From Theoria 49:87-111 , 1983

Explaining differences and weighting causes

by

GERMUND HESSLOW

(Lund University)

1. Introduction

Although explanations of individual events invariably mention causes, some causes seem to lack explanatory power. Suppose, for instance, that a fire broke out in a barn because some careless person dropped a burning cigarette in the hay. The cigarette, we may assume, was clearly a cause according to common sense but also according to all reasonable definitions of ’cause’. It was both necessary and sufficient in the circumstances; it was a part of a universally sufficient condition; it raised the probability of the fire, etc. The same, however, is also true of several other conditions such as the presence of oxygen and inflammable material and the absence of dampness and an automatic fire extinguishing system. All of these factors are causes in a wide sense, but none of them can explain the fire. It would clearly be absurd to try to explain the fire by pointing out the abundance of oxygen in the air.

There is no physical law that could justify this discrimination of the oxygen. Suppose that we have been living for some time on a distant planet, where the temperature is extremely high and where there is no oxygen. Occasionally there is a leak in one of our oxygen tubes, and whenever this happens, inflammable objects in the vicinity catch fire. If asked why such a fire occurred, we would probably point to the oxygen. From a physical point of view the two situations are identical – both contain a heat source, oxygen, combustible material, etc.– but we select different factors to account for the fire.

It would seem, then, that we must distinguish between causes and explanations, or, perhaps better, between determining causes and explanatory causes, where the determining causes include the usually quite large set of conditions which contribute to the occurrence of an effect, and explanatory causes make up that subset of conditions which are selected from the determining causes because of their special explanatory power. It is this selection process which is my main concern in this paper.

Before I proceed, I would like to note that several common accounts of explanation fail to solve our problem. According to the ’covering law model’ ¹ neither the oxygen nor the cigarette will be explanations, since neither is sufficient for the fire, and, if the model is extended so as to include parts or components of sufficient conditions, both factors will be explanations. Nor can the statistical relevance model² help us, for both burning cigarettes and the presence of oxygen are statistically relevant to fires. There are, however, a number of other theories that treat different causes differently. According to a classical work by H. L. A. Hart and A. M. Honoré (1959) we select from the class of determining causes those that are abnormal or deviate from the normal course of affairs. According to Dray (1957), who focuses on the explanation of actions in history, we often select actions to which we have moral objections. Many have noted the importance of ’precipitating’ causes (e.g., MacIver (1952)) and others of manipulable causes (Collingwood (1940)). Toulmin (1961), in criticising the covering law model, claims that scientists typically explain deviations from ’ideals of natural order’. And so on.³

Although all of these theories contain an element of truth, and some of the proposed selection criteria could deal with our introductory example, none of them can deal with all cases of selection, and in specific cases they may give contradictory results. Furthermore, I think that the bewildering variety of suggestions is in itself a source of dissatisfaction, and I therefore also aim at finding some underlying order in this variety.

2. An intuitive solution

The figure below shows a number of stylised fruit flies, each one the result of a specific combination of causal factors. Let us to begin with concentrate on the two in the first row, marked M 1 and N 1. N 1 is a genetically normal fly, while M 1 belongs to a mutated strain. They have been raised in identical environmental conditions. If we now put the question “why does M 1 have such short wings?“ the intuitive answer seems obvious. The wings are short because M 1 is a mutation; the explanation is genetic.

But suppose that we had never seen N 1, and that all flies had been of the mutated kind in the left column. These are all genetically identical but have been raised in different temperatures (22 °C, 27 °C and 32 °C). If we now ask again why M 1 has such short wings, the answer will again seem obvious, although it will now be a different one. The wings are short because M 1 was raised in a low temperature, and the explanation is environmental.

The example shows, I think, that the explanation of the short wings must involve a comparison with other flies. The facts about how genes and temperature affect wing-length are the same in both cases, just as the question. The reason for the answer’s being different is that we chose different objects of comparison. Comparing with genetically different individuals will make the genes important, and if we compare with individuals, who have had a different environment, the temperature will be decisive.

In a pretty obvious sense we could say that the two explanations, genes and temperature, really answer two different questions and explain two different facts. In the first case we did not really explain “M 1 has short wings“ but rather “M 1 has shorter wings than N 1“, that is, we explained the difference between two individuals. We are generally rather careless when formulating questions. We talk about the explanation of a’s having the property F, although the question is really why a has F while b does not. Normally this is harmless, since the objects of comparison, which are understood, are relatively constant and ’natural’. If genetically normal flies occur in a much higher frequency than the mutations, and if temperatures above 22’ are rare, one would expect everyone to compare with flies of type N1. It is important to note however, that even in this case it would be perfectly permissible to compare with M 2 or M 3, as long as we specify the question.

