Microsoft Word - Objective Probability in EQM for web.docx


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This is a preprint of a paper to appear in British Journal for Philosophy of 

Science. 

Objective Probability in 

Everettian Quantum Mechanics 

Alastair Wilson 

University of Birmingham & Monash University 

a.j.wilson@bham.ac.uk 

ABSTRACT: 

David Wallace has given a decision-theoretic argument for the Born 

Rule in the context of Everettian quantum mechanics (EQM). This 

approach promises to resolve some long-standing problems with 

probability in EQM, but it has faced plenty of resistance. One kind of 

objection (the ‘Incoherence problem’) charges that the requisite notion of 
decision-theoretic uncertainty is unavailable in the Everettian picture, so 

that the argument cannot gain any traction; another kind of objection 

grants the proof’s applicability and targets the premises. In this paper I 
propose some novel principles connecting the physics of EQM with the 

metaphysics of modality, and argue that in the resulting framework the 

Incoherence problem does not arise. These principles also help to justify 

one of the most controversial premises of Wallace’s argument, ‘branching 

indifference’. Absent any a priori reason to align the metaphysics with 
the physics in some other way, we can adopt the proposed principles on 

grounds of theoretical utility. The upshot is that Everettians can, after 

all, make clear sense of objective probability. 

 

1. Introduction 
2. Setup 
3. Individualism vs. Collectivism 
4. The Ingredients of Indexicalism 
5. Indexicalism and Incoherence 

 i) The Trivialization Problem 

 ii) The Uncertainty Problem 

6. Indexicalism and Branching Indifference 
 i) Introducing Branching Indifference 

 ii) The Pragmatic Defence of Branching Indifference 

 iii) The Non-Existence Defence of Branching Indifference 

 iv) The Indexicalist Defence of Branching Indifference 

7. Conclusion 



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1. Introduction 

In recent work, David Wallace ((Wallace 2003), (Wallace 2006), (Wallace 

2007), (Wallace 2010), (Wallace 2011)) has argued that a proof of the Born rule 

is available in the context of Everettian quantum mechanics (EQM) using 

techniques from decision theory, and that the proof involves no special 

metaphysical or epistemological commitments1. In particular, Wallace claims 

that issues surrounding the connections between probability and uncertainty and 

between probability and expectation can be bracketed while assessing the claim 

that the modulus-squared amplitudes of components of the quantum state 

(which I will refer to as their weights) should be identified with objective 

probabilities. 

 

More is needed for a full account of objective probability in EQM, I shall 

argue, than Wallace acknowledges. Without the right connection between EQM 

and the metaphysics of modality, the argument fails to get a grip on its target, 

and a crucial premise lacks adequate motivation. The moral is that Wallace’s 
hoped-for metaphysical neutrality cannot be sustained: a non-eliminative 

solution to the Everettian probability problem requires us to adopt substantive 

metaphysical principles. I will argue, however, that the principles in question are 

just as plausible as the most prominent alternatives; and that once they are 

granted, Wallace’s argument can be vindicated, and the Everettian project 
strengthened. 

 

The plan is as follows. Section 2 provides some background on the debate 

over probability in EQM, and on Wallace’s argument. Section 3 describes two 
very different ways of combining the metaphysics of modality with the physics of 

EQM, which I call Individualism and Collectivism; I argue that we do not have a 

priori grounds for preferring one over the other, but should choose between them 

by considering their broader theoretical consequences. Individualism is further 

developed in section 4, where it is embedded in a metaphysical ‘package deal’ 
which I call Indexicalism. In section 5, I distinguish two versions of the 

‘Incoherence Problem’ for probability in EQM, and argue that both versions of 
the problem rely on a tacit assumption of Collectivism. In the context of 

Indexicalism, the problems do not arise. Section 6 applies Indexicalism to the 

problem of justifying one of the key premises of Wallace’s argument, Branching 
Indifference; I argue that Indexicalism provides significant additional support for 

this premise. Section 7 is a conclusion. 

                                            
1 Wallace’s argument was inspired by a pioneering paper by David Deutsch 

((Deutsch 1999), although there are several differences between their approaches. I 

will be focusing on the version of the argument that appears in (Wallace 2006), 

(Wallace 2007), and (Wallace 2010). 



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2. Setup 

The ‘probability problem’ in EQM can be factored into three components: 
 

• The Incoherence problem: what does objective probability consist in, 
according to Everettian quantum mechanics? 

• The Quantitative problem: why should objective probabilities be 
given by the Born rule, according to Everettian quantum 

mechanics? 

• The Epistemic problem: how does our statistical evidence confirm 
quantum mechanics, according to Everettian quantum mechanics? 

 

The Incoherence problem has historically been the focus of discussion of 

Everettian probability, and it is easy to see why. In EQM, the fundamental laws 

of nature are deterministic. An initial quantum state, in combination with the 

unitary evolution, determines a complete emergent multiverse. What can 

objective probability possibly amount to in a theory like this? 

 

Wallace’s decision-theoretic argument presupposes a solution to (or a 
dissolution of) the Incoherence problem, and aims to solve the Quantitative 

problem. To be sound, the argument therefore requires not only that the 

premises it uses be true, but also that a successful response to the Incoherence 

problem be available. In this paper, I will first propose a solution to the 

Incoherence problem; I will then seek to provide a defence of what I take to be 

the most problematic of Wallace’s premises in the light of that solution. 
 

How the Incoherence problem and Quantitative problem are related to the 

Epistemic problem is a matter of controversy. Hilary Greaves and Wayne 

Myrvold (Greaves and Myrvold 2010) have attempted to solve the Epistemic 

problem independently of any solution to the other problems. Early work by 

Wallace (Wallace 2006) claimed that once the Incoherence and Quantitative 

problem are solved, the Epistemic problem is no more pressing for EQM than is 

the equivalent problem for any other probabilistic physical theory. More recent 

work by Wallace (Wallace 2011) has aimed to solve the Epistemic problem at 

the same time as solving the Quantitative problem. I favour Wallace’s original 
response (Wallace 2006) to the Epistemic problem, but I will not discuss the 

issue further here; nor will I consider Wallace’s more recent version of the 
decision-theoretic argument, which addresses the Epistemic and Quantitative 

problems together. 

 Wallace’s strategy in addressing the Quantitative problem (as characterized 
in (Wallace 2010)) is to prove that a rational agent who knows that the 

quantum state has the structure that EQM attributes to it must allow weights 

to play the role of objective probabilities in his deliberations. This is a form of 



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probability coordination principle. Such principles connect subjective probability 

(or credence) for rational agents, conceived of as a state of mind, with objective 

probability (or chance), conceived of as a state of the world. David Lewis’ 
Principal Principle (Lewis 1980) and New Principle (Lewis 1994) are the best-

known examples of probability coordination principles; but there are many 

others, usually adapted to different metaphysical conceptions of objective 

probability. For example, Carl Hoefer (Hoefer 2007) argues for one such principle 

in the context of his ‘Third Way’ Humean chances. 
 

Some terminology: I will use the terms ‘Everett world’ and ‘branch’ 
interchangeably, to refer to the entities represented by decoherent histories that 

are part of a quasi-classical domain2. So far as possible I will remain neutral on 

whether Everett worlds overlap3, in the sense of having segments in common; 

more on this below. It will be useful to refer to sets of Everett worlds meeting 

some macroscopic criterion (such as ‘the device reads spin-up’) as outcomes. 
 

Wallace’s argument aims to establish the following conclusion: that an 
agent who believes EQM and who is given a choice between two quantum 

interactions, defined as completely-specified unitary transformations of the 

quantum state, should choose the transformation which maximises total utility 

(as far as the agent now is concerned) over all the outcomes which result after 

the interaction, weighted according to the squared-amplitude (weight) of each 

outcome. This is not necessarily to say that the agent should choose whichever 

transformation maximises expected utility; whether squared-amplitude-weighted-

total-utility amounts to the same thing as expected utility depends on how we 

think about expectation and uncertainty in the context of quantum interactions. 

