succeeds, it succeeds because it reintroduces into the Everett
interpretation a notion of uncertainty, to which we can apply our
existing decision and confirmation theories. Whether it is introduced
via subjective uncertainty or via the Vaidman method, uncertainty of
outcome result seems to be an essential component of the epistemology of
the Everett interpretation.



\section{Justifying the axioms of decision theory}\label{justification}

\subsection{The primitive status of the decision-theoretic axioms}

At this point a sceptic might ask: 

\begin{quote}
This house of cards that you have constructed makes essential use at
many points of the axioms of decision theory. What right have you to
assume that those axioms hold in a branching Universe? And don't respond
by reference to subjective uncertainty, please. Who is to say that the
axioms apply to \emph{this sort} of uncertainty, and not just to the
more conventional sorts that Savage \emph{et al} no doubt had in mind?
\end{quote}

I think that there is a certain amount of force to this objection, but
that as stated it misses the point. For underlying it is an
epistemological story that goes like this: 

\begin{quote}At one time, we had a
metaphysical framework which included an analysed notion of uncertainty
(analysed in the `conventional'\footnote{I use
the term reluctantly! If the Everett interpretation is true, then
branching-type uncertainty is as conventional as can be --- most of our
ordinary uncertainty talk refers at least in part to it.} way as
tenseless ignorance of the state of the entire universe and/or of our
location within it); at this time, though, we had no decision-theoretic
axioms whatsoever. We then considered what behavioural principles would
be rational for beings in our situation, and hence derived the axioms
of decision theory.
\end{quote}
But this is wildly wrong. Nothing like this actually underpins our
historical propensity to conform to the decision-theoretic axioms, and
in fact our reason to believe them is not based on any such argument
either.
Consider, for instance, why we believe an axiom like the `Dominance' axiom 
mentioned in section \ref{fission}.
(Recall that it says, roughly, that an agent should regard $A$ as preferable to $B$ 
whenever $A$ rewards him better than $B$ irrespective of how the future
turns out.  Suppose we consider trying to justify the following special case:
\begin{description}
\item[S1]A coin is definitely going to be flipped. You have a choice of
accepting a bet which pays you ten dollars if the coin lands heads, and
nothing if it doesn't. You are not certain that the coin will not land
heads. It is rational to take the bet.
\end{description}
Or consider trying to justify the following (a special case of what I call
\emph{constancy}, which is required in some versions of the decision-theoretic proofs of
the quantum probability rule \cite{wallace3kinds}):
\begin{description}
\item[S2]You have to choose whether or not to have a coin flipped. You
don't care whether the coin lands heads or tails (if it is flipped), and
whether or not it is flipped you'll receive ten dollars. It is rational
not to care about whether or not the coin is flipped.
\end{description}

I think the response of most people as to why they should accept these
principles would be bemusement: they are, I hope, blindingly obvious. If
we really pressed someone for a defence of the principles, they
\emph{might}, if they had sufficient patience, come up with something
like
\begin{description}
\item[S1]The coin will either land heads or tails. It's at least
possible that it will land heads, in which case if you accepted the bet
you'll be better off. And if it lands tails, it won't make any
difference. So take the bet! You won't do worse, and you might do
better.
\item[S2]If you choose to flip the coin, then it will come down either heads or tails. 
If it comes down heads, then you won't care whether it
was flipped, because you don't care what's on the coin and you get the
money anyway. If it comes down tails, likewise. So whatever happens, you
won't care about whether the coin was flipped.
\end{description}
Note two things about these explanations. Firstly, 
they don't break out of the set of interconnected terms like
`will', `might', `uncertain', and the like. As such, if subjective
uncertainty is defensible then they are just as applicable to quantum
branching as to any other kind of uncertainty.

Secondly, they aren't really explanations at all, in the sense that 
the explanations don't involve concepts or ideas that are more basic or
obvious than those used in stating the principles themselves. It's
rather like trying to justify the laws of logic: if I try to justify ``if ($A$
and $B$), then $A$'' by saying: ``suppose `$A$ and $B$'; then in
particular, $A$; therefore, $A$'' I similarly haven't really explained
anything, just shown some (in this case rather shallow) interconnections between equally basic
concepts.