The thesis I now wish to defend is that all explanations of individual facts of the form Fa – that is, where an object a has a certain property F – involve a comparison with other objects which lack the property in question. 1f this is correct, it would give us a simple answer to our introductory problem. If a barn burns down, the natural objects of comparison would be other barns that did not burn. Since oxygen is present in all barns, this factor cannot explain the difference between those barns that burn and those that do not. Since heat was present everywhere on our distant planet, a heat source cannot account for the difference between those objects that catch fire and those that do not.

The idea that an understanding of explanations requires a finer analysis of the explanandum has been made before.⁴ Van Fraassen, for instance, has introduced the concept of a ’contrast class’ with which an explanandum is to be compared. Van Fraassens main concern, however, is with problems, which in the terminology adopted here pertain to determining causes and the causal relation itself, rather than to the problem of selection.

3. Explaining differences

By an explanandum fact Ea I will understand simply the fact that the object a has the property E as, for instance, the barn’s being on fire. The corresponding explanandum is slightly more complicated. As we saw in the previous section, it involves not only the object under consideration, but also those objects with which we make a comparison. We will say that the objects of comparison belong to a reference class R.⁵ The only restriction put on R is that its members must not possess the explanandum property. Clearly, explaining why this barn caught fire, when other barns did not, would not make sense if the other barns had in fact caught fire. The real explanandum, then, can be construed as a three-place relation between an object (e.g., the barn), a property (being on fire) and a reference class (other barns)

<a, E, R >

When R has only one member, say my neighbour’s barn, it will be convenient to mark this by writing <a, E, r>.

It should be noted here that for each explanandum fact there will correspond a large number of explananda. We may compare this barn with the neighbour’s barn, with other barns in the district, with other wooden buildings, etc. A special case is when R consists of every object except those possessing the explanandum property.

Since we are not concerned here with defining determining causes, but rather with what I have called the selection problem, i.e., what distinguishes the narrower set of explanatory causes from determining causes, it will be assumed that we already know how to recognise the latter. This is not to imply that the problem of causation is in fact solved, or easy to solve, but only that I am concerned with a different problem here. In the definitions in the sequel it will always be assumed that the explanans is known to be a determining cause of the explanandum fact.

If a determining cause is to explain for instance the fire, I think that it is reasonable to require this: an explanatory condition should be such, that when it is added to the non-burning barn, this will be sufficient to bring about the fire, and if it had not been added, the barn would not have caught fire. ’This is what is meant when, as David Lewis (1973) has pointed out, “we think of a cause as something that mikes a difference“. We can put this more generally in the following way:

D 1 Ca is an adequate explanation of < a, E, R> iff,

(i) for all x in R, if Cx had been true, then Ex would have been true, and

(ii) if ~Ca had been true, then ~Ea would have been true.

(The reason for the use of the qualifying term ’adequate’ will become clear later.)

Let us now see how this solves our original problem. (i) requires of the cause that it’s being added to one of the barns with which we compare this one must be sufficient for a fire. If the cigarette is an adequate explanation, it must be true of all barns in R, that if a lighted cigarette had been dropped in one of them, it would have caught fire. This is obviously not true of the presence of oxygen, for oxygen was present in all barns but did not bring about any fires.

It should be observed that D 1 does not simply require that the cause be sufficient and necessary in the actual circumstances. In the circumstances of our example, which include for instance the presence of the lighted cigarette, oxygen was sufficient. What is required is that the cause be sufficient in those circumstances exemplified by the objects of comparison., and this depends on our choice of such objects. For instance, the low temperature was sufficient for, and explains, the short wings of a mutated fruit fly, but it would have had no effect on a genetically normal fly, and it cannot explain the short wings of M l when this is compared with N 1.