This issue has generated much controversy, and I discuss it in section 5. 

 

This concludes my brief sketch of the problems with probability in EQM. In 

the next section, I distinguish two approaches to combining the metaphysics of 

modality with the physics of EQM. The choice between these approaches will 

turn out to have significant consequences for the probability problem. 

3. Individualism vs. Collectivism 

The ontology of EQM can be aligned with the ideology of modal 

metaphysics in more than one way. In this section I contrast the prevalent 

approach in the literature, which I call Collectivism, with an approach – 

                                            
2 See (Saunders 2010a) for an introduction to the decoherent histories formalism 

and to its application to EQM. 
3 Note that this usage differs from that in (Wilson 2011a) where ‘branch’ is applied 

to mereologically overlapping entities, and ‘world’ only to non-overlapping entities. 

The present usage is closer to that of most other recent authors on EQM. 



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Individualism – which I think has been unfairly neglected. The presumption of 
Collectivism, I suspect, derives from a more general assumption about the 

relationship of physics and metaphysics, which I call the One-Model-One-World 

Principle, or OMOW for short. I argue that OMOW is unmotivated in the 

many-worlds context, and that nothing in the physics compels us to adopt either 

Individualism or Collectivism. That distinctively metaphysical choice should be 

made instead on grounds of overall theoretical utility. 

 

Consider the following two principles connecting EQM with modal 

metaphysics: 

 

Individualism: If X is an Everett world, then X is a metaphysically 

possible world. 

Collectivism: If X is an Everettian multiverse, then X is a 

metaphysically possible world. 

 

I take it that neither of these principles enjoys any obvious intuitive 

support. They both contain distinctively metaphysical technical terms, and it is 

implausible to think that we can determine the truth-values of such principles by 

direct inspection. However, Collectivism is the standard position among 

philosophers writing about EQM; why might this be? 

 

Although it is not to my knowledge anywhere made explicit, I suspect that 

the following rather general principle about how physical theory relates to modal 

metaphysics has been influential in producing a presumption in favour of 

Collectivism: 

 

OMOW Each model of a physical theory represents exactly one 

metaphysically possible world. 

 

I don’t suggest that everyone who – implicitly or explicitly – adopts Collectivism 
must be committed to this principle. Other reasons for adopting Collectivism 

might involve the presumed causal or spatio-temporal connectedness of Everett 

worlds. But I do think that OMOW is lurking in the background of many 

discussions4. 

 

OMOW entails that each model of EQM corresponds to exactly one possible 

world. But models of EQM are complete multiverses; each model includes many 

histories, structures that we naturally want to think of as representing 

alternatives to one another. OMOW entails that these histories represent 

                                            
4 Tim Maudlin gives a clear statement of OMOW in a different context: ‘Let us 

suppose (and how can one deny it) that every model of a set of laws is a possible 

way for a world governed by those laws to be.’ (Maudlin 2007), p.67. 



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different parts of one single possible world; it precludes thinking of each history 

as representing a possible world in its own right. 

 

I think that OMOW, while perhaps a natural assumption in the context of 

one-world physical theories such as classical mechanics, is false in the context of 

many-world physical theories such as EQM. By way of analogy, consider a 

physical theory that has as a model a Lewisian plurality of worlds, of the sort 

described by David Lewis (Lewis 1986). OMOW entails that the plurality would 

correspond to a single possible world; so taking OMOW to be an a priori truth 

requires taking the ontological postulates of Lewisian modal realism to be a 

priori false. 

 

The multiverse of EQM has a strong structural resemblance to the 

pluriverse of Lewisian modal realism5. Since the natural way to interpret the 

connection between an ontology of Lewis-style worlds and modality involves a 

kind of modal realism, so the natural way to interpret the connection between an 

Everettian ontology and modality should involve a kind of modal realism. Hence, 

a proponent of EQM should prefer Individualism to Collectivism. 

 

I don’t take the argument just given against OMOW to be conclusive. But I 
do think that it gives us reason to be doubtful about the route to Collectivism 

which goes via OMOW, and hence that it gives us reason to consider further the 

potential merits of Individualism. In the next section, I supplement 

Individualism with some additional principles relating metaphysics to the physics 

of EQM; this package of views will then be applied to the probability problem in 

subsequent sections. 

4. The Ingredients of Indexicalism 

Individualism is silent on whether Everett worlds mereologically overlap one 

another; it leaves open the question of the status of propositions in EQM; and it 

says nothing about how actuality fits in to the picture. In this section, I combine 

Individualism with some additional principles connecting metaphysics with the 

physics of EQM. The resulting package deal, which I call Indexicalism, will 

provide all the resources that we need to tackle the probability problem in EQM. 

 

                                            
5 Exactly how close this resemblance is depends on whether we consider a diverging 

version of EQM or an overlapping version of EQM. (See section 4 for discussion of 

this distinction.) But the argument of this paragraph is in fact independent of the 

question of divergence versus overlap. (Lewis 1986) explicitly considered an 

overlapping version of his own modal realism. He rejected it, for much the same 

reasons that (Saunders 2010b), (Wilson 2011a) and (Wilson forthcoming) reject an 

overlapping version of EQM. But an overlapping modal realism is defensible (a 

position of this sort is defended by (McDaniel 2004) and the point remains that 

OMOW would rule out such a view a priori. 



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The first additional ingredient is the conception of Everett worlds proposed 

in (Saunders 2010b), in (Wilson 2011a) and in (Wilson forthcoming); these 

authors embrace a picture of Everett worlds as diverging rather than as 

overlapping, in the sense of David Lewis (Lewis 1986: 206). Overlapping worlds 

have initial segments in common; diverging worlds have qualitatively identical 

but numerically distinct initial segments. Whilst the ‘branching worlds’ 

metaphor is apt in the case of overlap, the ‘parallel worlds’ metaphor is more 
germane to divergence. 

 

The diverging picture arises from a non-standard interpretation of the 

consistent histories formalism. The projection operators which feature in the 

consistent-histories formalism are normally interpreted as representing token 

property-instantiations. This allows that objects or events in two different 

histories can be numerically identical, resulting in a metaphysic of overlapping 

Everett worlds. But if the projection operators are instead interpreted as 

representing types of property-instantiations, then it becomes possible for events 

in distinct histories to only ever be qualitatively identical to one another, 

generating a metaphysic of diverging Everett worlds. 

 

The choice between divergence and overlap is thus a choice between 

different ways of taking the formalism to represent reality; this choice is 

underdetermined by the formalism itself. As a result, neither an overlapping nor 

a diverging picture is forced on us by the physics; we must make the choice 

between them on grounds of overall theoretical utility and coherence. 

 

I will argue in section 5 that, given that the choice between divergence and 

overlap turns on such general considerations, the diverging conception is to be 

preferred: the overlapping metaphysic gives rise to problems in accounting for 

the truth-values of future contingent statements. These arguments are also 

developed in more detail, by (Saunders 2010b), (Wilson 2011a), and (Wilson 

forthcoming). For clarity, however, I will build divergence into my framework 

from the outset. 

 

The diverging version of EQM is relatively new; and it will no doubt be one 

of the more controversial of the assumptions appealed to in the course of this 

paper. Readers sceptical of diverging EQM, believing it to be incompatible with 

the mathematical formalism of QM,  are directed in particular to (Saunders 

2010b). Saunders there provides an explicit construction of a ‘vector mereology’ 

according to which superpositions of quantum histories – as represented by 

Hilbert-space vectors – come out as non-overlapping. I take this construction to 
demonstrate the consistency of divergence with EQM. In what follows, I will 

simply take it for granted that a diverging interpretation of the consistent 

histories formalism is legitimate. 