\subsection{Holistic scepticism}

We could conclude from the above that the decision-theoretic axioms
should just be taken as primitively obvious, and left entirely
unexplained; but this would be too quick. By analogy, consider the Soundness Theorem of first-order 
logic, which demonstrates that the
rules of deduction produce only semantically valid arguments. No-one who
ever doubted this fact would be convinced by the Soundness Theorem: our
confidence in the laws of logic is far stronger than our confidence that
we got the proof right, especially as the proof itself involves heavy
use of the very logical notions which we wish to explain. Nonetheless the 
Soundness Theorem, showing as it
does the interconnections between semantic and syntactic notions of
validity, gives (some) insight into why the rules of deduction are as
they are. \citeN{dummettdeduction}, in making this point,
refers to the theorem as an \emph{explanatory argument} for the laws of
deduction; he contrasts it with \emph{suasive arguments}, the sorts of
arguments which can convince the unconvinced, and claims plausibly that
no suasive argument is available here.

Further insight into the relevance of the Soundness Theorem can be
gained if we imagine what we should conclude if --- \emph{per
impossibile} --- that theorem turned out to be false (and not false due
to some isolated error, but irresolvably false). We would not, of
course, simply shrug our shoulders and stop using deduction! Rather, our
entire intellectual framework would be in ruins. To countenance the
possibility of such a failure would be to countenance a particularly
strong form of scepticism, according to which we are not merely mistaken
about many features of our world, but furthermore that world is set up
so as to prevent us reasoning about it in any justified way. (I will
call this `holistic scepticism').


As another, and more naturalistic, example, suppose that we are
interested in explaining the veridical nature of vision. That is, we would like to explain why
are we (usually) justified in assuming that the three-dimensional world
around us is as it appears to our sight.
This would (conceptually, at least) be a reasonably straightforward task
provided that we were studying some \emph{other} species: a satisfactory theory
of how their perceptual apparatus and their brains function will allow
us to determine whether their perceptions match the outside world
totally, partially, or not at all --- although in the latter case the
holistic framework by which we ascribe mental states to them may again
start to break down and to give wildly indeterminate results.

Justifying our \emph{own} visual capacities is more complex. We can use
exactly the same scientific methodology as we might for other species,
but with the proviso that we are assuming at start that our perceptions
are normally accurate --- else we could trust neither the
readings we ``observe'' on our apparatus, nor the results communicated
to us by co-workers. Nonetheless it seems that we can
``bootstrap'' our way to a satisfactory justification of our perceptual
accuracy, simply because we can ourselves find the scientific
explanation that third-party students of our species are able to find 
--- and which, we have already argued, constitutes an explanation if
anything does. 

Furthermore, the requirement that we assume \emph{ceteris paribus} that our 
perceptions are accurate does not prevent us from identifying relatively
isolated flaws in that assumption, again in the same manner that 
third-party observers would use. Suppose for instance that everyone on Earth
had vivid perceptions of winged snakes flying out of the Sun at noon.
Nothing in the picture thus far developed prevents scientists from
concluding that \emph{those} perceptions are not veridical, and indeed
from identifying the neural mechanisms which cause such false
perceptions. 

What is \emph{not} possible in this picture is for us to identify in
\emph{ourselves} the really widespread failure of visual veridicality
which we might in principle discover for another species. Such a
``discovery'' would so thoroughly undermine our starting assumptions as
to be worthless --- but in doing so it would thoroughly undermine our
entire worldview. Vision, as much as deduction, is sufficiently
essential to our epistemological project that it is another form of
holistic scepticism to suppose that we might find that it is flawed in too
widespread a way.\footnote{What about `virtual reality' scenarios such
as those described in films like \emph{The Matrix} (and more soberly in
some variants of the brain-in-a-vat thought-experiment)? These seem
comprehensible (indeed, theoretically possible) despite the apparent
widespread failure of sensory reliability which they imply. I am myself
persuaded by the analysis of \citeN{chalmersmatrix} (itself rather
reminiscent of Putnam's \citeyear{putnamrth} treatment of the brain in
the vat): no `failure of sensory reliability' actually occurs. Rather,
the various objects represented to us in a sufficiently all-encompassing virtual reality 
should genuinely be taken to exist. Our error is not in believing them
to exist, but in believing them not to be computer-generated entities
instantiated in some underlying hardware. (There is actually something of
an analogy to the charity argument presented in section \ref{subjective2}.)
Further discussion, though, would take us too far afield.}

But now suppose the scenario to be modified slightly. Suppose that our
argument for holistic scepticism relied on some
auxiliary hypothesis $H$: something (let us suppose) intensely
plausible, but not so conceptually indispensable as deduction or sight.
Then our conclusion would be clear: $H$ (or some other such auxiliary hypothesis) must be abandoned, however
plausible it may have seen.