In what sense can we say of a cause, which satisfies D 1 that it explains the difference between objects’? The intuitive point I tried to make in section 2 was that the solution to our problem lay in the fact that we do not really explain for instance why a fly has short wings, but rather why this fly has short wings when that one has not, or why this fly has shorter wings than that one. Now, if a certain condition explains why a has shorter wings than b, then the absence of the same condition must surely explain why b has longer wings than a. It is this symmetry which makes it natural to talk about the difference between two objects concerning a certain property rather than simply one object’s possessing the property. The concept of a ’difference’ is neutral concerning which object is compared with which. This ’symmetry requirement’ is satisfied by D 1, from which it follows that:

SR Ca is an adequate explanation of <a, E, R >, iff it is true for every x in R that
~Cx is an adequate explanation of <x, ~F, a>.

This follows because if (i) in D 1 is satisfied by Ca, then (ii) will automatically be satisfied by ~Cx, and if (ii) is satisfied by Ca, (i) will automatically be satisfied by ~Cx.

4. Determining the true explanandum

So far we have taken for granted that there is always a natural reference class, as if the choice of objects of comparison were somehow self-evident. This, of course, is not so. We may request an explanation for any explanandum we wish, and since the reference class is implicit in the explanandum, we may construct a reference class in any way we wish. There is a problem involved here, however, because our questions typically mention only the explanandum fact and are thus elliptic for the intended, or ’true’ explanandum. The use of the word ’intended’ should not be taken too literally here, but I do think that, when pressed, we can generally improve a question that only mentions an explanandum fact. If I ask why a certain bridge collapsed and someone answers ’gravity’, I would say something like ’I didn’t ask why bridges fall downwards rather than upwards. I want to know why this particular bridge collapsed, when that one did not’. Objects of comparison often ’exist’ only ’in the hack of our minds’ and are shaped by arbitrary factors like personal experience and emotional preferences. The fact that we are generally able to determine the true explanandum is due to the regularity with which certain kinds of objects of comparison nevertheless occur.

It will be useful to look briefly at some typical ways of forming reference classes, partly because this will enable us to explain some of the selection criteria which have been proposed in the literature.

(a) R as the normal case. Although a barn can be identified in many different ways, as a wooden object, a building, etc., it will typically be explicitly identified as a barn. We ask why the barn caught fire, not why this object did. We often delimit the reference class further, however, by excluding barns not made of normal materials, not used for normal purposes, etc., just as among the fruit flies we tend to compare with the most frequent type. If a certain causal condition were normal, it would occur among the members of R, but then it could never explain the difference between the explanandum object and those in R. It follows that, when R is chosen as the statistically normal, the explanatory cause must be abnormal, which explains the view, expressed by Hart and Honoré (1959) and others, that we select from the determining causes those that are abnormal.

(b) R as the temporally normal case. If the question is ’why did the barn catch fire at this particular time’, we seem to be asking about a temporal difference, and the proper object of comparison will not be another barn, but this barn at an earlier time. This accommodates the rule that we prefer ’precipitating’ causes to ’standing conditions’ (e.g., MacIver (1952), p. 161). (a) and (b) will often coincide, and both will sometimes coincide with other selection criteria as, for instance, the view that we prefer variable conditions to constant ones (e.g., Nagel (1961), p. 584). Obviously, a condition that is constant in the sense that it does not vary over time or between individuals can never explain differences between time-points or between individuals.

(c) R as a theoretical ideal. In many sciences, particularly those that are theoretically advanced, it is a common procedure to use as an object of comparison a hypothetical object or state of affairs, which is defined by some theory. Such theoretical ideals, as we may call them, have the obvious advantage of providing the scientist with a constant object of comparison, thus facilitating systematisation of the field covered by the theory. Typical examples of theoretical ideals are Weber’s ’ideal types’, the equilibrium models of the perfectly working market economy in neo-classical economics, the definition of a ’wild type’ in bacterial genetics, the physiology of the healthy human organism in medicine, etc.

If I understand him correctly, theoretical ideals are pretty much the same as what Toulmin ((1961), pp. 56 ff.) calls ’ideals of natural order’, that is ’natural courses of nature’ that do not call for any explanations. For Aristotle as well as for common sense, such an ideal would be exemplified by a body at rest. For Newton, the ideal motion was that stated in the first law, ’a state of rest, or of uniform motion in a right line’. The theory does not explain this ideal, only deviations from it. Thus, a force in Newtonian mechanics cannot explain such things as why the moon circles round the earth, but only why it circles round the earth rather than going on along a tangent.