 



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Once we adopt Individualism, it is plausible to think that ordinary, 

contingent propositions – the sort of propositions that have non-trivial objective 

chances – can be identified with (or are at least isomorphic to) sets of Everett 
worlds. An initial plausibility argument for this identification goes as follows6: 

 

1. Objective chances attach to propositions. 
2. If objective chances are given by weights, then the bearers of 

objective chances are (or are isomorphic to) sets of Everett worlds. 

3. Objective chances are given by weights. 
4. Propositions bearing objective chances are (or are isomorphic to) 

sets of Everett worlds. 

 

A semantics identifying chance-bearing propositions with sets of Everett 

worlds obviously has much in common with Lewis’ treatment of propositions as 
sets of Lewisian worlds. For a proposition P to be true at the actual world, in 

the picture incorporating Individualism which I will develop, is for the actual 

Everett world to be a member of the set of Everett worlds P. 

 

This theory of propositions is compatible both with conceptions of actuality 

according to which it is an absolute property of worlds (see e.g. (Bricker 2001)) 

and with the Lewisian conception of actuality as indexical in nature (see e.g. 

(Lewis 1970), (Lewis 1986)). In my view, the latter conception fits much more 

naturally with EQM. 

Although Everett memorably insisted (in a note added in proof to his 

original article) that all branches were “actual”, it is clear from the context that 

he took the terms ‘actual’ and ‘real’ to be synonymous with one another: 
 

From the viewpoint of the theory all elements of a superposition (all 

“branches”) are “actual”, none any more “real” than the rest. 
(Everett 1957) 

 

It is entirely unsurprising that Everett did not consider the possibility of an 

indexical conception of actuality; this idea was only introduced into academic 

philosophy by Lewis, over a decade later. Consequently, now that we have such 

a conception, there is no reason to abjure it on Everett’s authority. In what 
follows, I will argue that indexical actuality is in fact a key component in a 

satisfying Everettian response to the probability problem. 

 

The following principles jointly comprise the view that I call Indexicalism: 

 

                                            
6 This formulation was suggested to me by Cian Dorr. 



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Divergence: Everett worlds do not overlap; each macroscopic 

object and event exists in one Everett world 

only.  

Individualism:  Distinct Everett worlds comprise alternative 

metaphysical possibilities.  

Propositions-as-sets-of-worlds: Ordinary contingent propositions are sets of 

Everett worlds – a proposition P is true at an 
Everett world w if and only if w is a member of 

P. 

Indexicality-of-actuality:  Each Everett world is actual according to its 

own inhabitants, and only according to its own 

inhabitants. 

 

In the next section, I will show how Indexicalism can dissolve the 

Incoherence problem in EQM.  

5. Indexicalism and Incoherence 

The Incoherence problem is the problem of making sense of non-trivial 

objective probabilities in a many-worlds scenario. In this section, I will divide the 

problem into two, and show how Indexicalism resolves each sub-problem. The 

first sub-problem concerns the application of objective probabilities to 

propositions; I will refer to this as the Trivialization problem. The second sub-

problem has been more extensively discussed in the literature, and concerns the 

alleged unavailability in EQM of an adequate notion of decision-theoretic 

uncertainty. I think that this second sub-problem, which I call the Uncertainty 

problem, is in fact a special case of the Trivialization problem; but given its 

recent prominence, it will be helpful to treat the special case separately. I will 

argue that both problems are generated by a tacit assumption of Collectivism, 

and that they can both be resolved by a proper application of the Indexicalist 

framework.  

i) The Trivialization Problem 

At the heart of the Trivialization problem is the question of whether, in 

EQM, we can coherently assign non-trivial7 objective probabilities to 

propositions, as we do in the context of an orthodox one-world metaphysics of 

objective probability. 

 

Probabilities are necessarily probabilities of propositions. But, so the 

objection runs, given any quantum state, and the deterministic unitary state 

evolution, the future will be described, with objective probability one, by a 

                                            
7  By ‘a non-trivial objective probability’ I mean an objective probability other than 

zero or one. 



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highly specific proposition which entails the existence of countless Everett 

worlds. The total state of reality at any time and the laws of nature therefore 

jointly determine the whole of the multiverse, with probability one. All 

propositions entailed by this maximal proposition have probability one; all 

propositions inconsistent with it have probability zero. Objective probability is 

trivialized. 

 

This objection to EQM has been around for a long time, and it has been 

made eloquently by Barry Loewer, amongst others: 

 

How are we to understand this measure? What does the measure 

measure? None of the familiar notions of probability seem appropriate. 

Clearly it cannot be construed as measuring the chances in a 

stochastic law... since [EQM]8 is a deterministic theory. 

(Loewer 1996) p.230 

 

According to the Trivialization Problem, weights – which attach to individual 

Everett worlds and not to whole multiverses – are simply the wrong type of 

quantity to play the role of objective probabilities. If this is right, then Wallace’s 
decision-theoretic argument, whatever else it might show, cannot be in any way 

relevant to objective probability. 

 

Wallace has recently expressed moderate sympathy with the following swift 

dismissal of arguments of this sort: 

 

[F]ormally speaking the measure defined by mod-squared amplitude on 

any given space of consistent histories satisfies the algorithms for a 

probability. Indeed, mathematically the setup is identical to any 

stochastic physical theory, which ultimately is specified by a measure 

on a space of kinematically possible histories. 

(Wallace 2010) p.228 

 

Unfortunately, critics of Everettian probabilities will not be prepared to go along 

with this. According to the line of thought which drives the Trivialization 

problem, weights are categorically not equivalent to a measure on a space of 

kinematically possible histories – since, so it is argued, any initial condition 
combined with the unitary evolution gives only a single (rather oddly-shaped) 

kinematically possible history – the multiverse itself. 
 

Collectivism is essential to the Trivialization Problem. Since, according to 

Collectivism, distinct Everett branches are not alternative possibilities but co-

                                            
8 Loewer actually addresses this criticism at the ‘instantaneous minds view’, a 

‘many-minds’ theory due to Lockwood (Lockwood 1989). But it applies with equal 

force to an unmodified version of EQM. 



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actualities, objective probabilities are probabilities for ways that the entire 

multiverses might go, rather than probabilities for ways that individual Everett 

worlds might go. Collectivism therefore undercuts any attempt to bypass the 

Trivialization problem; to identify Everett worlds with kinematically possible 

histories is in effect just to adopt Individualism9.  

 

Individualism is not the only component of Indexicalism which is required 

to provide a proper response to the Trivialization problem. Since the bearers of 

objective probabilities are propositions, we require an account of propositions 

according to which the squared-amplitude measure is defined over them. Such an 

account is provided by the principle Propositions-as-sets-of-worlds, which is 

incorporated into Indexicalism. Propositions bear objective probabilities in virtue 

of the squared-amplitude measure being defined over the sets of Everett worlds 

with which the propositions are identified. The probability that P, on this 

picture, can be thought of as the probability that the actual Everett world is a 

member of the set of Everett worlds P. 

 

The Indexicality-of-actuality component of Indexicalism is likewise required 

if objective probabilities are to be assigned to propositions in full generality. 

Among the propositions to which we need to assign objective probabilities are 

such propositions as ‘the actual outcome will be spin-up’. But if ‘actual’ applies 

to the entire Everettian multiverse, then the phrase ‘the actual outcome’ fails to 
refer uniquely, since the multiverse contains many outcomes. According to 

Indexicality-of-actuality, ‘the actual outcome’ refers uniquely to the outcome in 
the Everett world of the speaker; consequently, the Everett worlds in which 

tokens of ‘the actual outcome is spin-up’ are true are just those Everett worlds 
which contain spin-up outcomes, as required. 

