Abstract
I have two main aims. The first is general, and more philosophical (§2). The second is specific, and more closely related to physics (§§3 and 4). The first aim is to state my general views about laws and causation at different ‘levels’. The main task is to understand how the higher levels sustain notions of law and causation that ‘ride free’ of reductions to the lower level or levels. I endeavour to relate my views to those of other symposiasts. The second aim is to give a framework for describing dynamics at different levels, emphasizing how the various levels' dynamics can mesh or fail to mesh. This framework is essentially that of elementary dynamical systems theory. The main idea will be, for simplicity, to work with just two levels, dubbed ‘micro’ and ‘macro’, which are related by coarsegraining. I use this framework to describe, in part, the first four of Ellis' five types of topdown causation.
1. Introduction
I have two main aims. The first is general, and more philosophical (§2). It concerns not just this Theme Issue's topic, topdown causation, but the general relations between ‘levels’. The second aim is specific, and more closely related both to topdown causation and to physics, in particular dynamical systems theory (§§3 and 4).
In discussing relations between levels, I will take it that the overall task is to understand how the higher levels sustain notions of law, causation and explanation that are ‘autonomous’, or ‘ride free’, from whatever reductions there might be to the lower level or levels (or at least: notions that seem to be autonomous or to ride free!). This is a large task, with a large literature of controversy, both nowadays and in the past. The reasons for the controversy are obvious. People disagree about how to understand the notions of level and reduction, and also those of law, causation and explanation. They disagree about the extent to which, and the sense in which, the higher levels are autonomous or ride free. And these disagreements are fuelled by having different sets of scientific examples in mind.
These disagreements become more vivid (and more comprehensible) when one considers historical changes in the disputants' scientific examples. One broad example is the demise of vitalism. Before the centurylong rise of microbiology, biochemistry and molecular biology (from say 1860 to 1960), it was perfectly sensible to believe that biological processes depended on certain ‘vital forces’; and therefore that, as the slogan puts it, ‘biology is not reducible to chemistry and physics’—in a much stronger sense of ‘not reducible’ than one could believe today. Other broad examples extending over many decades are: (i) the rise of atomism and statistical mechanics in explaining irreversible macroscopic processes and (ii) the rise of atomism, the periodic table and then quantum chemistry, in explaining chemical bonding. Again: before these developments, one could believe in irreducibility, e.g. of chemical bonding to physics, in a much stronger sense than one can today. In short: what historians now call ‘the second scientific revolution’ from 1850 onwards has given us countless successful reductions of behaviour (both specific processes and general laws) at a higher (often macroscopic) level to facts at a lower (often microscopic) level.
Thus, the overall philosophical task, both nowadays and in yesteryear, is: first, to state and defend notions of level and reduction, and of law, causation and explanation; and second, to use them to assess, in a wide range of contemporary scientific examples, the extent to which, and the sense in which, the higher levels are autonomous, or ride free, from the lower levels. But nowadays, after the triumphs of the second scientific revolution, we must expect the extent of, and/or senses for, such autonomy of the higher levels to be more restricted and/or more subtle. One aspect of this overall task is the topic of this special issue: assessing the prospects for topdown causation.
My own contribution will proceed in two stages. First (§2), I will summarize some of my own views about the overall task, relating them to topdown causation and the views of some other authors. For example, I will briefly endorse some views of Sober's about reduction and causation, and List & Menzies' recent defence of topdown causation (§2.2 and 2.4). My overall views are defended in detail elsewhere [1,2]. They mostly concern emergence, reduction and supervenience (§2 will report my construals of these contested terms). I should admit at the outset that I will have nothing distinctive to say about the notions of law, causation and explanation. But, in fact, I take a broadly Humean view of all three; and it will be clear that this will fit well with my views on emergence, reduction and supervenience.
Second, in §§3 and 4, I will give a framework for describing dynamics at different levels, emphasizing how the various levels' dynamics can mesh, or fail to mesh. This framework is essentially that of elementary dynamical systems theory. The main idea will be, for simplicity, to work with just two levels, dubbed ‘micro’ and ‘macro’, which are related by coarsegraining. I then consider two topics, in §§3 and 4, respectively.
First, there is the question whether a microdynamics, together with a coarsegraining prescription, induces a welldefined macrodynamics (§3). I describe how physics provides some precise and important examples of such ‘meshing’ (e.g. in statistical mechanics), as well as examples of where it fails. I will stress that failure of meshing need not be a problem, let alone a mystery: the pilotwave theory, in the foundations of quantum mechanics, will provide a nonproblematic example. I also discuss how to secure meshing by redefining the coarsegraining; and relate the topic to the philosophical views of Fodor & Papineau on multiple realizability, and of List on free will.
Second, I use the framework to describe, in part, the first four of Ellis' [3,4] five types of topdown causation (§4). There are various choices to be made in giving such a dynamical systems description of Ellis' typology; but I maintain that the fit is pretty good. In particular, I note that Ellis calls my first topic above, i.e. the meshing of micro and macrodynamics, ‘coherent higher level dynamics’ or ‘the principle of equivalence of classes’; and he takes it as a presupposition of his typology of topdown causation.
Finally, a clarification. This paper has some ‘reductionist’ features, which might be misleading. Thus, in §2 I will join Sober & Papineau in rejecting the multiple realizability argument against ‘reductionism’. And in §4, I will not try to articulate the differences between my formal descriptions, in the jargon of dynamical systems, of Ellis' types of topdown causation, and Ellis' own informal and richer descriptions. These features might suggest that I deny any or all of the following three claims:

There are, or can be, laws and/or explanation and/or causation at ‘higher levels’, or in the special sciences.

There is a good notion of causation beyond that of functional dependence of one quantity on another.

Topdown causation, at least of Ellis' types, needs more than my dynamical systems framework.
But, in fact, I endorse (i)–(iii). It is just that they are not centrestage in my discussion.
2. Reduction, supervenience and causation
My first aim is to summarize some of my views about the relations between levels. Sections 2.1 and 2.2 discuss reduction and ‘multiple realizability’, and §2 discusses supervenience. Broadly speaking, I will deny the widespread (perhaps even orthodox?) views that multiple realizability prevents reduction, and that levels are typically related by supervenience without reduction. Section 2.4 concerns causation: here I will endorse Shapiro & Sober's, and List & Menzies', recent arguments for topdown causation.
2.1. Reduction
I have analysed the relations between reduction, emergence and supervenience, elsewhere [1,2]. In short, I construe these notions as follows. I take emergence as a system's having behaviour, i.e. properties and/or laws, that is novel and robust relative to some natural comparison class. Typically, the behaviour concerned is collective or macroscopic; and it is novel compared with the properties and laws that are manifest in (theory of) the microscopic details of the system. I take reduction as a relation between theories: namely, deduction using appropriate auxiliary definitions. (As we will see, this is in effect a strengthening of the traditional Nagelian conception of reduction.) And I take supervenience as a weakening of this concept of reduction; namely, to allow infinitely long definitions (more details in §2.3).
Then my main claim was that, with these meanings, emergence is logically independent both of reduction and of supervenience. In particular, one can have emergence with reduction, as well as without it. Physics provides many such examples, especially where one theory is obtained from another by taking a limit of some parameter. That is, there are many examples in which we deduce a novel and robust behaviour, by taking the limit of a parameter.^{1} And emergence is also independent of supervenience: one can have emergence without supervenience, as well as with it.
Broadly speaking, this main claim gives some support to the ‘autonomy’ of higher levels (cf. claim (i) at the end of §1), namely by reconciling such autonomy with the existence of reductions to lower levels. Some of my other claims had a similar reconciling intent: e.g. my joining Sober & Papineau in holding that multiple realizability is no problem for reductionism.^{2} I shall develop this position a little by discussing Nagelian reduction (this subsection), multiple realizability (§2.2) and supervenience (§2.3).
Nagel's idea is that reduction should be modelled on the logical idea of one theory being a definitional extension of another.^{3} Writing t for ‘top’ and b for ‘bottom’, we say: T_{t} is a definitional extension of T_{b}, iff one can add to T_{b} a set D of definitions, one for each of T_{t}'s nonlogical symbols, in such a way that T_{t} becomes a subtheory of the augmented theory T_{b} ∪ D. That is, in the augmented theory, we can prove every theorem of T_{t}. Here, a definition is a statement, for a predicate, of coextension; and for a singular term, of coreference. To be precise: for a predicate P of T_{t} it would be a universally quantified biconditional with P on the lefthand side stating that P is coextensive with a righthand side that is an open sentence ϕ of T_{b} built using such operations as Boolean connectives and quantifiers. Thus, if P is nplace: (∀x_{1}) … (∀x_{n})(P(x_{1}, … , x_{n}) ≡ϕ(x_{1}, … , x_{n})). (The definitions are often called ‘bridge laws’ or ‘bridge principles’.)
A caveat. I said that Nagel held that reduction ‘should be modelled on’ the idea of definitional extension, because definitional extension is (a) sometimes too weak as a notion of reduction and (b) sometimes too strong.
As to (a): Nagel [5, pp. 358–363] holds that the reducing theory T_{b} should explain the reduced theory T_{t}; and following Hempel, he conceives explanation in deductive–nomological terms. Thus he says, in effect, that T_{b} reduces T_{t} iff:

