1 Introduction

Mental events and properties seem to be able to cause other events and properties. How can we account for this seeming fact? It has become popular to appeal to the interventionist theory of causation to explain mental causation. According to interventionism, one event or property causes another if and only if an isolated manipulation of the first event or property would lead to a change in the second event or property while other factors are appropriately held fixed.

What other factors precisely should be held fixed, and at what values? This question becomes especially tricky if we try to apply interventionism to mental causation and at the same time assume non-reductive physicalism, that is, the view that mental properties are not identical to, but supervene on, physical properties. If we represent both a mental property and the physical property that actually realizes it, should we hold the physical property fixed while intervening on the mental property? The original interventionist theory does not give us an answer, because it eschews representing properties that stand in necessary relations, such as the mental property and its physical realizer(s). It has been argued that, if the original interventionist theory is applied to cases of supervenient mental properties in spite of this eschewal, we get the result that the mental is causally inert, because we would have to hold fixed the physical realizer while intervening on the mental property, which is impossible (see Baumgartner, 2009).

How can we apply interventionism to supervenient properties, such as mental properties, in a way that accommodates their efficacy? One strategy is to uphold the ban on variables that stand in necessary connections. Thus, the strategy imposes conditions for the aptness or adequacy of causal models by restricting the variable set that a model may contain.Footnote 1 One way of implementing this first strategy is to introduce a variable for a mental cause and a variable for its physical effect while omitting a variable for the supervenience bases of the mental cause (see Eronen, 2012). Another strategy is to allow variables that stand in necessary connections while stipulating that we must not hold the physical realizer of a mental property fixed if we intervene on the mental property.Footnote 2 There have, however, been two recent attempts to reconcile the holding-fixed requirements of original interventionism with mental causation while also allowing variables that stand in necessary connections: one by Zhong (2020) and one by myself (Kroedel 2020). In this paper, I argue that these attempts create more problems than they solve or at least entail severe constraints on the construction of causal models.

2 Interventionism and non-reductive physicalism

The debate about mental causation and interventionism is primarily concerned with the question of whether mental properties or types, such as the property of being in pain, can be causally efficacious. It is a tricky question how exactly causation between properties or types relates to causation between particular events, such as when we say that a specific pain of mine caused me to wince on a particular occasion. I am going to set this question aside for the purposes of this paper, however. I am going to assume, without necessarily endorsing, non-reductive physicalism about the mind. According to non-reductive physicalism, mental properties supervene on (i.e., are necessitated by) physical properties, but not symmetrically so, since one and the same mental property can have different physical realizers, that is, different physical supervenience bases,Footnote 3 on different occasions.

According to what I will call orthodox interventionism, that is, the original version of interventionism that does not yet take into account supervenient properties,Footnote 4 identifying the causal relations between types or properties is a multi-stage process (see Woodward, 2003: 25–93). First, we represent the properties in question by variables. In the simplest case, the variables are binary, with one value representing that the corresponding property is instantiated and the other value representing that the property is not instantiated. Second, we determine the relations of direct causation between the variables in our model. To check whether variable X is a direct cause of variable Y, we need to check whether there is any intervention on X with respect to Y (that is, a manipulation of X that does not change Y except possibly via X) that would change both X and Y if all other variables are held fixed at some values. If there is such an intervention (actual or merely nomologically possible), then X is a direct cause of Y; otherwise it is not. Once the relations of direct causation between our variables are identified, we can use them to identify causal paths as chains of direct causation and to give structure to the causal graph of our model by drawing an arrow from one variable to another just in case the former variable is a direct cause of the latter. The third stage is to determine the relations of contributing causation: X is a contributing cause of Y just in case there is an intervention (actual or merely nomologically possible) on X with respect to Y that would change both X and Y if all the variables outside of some causal path from X to Y are held fixed at some values. The fourth stage concerns actual or token causation, that is, causation not between the variables (and the properties they represent) as such, but between (the events represented by) specific values of those variables. To determine whether X’s having a specific value x is an actual cause of Y’s having a specific value y, we again have to check for possible interventions on X with respect to Y while holding off-path variables fixed. The main difference (apart from the new category of causation, namely actual causation) is that it now becomes important at what values precisely the off-path variables are held fixed. For certain textbook cases of redundant causation, such as early pre-emption, we should hold the off-path variables fixed at their actual values. For certain other cases, such as symmetrical overdetermination, it seems that we should hold them fixed at counterfactual values (see Woodward, 2003: 83–84).