 

This indexical treatment of propositions and of actuality faces a difficulty 

involving future contingents, which has featured prominently in the recent 

literature on Everettian probability10. On the overlapping metaphysical picture 

usually combined with EQM, distinct worlds have initial segments in common: 

agents facing impending quantum interactions are located in every world which 

emerges from the interaction. An agent about to conduct a spin measurement, it 

seems, is then located both in an Everett world where the result is spin-up and 

in an Everett world where the result is spin-down. Will the result be up or 

down? Everettians seem unable to make sense of this question: both results will 

occur. (Or perhaps better: each result will occur.)  
                                            

9 A parallel step is taken at the level of decision theory by Wallace and Greaves. 

Thus e.g. (Greaves 2007) stresses the importance of identifying ’states’ in decision 

theory with histories (individualism), rather than with centered worlds (collectivism). 

My approach is to tinker with the metaphysics rather than with the decision theory. 
10 See, e.g., (Saunders 1998), (Greaves 2004), (Saunders and Wallace 2008), 

(Wilson 2011a). 



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This problem has no analogue in Lewisian modal realism, according to 

which agents are always world-bound and worlds do not mereologically overlap. 

And it seems to undermine the use of indexical actuality to ground objective 

probabilities for future contingent propositions. If I am both part of a spin-up 

Everett world and part of a spin-down Everett world, why aren’t the 
probabilities of the actual outcome being spin-up and of the actual outcome 

being spin-down both equal to one? 

 

Various solutions to the problem of future contingents in EQM have been 

proposed. Vaidman (Vaidman 1998) and Tappenden (Tappenden 2010) have 

sought to ground our present assignment of probabilities to future contingents in 

a future assignment of probabilities to past contingents. (Saunders and Wallace 

2008) proposed two distinct solutions to the problem (although they did not 

clearly distinguish between them) – one involving a specific (‘worm-theoretic’) 
metaphysics of personal identity, the other involving a revisionary metaphysics 

of macroscopic objects in general. However, Wilson (Wilson 2011a) argues that 

both of the solutions suggested by Saunders & Wallace face significant 

difficulties. My favoured solution to the problem with future contingents is that 

proposed in (Saunders 2010b), in (Wilson 2011a) and in (Wilson forthcoming) 

which embraces a picture of Everett worlds as diverging rather than as 

overlapping. It is for this reason that Divergence has been incorporated into 

Indexicalism. 

ii) The Uncertainty Problem 

Although I think the Indexicalist response to the Trivialization problem is a 

fully general one, a special case of the problem is of particular interest: the case 

of our epistemic attitudes to chance-bearing propositions. A common objection 

made against EQM is that that it cannot allow for a kind of status that chance-

bearing propositions about the future are alleged to have, dubbed ‘objective 

uncertainty’11 by Belnap & Mueller (Belnap and Müller 2010). Objective 
uncertainty stands to subjective uncertainty as objective chance stands to 

subjective credence; so if an agent knows that it is objectively uncertain whether 

P, then the agent is rationally required to be subjectively uncertain whether P 

even if the complete current12 state of reality, and the dynamics governing the 

evolution of reality, are both known with certainty. 

 

                                            
11 Confusingly, Wallace and Greaves have used the term ‘subjective uncertainty’ for 

essentially the same notion; but I find Belnap & Mueller’s terminology more 

appropriate, as it underlines that the uncertainty in question is not eliminated by an 

agent’s non-indexical knowledge. 
12 Setting relativistic worries about absolute simultaneity aside makes no difference 

to the philosophical issue, and simplifies the presentation. 



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Objective uncertainty, it is usually claimed, is a straightforward feature of 

one-world stochastic theories, such as the GRW approach to quantum theory 

(Ghirardi et al. 1986); because of the stochasticity of the dynamics, in a GRW 

world even certain knowledge of the dynamics and the current state of reality 

could not provide us with certainty about what will happen next. The subject-

matter of objective uncertainty in stochastic theories is just the actual future. 

 

It has often been thought that objective uncertainty is unavailable in 

EQM, because of the deterministic nature of the unitary dynamics. For example, 

Hilary Greaves has argued as follows13:  

 

The [Subjective Certainty] intuition is that if an observer-stage knows 

both the relevant aspects of the objective description of the universe, 

and his own location within that universe, there is no room for 

uncertainty, subjective or otherwise; and, further, that (due to the 

determinism of the Everettian quantum dynamics and our assumption 

that the pre-fission observer-stage knows the initial state) Everettian 

branching meets this condition. 

(Greaves 2004) 

 

In a similar vein, Huw Price (Price 2010) argues that without objective 

uncertainty Wallace’s decision-theoretic strategy fails to gain any traction:  
 

I think it is worth stressing that the availability of an appropriate 

notion of uncertainty doesn’t emerge from the DW argument, but is 
presupposed by it. So if we sceptics are challenged to say which of 

these axioms we disagree with, we have at least the following answer: 

we’re sceptical about any axiom – e.g., Wallace’s Dominance, for one – 

that presupposes an analogue of uncertainty. And we’ll remain 
sceptical, until our opponents convince us that they have a notion that 

will do the job. 

(Price 2010) p.375 

 

Thoughts such as this are frequently expressed by critics of EQM. And 

taking themselves to have established that there is no room for objective 

uncertainty in EQM, the critics typically go on to argue that there is no sense in 

which a fully-informed Everettian agent about to make a measurement can 

expect either to see one outcome or the other, but not both. Greaves, for 

instance, argues that an observer about to make a measurement should expect to 

see each outcome. 

 

Wallace’s preferred response to such worries is simply to deny that there is 
a conceptual connection between objective probability and objective uncertainty. 

                                            
13 Because she is presupposing Collectivism, Greaves uses the phrase ‘the universe’ 

to refer to an entire Everettian multiverse. 



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Although in earlier writings ((Wallace 2003), (Wallace 2006), (Wallace 2007), 

(Saunders and Wallace 2008)) he sought to provide a philosophical grounding for 

objective uncertainty in the face of quantum interactions, more recently 

(Wallace 2010), (Wallace 2011) he has concluded that no such grounding is 

necessary. Although he admits that it seems obvious that objective probability 

requires the possibility of objective uncertainty, this is taken to be a case where 

our pre-theoretic beliefs are unreliable: 

 

It is, to be sure, extremely intuitively plausible that probability 

requires [alternative possibilities, only one of which is actualised, and 

genuine uncertainty as to what the outcome is]. But it is hard to see 

why this matters. Intuition, absent supporting arguments, is no guide 

to truth, as twentieth century physics has made clear.  

 (Wallace 2011) p.11614 

 

I find this response unsatisfying. Although our pre-theoretic beliefs are not 

sacrosanct, and scientific discoveries have overthrown many of them, such cases 

tend to be cases where we have illegitimately extrapolated a correct belief about 

a familiar domain to an unfamiliar domain. An example: given that permuting 

two (discernible) cups of tea generates a distinct alternative possibility, we infer 

that permuting two (indiscernible) electrons in an antisymmetrized state 

generates a distinct alternative possibility. Because of the significant differences 

between cups of tea and electrons, it turns out that this inference is invalid 

despite being ‘intuitive’. Despite what we would naively have assumed, 
indiscernible quantum particles do not manifest the same statistical behaviour as 

discernible cups of tea. 

 

I suggest that this sort of extrapolation from the familiar to the unfamiliar 

is characteristic of cases where our pre-theoretic beliefs are unreliable. But no 

such illegitimate extrapolation seems to be going on in the case of the conceptual 

connection between objective probability and objective uncertainty. If EQM is 

correct, then what we ordinarily think of as objective probability is – and has 

always been – constituted by weights of Everett worlds15. And if this is so, then 
weights are not a new and unfamiliar phenomenon, in the face of which we must 

inevitably accept the breakdown of our probabilistic concepts; rather, they are a 

familiar phenomenon under a new and unfamiliar mode of presentation. As a 

result, denying that objective probability requires objective uncertainty involves 

                                            
14  References to Wallace’s book manuscript are to the draft of 09/04/2010. 
15 By this I do not mean to exclude the possibility that processes not essentially 

involving branching - deterministic slot machines, for example - can also give rise to 

objective chances. The claim is only that many of the objective probabilities we 

recognise in ordinary life (such as those in weather forecasts looking many weeks 

ahead, for example) are at least partially constituted by Everettian branching. For a 

technical discussion of this claim, see (Zurek and Paz 1994). 