T_{t} is a definitional extension of T_{b}; and

in each of the definitions of T_{t}'s terms, the definiens in the language of T_{b} must play a role in T_{b}; so it cannot be, for example, a heterogeneous disjunction.
As to (b): definitional extension is sometimes too strong as a notion of reduction; as when T_{b} corrects, rather than implies, T_{t}. Thus, Nagel says that a case in which T_{t}'s laws are a close approximation to what strictly follows from T_{b} should count as reduction, and be called ‘approximative reduction’.
More important for us is the fact that definitional extensions, and thereby Nagelian reduction, can perfectly well accommodate what philosophers call functional definitions. These are definitions of a predicate or other nonlogical symbol (or in ontic, rather than linguistic, jargon: of a property, relation, etc.) that are secondorder, i.e. that quantify over a given ‘bottom set’ of properties and relations. The idea is that the definiens states a pattern among such properties, typically a pattern of causal and lawlike relations between properties. So an ntuple of bottom properties that instantiates the pattern in one case is called a realizer or realization of the definiendum. And the fact that, in different cases, different such ntuples instantiate the pattern is called multiple realizability. Examples of functional or secondorder properties, and so of multiple realizability, are legion. For example, the property of being locked is instantiated very differently in padlocks using keys, combination locks, etc.
2.2. The multiple realizability argument refuted
Multiple realizability is undoubtedly a key idea, philosophically and scientifically, for our overall task: understanding relations between levels, and especially how higher levels can be autonomous, or ride free, from lower levels. Admittedly, there is not much to be said by way of a theory about being locked and similarly for countless other multiply realizable properties, like being striped, or being mobile, or being at least 50 per cent metallic or …. For being locked, being striped, etc. do not define, or contribute to defining, significant levels. But some multiple realizable properties do so: cf. claim (i) at the end of §1.
Here is a schematic biological example (my thanks to a referee). Fitness is multiply realized by the different morphological, physiological and behavioural properties of organisms. Indeed, very multiply realized, what makes a cockroach fit is very different from what makes a daffodil fit. Thus, fitness is a higher order or ‘more abstract’ similarity of organisms (as are its various degrees). And unlike being locked, etc., it contributes to defining a significant level: there are general truths about it and related notions to be expressed and explained. For this it is not enough to have a theory about cockroaches, and another one for daffodils, etc. Rather, we need the theory of natural selection.
In short, multiple realizability is undoubtedly important for understanding relations between levels. But many philosophers go further than this. Some think that multiple realizability provides an argument against reduction. The leading idea is that the definiens of a multiply realizable property shows it to be too ‘disjunctive’ to be suitable for scientific explanation, or to enter into laws. And some philosophers think that multiple realizability prompts a nonNagelian account of reduction; even suggesting that definitional extensions cannot incorporate functional definitions.
I reject both these lines of thought. Multiple realizability gives no argument against definitional extension; nor even against stronger notions of reduction like Nagel's, which add further constraints additional to deducibility, e.g. about explanation. That is, I believe that such constraints are entirely compatible with multiple realizability. This was shown very persuasively by Sober [8]. But as these errors are unfortunately widespread, it is worth first rehearsing, then refuting, the multiple realizability argument.
We again envisage two theories, T_{b} and T_{t}, or two sets of properties, ℬ and 𝒯, defined on a set 𝒪 of objects. The choice between theories and sets of properties makes almost no difference to the discussion; and I shall here mostly refer to ℬ and 𝒯, rather than T_{b} and T_{t}. So multiple realizability means that the instances in 𝒪 of some ‘top’ property P ∈ 𝒯 are very varied (heterogeneous) as regards (how they are classified by) their properties in ℬ.
The multiple realizability argument holds that, in some cases, the instances of P are so varied that even if there is an extensionally correct definition of P in terms of ℬ, it will be so long and/or heterogeneous that:

explanations of singular propositions about an instance of P cannot be given in terms of ℬ, whatever the details about the laws and singular propositions involving ℬ; and/or