3 Zhong’s suggestion

The first attempt at applying orthodox interventionism to the case of non-reductive physicalist mental causation that I will discuss is due to Lei Zhong. Zhong holds that we should apply interventionism by way of the following principle (2020: 294; see also Zhong, 2014: 344):

(N) A property X causes another property Y on a particular occasion if and only if

 (N1) If an intervention that sets X = present were to occur, then Y = present;

 (N2) If an intervention that sets X = absent were to occur, then Y = absent.

Zhong calls (N1) the presence condition and (N2) the absence condition. What is claimed to be absent and present in (N1) and (N2) are not the variables X and Y but rather the corresponding properties. The instantiation and non-instantiation of these properties is represented, respectively, by the values present and absent of variables X and Y. Zhong holds that even if X and Y actually both have the value present or both have the value absent, it does not follow that (N1) or (N2) is automatically true, because he denies that counterfactuals with true antecedents and consequents are automatically true (see 2020: 303).

Now suppose that mental or other higher-level properties are in play and that the question arises whether they or their actual realizers (or both) have certain effects. We might be dealing with a mental property M, its actual realizer P and a putative effect E, or we might be dealing with a determinable property such as red, one of its determinate properties such as scarlet, and the question of which of these two (if any) causes a certain pigeon to peck (see Fig. 1). At the moment, we are only dealing with binary variables that represent whether or not certain properties are instantiated or present, so for simplicity I will not always distinguish between properties and variables.

Fig. 1
figure 1

Potential mental and physical causes (left) and potential determinable and determinate causes (right)

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For the time being, let us focus on the case of a mental property M, its actual realizer P and the putative effect E. Zhong holds that in order to determine whether M is a causeFootnote 5 of E, we should hold the other variables, including P, fixed; similarly, he holds that in order to determine whether P is a cause of E, we should hold M fixed (see 2020: 296). Of course we cannot hold P fixed at its actual value, present, and still change M, because P’s being present entails that M is present as well. Zhong suggests that we hold P fixed at its counterfactual value, absent, instead (see 2020: 299–300). If we hold P fixed at absent and intervene on M, then E might or might not change, as the case may be. Thus, the presence and absence conditions might or might not be satisfied for M and E, and correspondingly M might or might not come out as a cause of E (see 2020: 304). As for the question of whether P is a cause of E, Zhong holds that we should hold M fixed at its actual value, present. If we hold M fixed at present and intervene on P, then E might or might not change, as the case may be. Thus, the presence and absence conditions might or might not be satisfied for P and E, and correspondingly P might or might not come out as a cause of E. While, according to Zhong, M might cause E in some cases, and P might cause E in some cases, there is no case where both M and P cause E, which has interesting repercussions for the exclusion problem about mental causation (see 2020: 311–313).

4 Model-building and holding fixed

On the face of it, Zhong’s suggestion offers a nuanced account of mental and other supervenient causation. But if we take a closer look, problems emerge. Some of these problems have to do with the way in which Zhong’s model represents realizers. Others have to do with Zhong’s policy for holding variables fixed during the interventions that are relevant for his presence and absence conditions.

It can happen that we have to represent not just a supervenient property and its actual realizer in a model but alternative realizers too. It might even happen that we have to represent all possible realizers of a supervenient property. In the case of determinable and determinate colour properties, such a case might seem unlikely to arise because there are (too) many ways of, say, being red apart from being scarlet. And who knows how utterly numerous the possible realizers of a given mental state might be, which might make it even less likely that each of them is relevant in a given case. But there are supervenient properties that have only a few possible realizers, which might all be relevant at once. Being made of jade is such a property. Being made of jade can be realized in just two ways: by being made of jadeite or by being made of nephrite.Footnote 6 I am going to assume, for simplicity and for continuity with Zhong’s determinable/determinate examples, that these two realizers are mutually exclusive, such that it is impossible for the property of being made of jade to be realized both by being made of jadeite and by being made of nephrite at the same time.Footnote 7