15 
 

denying obvious claims about the macroscopic realm; for example, that rational 

certainty that it will rain a year to the day from now is incompatible with 

rational belief that there is non-zero objective probability that it will stay dry. I 

therefore find Wallace’s appeal to the unreliability of our pre-theoretic beliefs 
about probability unconvincing. 

 

Accordingly, I think it was a mistake for Wallace to retreat from his earlier 

view that objective uncertainty is a pre-condition of making sense of objective 

probability. In holding that objective uncertainty is necessary for objective 

probability, I align myself both with many critics of EQM (e.g. (Albert and 

Loewer 1988), (Maudlin 1994), (Loewer 1996), (Price 2010)) as well as with 

many of its friends (e.g. (Saunders 2010b), (Greaves 2004), (Greaves 2007), 

(Greaves and Myrvold 2010)). All of these authors have seen an incompatibility 

between objective probabilities and EQM. Not all of them have responded to it 

in the same way, however; while critics have typically seen a fatal difficulty for 

EQM here, Saunders and Wallace have sought to provide an account of objective 

uncertainty in EQM adequate to underwrite objective probabilities. And Greaves 

has argued that abandoning objective probability need not spell disaster for 

EQM; in her own work, and in collaboration with Myrvold, she has argued that 

– despite not being chances – branch weights can play the role in inference and 
decision-making in branching theories that chances play in non-branching 

theories. 

 

Not all Everettians take objective probability to require objective 

uncertainty. A line of thought stretching back to (Papineau 1996) maintains that 

the supposed conceptual link between uncertainty and probability is in fact 

inessential to the concept of probability; its abandonment it is seen as a 

legitimate – even an admirable – example of empirical discoveries leading us to 
revise our metaphysics16. 

 

If there really were no way to make sense of objective uncertainty in EQM, 

and if our choice was between abandoning EQM and abandoning certain 

apparently plausible claims about the nature of objective probability, then it 

would seem like a dangerous philosopher’s conceit to urge that the physics, and 
not the metaphysics, should be modified. But the dilemma is a false one: 

Everettians can after all provide for objective uncertainty, by adopting 

Indexicalism and taking objective uncertainty to be indexical uncertainty. 

 

Together, the principles incorporated in Indexicalism characterize coherent 

notions of uncertainty and expectation which pertain to ordinary contingent 

matters of fact. The subject-matter for objective probability and objective 

                                            
16 Views of this sort have recently been endorsed by (Wallace 2010) and 

(Tappenden 2010), and re-affirmed by (Papineau 2010). 



16 
 

uncertainty, for any agent A, is the nature of the Everett world of which A is a 

part – that is, of the Everett world which is actual for A. Even if A knows all 

the facts about the quantum state, A can retain indexical uncertainty – 
uncertainty about where in the multiverse A is. And since there are many 

Everett worlds which match A’s Everett world perfectly up to any given time, 

even a complete specification of the past of A’s Everett world does not fix its 
future. For realistic agents, Indexicalism renders objective uncertainty about the 

actual future ubiquitous in EQM, and thereby dissolves the worries about 

objective uncertainty and expectation that have been voiced by Loewer, Greaves, 

Price and others. 

 

Of course, a truly omniscient agent, with complete knowledge of the 

quantum state and complete self-locating knowledge, would retain no 

uncertainty at all17; but Indexicalism at least allows that such an agent could 

have coherent thoughts about the future. A truly omniscient agent would still 

expect to see one course of events alone, the course of events in the Everett 

world she knows to be her own. Such agents would, it is true, have no need of 

objective probability according to the Indexicalist picture; but since they are 

physically impossible, and occur in no Everett world, this need not worry us 

unduly. Objective probability, we may say, is irreducibly self-locating. 

 

What the Incoherence problem shows us, I think, is that we need to adopt 

the full package of Indexicalism in order to make proper sense of the application 

of objective probabilities to propositions describing distinct Everett worlds. Once 

we do so, we can agree with Wallace that weights play the role of a measure 

over a space of possible histories. If instead we presuppose Collectivism, as critics 

of EQM usually do (at least tacitly), then the only objective probabilities which 

could make sense in the context of EQM would be probabilities assigned to 

whole multiverses. 

 

I argued in section 3 that there is no clear a priori reason to prefer 

Collectivism to Individualism. Since the former but not the latter leads to 

difficulties making sense of objective probabilities, Everettians who wish to make 

sense of non-trivial objective probabilities for outcomes of quantum interactions 

ought to prefer Individualism to Collectivism. And (assuming that Divergence is 

acceptable) it is a relatively natural step from Individualism to the full 

Indexicalist package. 

                                            
17 Note that this is true even for a one-world stochastic theory like GRW; a truly 

omniscient agent in a GRW world knows how all stochastic processes will in fact 

turn out, and so has no need for objective probabilities. 



17 
 

6. Indexicalism and Branching Indifference 

Once we adopt Indexicalism, the first major obstacle to using Wallace’s 

argument to ground objective probability in EQM – the Incoherence problem – 
is solved. Objective probabilities attach to propositions, as I have argued they 

must; and objective uncertainty can be cashed out as indexical uncertainty. 

What remains is to justify each of the axioms used in the proof. Wallace argues 

for these axioms in (Wallace 2010), and I think most of his defences of them are 

persuasive – with one notable exception. A crucial assumption in the proof – and 
one which is violated by some of the most influential proposed counterexamples 

to it – is branching indifference. In this section, I will explain this axiom and 

raise some objections to Wallace’s arguments for it; I will then provide an 
argument for Branching Indifference which makes essential appeal to 

Indexicalism. 

i) Introducing Branching Indifference 

Here is Wallace’s statement of branching indifference18, which is supposed to 
be a principle of rationality in the Everettian context: 

 

BRANCHING INDIFFERENCE:  An agent doesn’t care about 
branching per se19: if a certain measurement leaves his future self20 in 

N different microstates but doesn’t change any of their rewards, he is 
indifferent as to whether or not the measurement is performed. 

(Wallace 2010) p.238 

 

                                            
18 Wallace intends the terminology of ‘branching’ to be neutral between overlap 

and divergence. For the purposes of this paper, I follow his usage. 
19 Wallace’s use of the term ‘per se’ here might seem worrying - even if we don’t 

care about branching for its own sake, might we not nevertheless care about it 

because we believe it to be correlated with something we do care about, such as 

pleasure or pain? In fact, this possibility need not violate branching indifference; 

given Wallace’s characterization of rewards, an agent who believes that branching is 

(somehow) correlated to pleasure or pain within branches will assign different 

distributions of rewards to branches in branching setups which differ physically only 

according to the amount of branching they involve. Such an agent will satisfy 

branching indifference, but will violate other axioms of Wallace’s proof, such as 

‘diachronic consistency’ or ‘state supervenience’. So, despite the appearance of ‘per 

se’ in its statement, branching indifference does not merely rule out taking branch 

number to be an end in itself; branching indifference rules out caring about branch 

number at all. 
20 This mention of ‘future selves’ is dispensable. Wallace’s proof is intended to 

apply equally to acts where some of the post-interaction branches do not contain any 

‘future selves’ of the agent; such branches can still be assigned ‘rewards’ by the agent 

at the present time. 



18 
 

To get a sense of what branching indifference involves, see Figure 1. 