P cannot be a natural kind, and/or cannot be a lawlike or projectible property, and/or cannot enter into a law, from the perspective of ℬ.
Usually an advocate of (a) or (b) is not ‘eliminativist’, but rather ‘antireductionist’. P and the other properties in 𝒯 satisfying (a) and/or (b) are not to be eliminated as cognitively useless. Rather, we should accept the taxonomy they represent, and thereby the legitimacy of explanations and laws invoking them. Probably the most influential advocates have been: Putnam [9] for version (a), with the vivid example of a square peg fitting a square hole, but not a circular one; and Fodor [10] for version (b), with the vivid example of P = being money.
I believe that Sober [8] has definitively refuted this argument, in its various versions, whether based on (a) or on (b), and without needing to make contentious assumptions about topics like explanation, natural kind and law of nature. As he shows, it is instead the various versions of the argument that make contentious assumptions! I will not go into details. Suffice it to make three points, by way of summarizing Sober's refutation.^{4} The first two correspond to rebutting (a) and (b); the third point is broader and arises from the second.
As to (a): the antireductionist's favoured explanations in terms of 𝒯 do not preclude the truth and importance of explanations in terms of ℬ. As to (b): a disjunctive definition of P, and other such disjunctive definitions of properties in 𝒯, is no bar to a deduction of a law, governing P and other such properties in 𝒯, from a theory T_{b} about the properties in ℬ. Nor is it a bar to this deduction being an explanation of the law.
The last sentence of this refutation of (b) returns us to the question whether to require reduction to obey further constraints apart from deduction. The tradition, in particular Nagel himself, answers Yes (as I reported in caveat (a), §2.1). Nagel in effect required that the definiens play a role in the reducing theory T_{b}. In particular, it cannot be a very heterogeneous disjunction. (Recall that the definiens is the righthand side of a bridge principle.) The final sentence of the last paragraph conflicts with this view. At least, it conflicts if this view motivates the nondisjunctiveness requirement by saying that nondisjunctiveness is needed if the reducing theory T_{b} is to explain the laws of reduced theory T_{t}. But I reply: so much the worse for the view. Sober puts this reply as a rhetorical question [8, p. 552]: ‘Are we really prepared to say that the truth and lawfulness of the higher level generalization is inexplicable, just because the … derivation is peppered with the word “or”? I agree with him: of course not!
2.3. Supervenience? The need for precision
So far my main points have been that reduction in a strong Nagelian sense is compatible with both emergence (§2.1) and multiple realizability (§2.2). But these points leave open the questions how widespread is reduction, and what are the other, perhaps typical or even widespread, relations between theories at different levels. My view is that, within physics and even between physics and other sciences, reduction—at least in the approximative sense mentioned in caveat (b) of §2.1—is indeed widespread. I develop this view in Butterfield [1, §3.1.2] and [2, §4f], partly in terms of the unity of nature (cf. opening remarks about the second scientific revolution in §1).
As to what relation or relations hold when reduction fails, philosophers' main suggestion has been: supervenience. Roughly speaking, this notion is a strengthening of the idea of multiple realizability. I will first explain the notion, and then state four misgivings about it.
Again we envisage two sets of properties, ℬ and 𝒯, defined on a set 𝒪 of objects. We say that 𝒯 supervenes on ℬ (also: is determined by or is implicitly defined by ℬ) iff any two objects in 𝒪 that match in all properties in ℬ also match in all properties in 𝒯. Or equivalently, any two objects that differ in a property in 𝒯 must also differ in some property or other in ℬ. (One also says that ℬ subvenes 𝒯.) A standard (i.e. largely uncontentious!) example takes 𝒪 to be the set of pictures, 𝒯 their aesthetic properties (e.g. ‘is wellcomposed’) and ℬ their physical properties (e.g. ‘has magenta in topleft corner’).
It turns out that supervenience is a weakening of the notion of definitional extension given in §2.1; namely to allow that a definition in the set D might have an infinitely long definiens using ℬ. The idea is that, for a property P ∈ 𝒯, there might be infinitely many different ways, as described using ℬ, that an object can instantiate P: but provided that, for any instance of P, all objects that match it in their ℬproperties are themselves instances of P, then supervenience will hold.
Thus, many philosophers have held that in cases where one level or theory seems irreducible to another, yet to be in some sense ‘grounded’ or ‘underpinned’ by it, the relation is in fact one of supervenience. They say that the irreducible yet grounded level or theory (specified by its taxonomy of properties 𝒯) supervenes on the other one. That is, there is supervenience without definitional extension: at least one definition in D is infinitely long.
At first sight, this looks plausible: recall from the start of §2.2 that examples of multiple realizability are legion. But we should note four misgivings about it. The first two are widespread in the literature; the third and fourth are more my own. The first and fourth are philosophical limitations of supervenience; the second and third, scientific limitations.
First: philosophers of a metaphysical bent who discuss reduction, emergence and related topics find it natural to require that, in reduction, the ‘top’ properties 𝒯 are shown to be identical to properties in (or perhaps composed from) ℬ; and that this is so whether the reduction is finite, as in definitional extension, or infinite, as in supervenience. But the identities of properties (and the principles for composing properties) are controversial issues in metaphysics; and the holding of a supervenience relation is not generally agreed to imply identity. So, for such philosophers, supervenience leaves a major question unanswered.
Second: although the distinction between finite and infinitely long definitions is attractively precise, it seems less relevant to the issue whether there is a reduction than another, albeit vague, distinction: namely, the distinction between definitions and deductions that are short enough to be comprehensible, and those that are not. Recall that, according to the notion of a definitional extension given in §2.1, a definition in D can be so long as to be incomprehensible, e.g. a million pages—to say nothing of the length of the deductions!
Thus, the remarks usually urged to show a supervenience relation in some example, e.g. that no one knows how to construct a finite definition of ‘is wellcomposed’ out of ‘has magenta in topleft corner’ and its ilk, are not compelling. Our inability to complete, or even begin, such a definition is no more evidence that a satisfactory definition would be infinite than that it would be incomprehensibly long. In other words, we have no reason to deny that the example supports a definitional extension, albeit an incomprehensibly long one. And so far as science is concerned, definitional extension with incomprehensibly long definitions and deductions is useless: that is, it may as well count as a failure of reduction. Philosophers, including Nagel himself, have long recognized this point: recall the caveat (a) in §2.1.^{5}
The third misgiving is similar to the second, in that both accuse supervenience of having—for all its popularity in philosophy—limited scientific value. But where the second sees supervenience's allowance of infinite disjunctions as a distraction from the more important issue of comprehensibility, the third sees supervenience's allowance of infinite disjunctions as a distraction from the more important issue of the limiting processes that occur in the mathematical sciences, and in particular in examples of emergence in physics. That is, because supervenience's infinity of ‘ways (in terms of ℬ) to be P ∈ 𝒯’ bears no relation to the taking of a limit (e.g. through a sequence of states, or of quantities, or of values of a parameter), it sheds little or no light on such limits, in particular on the emergent behaviour that they can produce. Agreed, this sort of accusation can only be made to stick by analysing examples: suffice it to say here that Butterfield [2] analyses four such.
The fourth misgiving concerns philosophers' appeal to supervenience, not as a relation between two independently specified levels or theories, but as a tool for precisely formulating physicalism: the doctrine that, roughly speaking, all facts supervene on the physical facts. Here, my complaint is for physicalism to be precise, you need to state precisely what are ‘the physical facts’ (or what is ‘the physical supervenience basis’). Sad to say, in the philosophical literature, both proponents and opponents of physicalism tend to be vague about this. Here is one example which has been discussed widely (more details in [1, §5.2.2]). (I also recommend Sober's very original discussion of how the definition of, and our reasons for, physicalism are usefully cast in terms of probability, especially the Akaike framework for statistical inference [16].)
If there had been fundamental manybody forces (called by C. D. Broad, the British emergentist of the early twentieth century, ‘configurational forces’), then the theory of a manybody system would not be supervenient on (let alone a definitional extension of) a theory of its components that used only twobody interactions.^{6} Of course, if there had been such forces, physicists would have made it their business to investigate them, so that a phrase like ‘the physical facts’ would have come to include facts about such configurational forces, as well as facts about the familiar twobody forces. At least it would have come to include such facts if the configurational forces turned out to fit into the familiar general frameworks of (classical or quantum) mechanics, e.g. having a precise quantitative expression as a term in a Hamiltonian. But this says more about the elasticity of the word ‘physics’, or about universities' departmental structure, than about the truth of a substantive doctrine of ‘physicalism’!
2.4. Causation
So far, I have ignored issues about timeevolution, and in particular causation: I have stressed what one might call ‘synchronic issues’, rather than ‘diachronic issues’. But, from now on, diachronic issues will be centrestage. As I mentioned in §1, I take a broadly Humean view of causation, but do not advocate a specific account. Nor will I need such an account for the rest of this paper's aims. There are three such aims. In this subsection, I will report and recommend two recent arguments broadly in favour of topdown causation. My final aim, in later sections, will take longer: it is to describe Ellis' types of topdown causation, in terms of functional dependence (cf. claims (ii) and (iii) at the end of §1).
Of course, there is much to say about topdown causation apart from what follows in the rest of this paper; and even apart from my fellow symposiasts—in a large literature, I recommend Bedau [14, pp. 157–160, 175–178]. And I cannot trace the consequences of what follows, for other authors' views. But I commend what follows to advocates of topdown causation, such as that of Ellis and Atmanspacher, Auletta, Bishop, Jaeger and O'Connor. For I think it makes precise some of their claims: such as that higher level facts or events constrain, modify or form a context for the lower level, which is therefore not independent of the higher level [22]; or that lower level facts or events are necessary but not sufficient for higher level ones [21].
So I turn to reporting and recommending the two arguments. The first is Shapiro & Sober's argument, not so much for topdown causation, as against a contrary position, namely epiphenomenalism. This is the doctrine that higher level states cannot be causes, i.e. they are causally ineffective. Thus, the idea of epiphenomenalism is that such states are preempted, as causes, by lowerlevel states. (Here, we could say ‘property’, ‘fact’ or ‘event’, instead of ‘state’: it would make no difference to what follows.) Shapiro & Sober rebut this, by adopting an account of causation in terms of intervention (or, in another jargon: in terms of manipulation). On the other hand, the second argument is List & Menzies' positive argument for topdown causation; it is based on an account of causation in terms of counterfactuals. Fortunately, both accounts of causation are plausible, and I will not need to choose between them. Nor will I need to develop the accounts' details. For the argument by each pair of authors needs only the basic ideas of the account.
I can present both arguments in terms of the same example of a pair of levels: the mental and the physical. Both arguments are very general, and apply equally to other examples of pairs of levels. But this example has various advantages. It is vivid and widely discussed in philosophy. All these four authors use it. Although Shapiro & Sober also discuss higher level causation in biology, especially evolutionary biology, for List & Menzies, this example is the main focus. So their central case of topdown causation is mental causation, e.g. my deciding to raise my arm causing it to go up. Besides, all these authors rebut various formulations and defences of epiphenomenalism about the mental with respect to the physical by Kim, who is probably the most prolific recent writer on the mental–physical relationship.
Thus here, in terms of the mental and the physical, is what Shapiro & Sober [23] call ‘the master argument for epiphenomenalism’:
How could believing or wanting or feeling cause behavior? Given that any instance of a mental property M has a physical microsupervenience base P, it would appear that M has no causal powers in addition to those that P already possesses. The absence of these additional causal powers is then taken to show that the mental property M is causally inert. ([23, p. 241]; notation changed)
In addition to Kim's formulations of this argument that Shapiro & Sober go on to document, compare Kim [24, p. 19]. The argument is a cousin of what is often called ‘the exclusion argument’, also often advocated by Kim: for a discussion, see, for example, Humphreys [25].
Assessing this argument depends of course on one's account of causation. (And, one might guess, on one's account of realization or supervenience—but, in fact, the varieties of these notions turn out not to matter.) But the argument fails utterly, on each of two plausible accounts of causation: the interventionist account adopted by Shapiro & Sober, and the counterfactual account adopted by List & Menzies. And the failure follows from just the basic features of each pair of authors' account.
Thus the leading idea of Shapiro & Sober's rebuttal is:
To find out whether M causally contributes to N, you manipulate the state of M while holding fixed the state of any common cause C that affects both M and N; you then see whether a change in the state of N occurs. … It is not relevant, or even coherent, to ask what will happen if one wiggles M while holding fixed the microsupervenience base P of M. … Because a supervenience base for M provides a sufficient condition for M, where the entailment has at least the force of nomological necessity, asking this question leads one to attempt to ponder the imponderable—would N occur if a sufficient condition for M occurred but M did not? ([23, pp. 238–240]; notation changed)
Besides, Shapiro & Sober strengthen this rebuttal by augmenting their account of causation with probabilities (see also Sober [16, pp. 145–149]. There is a happy concordance here between interventionist and probabilistic accounts of causation. But, again, I will not need to give details.
I turn to List & Menzies [26] and Menzies & List [27]. In short, they make two points: (i) a general point, in common with other authors, which supports the idea of causation occurring at higher levels, and between levels (cf. claim (i) at the end of §1), and (ii) the specific argument in favour of topdown causation, and against epiphenomenalism. Broadly speaking, their first point supports the idea that we should understand causation in terms of counterfactuals (especially ‘if C had not occurred, then E would not have occurred’). On the other hand, their specific argument assumes some such counterfactual account.^{7}
The general point is that a cause needs to be specific enough to produce its effect—but not more specific. The point is clear from how we think about causes in countless examples. What caused the bull to be in a rage? Answer: the bull's seeing the matador's red cape nearby. If the cape happens to be crimson and the matador to be standing 3 m from the bull, nevertheless, the cause is as stated. It is not the more specific fact of the bull's seeing the matador's crimson cape 3 m away.
Nor is this point just a matter of our intuitive verdicts in countless examples. It is upheld by plausible accounts of causation, in terms of counterfactuals (stemming from Lewis [28]). The key idea is that C's causing E requires that if C were not to hold, then E would not hold. And, indeed, if the cape were not red, but say green, then the bull would not be enraged. But if C is too specific, this requirement tends to fail. We cannot conclude that, if the cape were not crimson, the bull would not be enraged—for if the cape were not crimson, it might well have been some other shade of red, and then the bull would still have been enraged.
This point applies in countless examples where the contrast between appropriate and toospecific causes corresponds to a contrast between levels, such as the mental and the physical. Witness the bull example (which is adapted from Yablo [29]); or in the other ‘direction’, a mental state causing a physical one, as in the timehonoured example of armraising. What caused my arm to go up? My deciding to raise it.
To sum up: this point teaches us that more specific information about the facts and events in some example is not always illuminating about causal relations; and (a fortiori), that it is wrong to think that the ‘best’ or ‘real’ causal explanation of the facts and events always lies in the most specific and detailed information about them. (Of course, it is ‘reductionists’ rather than others who are most likely to suffer these bad temptations.) And so this point supports the idea that there are causal relations between facts or events at higher levels, or between different levels.^{8}
List & Menzies build on this general point so as to refute the ‘master argument’—that a mental state M cannot cause a later physical, say neural, state N, as its realizing or subvening neural state P preempts it as a cause. They show that, on plausible accounts of causation invoking counterfactuals, the argument fails. Indeed, they state a causal assumption about how M causes N, which in many actual cases we have good reason to believe true, and which implies that P is not a cause of N. (This assumption is that, even if M were realized by a physical state other than its actual realizer P, N could still obtain. List & Menzies' jargon is that M's causing N is ‘realizationinsensitive’; see [26, §V].)
To sum up, here is excellent news for advocates of topdown causation, at least if they like interventionist or counterfactual accounts of causation. More power to these authors' elbows. Or, expressed in the spirit of their results: more power to their wills, so as to cause their elbows to move!
3. Dynamics at different levels
I now turn to this paper's second main aim. In this section, I give a framework for describing dynamics at different levels, emphasizing how two levels' dynamics can mesh or fail to mesh. Section 4 then applies the framework to describe some of Ellis' [3] types of topdown causation.
There will of course be some connections with topics addressed in §2. Here are two examples. (i) Section 3.2 describes how Papineau's critique of Fodor's version of the multiple realizability argument (see §2.2) is a matter of two levels' dynamics failing to mesh. (ii) I admit at the start of §4 that my descriptions of Ellis' types are partial, because they avoid controversies about what is required for causation, beyond functional dependence of quantities (cf. claims (ii) and (iii) at the end of §1).
3.1. The framework introduced
For simplicity, I will work with just two levels, dubbed ‘micro’ and ‘macro’, which are related by coarsegraining. There will be several other simplifying assumptions, as follows.