We can easily imagine cases where the properties of being made of jade, of being made of jadeite, and of being made of nephrite are all at issue. For example, there might be a detector device for minerals that flashes a light on detection but about which we yet have to determine whether it detects jade, jadeite only, or nephrite only. How should we model the case of this detector? We need to represent whether our sample is made of jade, whether it is made of jadeite, whether it is made of nephrite and whether the light flashes. If we follow Zhong’s approach, we need to introduce separate binary variables for the sample’s being made of jadeite and its being made of nephrite. Overall, we get four binary variables. Let us call them Jade, Jadeite, Nephrite and Flash. Let us assume that, like in Zhong’s models, each variable can either assume the value present or absent. The value present for either Jadeite or Nephrite necessitates the value present for Jade, so we get wide arrows to Jade like in the earlier examples. The causal relations from Jade, Jadeite, and Nephrite to Flash remain to be determined (see Fig. 2).Footnote 8

Fig. 2
figure 2

The structure of the jade example

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The obvious alternative to the model just described would be a model that has only one variable for the several realizers of jade that is multi-valued, such that one value corresponds to each of the possible realizers (and one to the absence of all realizers). More specifically, instead of Jadeite and Nephrite, we could use a single variable, called Realizer, which has three possible values that represent, respectively, the presence of jadeite, the presence of nephrite, and the absence of both jadeite and nephrite.Footnote 9 Zhong eschews such a way of modelling realizers. He holds that a multi-valued realizer-variable would not correspond to a genuine property (see 2020: 297–298). Going with one variable for each specific realizer is not merely an optional feature of Zhong’s approach. With a multi-valued realizer variable, his idea of holding the realizer-variable fixed at absence while intervening on the supervenient property would no longer be feasible: if a multi-valued realizer-variable of the sort described were set to the absence of all realizers, one could no longer intervene on the supervenient property, because the absence of all realizers would entail the absence of the supervenient property.

A model with separate variables like Jadeite and Nephrite has significant drawbacks. We will turn to issues about holding variables fixed in an instant, but even before these issues arise, the Zhongian model for the jade case has a disadvantage. Orthodox interventionism forbids the variables in a model to have non-causal necessary connections. Now of course one cannot fully comply with this ban if both supervenient properties and their realizers are represented in some form or other. But if one introduces separate binary variables for each of the realizers that are represented, one gets extra and, it might seem, gratuitous necessary connections between those variables. For we assumed that at most one realizer can be instantiated at once, such that (say), setting Jadeite to present forces Nephrite to assume the value absent in a non-causal way. The more realizer-variables we have, the more non-causal connections of this sort we get. And although we chose an example with only two possible realizers for simplicity, there is no limit in principle to the number of possible realizers for a given supervenient property. Consequently, there is no limit in principle to the number of non-causal connections between variables. One might tolerate necessary connections between variables in principle, but still require that they not be multiplied beyond necessity.Footnote 10 The Zhongian model for the jade example does not fare very well with respect to this requirement.

This is only the beginning of the trouble with our Zhongian model, however. Let us consider how we should assess which (if any) of the variables Jade, Jadeite, and Nephrite are candidates for being causes of Flash according to Zhong’s conditions (N1) and (N2). Since Jade, Jadeite, and Nephrite are all binary, there are eight combinations of values of these variables (see Table 1). Only three of these combinations are metaphysically possible (these are printed in boldface in the table). The remaining combinations are impossible either because of the supervenience relation between jade and its two realizers or because of our assumption that at most one realizer can be present at once.

Table 1 Combinations of values of the variables in the jade example
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Each of the three possible combinations is uniquely specified by two of the three values it contains. In other words, once the values of two of the variables Jade, Jadeite, and Nephrite are set, this determines the value of the third variable. This relationship between the three variables is obvious in the direction from Jadeite and Nephrite to Jade, because it is in the nature of supervenience that fixing the subvenient properties fixes the supervenient property. The relationship is perhaps a bit less obvious in the direction from Jade and one of Jadeite and Nephrite to the other realizer variable, but it holds this way, too: once we set the values of Jade and (say) Nephrite, there is only one possible value left for Jadeite.