(d) R as the subjectively expected. Closely related to the last case are explanations which purport to show why an observed course of events differed from an expected one. ’To explain a thing’, according to William Dray, ’is sometimes merely to show that it need not have caused surprise’ ((1957), p. 157).⁶ Such ’how-possibly explanations’ explain ’how some later event or condition could have come to pass in spite of known earlier conditions which give rise to a contrary expectation’ (p. 162). It seems to me that these explanations explain the difference between actual and expected events and differ from other explanations only in their choice of reference class.

(e) R av a moral ideal. Causes are sometimes said to be ’responsible’ for their effects, and events are ’blamed’ on this or that condition. Many philosophers have seen this blending of causal and moral jargon as a symptom of confusion, but Dray regards it as a selection criterion. ’A cause’, he claims, ’will often be an omission which coincides with what is reprehensible by established standards of conduct’ ((1964), p. 56). That is, we select conditions, which should not have been present. I think that we can construe the explanandum in such cases as the difference between an actual course of events and a hypothetical course of events in a morally ideal world. If the actual world differs from the morally ideal one, the difference can only be explained by events or actions, which ’deviate from established standards of conduct’.

The fact that we can ’explain’ selections of causal conditions in this way considerably undermines the idea of selection rules and the currently popular conception of explanations that goes along with it.

In a discussion of the selection problem Dray (1957) introduced the term pragmatic for referring to those aspects of an explanation which are relative to an individual or a context and which are thus arbitrary from a logical point of view. A recent example of an elaborate ’pragmatic theory of explanation’ is van Fraassen (1980). Implicit in the pragmatic viewpoint is the idea that all the conditions making up the ’complete’ cause have the same logical standing, and that only ’pragmatic’ (i.e., subjective or non-logical) criteria can explain the selection of one of them as ’the’ cause.

This view is very misleading. If C₁a and C₂a are both conditions for Ea it would certainly be arbitrary to select C₁a as a determining cause. But if we ask about the difference between a and b, where ~C₁b, C₂b and ~Eb, it is clear that only C₁ is relevant since this is the only factor that varies between a and b. There is nothing arbitrary about this. On the contrary: given a specific question, the condition selected is precisely that which is capable of answering the question.

We do not prefer abnormal, precipitating, variable, etc., conditions because such conditions are particularly interesting. We select them because they are (logically) relevant to the questions we ask. The regularity with which certain kinds of conditions occur in explanations should thus not be interpreted as a selection of preferred conditions, but as reflecting a regularity in the kinds of explananda that interest us and their reference classes.

This point is not merely hair-splitting, for it enables us to eliminate the disturbing questions about selection criteria mentioned in the introduction. When the explanandum is clearly stated, the problem of contradictory selections does not arise, and neither does the problem of determining which criterion was operative in a case where more than one criterion is satisfied.

5. Adequacy and completeness

D 1 requires of an adequate explanatory cause that it would have been sufficient for producing E in a member of R. Since the majority of explanations fail to satisfy this requirement, although they presumably have some explanatory power, we must say something about these less than perfect, or, in the present terminology, less than adequate explanations. In doing this it will be convenient to adopt the framework of a well-known model of causality of the type proposed, for instance, by J. L. Mackie (1965).

It will be assumed that every event has what we may call a complete cause, that is, a set of predicates such that when all of them are exemplified in an object, this will be sufficient for the occurrence of the effect. Let us for the sake of simplicity assume that for the effect Ea there is only one complete cause Ca, where Ca means that a has all the properties in the set {A ₁, A ₂,... A_n}. Now, consider the reference class R which contains all objects with the properties in {A ₁, A₂ ,... A _n-1}. To give an adequate explanation of <a, E, R>, we must cite A_n, for this is the only factor that is missing in order to achieve a sufficient cause, and, since there is only one complete cause, A_n will also be necessary in the circumstances for Ea. If we now weaken the requirements on the reference class and consider the wider class R’, which contains all objects having all the properties in {A ₁ , A ₂,... A_n-2}, then both A _n and A_n-1 will have to be part of an adequate explanation. If we do not make any requirements at all on R, that is, if we want to explain Ea as compared with every object (except, of course, those that have E), the explanation will contain all the properties A ₁,... A_n . Thus, to explain an event without any implicit comparison with a set of reference objects will require a complete cause.

The previous remarks should not lead us to believe that reference classes are ever defined in the above way. When faced with explanations that are incomplete in the sense that they only mention a part of the complete cause, defenders of the covering law-theory have generally claimed that the conditions left out are tacitly understood or taken for granted, mainly because they are too trivial to be made explicit. But this does not seem very likely, for people are very often ignorant of these ’missing conditions’. Fires were adequately explained before anyone knew that oxygen existed. Furthermore, mentioning the presence of oxygen when explaining a fire would not make the explanation more complete – it would make it distinctly odd. The missing conditions enter into explanations on the present account too, however, but they do so by being true of the objects of comparison.