Branching indifference requires a rational agent to be indifferent between the 

two branching setups illustrated. A and B are the rewards which attach to each 

branch; the numbers in boxes are the weights of each branch. The only 

difference between these setups is the number of branches corresponding to 

reward B; the range of available rewards, and their relative weights, is the same 

in each setup. 

 
Figure 1 

 

Wallace’s defence of branching indifference has two strands. The first strand 

– the ‘pragmatic defence’ – denies that number of branches is something about 

which we could rationally have preferences, while the second strand – the ‘non-

existence defence’ – maintains that branch number is not even well-defined. 
After discussing these approaches and raising some difficulties for them, I will 

describe how Indexicalism grounds a third defence of branching indifference 

which avoids these problems. 

ii) The Pragmatic Defence of Branching Indifference 

The pragmatic defence of branching indifference appeals to ‘the limitations 

of any possible physically realizable agent’: 
 

a preference order which is not indifferent to branching per se would 

be in principle impossible to act on: branching is uncontrollable and 

ever-present in an Everettian universe. 

(Wallace 2010) p.238 



19 
 

 

Prima facie, this defence looks incompatible with Wallace’s further claim 
that there is no such thing as the number of branches. If branching occurs 

uncontrollably then it occurs to some degree or other. But it would be a mistake 

to push this point – the claim in question is phrased as a subjunctive, and 
Wallace is just attempting a reductio of the idea that there can be a preference 

order which is sensitive to amount of branching. 

 

It is not clear that the reductio succeeds. That branching is ever-present 

does not prevent it being present to different degrees in different physical 

processes. Decoherence still holds many unknowns, and although no 

unproblematic scheme for quantifying degree of or amount of branching in a 

natural way has yet emerged, it does not seem out of the question that one 

might emerge in the future. Nor does it seem impossible that, according to some 

such scheme, the sort of choices we actually make might affect the amount of 

branching in our future. A thorough investigation of the details of decoherence 

that are relevant to quantifying branching is beyond the scope of this paper; but 

recall that one of the main sources of decoherence is the operation of classically 

chaotic processes. In the light of this, we might expect choices which affect the 

distribution of such processes across the universe to produce differential amounts 

of branching. In general, if choices happen to impact significantly on macroscopic 

variety in our future light cones, it seems at least possible that they will impact 

on the amount of branching in our futures. 

 

Call the claim that there is no well-defined sense in which different 

physically possible acts produce different amounts of branching branching 

homogeneity. Even if branching homogeneity holds true, and our choices cannot 

in fact affect the total amount of branching, does it follow that we must be 

indifferent to branch number? Many things we value are out of our control, to a 

greater or lesser degree21. Preference orderings which are in principle impossible 

to act upon still appear to be coherent and possible preference orderings, pending 

positive arguments to the contrary. Wallace has in fact offered positive 

arguments of this sort in various places (e.g. (Wallace 2003), (Wallace 2007), 

(Wallace 2010). Here is an example: 

 

If we are prepared to be even slightly instrumentalist in our criteria 

for belief ascription, it may not even make sense to suppose that an 

agent genuinely wants to do something that is ridiculously beyond 

even their idealised capabilities. For instance, suppose I say that I 

desire (ceteris paribus) to date someone with a prime number of atoms 

in their body. It is not even remotely possible for me to take any 

action which even slightly moves me towards that goal. In practice my 

actual dating strategy will have to fall back on ‘‘secondary’’ principles 

                                            
21 (Albert 2010) emphasizes this point. 



20 
 

which have no connection at all to my ‘‘primary’’ goal—and since 
those secondary principles are actually what underwrites my entire 

dating behaviour, arguably it makes more sense to say that they are 

my actual desires, and that my ‘primary’ desire is at best an 
impossible dream, at worst an empty utterance. 

(Wallace 2007) p.328 

 

This passage endorses a functional conception of desire, according to which 

an agent’s desires constitutively depend on their dispositions to action across the 
whole range of possible circumstances. This conception has the consequence that 

in cases where it is physically impossible for the agent’s dispositions to action to 

be sensitive to some parameter – where that parameter is completely redundant 

in accounting for the agent’s dispositions – then the parameter does not feature 

in the agent’s preference ordering. 
 

To see the consequences of the functional conception of desire for Wallace’s 
defence of branching indifference, we must be sensitive to the distinction 

between the notion of a preference order over acts available to us – what we 
would prefer to do - and the more general notion of a preference order over 

future events – what we would prefer to happen. The functional conception of 
desire licenses us to restrict our attention to a preference order over physically 

possible acts. If some future event is a consequence of no possible act which is 

available to us, we can still have preferences defined over it in some (perhaps 

wistful) sense. But it is important to see that this would not be in conflict with 

branching indifference, which restricts only preferences over available acts. 

 

An agent who knows which acts are physically possible for him, and who 

knows that each of those acts results in exactly the same amount of branching, 

cannot rationally prefer one act to another on the grounds that it results in more 

(or in less) branching. As a result, the assumption of branching homogeneity – 
that no physically possible act leads to any more or less branching than any 

other act – combined with the functional conception of desire, does seem to 
establish that compliance with branching indifference is a rational requirement. 

 

However, if we do not grant the assumption of branching homogeneity, this 

defence of branching indifference will not work for us. Furthermore, not only will 

there then be difficulties with motivating branching indifference as a rational 

requirement on preference orders, but there will be some prima facie reasons to 

think that preference orders violating branching indifference may in certain 

circumstances be rationally required. Here is the motivating thought: if a state of 

affairs involves significant additional branching and thereby involves the coming 

into existence of significantly many additional people very like myself, why is 

this not exactly the sort of thing I should care about? 



21 
 

 

Consider an analogy: suppose we have an offer from God to create multiple 

copies of our solar system in far-flung corners of the universe. Even if these 

copies would be causally isolated from us, many prominent systems of population 

ethics still entail that we have reason to create them (as long as their lives would 

be reasonably happy, of course). Why is the situation significantly different in 

the Everettian case, when the copies to be created are not spatially distant, but 

instead are in different Everett worlds? What gives the Everettian a rational 

requirement to value the number of duplicates of herself in her own Everett 

world, but not to value the number of duplicates of herself across other Everett 

worlds? Call this line of thought the ethical objection22. 

 

The ethical objection has lurked in the background of many discussions of 

probability in EQM. It can be traced back to (Graham 1973), and more recently 

it has been expressed in various more or less uncritical appeals to the notion of 

what an Everettian agent can care about (for sophisticated versions of this line 

of argument, see e.g. (Greaves 2004), (Price 2010), (Kent 2010), (Albert 2010)). 

Those making such appeals typically first point out that EQM introduces a 

multiplicity of the sort of people, acts, events, or artefacts to which we ascribe 

normative value; they then argue that the way in which this multiplicity 

depends on our actions may have consequences for a rational agent’s preference 
over acts. The argument is perhaps put most clearly by Huw Price: 

 

‘Where goes ontology, there goes possible preference.’ Decision theory 
places no constraint on what agents care about, other than that it be 

real. The new ontology of the Everett view – the global wavefunction 

itself – thus brings in its wake the possibility of an agent who cares 
about that. Hence the challenge, in its most general form: by what 

right do we assume that the preferences of Everettian agents are 

driven by ‘in branch’ preferences at all? 
(Price 2010) p.380 

 

Of course, the Born rule strategy that Wallace is attempting to prove 

allows that agents can sometimes care about global properties of the quantum 

state: the strategy is itself a rule for choosing between sets of branches generated 

by different unitary transformations. So it is not quite right to say that 

Wallace’s argument assumes that Everettian agents are driven only by in-branch 

                                            
22 Variants of the ethical objection are possible which do not question branching 

indifference directly, but which interfere with the decision-theoretic strategy at 

different points. For example, (Price 2010) suggests that even if an Everettian must 

be indifferent to branch number, they need not be indifferent to outcome number � 

that is, to the number of different types of world diverging from some interaction. In 

fact, this version of the ethical objection comprises a challenge to Wallace�s 

‘diachronic consistency’ axiom. 