We think of the microlevel as a state space , with the microdynamics as a map T: → (so time is discrete). Since T is a function, we assume a pasttofuture microdeterminism.

But T need not be invertible, so futuretopast determinism can fail. Besides, most of what follows applies if T is just a binary relation, i.e. one–many as well as many–one, so that there is pasttofuture microindeterminism. (This indeterminism need not reflect quantum mechanics—under the orthodox interpretation! It could reflect the system being open.)

We will not need to discuss in detail the set ℚ of quantities. But we envisage that each Q ∈ ℚ is a realvalued function on . For a classical physical state s, Q(s) would be thought of as the system's intrinsic or possessed value for Q when in s. But, in quantum theory, Q(s) would naturally be taken to be a Bornrule expectation value, e.g. s is a Hilbert space vector, and . In either classical or quantum physics, we naturally think of ℚ as separating in the sense that, for any distinct s_{1} ≠ s_{2} ∈ , there is a Q ∈ ℚ such that Q(s_{1}) ≠ Q(s_{2}).

We think of the macrolevel as given by a partition P of , i.e. a decomposition of into mutually exclusive and jointly exhaustive subsets C_{i} ⊂ . The C_{i}, i.e. the cells of the partition P = {C_{i}}, are the macrostates. So C stands for ‘cell’ or ‘coarsegraining’.

