Indeterminacy of world number in EQM

One of the most puzzling features of the decoherent multiverse of Everettian quantum mechanics is that the number of branches it contains is typically indeterminate. David Wallace describes the situation as follows:

There is no sense in which [chaotic] phenomena lead to a naturally discrete branching process: … while a branching structure can be discerned in such systems it has no natural “grain”. To be sure, by choosing a certain discretisation of (configuration-)space and time, a discrete branching structure will emerge, but a finer or coarser choice would also give branching. And there is no “finest” choice of branching structure: as we fine-grain our decoherent history space, we will eventually reach a point where interference between branches ceases to be negligible, but there is no precise point where this occurs. As such, the question “how many branches are there?” does not, ultimately, make sense.
(Wallace, in Saunders et. al, 2010, Many Worlds?, p.67-68)

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Advocates of EQM use a variety of terminology to express this feature of the theory: ascriptions of branch number are ‘interest-relative’ (Saunders), are ‘arbitrary conventions’ (Saunders), are subject to ‘some indeterminacy’ (Wallace), are ‘not well-defined’ (Greaves) or ‘presuppose a piece of structure which is not present in the theory’ (Greaves). The question ‘how many branches?’ is said to be ‘a non-question’ (Wallace), with ‘no good answer’ (Saunders) or simply with ‘no answer’ at all (Wallace). Branch number ‘has no categorical physical significance; it is not part of what is really there.’ (Saunders)

This treatment of an apparently meaningful question as lacking any good answer is reminiscent of Carnapian, Wittgensteinian and neo-verificationist rejections of metaphysics. One of the best responses to these critics of metaphysics is the appeal to classical logic. If mountains have no boundaries, but it is not the case that everything overlaps with every mountain, it follows that there are no mountains. And if there is no positive number N such that N is the number of branches, it follows that there are no branches. Of course, critics of metaphysics do not deny that there are mountains; and Wallace, Greaves and Saunders do not deny that there is branching. Our puzzle, then, is how to make sense of how it can be that there is branching despite there being no positive N such that N is the number of branches.

The key to resolving the puzzle, I think, is to grant that there is something defective about the question ‘how many branches?’ but to deny that this requires it to be nonsensical or to lack a true answer. We can instead say that the number of branches is indeterminate: that there is some N such that N is the number of branches, but there is no N such that N is determinately the number of branches. The question ‘how many branches?’ is defective because it has no determinate answer, even though (determinately) it is meaningful and thus (determinately) does have an answer. This sort of response is hinted at by Wallace’s talk of ‘indeterminacy’ and by Greaves’ talk of ‘vagueness’, but it has not to my knowledge anywhere been made explicit; it also appears to be in tension with phrases like ‘non-question’ that Wallace and Greaves use elsewhere. I therefore propose it not as an interpretation of Greaves, Wallace and Saunders but as a friendly amendment to their strategy, designed to make decoherence-based EQM more palatable to the mainstream metaphysician.

The bivalent indeterminacy explanation of the defectiveness of questions about branch number is only satisfying if we can give an appropriate account of the determinacy operator. The contemporary literature on vagueness provides various frameworks for thinking about determinacy and indeterminacy that allow us to retain classical logic. I will discuss only the most prominent approaches, each of which makes use of the notion of admissible precisifications. A precisification is, as the name suggests, a way of making precise a vague expression. The characterization of admissibility varies from theory to theory.

Epistemicist approaches to vagueness say that some precisification is picked out by patterns in our community’s global linguistic usage as objectively the correct precisification, but that it is typically unknowable (for a distinctive sort of reason) which is the correct one. According to epistemicism, admissibility is epistemic in nature, and indeterminacy is a kind of ineliminable ignorance, the ineliminability of which derives from our lack of knowledge of the semantic values of the words we use.

Supervaluationist approaches  in contrast hold that admissibility is semantic in nature: admissible interpretations are those whose correctness is not ruled out by the various conventions we have (explicitly or implicitly) laid down to govern our language. For supervaluationists, indeterminacy derives from semantic indecision. Fine, Lewis, and McGee & McLaughlin give influential (though differing) versions of supervaluationism. I prefer McGee & McLaughlin’s version, since (like epistemicism) it allows for the preservation of bivalence.

Let’s see how these two views apply to the indeterminacy of branch number present in decoherence-only versions of EQM. We treat ‘the number of branches’ as a vague expression, and consider various ways of making it precise. Each of these ways corresponds to some particular coarse-graining of the decoherence-based decomposition of the quantum state into components. The coarse-graining we choose is subject to constraints: if it is too fine-grained, then interactions between the components will become non-trivial and decoherence will be lost. If we make it too coarse-grained, there will cease to be any branching at all. And no coarse-graining produces non-integral values for branch number. We can therefore place two extremely minimal constraints on admissible precisifications for ‘the number of branches’ in any episode of branching: every admissible precisification must correspond to some natural number of branches, and this natural number must be greater than 1.