In practice all explanations will be incomplete in the sense that they mention only a part of the complete cause and explain the explanandum event only relative to some reference class. It is important here not to confuse incompleteness with inadequacy. A large enough amount of rattlesnake poison will always cause, and so explain, the death of a man when he is compared with other human beings, but not when he is compared with King snakes (which are immune), so the poison is not a complete explanation. But since it is sufficient to kill a human being, it will be adequate when R consists of human beings. Smoking, on the other hand, is not sufficient for death and will not adequately explain a death when R consists of human beings. Non-adequate explanations usually call for supplement. Thus, although we regard smoking as an explanation of lung cancer, we should also want to know why only some smokers develop cancer. The supplementary factor would explain the further difference between those smokers who do and those who do not develop cancer. Even if smoking is not what I have called an adequate explanation of cancer, it is still an explanation. How should it be characterised? A natural possibility, and one which is currently receiving a great deal of attention, would be to require that the explanans be statistically relevant to the explanandum. If, as before, we assume that a relation of determining causality obtains, we could formulate this idea as follows:

D2 Ca is and adequate explanation of <a, E, R>, iff in R*

P(Ex/ Cx) > P(Ex/~Cx)

Since we have made the requirement on reference classes that they must not contain any objects with the explanandum property, we cannot, at least as long as we are dealing with objective probabilities, define the probabilistic relation over such reference classes. If we did, P (Ex/Cx) would always be zero. The probabilities must therefore be defined over a new reference class R*, consisting of R together with all objects of the same kind which have the explanandum property.

Although D 2 could be regarded as an approximation of D 1, it gives ’rise to a problem which seems to be due to its statistical nature. I have argued elsewhere that causes may sometimes lower the probability of their effects (Hesslow 1981b). Contraceptive pills, for instance, can cause thrombosis, but since they also lower the probability of pregnancy, which is a more efficient cause of thrombosis, they may actually lower the probability of thrombosis. A cause can thus have two opposite effects by working via two different mechanisms, but it cannot affect the probability in two directions.

It might be possible to rescue D 2, if we could somehow eliminate all pregnant women from the reference class, but this may be very difficult. We could not say, for instance, that Ca explains <a, E, R> if there is some reference class in which Ca raises the probability of Ea, for this will always be the case and everything would explain everything. Another possibility would he to make some kind of requirement of total evidence, but this would make explanations relative to our present knowledge, a consequence that will be hard to swallow for those who, like myself, think that explanations can be true or false regardless of what we know. Thus, although it would be rash to dismiss the possibility of modifying D 2 in an acceptable way, ⁸ I think that the difficulties are serious enough to warrant the consideration of an alternative.

Suppose for simplicity that the presence of a lighted cigarette in combination with the presence of dry hay is sufficient to make a barn catch fire. If our reference class consists of all barns (that did not catch fire), the presence of a lighted cigarette will not adequately explain a fire in this barn. There will, however, be a subset of the reference class (consisting of barns with dry hay) such that when this subclass is used as a reference class, a cigarette will adequately explain a fire. Thus, a definition, which fits nicely the general viewpoint taken in this paper, would be the following:

D 3 Ca is an explanation of <a, E, R>, iff there is a subset R’ of R such that

Ca is an adequate explanation of <a, E, R’> .

The blood clot example is clearly not a problem for D 3, for the latter allows us to say that the explanatory value of pills depends on whose clot we wish to explain and with which women we make the comparison. Since we can only compare with women who did not get blood clots, the frequency of women who got clots from other causes will be irrelevant. Note that if R has only one member, there can be no subset except the empty set, and D 3 will not be applicable. A ’unique’ difference <a, E, b> can only have an adequate explanation.