Filling out (ii)–(iv), we naturally think of each C_{i} as specified by a set of values for some subset of ℚ, the macroquantities ℚ_{mac} ⊂ ℚ. That is, each C_{i} is the intersection of levelsurfaces, one for each quantity in ℚ_{mac}. But we can downplay quantities, and just focus on the macrostates C_{i}, i.e. the cells of the partition P.
3.2. Meshing of dynamics: examples and counterexamples, in physics and philosophy
I now consider the way in which the dynamics at the two levels can mesh with each other, or fail to do so. Physics provides precise and important examples of such ‘meshing’ (e.g. in statistical mechanics), as well as examples where it fails. I also relate meshing to the views of Fodor & Papineau on multiple realizability, and to the views of List on free will.
The framework, especially assumptions (i) and (iv) of §3.1, yields natural definitions of:

how the microdynamics T defines a macrodynamics, and

whether these two dynamics mesh.
As to (a), recall that any function f:X → Y between any two sets X and Y defines a function between their power sets, by the obvious rule A ⊂ X ↦ f(A): = {y ∈ Yy = f(x) for some x ∈ A} ⊂ Y. We apply this idea to T : → , and then restrict to the partition . In general, the image of a cell C_{i} is not a subset of a single macrostate. That is, two distinct s_{1} ≠ s_{2} ∈ C_{i} are sent by T to distinct macrostates. In other words, we have macroindeterminism, despite T giving microdeterminism. The micro and macrodynamics do not mesh; see figure 1, where the union symbol, ∪, beside some of the upward lines indicates coarsegraining.
On the other hand, as to (b): if for every cell C_{i} ∈ P, its image is a cell in P, for some j, then the microdynamics T does induce a deterministic macrodynamics, and we will say that the micro and macrodynamics mesh. So in figure 1, the upward ∪ arrow from T(s_{1}) would not be ‘stray’, but would point to C_{j}. In mathematical jargon, coarsegraining and timeevolution commute. Another jargon: coarsegraining is equivariant with respect to the group actions representing timeevolution (i.e. for discrete time: actions of the group of integers ℤ, on and on P).
Physics provides many important examples of meshing dynamics. The most obvious examples concern conserved quantities, especially in integrable systems. Recall from (iii) and (v) of §3.1 the idea of macroquantities ℚ_{mac} ⊂ ℚ, i.e. the idea that each cell C_{i} is the intersection of levelsurfaces, one for each quantity in ℚ_{mac}. If we instead define ℚ_{cons} ⊂ ℚ as those quantities whose values are constant under timeevolution T (which could now even be indeterministic), then, obviously, the intersection of the levelsurfaces of these conserved quantities will be invariant under T. So there is meshing dynamics, though the macrodynamics is trivial, i.e. every quantity we are concerned with is constant in value. And one could go on to investigate the ‘integrable’ systems for which ℚ_{cons} is ‘rich enough’, in the sense that these intersections are the minimal sets invariant under timeevolution; see also §3.3.
But there are important examples of meshing, when the system is not integrable, and even has only a few conserved quantities. Indeed, that is putting it mildly! Several of the most famous and fundamental equations of macroscopic physics (such as the Boltzmann, Navier–Stokes and diffusion equations) are the meshing macrodynamics induced by a microdynamics. Or rather: they are the meshing macrodynamics once we make the framework of §3.1 more realistic by allowing that:

the meshing may not last for all times;

the meshing may apply, not for all microstates s, but only for all except a ‘small’ class;

the coarsegraining may not be so simple as partitioning ; and indeed

the definition of the microstate space may require approximation and/or idealization, especially by taking a limit of a parameter: in particular, by letting the number of microscopic constituents tend to infinity, while demanding of course that other quantities, such as mass and density, remain constant or scale appropriately.
This point, especially (d), returns us to the claim in §2.1 that emergent (i.e. novel, robust) behaviour may be deduced from a theory of the microscopic details, often by taking a limit of some parameter. As I said there, I gave four such examples [2]. The equations just listed—Boltzmann, etc.—provide others.^{9}
Having emphasized dynamical meshing, not least for its philosophical importance in reconciling emergence and reduction, I should add that, on the other hand, failure of meshing—in mathematical jargon, the noncommutation of the diagram in figure 1; in physical jargon, macroindeterminism—need not pose any difficulties or ‘worries’ for our understanding the situation. For there can be other factors that make the noncommutation, the macroindeterminism, well understood, and even well controlled and natural.
For a physicist, the obvious and striking example of this is the pilotwave theory in the foundations of quantum mechanics as follows. The deterministic evolution T on the pilotwave microstates s (comprising, for example, positions q_{i} ∈ IR^{3} of corpuscles i = 1, …, N, as well as an orthodox quantum state ψ) induces an indeterministic evolution on ψ by a precise version of the textbooks' projection postulate rule that, when ψ divides into disjoint wavepackets, it is to be replaced by whichever of the (renormalized) packets has support containing the actual configuration 〈q_{i}〉 ∈ IR^{3N}. Besides, a mathematically natural probability measure on the microstates s (dependent on the quantum state ψ: namely, ψ^{2}) makes the macroindeterministic dynamics probabilistic, with the induced probabilities being the orthodox Bornrule probabilities (which are empirically correct for myriadly many kinds of experiments). So, in this example, the macroindeterminism is entirely understood, wellcontrolled and natural (cf. Bohm & Hiley [33, especially ch. 3] and Holland [34, especially ch. 3]).
For a philosopher also, there is an obvious and striking example of macroindeterminism, i.e. noncommutation of figure 1. Namely, the venerable idea of free will (ah, the joys of interdisciplinarity!). More precisely, a philosopher attracted by compatibilism—i.e. the view that determinism and free will are compatible—can take this sort of macroindeterminism induced by a deterministic microdynamics to be exactly what free will involves. Or, still more precisely, what free will could be taken to involve in a world governed by such a microdynamics. For a full defence of this version of compatibilism (including the discussion of such background assumptions as nonreductive physicalism), I recommend List [35].
On the other hand, I agree that, in these less welldefined philosophical contexts about the relations between levels, failure of meshing can be ‘worrying’. One example of such a worry is Papineau's critique [12, pp. 180–185] of Fodor's [10] vision of special sciences as autonomous. Thus, I agree that Papineau is right to press failure of meshing as a problem for Fodor. He argues that Fodor just assumes without justification that there will be meshing: Fodor's discussion appeals to a diagram like figure 2, which (with a trivial change from my notation) pictures the microlaws as indeed preserving the macrocategories, i.e. as inducing a welldefined dynamics by coarsegraining.^{10}
Papineau goes on (p. 186f) to argue—surely rightly—that in many cases, in everyday life and the special sciences, the meshing shown in figure 2 is secured by the macrocategories being defined by selection processes.
I take up the topic of selection in the discussion of Ellis in §4. For the moment, I just note that Ellis also addresses the idea of meshing dynamics. He calls the commutation we see in figure 2 ‘coherent higher level dynamics’, ‘effective same level action’ and ‘the principle of equivalence of classes’ [3, pp. 7–8 and fig. 3b, p. 45]. And, at least as I read him, his typology of five kinds of topdown causation presupposes such meshing. So, before turning to that typology, it is appropriate to discuss, albeit briefly, how one might secure such meshing (see §3.3).
3.3. Meshing secured by redefining the macrostates
One response to the failure of meshing is to redefine the macrostates, so as to secure it. I shall briefly consider this response from a general, and so mathematical, perspective. This will amount to considering how to get macrodeterminism ‘by construction’: or as one might gloss it less charitably, ‘by fiat’! So I should emphasize that, in a real scientific context, one would usually invoke considerations about how to respond, which are much more specific than the general ideas (like taking unions of the cells of the given partition) which I now mention. For example, one might invoke considerations like those in §3.2: about conserved quantities, or about allowances (a)–(d), or about selection processes, as discussed by Papineau.
I will make three comments. (1) The first is a ‘false start’. (2) The second is a response to the false start; and (3) the third is a look at what one might call the ‘converse’ to failure of meshing. All three will be recognizable as, in effect, some of the first steps in anyone's study of discretetime dynamical systems.
3.3.1. A false start
Given a cell C_{i} that ‘gets broken up’ by the timeevolution T, there seem at first sight to be two possible tactics for changing the partition so as to secure meshing. But each runs into difficulties.