Given that the values of any two of Jade, Jadeite, and Nephrite box in the value of the remaining variable, it is impossible to hold the value of two of these variables fixed while changing the value of the third variable by an intervention. Thus, Zhong’s presence and absence conditions, (N1) and (N2), cannot both be true if we substitute ‘Jade’, ‘Jadeite’, or ‘Nephrite’ for ‘X’, substitute ‘Flash’ for ‘Y’, and require that, during the intervention that the antecedent of the conditions talk about, the remaining two variables must be held fixed at some value.Footnote 11 Notice that it does not matter at all at which values we hold the remaining variables fixed: as soon as we hold two of Jade, Jadeite, and Nephrite fixed at some possible combination of values, it is impossible to change the third variable.Footnote 12Footnote 13

Should we hold fixed two of Jade, Jadeite, and Nephrite when intervening on the third variable? In many places, Zhong seems to imply that we should. He professes adherence to “the general interventionist requirements on fixation” (2020: 296), which requires one to hold all other variables fixed if one checks whether X is a direct cause of Y. He also claims that “a suitable intervention on supervenient/subvenient properties should hold the other variables fixed at some value” (2020: 296). Lastly, he holds that, in the colour example, one of red and scarlet should be held fixed if one intervenes on the other, because the two properties are “competing causal candidates” (2020: 302 footnote 32). At least at the level of type causation, being made of jade, being made of jadeite, and being made of nephrite seem to compete with one another too. The upshot is that Zhong seems to be committed to holding two of our three variables Jade, Jadeite, and Nephrite fixed while intervening on the third variable, which makes his own conditions on causation impossible to satisfy.

Perhaps there is a loophole that Zhong could try to exploit. In response to an objection by McDonnell (2017) that is also about the issue of alternative realizers, Zhong makes the following suggestion: if we cannot hold two variables fixed at once, but only one of the corresponding properties is actually instantiated, then we should hold the (variable corresponding to) the actually instantiated property fixed and let the other property vary (2020: 308). This suggestion is something of a deviation from standard interventionist methodology, where, as we saw in Sect. 2, the actual values of variables do not come into play when type-level causal relations are determined, but perhaps the deviation is worthwhile.Footnote 14

Here is how the loophole could be exploited in the case of the jade example. Suppose that, in the actual situation, we are dealing with a sample that consists of jadeite (and, ipso facto, of jade). Thus, nephrite is not actually present, and we are allowed in principle to vary the variable Nephrite. For example, when assessing an intervention on Jadeite with respect to Flash, we need to hold Jade fixed, but we may vary Nephrite, which allows us to change Jadeite from present to absent while holding Jade fixed at present; thus, the intervention changes Nephrite from absent to present (and it might or might not change Flash).Footnote 15

The loophole will not be available in all troublesome cases, however. For there are cases that are relevantly similar to our jade example, but where the properties whose holding fixed in is question are all instantiated.

Consider the following example, which is due to Peter Spirtes and Richard Scheines (2004) and which is discussed in the context of Woodward’s interventionism in his 2015. In the human body, total cholesterol is composed of high-density cholesterol and low-density cholesterol.Footnote 16 To model the example, let us use, as Woodward does, the following variables: TC for total cholesterol, HD for high-density cholesterol, LD for low-density cholesterol, and D for heart disease. It is most natural to stipulate that TC, HD, and LD are multi-valued variables whose values represent the specific concentrations of the substances in question. I will make this stipulation too, but if one wanted to, one could model the case with binary variables only, in which case one would need a multitude of binary variables for each of total cholesterol, high-density cholesterol, and low-density cholesterol, with each variable corresponding to a specific concentration of the substance in question. Using binary variables would obviously be much more cumbersome, but would face the problem that I will present in an analogous way.Footnote 17

Given the preferable way of modelling the case, TC is the sum of HD and LD, so the values of HD and LD jointly necessitate the value of TC. If we extend the conventions from the previous figures slightly such that wide arrows may also represent jointly necessary connections between variables, we get the situation shown in Fig. 3 (see also Woodward, 2015: 331).