The constraints on precisifications described in the previous paragraph ensure the right results in the case of questions about world number. According to both epistemicism and supervaluationism, it makes perfect sense to say that there is some N such that N is the number of branches, but it is indeterminate which N is the number of branches. Despite the indeterminacy of branch number, it remains determinately true that there is branching.

As it happens, I expect that most Everettians, if they accept my account of branch number indeterminacy in terms of a precisification-based theory of vagueness, would prefer a supervaluationist reading of the indeterminacy operator. Epistemicism is widely considered implausible even in classical cases of vagueness – it is simply hard to believe in all those unknowable facts of the matter – and I see no particular reason why epistemicism should be any more or less plausible in the case of branch number than it is in classical cases of vagueness.

The claim that branch number is vague could be resisted in a number of ways. For example, van Fraassen suggests as a constraint on the usefulness of vague predicates that there be clear cases and clear counter-cases. While there are clear counter-cases for ‘is the number of branches’ (for example zero) there are no clear cases – no numbers N such that determinately N is the number of branches. This seems to me an overly restrictive conception of vagueness, however. For example, consider the predicate ‘is the smallest large number’. We might think that zero is a clear counter-case: it is as far from large as anything could be. But there are no clear cases – no number is such that it is clear that it is the smallest large number. Van Fraassen’s suggested characterization of vagueness accordingly looks inadequate. Vagueness should instead be thought of as the phenomenon of borderline cases; that characterization (which is favoured by Williamson) allows branch number to count as indeterminate.

Wallace prefers to say that it is not vague but arbitrary how many worlds there are. However, this seems to have highly implausible consequences. Arbitrariness is a property of choices, or of decisions: if a choice is arbitrary, then it is up to us how we make it. But it is not, in any coherent sense, up to us how many worlds there are. (It may in some sense be up to us what we mean by ‘world’: but given any choice of what is to count as a world, it is not up to us how many such objects there are.) Of course, it makes sense to say that for practical purposes we must pick some coarse-graining to analyse a particular physical interaction, and to say that the choice we actually make in any given situation is arbitrary; but this is perfectly compatible with (and indeed can potentially be explained by) the thesis that world number is indeterminate. Knowing that something is indeterminate but – for pragmatic reasons – picking a single precisification to work with invariably involves making an arbitrary choice.

The claim that world number is indeterminate an unsettling one. Quineans might well worry how we give adequate identity conditions to entities indeterminate in number; and prima facie such entities might give rise to a version of Evans’  argument against vague identity. There are many reasons of this general sort to doubt that vagueness is possible in fundamental reality; the Everettian, though, has a multi-purpose response. It is that, in EQM, worlds (like people and minds) are not fundamental; rather, they are emergent phenomena. Saunders and Zurek have emphasized that there need be no vagueness at the more fundamental level, the level of the quantum state. And Wallace  argues forcefully that approximation and indeterminacy are characteristic features of all emergence in the non-fundamental sciences. According to this line of thought, EQM is altogether unexceptional in using vague terms in its explanations.

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Indeterminacy of world number in EQM

Directions of explanation and of analysis

The other day U.Melbourne hosted an interesting talk by John Maier. In it he argued that a variety of ideas within philosophy of mind and action can be analysed in terms of the single primitive ‘X is an option for Y’. I’m just going to pick one strand out of the talk because it set me thinking about the nature of analysis. As an example, consider Maier’s suggested analysis of ability:

(1) S has the ability to A iff S normally, in virtue of her intrinsic properties, has A as an option.

Now, this biconditional looks true to me – indeed, necessarily true. Having an ability intuitively does entail there being something about you that is intimately related to the possible doing of A. When the ability to do some action A is lacking, it is (at least ‘normally’) the case that the intrinsic state of the agent does not meet a certain A-enabling condition.

True, perhaps, even necessarily true; but it doesn’t look like an analysis should. Several questioners expressed the feeling that ‘has as an option’ ought not to be analytically fundamental, but it wasn’t easy to pin down exactly why. Here’s my suggestion for why (1) is not a good candidate for an analysis of ‘ability’, regardless of its truth: it’s that the analysandum seems to be explanatorily prior to the analysans. That is, it seems that the having of an ability to A explains why S normally has A among her options, rather than that the presence of A amongst S’s options explains why S has the ability.

Maier wasn’t sure about this intuition about explanation, but he suggested that even if it is correct then that need not be a reason to abandon the analysis. That is, he thought that the direction of explanation over a bi-conditional need not match the order of analysis.

There is something disarming about this response. It reflects an unexpectedly modest conception of analysis. But there is also something unsettling about it. If explanatory considerations are orthogonal, what reasons could tell in favour of the claim that a given biconditional is or is not an analysis?