We might define a coarser notion of macrostate (a smaller partition with larger cells) by taking the union of the cells that contain images T(s_{1}), T(s_{2}) of states s_{1}, s_{2} in C_{i} that get sent to different cells. That is, writing as usual TC_{i} for the imageset, i.e. TC_{i}: = {s′s′ = Ts, some s ∈ C_{i}}, we might define 3.1However, the sets TC_{i}, as C_{i} runs through P, are not a partition of , as different TC_{i} can overlap. So to get a partition with a meshing dynamics, we have to define a chain of overlapping sets TC_{i}, i.e. a sequence 〈TC_{0}, TC_{1}, … ,TC_{N}〉, with TC_{0} ∩ TC_{1} ≠ ∅︀, TC_{1} ∩ TC_{2} ≠ ∅︀, etc; and then define the partition whose cells are maximal chains of imagesets TC_{i}. This partition has, by construction, a meshing dynamics. But it is liable to be ‘uninformative’: that is, the unions of maximal chains of imagesets TC_{i} are liable to be large.

We might define a finer notion of macrostate (a larger partition with smaller cells) by decomposing the cell C_{i} that gets broken up by T into subsets, according to which cells its elements s ∈ C_{i} get sent to. That is, we might define, for each j in the index set of the given partition P, 3.2and then consider the decomposition of C_{i} into its subsets (TC_{i})_{j}. Then the sets (TC_{i})_{j}, as i,j run through the index set of P, obviously form a partition of (perhaps with, harmlessly, various ‘copies’ of the empty set, i.e. cases where (TC_{i})_{j} = ∅︀). And since this partition has smaller cells than P did, this tactic is not in danger of being uninformative in the way that the first tactic, in (i), was. But now the trouble is that this partition need not have a meshing dynamics. For though indeed T maps all of (TC_{i})_{j} into C_{j}, T need not map all of (TC_{i})_{j} into some cell of the partition we have just defined, i.e. into some (TC_{k})_{l}.
3.3.2. A response
Rather than starting from a given partition P with a nonmeshing dynamics, and asking how to modify it so as to get meshing, it is easier to start with just the microdynamics and consider defining ab initio a partition with a meshing dynamics. Indeed, it is easy to address the stronger question of defining cells each of which is invariant under the dynamics, i.e. mapped into themselves by T.
Thus, for any state s ∈ , the set 3.3is obviously the smallest set invariant under T that contains s. This statement also holds true if T is not a function, but one–many, i.e. the timeevolution is indeterministic (so that many chains could start at s). If T is a function, then [Ts] either is a cycle, i.e. s ↦ s_{1} ↦ s_{2} ↦ … ↦s, or is an ωsequence.
By containing only ‘descendants’ of s, definition of [Ts] of equation (3.3) obviously favours the ‘initial time’. If T is a function, we can avoid this favouritism by instead defining 3.4which is obviously the smallest set invariant under T that contains both s and any of its ‘ancestors’. If T is not a function, then we need to allow for chains from ancestors of s that do not pass through s. Then 3.5is by construction the smallest set invariant under T that contains both s and any of its ‘ancestors’.
3.3.3. The converse scenario
So much by way of discussing the failure of meshing, i.e. the scenario in figure 1, and how one might respond to it by redefining the macrostates. That scenario also prompts one to consider the converse scenario: that is, microindeterminism inducing, by coarsegraining, macrodeterminism, as in figure 3.
Indeed, this scenario can happen in a tightly defined theoretical context, in a scientifically important way. This means in particular: the macrodeterminism is not ‘won on the cheap’, by having very large cells (in other words, by using a logically weak taxonomy of microstates). Brownian motion provides an example. Think of the Langevin equation's probabilistic description of Brownian motion. Different realizations of the noise (i.e. different, unknown, trajectories of the atoms bombarding the large Brownian particle) give the particle different spatial trajectories: microindeterminism. But averaging over the realizations gives a deterministic evolution of the probability density for the particle's position (this evolution is given by a Fokker–Planck equation).^{11}
4. Ellis' types of topdown causation
I turn to using the framework of §3 to describe, in part, the first four of Ellis' five types of topdown causation [3,4]. I think these descriptions are worth formulating, because they are precise. But I say ‘in part’, because this precision comes with a pricetag; indeed two pricetags.
First, I will face several choices about how best to formalize Ellis' ideas. For clarity and brevity, I will make simple choices: even to the extent of not representing microstates and microdynamics, and so also not representing topdown causation. But I shall suggest that my choices are innocuous: one can see how one could elaborate the descriptions so as to represent microstates, topdown causation, etc.
But second, and perhaps more important, I will also simplify by setting aside nonformal, indeed philosophical, questions about what, beyond ideas like functional dependence and coarsegraining, is needed for causation, in particular topdown causation. (Recall my concessions (ii) and (iii) at the end of §1.) For example, I set aside, the distinction, made by Ellis [3, p. 6, p. 8 et seq.] and Auletta et al. [22], between causal effectiveness and causal power; and their taking topdown causation to require the macrolevel to have causal power over the microlevel. And apart from the views of Ellis and kindred spirits like Jaeger & Calkins, I will not try to connect my descriptions of Ellis' third and fourth types, which concern adaptation and evolution and so biology, to biological details, or to this Theme Issue's other relevant authors [38–40].
These simplifications are evident already in my proposed description of Ellis' first type, which he calls algorithmic topdown causation. Ellis writes [3, p. 8]: ‘algorithmic topdown causation occurs when highlevel variables have causal power over lower level dynamics through system structuring, so that the outcome depends uniquely on the higher level structural, boundary and initial conditions’. In line with my simplifications, I will take this quotation to mean just meshing dynamics, or commutation, as in §3.2: the ‘Fodor's hope’ of figure 2. For meshing dynamics matches the quotation's idea of a higher level outcome being uniquely determined by higher level facts. But that this is indeed a simplification is clear from Ellis also calling meshing dynamics ‘coherent higher level action emerging from lower level dynamics’ and ‘the principle of equivalence of classes’ (ibid.), and his taking it as a presupposition of his typology of topdown causation.
I will devote §4.1–4.3 to describe each of Ellis' second, third and fourth types. For clarity, I will begin each subsection with Ellis' own description [3] of the type. For brevity, I omit his fifth type (called ‘intelligent topdown causation’), which involves the use of symbolic representation to investigate the outcome of goal choices. But, by the end of my description of his fourth type (§4.3), it will be clear how the fifth type might be partially described with my framework.
4.1. Topdown causation via nonadaptive information control
Ellis writes [3, p. 12]: ‘in nonadaptive information control, higher level entities influence lower level entities so as to attain specific fixed goals through the existence of feedback control loops, whereby information on the difference between the system's actual state and desired state is used to lessen this discrepancy … unlike (algorithmic topdown causation), the outcome is not determined by the boundary or initial conditions; rather it is determined by the goals’.
I will represent this notion of control in my framework using two simple ideas (which will be useful later): which I will call culling and conforming. In both culling and conforming, some macrostate C* (cell C* of the partition P) is designated as ‘desired’. The difference will be that, in culling, one simply sends to some ‘rubbish’ or ‘dead’ state, 0 say, all states other than the desired one C*. So this will be modelled with a characteristic function χ_{C*} (though with the ‘yes’ or ‘winning’ value being C* itself, rather than 1). In conforming, on the other hand, states other than the desired one are sent to the desired one: so this will be modelled by a constant map with value C*. The details are as follows, with due precision about the difference between maps at the macro and microlevels.
Culling. Writing 0 for the rubbish state at the macro or microlevel, and adjoining the rubbish macrostate to the partition P so as to define a partition P^{0} : = P ∪ {0}, we have:

at the macrolevel: a characteristic map ; with , unless C_{i} = C* in which case χ_{C*} (C_{i}): = C* ≡ C_{i};

at the microlevel: a characteristic map χ_{C*} : → {,0}, or if one prefers χ_{C*} : →{C*, 0}, with a similar ‘culling’ definition.
Conforming. We have:

at the macrolevel: a constant map κ_{C*}:P → P sending all cells in P to C*: κ_{C*}(C_{i}): = C*, for all C_{i};