22 
 

preferences. But branching indifference does at least rule out caring about one 

particular global property of the quantum state – branch number. Price’s 
complaint seems to apply in full force here: if there is such a thing as branch 

number in the Everettian ontology, how can there be something incoherent or 

irrational about an agent who cares about it23? 

 

The combination of a functional conception of intentional states and of the 

assumption of branching homogeneity does provide a way of ruling out 

preferences about branch number. To adopt the pragmatic defence of branching 

indifference is to insist that no physically possible agent’s behaviour could ever 
make it correct to ascribe a preference order to the agent that is sensitive to 

branch number. But both of the components of this defence of branching 

indifference are controversial; and it would be better not to have to rest too 

much weight on it. In the next sub-section, I consider the other strand of 

Wallace’s defence of branching indifference. 

iii) The Non-Existence Defence of Branching Indifference 

 The main defence of branching indifference that can be found in 

Wallace’s writings is that branch number is not well-defined, and hence that it 
cannot be a locus of normative value: 

 

There is no such thing as “branch count”: ... the branching structure 
emergent from unitary quantum mechanics does not provide us with a 

well-defined notion of how many branches there are. All quantum 

mechanics really allows us to say is that there are some versions of me 

for each outcome. 

(Wallace 2010) p.255 

 

This response can be combined with a wholesale rejection of the 

meaningfulness of talk about branch number24; alternatively, it can be combined 

with the thought that ascriptions of branch number, while meaningful in some 

                                            
23 The ethical objection can be made vivid by adapting Derek Parfit’s ‘repugnant 

conclusion’ argument (Parfit 1984). The repugnant conclusion is that, given minimal 

and apparently plausible assumptions, for any finite population, no matter how large 

or how well-off its members, there will be a larger population whose members have 

lives only barely worth living but which is more valuable overall. Informally, 

quantity can always be made to override quality. The analogue of the repugnant 

conclusion in the Everettian case is that it will generally be rational for an 

Everettian to induce as much branching as possible, even at significant personal cost, 

as long as enough additional people are produced with lives even barely worth living. 
24  As in (Saunders 2005), whose derivation of the Born rule was  explicitly 

premised on the stability of probabilities under changes in coarse-graining, and hence 

changes in branch number. 



23 
 

minimal sense, suffer from presupposition failure25. If the former route is taken, 

ascriptions of branch number are meaningless; if the latter route is taken, they 

are either meaningless or trivially false. Either way, they are never true. So, this 

strand maintains, once we properly grasp how branch number is an artefact of 

the mathematics, we will realize that there is simply no coherent way of 

incorporating it into our utility function. 

 

I have argued elsewhere ((Wilson 2011b)) for an alternative treatment for 

ascriptions of branch number, which takes it to be bivalently indeterminate. 

Claims about branch number are always either true or false, but they need not 

always be determinately true or determinately false. This approach is compatible 

with various interpretations of the ‘determinately’ operator. Prima facie, the 

most familiar theories of vagueness – semantic indecision, epistemicism, and 

ontic vagueness – are all live options for Everettians26. 
 

If such a treatment of branch number is rejected, and the unpalatable view 

that claims about branch number are never true is embraced, then I think there 

can be little argument with branching indifference. But the cost of this move, as 

I have already emphasized, is high. It threatens to destroy the fragile grip we 

have on the worldview provided by EQM, by undermining the metaphors of 

branching and divergence which seem indispensable in explicating it. Indeed, the 

claim that ‘how many people are there, unrestrictedly speaking?’ has no true 
answer seems dangerously close to a reductio of any version of EQM which 

entails it27. There is accordingly good reason to look at alternative routes to 

justifying branching indifference. 

 

An indeterminacy-based treatment of branch number does not by itself 

provide support for branching indifference. It might be tempting to think that 

when we know that it is indeterminate what the value of some quantity is, it is 

rationally required for us to be indifferent to that value. But this is not the case 

according to precisificational conceptions of indeterminacy, which allow us, for 

example, to determinately prefer being non-bald to being bald even if baldness 

admits of borderline cases. The point can be put as follows in the case of branch 

number: even if branch number is indeterminate, it is determinately greater than 

                                            
25 The classic examples of presupposition failure are cases of failed definite 

description, such as ‘the current king of France is bald’. Such sentences need not be 

taken as meaningless; on a Russellian theory of descriptions they are (trivially) false, 

for example. 
26 For semantic indecision theories, see (McGee and McLaughlin 1995), (Dorr 

2003); for epistemicism, see (Williamson 1996); for ontic vagueness, see (Barnes and 

Williams 2010). 
27 If there is no true answer to the question ‘how many worlds?’, then there is 

equally no true answer to the question ‘how many people?’, unless we are restricting 

our quantifiers to a particular world or worlds. 



24 
 

one28. This is already enough to generate a problem for branching indifference. If, 

ceteris paribus, copies of ourselves are to be valued, then a situation which 

determinately contains many copies of ourselves is (determinately) more valuable 

than one which contains fewer, even if it is indeterminate how many more copies 

the former contains than does the latter. This result follows straightforwardly 

from the logic of the determinacy operator, whether we appeal to epistemicism, 

to supervaluationism, or to any other bivalence-preserving precisification-based 

theory of vagueness to cash it out. 

 

We might think that, regardless of the general metaphysical concerns we 

might have about it, the ‘no-such-thing-as-branch-number’ response undermines 

the decision-theoretic strategy’s applicability. If claims about branch number are 
strictly meaningless, then can the idealized betting scenarios appealed to in 

Wallace’s argument, in which branch number is well defined, adequately 
represent real-life scenarios? Wallace is sensitive to this concern, and has 

recently attempted (in (Wallace 2011) p. 183) to address it. There he gives an 

extension of the argument which shows that, for a given quantum decision 

problem, the choice of coarse-graining of the consistent history space can be 

varied without altering whether a given preference order definable over acts 

satisfies all of the axioms. For the sake of argument I shall assume that this 

response succeeds. My aim in this section is to argue that treating ascriptions of 

branch number as literally meaningless, or as never true, is not necessary to 

motivate branching indifference; I am happy to grant that it is sufficient. 

 

If we supplement EQM with the functional conception of intentional states 

and with the assumption of branching homogeneity, or if we reject talk about 

branch number altogether, then we can motivate branching indifference. 

However, both of these routes to branching indifference may be found 

problematic. In the next section, I want to suggest an alternative way of 

defending branching indifference, a way which is compatible with an 

indeterminacy-based treatment of branch number and which takes Indexicalism 

as its main premise. 

iv) The Indexicalist Defence of Branching Indifference 

According to Indexicalism, other Everett worlds comprise alternative 

possibilities. Take any contingent proposition P that an agent might care about; 

according to Indexicalism, P corresponds to a set of Everett worlds. Introducing 

additional branching amounts to taking some Everett world w, which either is or 

is not a member of the set P, and generating multiple Everett worlds which are 

qualitatively identical to w up to some time t and different after t. But recall 

that the statement of branching indifference concerns a case in which all the 

                                            
28 Wallace tacitly recognises this when he acknowledges that there are ‘some 

versions of me’ for each outcome; note the plural. 



25 
 

things the agent cares about (i.e. the ‘rewards’) are the same at all of the post-
branching Everett worlds. Thus, all the post-branching Everett worlds are in the 

set corresponding to P if and only if w is in that set. Therefore introducing 

additional branching will not change the truth-value of P at any world; and 

since P was an arbitrary proposition that an agent might care about, introducing 

additional branching never changes anything that an agent might care about.  