One response would be to point to the hierarchical nature of analysis: since the relation of analysis is non-symmetric and transitive, successful analyses will form a tree, with the most primitive notions at the base. Thus there might turn out to be lots of necessarily true biconditionals with clauses ascribing options to some subject S on one side and simple action-theoretic claims on the other, and few true biconditionals connecting (say) claims about capacities with simple action-theoretic claims. Recognizing that this situation obtains would give us reason to take having as an option as analytically basic. That, at least, is how I can best reconstruct Maier’s picture.

The problem with this response seems to be that it undermines the motivation for seeking analyses. Call an analysis where the obtaining of the analysans does not explain why the analysandum obtains a ‘mere analysis’. If some biconditional is a mere analysis, then what makes it more interesting than a necessarily true biconditional which is not an analysis at all?

A more promising defence of Maier’s biconditional as an analysis might be to grant that the direction of explanation and the direction of analysis coincide in all cases, but to argue that the normal availability of the option to S in virtue of one’s intrinsic properties does explain the ability to S. We might say that since to have the ability to S just is to normally have S as an option in virtue of one’s intrinsic properties, of course the latter explains the former – it implies it by subsumption under a general rule, which is (according to the D-N model) a form of explanation. This response will work in every case, as far as I can see.

What’s going on here is that, granting the truth of the analysis, the analysans obtaining is explaining how the analysandum obtains. But (and this is to point to one of the limitations of the D-N model) the analysans is not explaining why the analysandum obtains. So here is my half-baked proposed analysis of analysis*:

X is an analysis of Y iff 1) (necessarily) Y iff X and 2) X explains why Y.

* This reminds me of the questions following David Chalmers’ first Locke lecture earlier this year, where an obstreperous Carnapian seemed to think he’d refuted Chalmers totally by posing the question: ‘Ah, but what’s your analysis of analysis?’

– call an analysis where the obtaining of the analysans does not explain the obtaining of the analysandum a ‘mere analysis’
Directions of explanation and of analysis

‘Interpretations’ of quantum mechanics

I’m going to alternate properly new posts with short extracts from my thesis. So here’s the first such extract, on why Everettian quantum mechanics is not just one ‘interpretation’ among many:

Often EQM is presented as an interpretation of quantum mechanics, for purposes of comparison with other ‘interpretations’; examples usually given are pilot-wave theory, the ‘Copenhagen interpretation’, and dynamical collapse theories such as that of Ghirardi, Rimini and Weber (Ghirardi, Rimini, and Weber 1986). I think that that this way of conceiving the foundational issues, where the equations of quantum mechanics are common to the different approaches and they are distinguished only by a quasi-metaphysical layer placed on top of the equations, is badly misleading.

The main reason for this is that that most ‘interpretations’ impose extra dynamical structure of their own onto the basic skeleton of the quantum mechanical formalism. For example, in addition to the unitary evolution of the quantum state, (non-relativistic) pilot-wave theory postulates point-like particles with definite trajectories, and a ‘guidance equation’ which tells the particles how to move based on the structure of the state.  As such, it is strictly speaking not an interpretation of quantum mechanics at all; it is an autonomous theory which happens to share much of the theoretical structure of quantum mechanics. (In recognition of this point, the name ‘Bohmian mechanics’ is often adopted by enthusiasts of pilot-wave theory.)

The Copenhagen interpretation, as preached by Bohr (Bohr 1934), (Bohr 1958), (Bohr 1963) is different from these explicit modifications of quantum mechanics; it takes the equations of quantum mechanics as they stand but accounts for their link with our experience of the macroscopic world in an non-standard way. In this respect it has much in common with EQM. The difference is that EQM is naturally thought of as a realist theory; it interprets the quantum state as a description of the physical properties of a system. The Copenhagen interpretation (at least as it is ordinarily understood[1]) is an instrumentalist theory; it tells us how to use quantum mechanics to predict behaviour in the macroscopic world (antecedently understood in terms of the theories and concepts of classical physics), but rejects any of talk of ‘correspondence’ between the quantum-mechanical description and any macroscopic reality. It is thus natural to group EQM and the Copenhagen interpretation together, treating the former as a form of scientific realism about quantum mechanics and the latter as a form of anti-realism[2] about quantum mechanics; we can then contrast them both with those theories which explicitly modify quantum mechanics.

The reasons for preferring EQM to Copenhagen are essentially instances of our more general reasons for preferring scientific realism to instrumentalism. EQM provides us with a picture of fundamental reality, albeit a strange one, while the Copenhagen interpretation rejects any such demands. According to the Copenhagen interpretation, questions about the fundamental nature of microscopic reality without reference to experimental context are simply misguided – there can be no informative answer to such questions.