\subsection{The role of an explanation of decision theory}

Returning to decision theory, we can now see what significance might be
played by an explanation of the decision-theoretic axioms. Such an
explanation will have little or nothing to do with our reasons to
believe decision theory, but it will give some insight into why decision
theory is nonetheless correct. Furthermore, if such an explanation leads
to holistic scepticism 
on the assumption that the world is branching, then we must abandon that
assumption. 

In more detail: let us suppose for a moment that the arguments of section \ref{everettsection} persuade
us to adopt the Everett interpretation, but then we decide that our
`rational' behaviour is completely unjustified in an Everettian
universe, and that the rational thing to do would be to curl up into a
ball. Then the entire argument sequence --- from our existing
rationality, through the Everett interpretation, to the wholesale
rejection of our existing rationality --- would be 
a \emph{reductio ad absurdum}, in effect, of
whatever got us started on that sequence in the first place. 

This last does not in any way suggest that the branching
hypothesis is more vulnerable to such a possible criticism than the 
non-branching hypothesis. If we were in fact to find that it is the 
non-branching assumption which undermines decision theory, then
branching would be forced upon us. 

I have not the slightest wish to argue for such
a radical and implausible conclusion; nonetheless, the non-branching case (being more familiar) provides a
good starting point in any investigation of how to explain the
decision-theoretic axioms. Recall that, in the non-branching picture,
\begin{enumerate}
\item `Ignorance' is ignorance of the (tenseless) truth of certain
facts about the (tenseless) state of the
Universe. An agent, in being ignorant of some future event, simply lacks
a certain item of objectively-describable knowledge.\footnote{Actually, this
simplified picture fails to take into account the need for indexical
knowledge.}
\item An agent (regarded as a person-stage) cares about future versions of himself
(that is, future person-stages with the appropriate structural and
causal relations to him.)
\end{enumerate}
A set of defences of \textbf{S1--2} might then be:
\begin{description}
\item[S1] There's something the agent doesn't know, and which
he is choosing to bet on. On bet 1, his future descendant does better
than on bet 2 irrespective of how that something turns out. So he isn't
ignorant of which bet will be better for his descendant, even though he
is ignorant about the outcome of the bet.
\item[S2] The agent can make one of two bets, and whichever bet he makes his future
descendant receives a a certain fixed reward. According to the bet he chooses, 
that descendant's environment has certain properties about which he is ignorant, but 
to which he is indifferent. Therefore, the agent himself 
should be indifferent as to those properties; therefore,
he should not care which bet he takes.
\end{description}
I hope that it is clear that no-one comes to believe $S1$ or $S2$ on the
basis of arguments such as these! They are explanatory only in the sense
that they help us come to a better understanding of why the principles
are true, not in the sense that they would convince someone who did not
already accept them that they should reconsider.

Can we construct analogous arguments in the case of branching? 
It is here, I think, that ``Parfittian'' arguments about caring for our successors, such as
those involved in the `fission program', come into place. Recall, for
instance, the fission-program reinterpretation of Dominance: an agent is
justified in preferring action $A$ over action $B$ iff all his
successors do better under $A$ than under $B$. If this is regarded not
as a \emph{reinterpretation} of the Dominance axiom but as an
\emph{explanation} of it, it is (I claim) every bit as reasonable an
explanation as the analogous non-branching explanation of Dominance.

Arguments such as these aim
to establish, in effect, that even if we knew that
future-directed uncertainty was really branching, we should rationally 
continue to behave exactly as we did before we knew this. Our existing
attitude to decision-making in the face of branching just \emph{is} to 
treat it as uncertainty, because that's what words like `uncertainty' 
actually refer to if the Everett interpretation is true. But the
arguments of the fission program show us why the decision-theoretic
axioms are justified on the assumption that it is true.