 

Introducing additional branching of Everett worlds, where each newly-

created Everett world contains the same reward as the parent Everett world, is a 

matter of multiplication of alternative token possibilities; but the number of 

alternative possibility types is unaffected. Given that the actual Everett world is 

in some set of Everett worlds characterized by a particular reward – that is, that 

a certain reward-specifying proposition is true at the actual world – we should 
be indifferent to how many other Everett worlds the salient set happens to 

contain. That is, we should be indifferent to how many ways there are for some 

reward-specifying proposition to be true. What we care about is invariably 

whether that proposition is true at the actual Everett world; about whether the 

reward in question is actual. 

 

Wallace himself, in an earlier incarnation, has used an argument for 

branching indifference which is very similar to the one just given. There he 

presented it as depending not on Indexicalism, but on ‘subjective uncertainty’, 
which is essentially the thesis that objective uncertainty about the results of an 

upcoming measurement is available for an Everettian agent: 

 

Suppose that someone proposes to increase a million-fold the number 

of the agent’s descendants who see heads: say, by hiding within the 
measurement device a randomizer that generates and displays a 

number from one to 1 million, but whose output the agent doesn’t care 
about and probably never sees. Then from the SU viewpoint, this just 

corresponds to introducing some completely irrelevant extra 

uncertainty. For it is the central premise of the SU viewpoint that [a] 

process which from an objective standpoint involves branching, may 

be described subjectively as simply one with uncertain outcomes. In 

this case the objective description is “the agent branches into a million 

copies who see heads, and one copy who sees tails”; the correct 

description for the agent himself is “I will either see heads or tails, and 
I am uncertain as to which; if I see heads then I am further uncertain 

about the result of the randomiser reading — but I don’t care about 

that reading”. 
But it is a (trivially) provable result of decision theory that 

introducing “irrelevant” uncertainty of this kind is indeed irrelevant (it 
is essentially the statement that if we divide one possible outcome into 

equally-valuable suboutcomes, that division is not decision-



26 
 

theoretically relevant). As such, from the SU viewpoint branching 

indifference follows trivially. 

(Wallace 2006) (emphasis added) 

 

This argument is plausible; but it trades both on the treatment of branches 

as alternative possible outcomes, and on the availability of a notion of objective 

uncertainty. I argued in section 5 that appeal to these resources requires 

Indexicalism. Much of the work in this argument is done by Individualism: 

without this principle the ‘equally-valuable suboutcomes’ are not alternative 
possibilities, so it makes no sense to be uncertain about which of them will 

occur, and the argument lapses29. 

 

The analogy between our choice of quantum act and God’s offer to create 
copies of the solar system rested on a tacit assumption of Collectivism. The 

copies God proposes to create are in far-flung corners of the universe and so they 

would – if created – be part of the actual world. Collectivism entails that the 
people brought into existence by additional branching processes are likewise part 

of the actual world, and hence it seems that just as it is rationally possible (and 

indeed, perhaps rationally required) to value the copies God creates, it is 

rationally possible to value the additional people which result from additional 

branching of Everett worlds. 

 

If we adopt Indexicalism, then the analogy between a choice of quantum act 

and God’s offer lapses. The disanalogy between the cases is that the people God 
offers to create would be actual, while those brought into existence by additional 

branching would be (according to the Indexicality-of-actuality component of 

Indexicalism) merely possible. Since we should only care about actual things, and 

since Indexicalism entails that other Everett worlds contain only non-actual 

things, Indexicalism entails that we should not care about the other Everett 

worlds and their inhabitants. 

 

Of course, even if Indexicalism is granted, branching indifference may still 

be rejected on the ground that an act’s consequences for actuality do not exhaust 
its normative significance. The most plausible motivation for taking this line of 

resistance, I think, flows from the attractions of what Lewis (Lewis 1986), 

following Smart (Smart 1984), called a ‘truly universalistic ethics’. That is a 
version of consequentialism according to which: 

 

                                            
29 Since the argument involves future branching, Divergence also plays a role: 

without divergence, we cannot coherently wonder about which future we will see, 

given that according to overlap we will see both futures. However, this feature of the 

argument appears to be dispensable; the argument works just as well when applied 

to a past branching event as to a future branching event. 



27 
 

‘morality consists of maximising the total of good, absolutely 

regardless of where and to whom the good may accrue.’ 
(Lewis 1986) p. 128 

 

Lewis accepts that a truly universalistic ethics is undermined by his own modal 

realism, and it is likewise undermined by EQM. But so much the worse, declared 

Lewis, for a truly universalistic ethics: 

 

[I]f modal realism subverts only a ‘truly universalistic ethics’, I cannot 

see that as a damaging objection. What collapses is a philosopher’s 
invention, no less remote from common sense than modal realism 

itself. 

(Lewis 1986) p.128 

 

I think this response is exactly right30. Normative ethics is not a discipline 

which ought to be done entirely a priori – the possible shapes that an ethical 
theory can take will depend significantly on the background metaphysic with 

which a theorist is working. For example, metaphysical views such as idealistic 

solipsism or ontological nihilism seem likely to mandate a moral error-theory. 

 

When applied to highly theoretical claims, such as candidate analyses of 

modality, our intuitions bear very little weight; what matters is how well the 

analyses perform (in conjunction with the rest of our worldview) in recovering 

our ordinary judgments. As with intuitions about the shape of an analysis of 

modality, our intuitions about the shape of an ethical theory are just not the 

right sort of considerations to trump metaphysical considerations. It is enough 

that some appropriate ethical framework can be provided which (in conjunction 

with the rest of our worldview) adequately underwrites our first-order moral 

judgments. 

 

Global metaphysical theories should be allowed the use of whichever ethical 

theory fits best with their other elements, rather than having any such theory 

imposed a priori. For example, a form of consequentialism which ascribes 

normative value only to actual states of affairs seems perfectly viable in the 

context of an Indexicalist version of EQM, and it gives exactly the same first-

order ethical consequences as the combination of a consequentialist truly 

universalistic ethics with a one-world physical theory. Everettians who adopt 

Indexicalism can therefore safely reject the ethical objection and uphold 

                                            
30 This is a defensive move. But Lewis thinks that modal realism actually provides 

a more natural home for our moral thought than a theory incorporating ‘absolute 

actuality’: ‘An ethics of our own world is quite universalistic enough. Indeed, I dare 

say that it is already far too universalistic; it is a betrayal of our own particular 

affections. If my modal realism has any bearing at all on matters of value and 

morality, it pushes me towards common sense, not away.’ (Lewis 1986), p.128. 



28 
 

branching indifference, without being obliged to commit both to branching 

homogeneity and to the functional conception of desire. 

7. Conclusion 

The conclusions which have emerged are that Indexicalism allows us to 

ground objective uncertainty in EQM; that Indexicalism allows us to vindicate 

treating the weights as an objective probability measure; and that Indexicalism 

provides new support for branching indifference. Absent any a priori argument 

for Collectivism, we are free to adopt Indexicalism on grounds of theoretical 

utility.  

 

Of course, some will think that the cure that I propose (modifying principles 

connecting EQM with modal metaphysics) is worse than the disease (the 

apparent incompatibility of EQM with objective probability). Such people will 

be best off embracing the ‘Fission Programme’ endorsed by Greaves (Greaves 
2004), by Papineau (Papineau 2010), and by Greaves and Myrvold (Greaves and 

Myrvold 2010). But I hope to have shown that the Fission Programme is not the 

only option for Everettians. 

 

Acknowledgements 

This research was supported by the Arts and Humanities Research Council; by 

University College, Oxford; by the University of Oxford Vice-Chancellor’s Fund; 
and by the Australian Research Council. Thanks to Cian Dorr, Toby Handfield, 

John Hawthorne, Eleanor Knox, Oliver Pooley, Paul Tappenden, David Wallace, 

Robert Williams, and an extremely helpful referee for this journal. Especial 

thanks to Simon Saunders. 

  



29 
 

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