Fig. 3
figure 3

The structure of the cholesterol example

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While there certainly is room for debate which of TC, HD, and LD qualify as causes of D in general as well as in a given case, it seems uncontroversial that at least some of these variables can be causes of D. Zhong is committed to denying efficacy to all of TC, HD, and LD, however. Here is why. The variables, TC, HD, and LD are clearly competing causal candidates. Further, all properties in question are instantiated, because (we may assume) total cholesterol, high-density cholesterol, and low-density cholesterol are all present in the case at hand. Thus, one cannot invoke the exemption from the holding-fixed requirement for uninstantiated properties that opened the loophole in the jade example. By Zhong’s own principles (instantiation and competition), the remaining two variables must be held fixed if we intervene on the third member of the trio TC, HD, LD with respect to the effect variable D. Such interventions are metaphysically impossible, however. For fixing the values of two of TC, HD, and LD necessitates a specific value for the third variable, which can no longer be changed by an intervention (see Woodward, 2015: 328). With the loophole no longer available, Zhong is committed to saying that neither total cholesterol nor high-density cholesterol nor low-density cholesterol causes heart disease. This seems too high a price to pay for his account of mental causation.

In response to the problems that arise in the cholesterol example, Zhong might claim that the rules about which variables to hold fixed are different in such a case. But then the burden would be on him to show why the cholesterol case should be treated differently, given that no relevant difference is obvious and given that he generally eschews exceptions from the general interventionist principles because he finds them ad hoc (see 2020: 296–297).

Alternatively, Zhong could respond that the cholesterol example should not be modelled with variables for all of high-density cholesterol, low-density cholesterol, and total cholesterol. He could claim, for instance, that to assess whether HD is a potential cause of D, we should use a model that merely has the variables HD, TC, and D, but lacks variable LD, which makes it possible to intervene on HD (with respect to D) while holding TC fixed. This alternative response would be even less convincing than the previous one, however, given that Zhong emphatically embraces introducing variables both for the necessitated and for the necessitating phenomena in his account of mental causation. The alternative response conflicts not merely with Zhong’s other commitments but also with general constraints on model-building. It seems plausible that a causal model should contain enough variables to represent the essential structure of a case (see Hitchcock, 2007: 503) In the cholesterol example, this structure seems to contain high-density cholesterol and low-density cholesterol as well as total cholesterol.Footnote 18 In sum, no quick remedy for the problems with Zhong’s approach seems to be in sight.

5 My 2020 suggestion

Another recent attempt at reconciling interventionist mental causation with non-reductive physicalism while maintaining the holding-fixed requirements of orthodox interventionism is due to myself (see Kroedel 2020: 148–150). Henceforth, I will refer to this attempt as “my 2020 suggestion”. According to the suggestion, we should model the case as follows. We should use a binary variable M that represents the presence or absence of the mental property in question. We should introduce a multi-valued variable (call it Pmulti) for the realizers, that is, supervenience bases, of our mental property. One specific value of Pmulti represents the absence all realizers, and the other values represent the presence of the different possible realizers. Lastly, there is again a variable E for the putative physical effect. (We will get to the question of how many values E has in a moment.) Because Pmulti represents the realizers of M, the value of Pmulti necessitates the value of M. Given what has been said so far, we get the situation depicted in Fig. 4.

Fig. 4
figure 4

Modelling mental causation with a multi-valued realizer-variable

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What are the causal relations between M, Pmulti, and E? According to my 2020 suggestion, we should start determining them by following the letter of orthodox interventionism about intervening and holding fixed. It will turn out that we thus get a number of causal relations between our variables according to orthodox interventionism. Some of these relations will turn out to be plausible, while some will turn out hold merely nominally; according to my 2020 suggestion, the latter will have to be pruned from the genuine causal relations by some additional principles about variables that stand in necessary connections.

But let us take one step at a time. To find out whether Pmulti is a direct cause of E according to the suggestion, we must check whether we can hold M fixed at some value while changing Pmulti by an interventionFootnote 19 that changes E. The only candidate value at which we can hold M fixed while having a chance to change Pmulti in the first place is the value that corresponds to the presence of the mental property, for the absence of the mental property necessitates the absence of any realizers and thus does not allow us to change the value of Pmulti. If we hold M fixed at the value present (to use Zhong’s terminology), we stand a good chance of finding an intervention that changes E along with Pmulti. If we represent the event that is the putative physical effect with a binary variable E, it is plausible that the event would not have occurred if some very exotic supervenience base of M had been instantiated instead of the actual realizer (see Kroedel, 2020: 149 note 59). Alternatively, we could represent E by a multi-valued variable whose values represent different ways in which the event in question could have occurred. Presumably, even if all possible realizers bring about E, at least some realizers would have brought about different versions of E. In sum, it seems plausible that Pmulti comes out as a direct cause of E even if we require that M be held fixed when we intervene on Pmulti.