at the microlevel: any microdynamics given by a function (or indeed relation, i.e. multivalued function) T on that sends all of into C*. That is, any T such that T() ⊂C* will induce κ_{C*}:P → P as its meshing macrodynamics.
Various combinations and liberalizations of these two ideas are possible. The simplest combination is to conform and then cull: χ_{C*} ○ κ_{C*}. Then no s ∈ ends up as ‘rubbish’. Instead, all s ∈ get sent to a microstate in the desired macrostate C*. This is a perfect control, with the system ending in the desired macrostate, whatever its initial s ∈ .
4.2. Topdown causation via adaptive selection
Ellis writes [3, pp. 14–15]: ‘Adaptive processes take place when many entities interact … for example individuals in a population, and variation takes place in the properties of these entities, followed by selection of preferred entities that are better suited to their environment or context. Higher level environments provide niches that are either favourable or unfavourable to particular kinds of lower level entities; those variations that are better suited to the niche are preserved and the others decay away. Criteria of suitability in terms of fitting the niche can be thought of as fitness criteria guiding adaptive selection. On this basis, a selection agent or selector accepts one of the states and rejects the rest; this selected state is then the current system state that forms the starting basis for the next round of selection …. Thus, this is topdown causation from the context to the system. An equivalence class of lower level variables will be favoured by a particular niche structure in association with a specific fitness criteria. Unlike feedback control, this process does not attain preselected internal goals by a specific set of mechanisms or systems; rather it creates systems that favour the metagoals embodied in the fitness criteria’.
In representing these ideas, one of course faces several choices about how much detail to represent explicitly. For example, should one represent explicitly:

the environment/context (even niches?), or just the adapting entities;

as regards these entities: individuals (and their variation?), or just the population;

as regards the environment and the entities: microstates, or just macrostates;

selection as a process with several possible outcomes, or just two (survival or death!), as in the notion of culling in §4.1;

several rounds of selection, or just one;

adaptation using the notion of conforming as in §4.1; and

adaptation within a round of selection, e.g. in the lifetime of an individual, or just over many rounds?
In answering these questions, I propose to keep things pretty simple. In terms of this list, I will represent explicitly:

(i) the environment;

(ii) individuals and their variation; and

(v) several rounds of selection.
But I will not represent:

(iii) microstates;

(iv) outcomes of selection other than survival or death, as in culling; and

(vi) and (vii) adaptation, in terms of conforming, in a single round or over many rounds.
But I admit that these are just choices of what to represent: no doubt, several other possible choices are equally (or more) useful.
As to (i), the environment: I will be simplistic. I take the environment to be unchanging, so that there is no coevolution. There is an environment statespace ^{e} on which there is a partition P^{e} = {C_{k}^{e}}. Niches will be represented only implicitly; namely, by the way in which the macrostates of the individuals (and so of the population) that are selected for depend on the macrostate C_{k}^{e} of the environment.
As to (ii), the individuals and their variation: again, I will be simplistic. I assume that, in each round of selection, there are N individuals, each with an individual statespace on which there is a partition P = {C_{i}}. So the population of N individuals has a Cartesian product statespace ^{N} (neglecting the tensor products of quantum theory!), with the product partition P^{N} whose cells are given by Ntuples 〈C_{i1}, C_{i2}, …, C_{iN}〉. So, variation consists in not all individuals being in one cell: i.e. the population macrostate is not an Ntuple with all components equal to some single C_{i}. (Here one could adopt the occupation number formalism from statistical physics.)
Finally, as to (v), several rounds of selection: again, I will be simplistic. Not only will I take selection to be essentially culling as in §4.1, albeit with a designated/desired macrostate C* that is a function of the environment's macrostate, i.e. C* = C*(C_{k}^{e}). Also, I will assume that after culling:

the M (M ≤ N) surviving individuals simply persist in the desired macrostate C* that they (are so lucky to!) have been in (or, if you prefer, each is replaced by an offspring in the macrostate C*); and

N − M new individuals spring up (as from dragons' teeth!), in some randomly selected combination of macrostates, to proceed to the next culling, along with the M individuals in C* who have survived from the last round.
Now it is obvious how this toymodel achieves adaptation. As I have assumed that the environment is unchanging, i.e. always in a certain macrostate C_{k}^{e}, the culling always favours the same macrostate, C* = C*(C_{k}^{e}), of individuals. Therefore, over sufficiently many rounds of selection, the random production, at the start of each round, of unfit variations, i.e. macrostates C_{i} ≠ C*(C_{k}^{e}), decays away. That is, over time, the population (and the individuals) achieves adaptation in the sense of notion of conforming as in §4.1: i.e. a constant map taking the desired macrostate C* as its value.
This scenario can be summed up in terms of functions on macrostates. First, we adapt to the partition P^{N} the notation we used in §4.1 for culling. That is,

we adjoin the rubbish macrostate 0 to the partition P^{N} so as to define a partition P^{(N,0)}: = P^{N} ∪ {0}; and

we adopt the obvious componentwise definition of the characteristic function χ_{C*} ≡χ_{C*}(C_{k}^{e}):P^{N} → P^{(N,0)}, defined on the partition P^{N} and with codomain P^{(N,0)}.
We also need to represent the birth, after each round of culling, of the new individuals. We do this by postulating that, after each round, we apply a function β : P^{(N,0)} → P^{N}, which is defined (i) to keep constant any component equal to C* and (ii) to replace any component equal to 0 by some ‘living’, nonrubbish macrostate C_{i} ∈ P (as hinted by β's codomain being just P^{N}). But I shall not formalize the idea that the new individuals' macrostates, given by (ii) of β, are randomly chosen: I shall simply imagine that, for each round of culling, the function β is in general different; so that we postulate a sequence of functions, β_{1}, β_{2}, β_{3}, …, each subject to the requirements (i) and (ii).
Thus, each round of selection and birth is an application of the characteristic culling function χ_{C*}(C_{k}^{e}), followed by an application of one of the β functions, representing the birth of new individuals. So the (macro)histories of all the individuals, and of the population, that are possible, for a given environment macrostate C_{k}^{e} and given random functions β_{n}, are encoded in the sequence of functions 4.1
These functions are defined so that a generic initial population macrostate is mapped eventually to a state where all individuals are in C*: adaptation! Thus, suppose that, in the first generation, the first, third and Nth individuals happen to be in the desired macrostate C*, while the second, fourth (and no doubt other!) individuals are not and so get culled. And suppose that, after many, say p, generations, the functions β_{n} have been ‘random enough’ to have thrown up at some time (i.e. for some n) the desired macrostate C* for every component. This would give a (macro)history as follows: 4.2
To sum up this section, here is a toymodel of adaptive selection. Agreed, it is very simple. Indeed, it does not represent microstates; so that, in terms of my framework, it cannot represent topdown causation. But I submit that, if we elaborated the model so as to include microstates, we would be in a situation like that in §4.1. We would face issues about whether the model's macrodynamics meshes with its microdynamics. For example, nonmeshing would be threatened by indeterminism, owing to each individual, and so the population as a whole, being an open system. But we could write down a meshing dynamics (perhaps, for realism, availing ourselves of allowances like (a)–(d) in §3.2); and thus get a formal description of Ellis' third type of topdown causation—at least in the sense of causation as functional dependence.
4.3. Topdown causation via adaptive information control
Ellis writes [3, p. 18]: ‘Adaptive information control takes place when there is adaptive selection of goals in a feedback control system, thus combining both feedback control (§4.1) and adaptive selection (§4.2). The goals of the feedback control system are irreducible higher level variables determining the outcome, but are not fixed as in the case of nonadaptive feedback control; they can be adaptively changed in response to experience and information received. The overall process is guided by fitness criteria for selection of goals, and is a form of adaptive selection in which goal selection relates to future rather than the present use of the feedback system. This allows great flexibility of response to different environments, indeed in conjunction with memory it enables learning and anticipation … and underlies effective purposeful action as it enables the organism to adapt its behaviour in response to the environment in the light of past experience, and hence to build up complex levels of behaviour’.
In representing these ideas, one again needs to make choices, and exercise judgement, about which details to be explicit about. I will again be simplistic. I will also build on choices of §4.2, and its ensuing notations. This means that, although I will represent the idea that the goal is not fixed but depends on history, I will not capture one prominent feature of control or feedback: namely, the system's timeevolution being guided towards the goal, by, for example, the system calculating the difference between its present state and the goalstate, and then engineering its change in the next timestep so as to reduce this difference.
Instead, I will postulate, like I did in §4.2, that a system evolves randomly, until it happens to hit its goal—after which it stays in that state. Agreed, that does not merit the names ‘control’, ‘feedback’ or ‘learning’. But it will be clear that, at the cost of more definitions and notation, my framework could equally well describe these notions, namely in terms of timeevolutions using such differencereducing rules. Thus, I submit that what follows, though simple, is enough for my purpose; namely, to show that one could modify a model of evolution such as that of §4.2 so as to represent Ellis' ‘adaptive information control’.
Recall that, in §4.2, the goal C* was a function of just the (timeconstant) environment macrostate C_{k}^{e}. We had C* = C*(C_{k}^{e}); and throughout the process of evolution (and adaptation), the same C* acted as the goal (the attractor macrostate for each individual). I now modify this so as to make each individual's goal a function of its history, and also the environment's history (recall Ellis' mention of memory and past experience).
So let us imagine N persisting individuals (despite the talk on offspring and births in §4.2), labelled by j = 1,2, … , N. Time is discrete, with the generic timepoint labelled n. So the environment passes through a sequence of macrostates 4.3and individual j passes through a sequence of macrostates 4.4
We postulate that the goal of each individual j at each timepoint n is given by a goalfunction C*_{n;j}, which takes as its argument a prior (macro)history both of j and of the environment (but for simplicity: not other individuals!), and as its value an individual's macrostate, i.e. an element C_{i} of the partition P of the individual statespace.
So let us write H_{n;j} for a (macro)history both of j and of the environment up to the timepoint n, i.e. H_{n;j} is a 2ntuple 〈C′_{ij}, C″_{ij}, …, C_{ij}^{(n)}; C_{k1}^{e}, C_{k2}^{e}, … , C_{kn}^{e}〉, and let us write ℋ_{n;j} for the class of these 2ntuples. Summing up, we have the goalfunction 4.5
We now adjoin these definitions to the scenario of §4.2. There, each timestep (‘round’) involved a culling and rebirth for those individuals who had not attained their goalstate, while those in the goalstate C* simply persisted in it. Now, using our present metaphor of N persisting individuals with an everlengthening history, each round must (i) keep those individuals that have attained their goalstate in that state and (ii) assign to any other individual some random macrostate as its next state.
The combination of (i) and (ii) means that, provided assignments of macrostates of (ii) are sufficiently random that for each individual they sometimes assign its present goalstate, then in the long run, all the individuals attain—and stay in—their goals. In short, again, we have adaptation.
As regards spelling out the formal details of (i) and (ii): I will skip the details about (ii), which are a straightforward adaptation of the β (‘birth’) functions of §4.2. As to (i), there are two features we need to require, the first being more important.