EQM and the Copenhagen interpretation do not just differ in the metaphysical picture they give us. EQM can form of a coherent and unified theory of nature in a way in which the Copenhagen interpretation cannot. EQM is a theory of closed systems, and thus can be applied to the entire universe, unlike the Copenhagen interpretation which restricts quantum mechanics to describing the behaviour of quantum systems embedded in classical environments. This is reflected in the way that quantum cosmology invariably proceeds, implicitly or explicitly, with EQM as a background assumption.


[1] Bohr himself can be interpreted rather differently, as a realist who took classical mechanics to be fundamental. See (Saunders 2005).

[2] By ‘anti-realism’ here I mean something stronger than van Fraassen’s constructive empiricism (Van Fraassen 1980), which takes physical theories about unobservable entities literally but advises a restrained epistemic stance towards them. The kind of anti-realism about the microscopic exemplified by the Copenhagen interpretation is semantic rather than epistemic in nature.

‘Interpretations’ of quantum mechanics

Conditions for causal autonomy

My doctoral thesis is finally (almost) out of the way, and I’ve started a one-year post-doc at Monash University. Australia is awesome – and it’s been nice to get back to doing some more varied philosophy over the last few weeks, after a year working almost exclusively on Everettian QM. To make up for lost time, I’m going to try to update this blog at least weekly, usually on Mondays.

This entry relates to a really interesting talk by Peter Menzies I heard a few days ago at a great little workshop in Wollongong on causation and explanation at multiple levels. Peter told us about some joint work with Christian List, recently published in J.Phil., in which they respond to Kim’s exclusion argument against non-reductive physicalism by appealing to a difference-making counterfactual theory of causation, but without Lewis’s strong centering requirement. Over the course of Peter’s talk lots of stuff slotted into place for me – exactly what sort of challenge multiple realizability poses for the physicalist, how best to divide up the philosophical labour between two notions of causation (as difference-making and as production), and how to go about connecting my own conception of counterfactuals as level-relative with the Lewisian similarity-based account. It was also great to see the possible-worlds semantics for counterfactuals being put to serious work; it gets a lot of criticism, but here it really helped to illuminate the importance of a subtle issue in the logic of counterfactuals for an important and controversial debate in the metaphysics of mind.

The most startling claim made by Menzies and List is that whether Kim’s exclusion principle holds will vary on a case-by-case basis, and is a contingent and empirical matter. They show that under certain conditions the causal relation between some psychological event and some piece of behaviour will be realization-insensitive; roughly, that the behaviour would have occurred even if the microphysical event which actually realized the psychological event had not occurred. If the dependency is realization-insensitive, they argue, then the psychological event and not the microphysical event counts as the difference-maker for the behaviour in question. In that case, the exclusion principle holds, but we have a case of downwards exclusion; the psychological event’s being the cause excludes the microphysical event which realizes it from being the cause. Thus the causal powers do not ‘drain away’ to the microphysical level; the psychological level is completely watertight. This, argue Menzies and List, is sufficient to ground the required causal autonomy of the mental.

One response to this, which I imagine will be fairly common, would be to say that the Menzies/List sort of difference-making causal autonomy is insufficient to ground ‘genuine’ causal autonomy. For example, fans of a production account of causation will argue that the microphysical event is still doing the productive work, and so the psychological event lacks any productive-causal autonomy. This seems right to me, but it also seems like something that the non-reductive physicalist need not be worried by. The non-reductive physicalist claim isn’t that psychological events can produce things that microphysical events can’t (that would amount to magic); it is only that psychological events can explain things that microphysical events can’t. Difference-making causes seem to ground a perfectly adequate kind of causal explanation.

Menzies and List suggest that realization-insensitivity will be a very widespread phenomenon. In particular, they suggest that it will generally be present in cases of multiple realization: where there is a high-level event which can be multiply realized, any causal relations in which that event stands to other events will be realization-insensitive. The rationale behind this is that, in the closest worlds where the low-level event which realizes the high-level event is absent, a different low-level event will realize that high-level event. Their (real-life) example is a monkey’s brain, in which multiple distinct neural states N1, N2, Ni… correspond to the same intention I to reach for food (action A). Suppose in the actual world, some monkey is in state N1, hence has intention I, hence performs action A. Menzies and List argue, plausibly,that had the monkey not been in state N1, it would have been in one of the other neural states Ni instead, so still would have had intention I, so still would have performed action A. Then the causal relation between I and A is realization-insensitive, and we have grounds for saying I is causally autonomous.

Menzies and List mention one clear case where a causal relation will not be realization-insensitive – the case where there is only one low-level event which can realize the high-level event which stands in the causal relation to some other event. In this situation it would obviously be false to say that, had the low-level event not occured, the high-level event still would have done – since they are one and the same events under different descriptions. Thus they say (in footnote 3): ‘One might regard realization-sensitivity as a plausible criterion for identifying higher-level properties with their physical realizers.’ This seems to me rather too quick (at least if it is regarded as a necessary and sufficient criterion, and not merely as a necessary criterion). That’s because we can imagine multiply realizable events which still give rise to realization-sensitive causal relations.