What about causal relations that involve M? If we have to hold fixed Pmulti when intervening on M with respect to E, we get the result that M is not a direct cause of E, because we cannot change M when we hold fixed Pmulti. But according to my 2020 suggestion, this result does not entail that M is causally inert altogether. For M turns out be a direct cause of Pmulti according to the orthodox definition: while holding E fixed at any value, we can change M (particularly from present to absent) and thereby change Pmulti as well.Footnote 20 Similarly, Pmulti turns out to be a direct cause of M: while holding E fixed at any value, we can change Pmulti (from a value that represents the presence of some realizer to one that represents the absence of all realizers, or vice versa) and thereby change M as well. Given that M is a direct cause of Pmulti and that Pmulti is a direct cause of E, there is a causal path from M to E via Pmulti. In our model, there are no variables outside of this causal path. So M is a contributing cause of E if there is an intervention that changes M and E; since Pmulti is on the causal path from M to E, there is no need hold Pmulti fixed during such an intervention. Indeed, it turns out that we can thus change M (from present to absent), which results (via a change in Pmulti) in a change of E. Let us take stock: if we apply interventionism in the way my 2020 suggestion recommends, M is not a direct cause of E, but M is a direct cause of Pmulti (and vice versa) according to the orthodox definition, and M is a contributing cause of E according to the orthodox definition.

Having determined these relations according to the letter of orthodox interventionism, my 2020 suggestion says that we should depart from orthodox interventionism by classifying those causal relations (direct or contributing) as spurious where there are necessary connections between the two relata. Thus, the relations between M and Pmulti are not genuinely causal; rather, they are relations of mere interventionist difference-making (see Kroedel, 2020: 149). M is a genuine (contributing) cause of E, however, because there is no necessary relation between M and E. Likewise, Pmulti is a genuine (direct) cause of E. The resulting picture is shown in Fig. 5.

Fig. 5
figure 5

Mere difference-making and causation in a model for mental causation

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6 More trouble from model-building

How does my 2020 suggestion deal with the examples from Sect. 4 that have proved troublesome for Zhong’s suggestion?

Let us consider the jade example first. If we model the example as it is depicted in Fig. 2, with separate variables Jade, Jadeite and Nephrite, my 2020 suggestion has implausible consequences.

Let us start by looking at the relations between Jade, Jadeite, and Nephrite; this will not yield implausible results yet, but it will set the stage for later problems to emerge. We can apply the orthodox definition of direct causation to each ordered pair of distinct variables from the trio Jade, Jadeite, and Nephrite. For each such ordered pair, we get the result that there is a relation of direct causation according to the orthodox definition. This can be read off from the boldface (i.e., metaphysically possible) lines of Table 1: for each of our ordered pairs we can find a value of the third variable such that changing the value of the first member of the ordered pair also changes the value of the second member. For instance, if we hold Jadeite fixed at absent, we can change Jade from present to absent by changing Nephrite from present to absent. The relations of (orthodox) direct causation between Jade, Jadeite, and Nephrite should be classified as spurious according to my 2020 suggestion, because each variable from the trio stands in a necessary connection to the two remaining variables. Thus, the relations between Jade, Jadeite, and Nephrite are not genuinely causal, but instead are relations of mere interventionist difference-making.

So far, so good. Trouble arises if we try to show that at least one of the three variables Jade, Jadeite, and Nephrite is a direct cause of Flash. To show this for one of Jade, Jadeite, and Nephrite, we must hold the remaining two variables from the trio fixed. But, as we saw in Sect. 4, if we hold two members from the trio fixed, we can no longer change the third member. So none of Jade, Jadeite, and Nephrite comes out as a direct cause (spurious or genuine) of Flash. It follows that none of Jade, Jadeite, and Nephrite comes out as a contributing cause (spurious or genuine) of Flash either. For in our model the only possible intermediaries on a causal chain from one of Jade, Jadeite, and Nephrite to Flash are the other two variables from the trio. But none of Jade, Jadeite, and Nephrite is on any causal chain to Flash, because none of them is a direct cause of Flash. In sum, we get the situation depicted in Fig. 6, in which Flash remains wholly uncaused.