If up to timepoint n + 1, the individual j and environment has experienced the joint history H_{n;j}, so that j's goal is then C*_{n;j} (H_{n;j}), and if j happens to be in the state C*_{n;j}(H_{n;j}), then we require that j will forever remain in C*_{n;j}(H_{n;j}). That is, we require that, once any individual enters its goal, it stays forever.

It is natural to have goalfunctions ‘stay consistent’, i.e. go on endorsing any goal that is attained. That is, it is natural to require that for any j and n, and joint history H_{n;j}, with initial segments H_{m;j} (m ≤ n): if the mth component of (the individualhistory firsthalf of) H_{n;j} is C*_{m;j}(H_{m;j}), then all the later initial segments of H_{n;j}, i.e. H_{p;j} with p > m, yield endorsements of this goal, i.e. C*_{p;j}(H_{p;j}) = C*_{m;j}(H_{m;j}). (Owing to (a), the system will in fact stay in C*_{m;j}(H_{m;j}) at all times p > m.)
To sum up, these two points encode the idea that a goalstate is an attractor.
Acknowledgements
I thank the John Templeton Foundation and George Ellis for the invitation; Harald Atmanspacher, Robert Bishop, Adam Caulton and especially Christian List and Elliott Sober for comments; the symposium participants for discussion; and, above all, George Ellis for the inspiration of his work, and for very helpful detailed comments on an earlier version.
Footnotes
One contribution of 15 to a Theme Issue ‘Topdown causation’.

↵1 I analysed four examples [2]. Footnote 6 and §3.2 will mention yet other examples.

↵2 There were other claims I will not need here, e.g. that emergence does not require limits, in particular not ‘singular’ limits.

↵3 The main source is Nagel [5, pp. 351–363]; see also Hempel [6, ch. 8]. Schaffner [7] is a masterly review not only of Nagel's position, but also of others' critiques, defences and modifications of Nagel.

↵4 Agreed, in philosophy, there is always more to say. I do not pretend that Sober's paper is the last word on the subject: in a large literature, I recommend Shapiro [11, especially §IVf. p. 643f] and Papineau [12, especially §4, p. 183f].

↵5 The idea of incomprehensible definitions and deductions, and thereby the need for higher level concepts and laws, is often illustrated with cellular automata such as Conway's Game of Life, with, for example, ‘glider’ as a higher level concept (cf. Dennett [13, pp. 196–200], Bedau [14, pp. 164–178] and O'Connor [15, pp. 1–3]). The idea is: these concepts and laws are definable and deducible from Life's basic rules, but only by processes so grotesquely long that you would be illadvised—mad!—to try and follow them, rather than investigating the higher level behaviour directly.
Besides: a theorem in logic (Beth's theorem) shows that, under certain conditions (namely firstorder finitary languages), the finite–infinite distinction collapses in the sense that if every term of T_{t} is implicitly definable in T_{b}, then T_{t} is a definitional extension (i.e. with finite definitions) of T_{b}. This point was first emphasized by Hellman and Thompson; more details are in Butterfield [1, §5.1].

↵6 Here I applaud Scerri's [17, pp. 3–4] critique of Hendry's surely maverick view [18, §3] that configurational forces are needed in modern quantum chemistry to explain molecular chirality and shape. On the contrary, I take it to be well established that the explanation lies in superselection (classical quantities) being rigorously emergent in appropriate limits (e.g. Primas [19], Amann [20] and Bishop & Atmanspacher [21]. So this is another example of the reconciling claim as in §2.1 that emergence and reduction are compatible.

↵7 But, again, that is a small price to pay, as such accounts are plausible, especially when compared with the traditional rival idea (called ‘nomological sufficiency’) that a cause, taken together with the laws, should imply the effect.

↵8 Agreed, it does not by itself imply such relations. But nor should we expect a general principle about causation to dictate on such specific matters as between which levels there are causal relations.

↵9 Uffink & Valente [30] expound and assess the principal deduction of the Boltzmann equation, namely Lanford's theorem. This example is especially topical in view of Villani's 2010 Fields medal (cf. Ambrosio [31] and Yau [32]).

↵10 Agreed, Fodor's and Papineau's jargon is different from mine. Both authors talk about laws, and/or causal relations, within each of the two levels, not about dynamics; and about realization or supervenience as the relation between the laws and their instances, not about coarsegraining. But I submit these are the only, or almost only, differences of jargon.
Another example of concern over failure of meshing in the context of the mental–physical relationship is Atmanspacher's discussion of the emergence of mental states from neurodynamics ([36, §§4.1, 6]; cf. Atmanspacher & beim Graben [37, §2]).

↵11 Probability theory also provides similar examples where timeevolution is downplayed, but which retain the key idea of combining variety of microstates with macroscopic uniformity. One main example is the method of arbitrary functions; where the probability of a macrostate is approximately the same for any of a wide class of probability density functions f on the microstates s ∈ , essentially because the partition P defining the macrostates is very intricate. Butterfield [2, §4] gives more details, including a discussion of emergence.
 Received May 31, 2011.
 Accepted July 26, 2011.
 This journal is © 2011 The Royal Society