An extremely general example of this sort of case arises from the symmetry between matter and anti-matter. It seems that every type of event above the atomic scale in our universe will be multiply realizable, if only because it could be realized by anti-matter rather than matter. Although it seems unlikely, cosmologists cannot rule out the possibility that some galaxies are actually made of anti-matter rather than of matter; and even if no actual galaxies are so constituted, anti-matter galaxies certainly seem to be physical possibilities. Imagine some bizarre high-level psychological event P which can be realized by only one very specific arrangement of atoms; still it will be multiply realizable, because the atoms could be matter atoms or anti-matter atoms. But this seems to be a case where multiple realization doesn’t lead to realization-insensitive causal relations. Suppose P is actually realized by matter atoms (event E), and that P causes some behaviour B. For P to be causally autonomous requires that, had E not happened, B would still have happened. Now there is of course a world where P is realized by anti-matter atoms (event A): in this world, B still happens. But it is quite plausible, if all the actual anti-matter is in some distant galaxy, that the closest not-E world isn’t an A-and-P world – it’s a not-P world. In this example, P is multiply realizable, but nonetheless the causal relation between P and B is realization-sensitive, and P is not causally autonomous. In such a case there is no exclusion in either direction – both P and A are causes of B.

Another example might be some mental event which in humans can be realized by only one very special neural event, but can also be realized by various states of a silicon chip. Supposing I were to instantiate that very special neural event, it might not be true to say that, had I not instantiated that neural event, I would still have instantiated the mental event – for that would have required me to be made of silicon.

[paragraph which follows edited to fix silly mistake]

These examples trade on multiple realizations of a high-level state that are very different in kind from one another. We can imagine other cases where the realizations are similar in kind, but where instead there are powerful factors suppressing the other realizers. Suppose we have a mental event which has three different realizations, but some tribe of people worships one of those realizers in particular, despising the other two; members of the the tribe set up mechanisms to ensure that, if any obtain, it would be the one the prefer. So they link precursor events of the despised realizers up to a number of independent mechanisms each of which is individually sufficient to destroy the system in question before the despised realizers occur. In this scenario, it might take what Lewis calls a ‘big miracle’ to get one of the despised realizers to obtain; in which cases the closest world where the preferred realizer does not obtain is a world in which none of them do.

These examples might all be seen as somewhat far-fetched, especially the third. (Some might see the third example as a reason to modify the Lewisian criteria for assessing counterfactuals rather than a reason to affirm realization-sensitivity for this causal relation). But I think the examples at least give an idea of how there can be realization-sensitive causal relations even in the presence of multiple realizability, and hence that multiple realizability is not a sufficient condition for high-level events to have autonomous causal powers. The additional condition they suggest is that all non-actual realizers of a high-level event must be sufficiently remote possibilities that worlds in which any of them realize the high-level event are less similar to the actual world than worlds in which the high-level event simply does not occur at all.

I don’t think Menzies and List need be too worried about this additional condition. Their aim, to rescue the causal autonomy of the mental, seems secured because the mental states of creatures like us are realizable in a colossal number of distinct ways, and not much of macroscopic importance will tend to hang on which realization is instantiated. Even if we count neural events rather than atomic events, and limit ourselves to some particular individual, there will still plausibly be millions if not billions of neurally distinct events which could realize a single psychological event. This number will reduce as we move to consider successively simpler creatures, or as we compare levels of description which are closer together – and that is exactly as it should be. Causal autonomy is something which we should expect to come in degrees.

The talk also triggered some interesting ideas about the context-sensitivity of causal ascriptions – but I’ll save those for another post.

Conditions for causal autonomy

Hiatus

Sorry, the third part of the post below will have to wait a while – there probably won’t be much on here until I’ve written up my D.Phil thesis. The blog will be active again by October most likely.

Hiatus

Against mushy credence [part 2]

In this post I’ll look at some cases which might motivate a mushy credence view. Which particular mushy credence we use to account for any particular case will depend on the details of the interpretation we place on mushiness – on which kind of indeterminacy, in particular, we take to be represented by the provision of a set of credence functions rather than a single function. For a survey of the different possible interpretations of set-valued credal states, see Bradley [Synthese, 2009]. In part 3 of this post I’ll consider these interpretations in more detail, and raise some problems for the use of sets of credence functions based on the sheer multiplicity of suggested interpretations. But let me anticipate the results of that discussion by restricting my attention in this post to what Bradley calls the ignorance interpretation of mushiness, and seeing how that interpretation is most naturally applied to our problem cases.