This is not to deny that \emph{some} isolated aspects of our view of rationality
might need revision if we accept the Everett interpretation (rather as the `winged snakes' of the
vision example above would imply a localised revision of our assumption that vision is reliable). One
possible example, long discussed informally among physicists and
recently analysed by \citeN{dlewisschrodinger}, is \emph{quantum Russian roulette}:  
bet a large sum on an unlikely quantum event, and arrange to be
instantly obliterated if you lose. It has been argued that if the
Everett interpretation is correct then it is rational to expect survival
with certainty in these experiments, and thus to agree to partake in
them. (I'm sceptical myself, but will not defend that scepticism here.) 
As always, we sail in Neurath's boat, and no part of our conceptual scheme is completely
insulated from criticism and revision. 
But it \emph{is} to deny that such revisions could be so widespread as
to undermine our original reasons for coming to accept the Everett
interpretation. 

We seem to find that the
epistemology of the Everett interpretation has a two-part structure. First we must ask: is the
interpretation explanatory of our current epistemic situation, on the
assumption that our existing approaches to decision-making and
uncertainty are basically rational? If the answer is yes, we must also
ask (on pain of holistic scepticism) whether that assumption remains valid if the
Everett interpretation is indeed true. I claim that we are now in a
position to give affirmative answers to both questions.

\section{Conclusions}\label{conclusion}

Probability enters our scientific theories through the Principal
Principle and only through the Principal Principle. As such, our term
`objective probability', if it picks out anything, picks out that
physical property which satisfies the functional definition given
implicitly by the Principle. Since regarding that functional definition
as true by postulate (\iec, primitivism) is deeply unattractive, the best attitude to
probability is a cautious functionalism whereby we assume that something
can be proved to satisfy the definition even though we do not currently
have any good candidates.

A solution of the measurement problem is an embedding of the `quantum
algorithm' --- that is, the algorithm by which we calculate the
objective probabilities of outcomes to experiments --- within a
complete, physically satisfactory theory. The Everett interpretation is
such a solution\footnote{On the assumption that decoherence solves the preferred basis problem.} 
provided that the problem of `subjective uncertainty'
can be solved: that is, provided that a way can be found to justify an
agent who believes the Everett interpretation nonetheless being
uncertain about the outcomes of a measurement. However, solutions to
this problem are available via interpretative charity and Lewisian treatments of identity 
(and possibly also via post-measurement ignorance, \iec Vaidman probability).

If a solution to the subjective uncertainty problem can indeed be found,
then the ``weights'' of branches are candidates for probability in the
sense that they are the right sorts of properties to fit into the
functional definition of probability offered by the Principal Principle.
If this were all that could be said, the Everett interpretation would be
no worse off than any other physical theory involving probability, since
no such theory has any argument for why its `probability' fits the
functional definition.

However, it is not all that can be said. Rather, the axioms of decision
theory combined with the mathematical structure of quantum mechanics
suffice to derive the Principal Principle with weights playing the roles
of probability; that is, according to the functional definition of
probability, we can prove that weights are probabilities.

An alternative approach to the Everett interpretation (which I have
called the `fission program') makes no use of subjective uncertainty, but instead
reinterprets the axioms of decision theory as axioms about how an agent
should care about his successors in the case of branching. This program
must be rejected in its full-strength
form, as it does not provide an epistemically acceptable account of how we can
come to accept the Everett interpretation. It does, however, have an
important role to play in the epistemology of the Everett
interpretation: after we have tentatively come to believe the
interpretation, the fission-program reinterpretations of the axioms
serve as `explanatory arguments' for the validity of those axioms,
forestalling worries that the Everett interpretation must be rejected
because it undermines our overall conceptual scheme too drastically.

A summary of the epistemic route to the Everett interpretation --- and,
perhaps, of the route to any substantial revision of our conceptual
scheme --- might be: first see whether your existing machinery for
theory appraisal recommends that you adopt the new theory. If it does,
see whether that `existing machinery' is still essentially valid after
adopting the new theory's viewpoint. If the first step fails, the theory
is straightforwardly to be rejected; if the second fails, the reasons
for rejecting the theory are more subtle but no less pressing. Happily,
it appears that neither fails in the case of the Everett interpretation: 
it solves the measurement problem in a fully satisfactory way.

\section*{Acknowledgements}

I am grateful to all the participants in recent discussions of these
matters in Oxford and via email, but in particular to Hilary Greaves, 
Wayne Myrvold and Simon Saunders. I am also grateful to Jeremy
Butterfield for his many constructive comments on an early version of
this paper.

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\end{thebibliography}

\end{document}