Fig. 6
figure 6

Mere difference-making and (the absence of) causation in a model of the jade example

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Now, it could be replied that we should not introduce binary variables for the presence or absence of jadeite and nephrite in the first place, because this multiplies variables that stand in necessary connections beyond necessity (see, again, Kroedel, 2020: 143–145). It could be suggested that we should instead model the jade example like the case of mental causation and introduce a single, multi-valued variable for the different supervenience bases of being made of jade.

Eschewing multiple binary variables for supervenience bases in favour of a single multi-valued variable does not solve all problems with my 2020 suggestion, however.

First, even if we introduce a single, multi-valued variable for all supervenience bases in cases like the ones we have discussed so far, we get “boxing-in” problems when there are more than two levels with supervenience relations between them. Suppose that instead of merely looking at a two-level situation involving a property (such the property of being made of jade) and its supervenience bases (such as the properties of being made of jadeite and being made of nephrite), we consider three levels. Schematically, let us introduce a binary variable, Macro, for the property at the top level; a multi-valued property, Meso, for the supervenience bases of Macro that constitute the intermediate level; and a multi-valued variable, Micro, for the supervenience bases of the properties represented by Meso. (Less schematically, we could think of Macro as representing the presence or absence of jade, of Meso as representing the presence or absence of jadeite and nephrite, and of Micro as representing the presence or absence of various isotope compositions or quantum states.) Let us also use a variable E (binary or otherwise) for the putative effect again.

With this variable set in place, we can now determine the causal (and mere interventionist difference-making) relations that hold according to my 2020 suggestion. That Micro is a direct cause of E is as plausible in the three-level example as it was in the earlier examples; the question is whether we can change E along with Micro if we hold Meso fixed at some level. Let us assume, at least for the sake of the argument, that there is such a way of changing both Micro and E. The relation between Micro and Meso turns out to be one of reciprocal direct causation according to the orthodox definition, that is (according to my 2020 suggestion) a relation of mere interventionist difference-making given the necessary connection between Micro and Meso. For if we hold Macro fixed at a suitable value, we can change Meso by changing Micro and vice versa.

So far, everything looks like in the earlier examples. But if we look at the macro level, it turns out that it is rendered causally inert by the lower levels. For suppose that we want to determine whether Macro is a direct cause according to the orthodox definition of Meso or vice versa. In order to do so, we must hold all the other variables fixed, particularly Micro. But once we hold Micro fixed, we can no longer change Meso or Macro. So Meso is not a direct cause according to the orthodox definition (which, according to my 2020 suggestion, translates into being a mere interventionist difference-maker here) of Macro or vice versa. For similar reasons, neither Meso nor Macro is a direct cause of E. Lastly, Macro is not a direct cause according to the orthodox definition of Micro or vice versa, because this would require the possibility of changing Macro while holding Meso fixed, which cannot be done. Meso might still qualify as a contributing cause of E because of the chain to E (via Micro). But Macro stands in no causal or mere interventionist difference-making relationsFootnote 21 to any other variable (see Fig. 7). The same would hold for variables at supervenience levels that we could stack on top of Macro. Thus, while my 2020 suggestion can accommodate the efficacy of supervenient properties in principle, it can no longer do so once more than two levels are in play.

Fig. 7
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Modelling three levels

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A second problem with my 2020 suggestion emerges when we try to apply it to the cholesterol example from Sect. 4. Assume that we model the example as above, that is, with variable TC for total cholesterol and two separate variables HD and LD for high-density cholesterol and low-density cholesterol (plus a variable D for heart disease). Then my 2020 suggestion runs into the same problems as it did when the jade example was modelled with separate variables Jadeite and Nephrite for the supervenience bases of jade. It turns out that there are pairwise relations of direct causation according to the orthodox definition, which are presumably classified as relations of mere interventionist difference-making according to my 2020 suggestion, between TC, HD, and LD. But none of TC, HD, and LD qualifies as a direct cause of D according the letter of orthodox interventionism. To thus qualify as a direct cause of D, there must be a possible intervention that changes the cause variable when all other variables besides D are held fixed. But once two of TC, HD, and LD are held fixed, it is impossible to change the third variable. Thus, D has no direct cause. Since it would need a direct cause in order to have a contributing cause, D ends up having no cause whatsoever.