Here’s what Bradley says about the ignorance interpretation:

“The agent may be unable to arrive at a judgement because she lacks the informational basis for doing so. This seems to be the kind of situation in which subjects find themselves, for instance, when placed in an Ellsberg paradox set-up in which the consequences of their decisions depend on the colour of a ball drawn from an urn containing an unknown proportion of balls of different colours. Many authors argue that in these kinds of situation the agent is not merely in a state of uncertainty in the sense that they don’t know for sure which colour ball will be drawn but can assign a probability to the prospect of each colour, but are rather in a state of ignorance in the sense that, such are the limits on what they know and can find out, that they have no non-arbitrary basis for assigning such a probability.”

So how can we apply this to the first of our cases?

[Percy] Someone you’ve never met emails to tells you that they use the name ‘Percy’ for some particular proposition; they don’t tell you anything about what Percy says. What is your credence that Percy is contingent? What is your credence that Percy is true?

Here the relevant ignorance is ignorance of which proposition Percy is. Therefore, applying the ignorance interpretation involves assigning a different credence function for each candidate for Percy’s identity. In particular, each candidate for Percy’s identity carries with it a credence distribution over contingency and non-contingency. For example, we are very confident indeed that ‘there are no round squares’ is noncontingent, while being equally confident that ‘there are no round windows in the White House’ is contingent. So it would seem that if we are to use a set of precise credences to represent our belief state concerning Percy’s contingency, then at least some of the credences in the set should be 1 or close to 1, and others 0 or close to 0. Perhaps there are some sentences whose status as contingent is controversial. ‘Other times exist’, ‘energy is conserved’, ‘ghosts exist’ might all fall into this category, and (if so) would contribute middling credences to the set. By aggregating the credences associated with all candidates of the relevant sort, we might hope to end up with a set spanning the [1,0] interval, or at least a set which comes very close to spanning that interval.

What about my state of belief in Percy’s truth? Again, different Percy-candidates come with a different credence distribution over truth and falsity. If Percy is ‘1=1’, then we might assign credence 1 to it; if Percy is 1=2, we might assign credence 0. And there are plenty of propositions to which we assign middling credence. So again, our state of belief in Percy’s truth will plausibly be represented by a set of credence functions spanning the [1,0] interval.

The case of Percy is relatively suitable for mushy treatment, as it’s plausible that there do exist continuum many propositions; this makes it at least a mathematical possibility that for any real number N between 0 and 1 we could find a proposition such that our credence in that proposition’s contingency or truth is N. But notice that even this is not guaranteed; it seems unlikely to be a requirement of rationality, for example, that an agent should have a credence in some proposition’s contingency equal to N, for any N in any range between 0 and 1. That is, it doesn’t seem out of the question that for no proposition whatsoever do I have a credence of 0.55551 in that proposition’s being contingent. But if so, then my state of belief in Percy’s contingency cannot be represented by an ignorance-interpreted mushy credence over an interval including 0.55551; the set which represents my state of belief in Percy’s  contingency would have to be ‘gappy’.

Another worry emerges when we notice that sets spanning the [1,0] interval are being used to represent our state of belief both in Percy’s contingency and in Percy’s truth. It follows straight away from this that our state of belief that Percy is contingently true must also be represented by a set spanning the [1,0] interval. This issue will be revisited later on.

[Quiz] In a tie-breaker round of the pub quiz you are asked how many times Australia have won the Ashes. You never watch sports or read the sports pages, and don’t know what the Ashes are or how often they are competed for. What is your credence that the number is larger than 10? What is your credence that the number is even?

In this case, the relevant ignorance concerns the properties of the Ashes competitions. ‘I don’t know enough about what the Ashes competitions are like’, the mushy credence lover might reason, ‘to assign a credence in their having been won any particular number of times by Australia’.  Suppose first that the Ashes are the prize in a daily card game between Australian and British airline crews, and that the card game is strictly a game of chance with a 50% likelihood of either side winning on any given occasion, and that they have been competed for on 10,000 occasions, and that neither side has ever cheated. Then the expectation value for Australian wins is 5,000; your credence that the number exceeds 10 is close to 1; and your credence that the number is even approaches 0.5. But now suppose that the Ashes refers to a single cricket game, played in 1882, and the rules were such that only one ball would be bowled, rendering it impossible for either side to win. Then the expectation value for Australian wins is 0, your credence that the number exceeds 10 is zero; and your credence that the number is even is close to 1.

It is easy to see that there are hypotheses about the rules and frequency of Ashes competitions which will result in any credence between 0 and 1 being ascribed to Australia winning more than 10 times, or to the number of wins being even. As in the case of [Percy], it seems that we must assign a set of credences spanning the [1,0] interval both to the proposition that the number is larger than 10, and to it being even.