In the jade example, my 2020 suggestion could be saved by using a single, multi-valued variable that represents all the supervenience bases. This manoeuvre could be justified on principled grounds, because it avoided gratuitous necessary connections between variables. One could try a similar manoeuvre for the cholesterol example. Instead of HD and LD, one could use a single variable, call it HLD, that represents both the concentration of high-density cholesterol and the concentration of low-density cholesterol. Variable HLD would have to be multi-valued, in a double sense. It would have to have a multi-value component that represents the concentration of high-density cholesterol and another multi-valued component that represents the concentration of low-density cholesterol. Technically, variable HLD could be an ordered pair of real numbers. If we use HLD instead of HD and LD, HLD could be a genuine direct cause of D. At least we could vary HLD while holding TC fixed. Introducing a variable like HLD would be pretty unusual, however, because typical multi-valued variables are monadic. It is also not clear whether one could invoke the same rationale for replacing HD and LD with HLD that was invoked for introducing a single variable for the supervenience bases in the jade and mental causation cases. For there is a necessary connection between HD and LD only if TC is held fixed, so it is doubtful that replacing HD and LD with HLD reduces the number of necessary connections between variables in any relevant sense.Footnote 22

The upshot of this section is that we can follow my 2020 suggestion to adhere to the orthodox interventionist requirements of holding variables fixed only if we impose constraints on the variable set. Generally speaking, it seems that we must not use more than two variables when we have necessary connections. Introducing further variables when there are necessary connection, either in the “horizontal” dimension (as in the case of Jade, Jadeite, and Nephrite) or in the “vertical” dimension (as in the case of Macro, Meso, and Micro) threatens the efficacy of all of the variables that stand in necessary relations. Similarly for cases in which the necessary connections do not hold between pairs of variables, but more holistically, as in the cholesterol example.

Of course it should not come as a surprise that we need some constraints on variable sets if there are necessary connections between variables.Footnote 23 Suppose that we want to model a case involving water and we introduce separate variables for the presence of water and for the presence of H2O. Given that it is metaphysically necessary that water is present if and only if H2O is present, we can never change the water-variable while holding fixed the H2O-variable and vice versa. Thus, neither variable can be a direct cause of a third variable (not even in the sense of being a mere interventionist difference-maker). That one should not introduce separate variables for the presence of water and the presence of H2O while adhering to the orthodox holding-fixed requirements looks like a corollary of the requirements rather than an objection to upholding them. If one were keen on adhering to the holding-fixed requirements in the way my 2020 suggestion recommends, one might likewise regard the consequences for variable choice as a corollary rather than a problem. There is dialectical room for manoeuvre. Yet it cannot be denied that the holding-fixed requirements come at a cost. In particular, if we stick to standard monadic variables, the holding-fixed requirements come at the price of rejecting Hitchcock’s requirement of using enough variables to represent the essential structure of a case (which was discussed at the end of Sect. 4): the essential structure of the cholesterol example seems to contain all three varieties of cholesterol, and the essential structure of the three-layers example would be impoverished by representing only two of the layers.

7 Conclusion

I have critiqued two attempts that try to reconcile higher-level causation with the holding-fixed requirements of orthodox interventionism. Lei Zhong suggests that, when intervening on a variable for a supervenient or subvenient property with respect to a putative effect, we should hold the variables for the corresponding subvenient or supervenient property fixed. Zhong also suggests that we represent individual supervenience bases by individual variables. I have argued that these two suggestions are not tenable together. Once we commit to holding variables fixed that stand in necessary connections, we have to hold too many of them fixed, which makes interventions impossible. No principled way of allowing only some variables to vary that stand in necessary connections to the variable that is intervened on is forthcoming. My 2020 suggestion also recommends that we hold variables fixed as orthodox interventionism requires, but allows a relation of mere interventionist difference-making between variables with necessary connections which can stand in for the relation of genuine direct causation. My 2020 suggestion also differs from Zhong’s in that my 2020 suggestion eschews variables for individual supervenience bases in favour of a multi-valued variable for all the supervenience bases. It turns out that my 2020 suggestion can accommodate higher-level causation in principle but only at the price of accepting significant restrictions on the choice of variables. It remains an open question whether holding on to the other parts of the interventionist orthodoxy is worth this price.