Here a further question may occur to us. Surely some of the candidates for the nature of Ashes competition are less plausible than others; if the rules were as given in either of my toy examples, then the Ashes competitions would be highly unlikely to feature in a pub quiz, or even to have been held at all. With this in mind, it would be natural to weight some of the candidates more strongly than the others; but this is not part of the standard mushy credence machinery. Without weighting, the only way that we can restore some plausibility to the application of ignorance-interpreted mushy credences in this case is if there is some natural partition of possible rules-candidates, such that each is equi-probable. But such a partition looks like it would be highly problematic. I will return to this issue in part 3 of this post.

[Constant] Fundamental physics reveals that the value of a  certain ‘fundamental constant’ Q of nature is around 75. It turns out that a value of 74 or lower for Q would have resulted in a failure for stars to form; a value of 76 or higher would have resulted in a supergiant black hole sucking in the entire universe. Whether or not we take these results to be evidence for eg a benevolent God or multiple universes will depend on how antecedently unlikely we take a value of 75 for Q to be. What prior credence should we have had in Q being between 74 and 76?

In this example, it is less clear how to apply ignorance-interpreted mushy credences. My best stab at characterizing the ignorance involved is ignorance of the range of metaphysically possible worlds in which Q takes any value at all. If all metaphysically possible worlds that instantiate Q have a Q-value of around 75, then the appropriate prior credence to have in Q being around 75 is 1; if there are metaphysically possible worlds featuring all values of Q from 0 to 100, then the appropriate prior credence to have in Q being around 75 is about 0.01; if there are metaphysically possible worlds featuring all real-numbered Q-values, then our prior credence in Q being around 75 is 0. Once again, then, we are led to ascribe a set of credence functions covering the [1,0] interval as the correct representation of a rational belief state in Q being between 74 and 76.

[Cube] Bas van Fraassen locks you in a mystery cube factory. You discover that no cubes produced have edges longer than 2 metres.  You have no other evidence about the distribution of cube size. What is your credence that the next cube produced will have a volume larger than a cubic metre?

Here the relevant ignorance is ignorance of the distribution of cube size. One hypothesis has it that all cubes produced by the factory have edge lengths of 10cm. This would result in a credence of 0 in the next cube produced having a volume larger than a cubic metre. Similarly, the hypothesis that all cubes have edge lengths of 1.5m would result in a credence of 1 in the next cube having a volume larger than a cubic metre. And it’s easy to see that various hypotheses about edge length distribution could make appropriate any real-valued credence in the next cube having a volume larger than a cubic metre.  So once again, we find a set spanning the [1,0] interval mandated to represent our belief state that the next cube produced will have a volume larger than a cubic metre. Anyone spot a theme developing here?

[Ignorant tennis] You sit down to watch a tennis match. You have never heard of either player, they appear evenly-matched in fitness and physique. What is your credence that player A will win?

Our ignorance concerns the capacities of the two players. As part of this ignorance, we don’t want to rule out the hypothesis that player A has a heart pacemaker with a faulty battery which is about to expire, and which will force him to retire on medical grounds, handing victory to B. Nor do we want to rule out that player B is similarly afflicted. On the former hypothesis, your credence in A winning should be 0; on the latter, it should be 1. And there are plenty of intermediate hypotheses about the relative ability of the two players according to which the credence in A winning can take any value between zero and one.  So, once again, your credence that player A will win ought to be represented by a set spanning the [1,0] interval.

[Knowledgeable tennis] You sit down to watch a tennis match. You have coached both players, and have an detailed knowledge of their abilities and playing style. You consider them exactly evenly-matched. What is your credence that player A will win?

We presume that the ‘abilities’ referred to here includes all eventualities, such as faulty pacemakers. There is no ignorance involved in the case ex hypothesi – therefore, the correct credence in A’s winning is 0.5. The friend of mushy credence will presumably point to the difference between 0.5 and [0,1] as representing the intuitive difference between Ignorant tennis and Knowledgeable tennis.

From the above treatment, every single case which seems amenable to mushy treatment (under the ignorance interpretation of mushy credence) seems to land us with a set of credences spanning the [1,0] interval. This does not lead to any immediate contradiction, just as we can have credence 0.5 in many distinct propositions without contradiction. But it does lead to a number of worries, which I will list briefly here and examine in more detail in part 3 of this post:

  • Sets of credences spanning the [0,1] interval are immovable, assuming that we update by conditionalization of each function within the set. If we start with a set spanning [1,0] and conditionalize, no further evidence will shift us from this position.
  • Representing belief states with sets spanning [0,1] intervals seems to wash out relevant epistemic differences. For example, we are presumably more confident that Percy is true than that Percy is contingently true, as the latter claim is logically stronger. But it our belief state in Percy’s truth and our belief state in Percy’s contingent truth must both be represented by a set spanning the [0,1] interval.
  • Although the ignorance-interpreted mushy credence machinery has the scope to represent partial suspension of judgement (for example, by a set spanning [0.4-0.6]), it seems that in a range of straightforward cases (all those surveyed above, in any case) this partial suspension of judgement is inapplicable, and we are stuck with maximal suspension of judgement.
  • It is not clear that the sets resulting from an ignorance interpretation of mushiness will not be ‘gappy’ – that is, it is not clear that there will always be a function in the set corresponding to every real number in some interval. Gappiness significantly detracts from the intuitive appeal of mushy credence functions, and complicates the mathematics needed to apply them.
  • Applying the ignorance-interpreted mushy credence machinery in the way described above seems to wash out differences between the plausibility of the various hypotheses (the Ashes resting on the results of a normal cricket match vs a 1-ball cricket match). To avoid this, we would require either a) weighting of different credence functions within the set or b) a natural partition of hypotheses into equiprobable hypotheses. Applying either of these solutions would significantly undermine the simplicity and intuitive appeal of the mushy credence approach.

In part 3, I will reassess motivations for using mushy credences, and tie various loose threads into a concerted case against the use of mushy credences in a probabilist epistemology.

[To be continued]

Against mushy credence [part 2]

Against mushy credence [part 1]

Recently I’ve been puzzling over the ‘coin puzzle’ recently spotlighted by Roger White. White uses it to raise trouble for a view of personal probabilities (variously called ‘mushy credences’, ‘fuzzy credences’, ‘imprecise probabilities’, ‘vague probabilities’, ‘thick confidences’) which characteristically represents personal probability states using sets of credence functions, such that each of the functions in the set individually conforms to the probability calculus and updates by conditionalization. The mushy credence view is often motivated by the desire to account for the difference between judgements of equiprobability and suspension of judgement.

I think White’s coin puzzle does serious damage to the mushy credence view. Properly understood, it is a way of making vivid the phenomenon of dilation, which results when mushy credences interact with sharp credences. Dilation isn’t news to the proponents of mushy credences, but as the coin puzzle shows, in combination with objective chance it becomes problematic. A known correlation of some p in which we have mushy credence with the result of some coin toss for which we know the sharp chance of heads leads to a mushy credence in heads, even when we knew the pre-toss chance. This mushification of our credence in chancy outcomes is repugnant and provides us with a reason to reject one of the premises that led us to it. But the only controversial premise was the mushy credence view.

This is bad news for proponents of mushy credences. Some I have spoken to would bite the bullet and accept the counter-intuitive consequences of known chance dilation, seeking to soften the impact by emphasizing that the coin case is unrealistic. Others (here I am thinking of Scott Sturgeon’s forthcoming paper in OSE) take this as reason to abandon the sets-of-credence-functions-individually-updated model. Others, like White, take it as reason to abandon mushy credences altogether and explain suspension of judgement in a different sort of way. Which route is the most promising?

I suspect that biting the bullet would prove too painful, and that once we abandon the formal model there wouldn’t be much left of the mushy credences view; but I won’t defend these claims here. Rather, my plan is to support the case against mushy credences by examining some cases which might have seemed amenable to mushy treatment, and providing an alternative non-mushy explanation of what is going on in these cases. So here are some example cases to ponder over. In the next part of this post, I’ll apply the mushy treatment to these cases and ask how well it fares with them.

[Percy] Someone you’ve never met emails to tells you that they use the name ‘Percy’ for some particular proposition; they don’t tell you anything about what Percy says. What is your credence that Percy is contingent? What is your credence that Percy is true?

[Quiz] In a tie-breaker round of the pub quiz you are asked how many times Australia have won the Ashes. You never watch sports or read the sports pages, and don’t know what the Ashes are or how often they are competed for. What is your credence that the number is larger than 10? What is your credence that the number is even?

[Constant] Fundamental physics reveals that the value of a  certain ‘fundamental constant’ Q of nature is around 75. It turns out that a value of 74 or lower for Q would have resulted in a failure for stars to form; a value of 76 or higher would have resulted in a supergiant black hole sucking in the entire universe. Whether or not we take these results to be evidence for eg a benevolent God or multiple universes will depend on how antecedently unlikely we take a value of 75 for Q to be. What prior credence should we have had in Q being between 74 and 76?

[Cube] Bas van Fraassen locks you in a mystery cube factory. You discover that no cubes produced have edges longer than 2 metres.  You have no other evidence about the distribution of cube size. What is your credence that the next cube produced will have a volume larger than a cubic metre?

[Ignorant tennis] You sit down to watch a tennis match. You have never heard of either player, they appear evenly-matched in fitness and physique. What is your credence that player A will win?

[Knowledgeable tennis] You sit down to watch a tennis match. You have coached both players, and have an detailed knowledge of their abilities and playing style. You consider them exactly evenly-matched. What is your credence that player A will win?

(thanks to John Cusbert for this last pair of examples.)

[To be continued]

Against mushy credence [part 1]