6. The relative importance of causes

So far we have treated causal selection as if there were only cne cause, or only one part of a complete cause, with any explanatory power, for instance by asking if genes or environment was the cause of the fly’s short wings. This way of putting the question was possihle because we chose the reference class in such a way that only one cause could explain the difference between flies But suppose that we did not restrict R to flies with similar genes or with a similar environment and let R consist of all flies with long wings. This would give us two causes, and it is an interesting question if there is some way of assigning relative weight to them. Although it has sometimes been claimed that this is meaningless, it is clear that we often make statements about the relative importance of causes and that we are fairly consistent when we do so. Everyone would agree, for instance, that eye colour is mainly genetic, even if the environment contributes, or that economic depression was more important for the support of the Nazis than were ideological convictions. But what does it mean to say that one cause was more important than another? For reasons that will become clear later the question should be understood as concerning relative explanatory power rather than determining power.

If we regard D l as a characterization of the ideal case, it would seem natural to say that one cause has greater explanatory power than another, if the first is a better approximation of the ideal than is the second. If we first consider D 2 we would have for instance the following:

D4 C₁ a has greater explanatory power than C₂a with respect to <a, E, R>, iff in R*

P (Ex/C₁x) > P (Ex/C₂x)

Definitions of this sort have been proposed and defended with the help of examples by Martin (1972) and Nagel (1961), and we shall see that D 4 works quite well in many, if not all cases. It raises some questions, however., concerning the underlying rationale. Assuming that we already know that in this individual case both the causes and the effect occurred, the additional information that one of the causes raises the probability of the effect more than the other cause does, will mainly tell us something about the statistical distribution of these causes and their effects in other cases. If it is accepted that it is meaningfu1 to talk about the relative importance of causes as a property of an individual case, it is difficult to understand the relevance of a statistical relation, which is a property of a population.

I think that the analysis of the explanandum as a difference gives us a clue to the answer. Since, on this view, the individual explanandum is not an ’individual case’ in the sense of a singular fact or event but a relation which incorporates several cases, the importance of other cases should not appear unnatural. Let us explore this idea and try to relate D 4 to the theory developed in this paper.

We will consider first a somewhat idealized, but by no means untypical example. Let C₁ and C₂ denote mutation and low temperature respectively, and let E denote short wings. We will assume that C₁ and C₂ are individually necessary and jointly sufficient for E. In the following imagined distribution of flies the figures can be interpreted both as absolute numbers and as percentages.

	C1	~C1
C2	E   10	5
~C2	30	55

All flies in the upper left cell are short-winged and all the rest are long-winged. P(Ex/C₁x) = .25 and P(Ex/C₂x) = .67, so according to D 3 C ₂ should have greater explanatory power than C₁ .

Since Ex is true iff C₁x and C ₂x, we have

(1) P(Ex/C₁x) = P(C₁x & C₂x)

P (C₁x)

(2) P(Ex/C₂x) = P(C₁x & C₂x)

P (C₂x)

and consequently,

(3) P (Ex/C₁x) < P(Ex/C₂x) iff P(C₁x) > P(C₂x)

Now P(C₁x) = P(C₁x & C₂x) + P(C₁x & ~C₂x) and P(C₂x ) = P(C₁x & C₂x) + P(~C₁x & ~C₂x), so the actual comparison will be between P(C₁ & ~C₂x) and P (~C₁x & C₂x), that is, between the lower left and the upper right cells in the example. These correspond to mutated flies in high temperature and genetically normal flies in low temperature. If flies of the former kind were the more common, it would be more natural to use these as objects of comparison and, hence, to ascribe to low temperature (C₂ ) a greater explanatory power. The right inequality in (3) corresponds to our tendency to prefer more common objects as objects of comparison. Given the view of the explanandum as a difference, (3) may thus be taken as supporting D 4.

But this analysis can be taken a step further. An explanandum of the form <a, E, R> is an individual event only in a narrow sense, since for every element x_i of R there will correspond a unique event, or difference, <a, E, x_i >, and in a very natural way we could regard < a, E, R> as a composite event constructed from the constituents < a, E, x₁>, <a, E, x₂>, etc. Thus, the short wings of an arbitrarily selected fly in our example involve a comparison with 90 long-winged flies, and the ’individual’ event is actually composed of 90 distinct differences. These can now be assigned to three different groups, corresponding to a partition of the reference class. (a) Differences between the selected fly a and flies having C₁ and ~C ₂, that is mutated flies in high temperature. These differences are adequately explained by C₂ . We will denote the subset of R corresponding to these differences R ₂, where the subscript indicates that C ₂ is the only property absent from the set and also the property that explains the differences involving these elements. The number of elements in R₂ , which is also the number of differences explained by C₂ , is denoted c ₂

(b) Differences between a and flies having ~C₁ and C₂, that is the genetically normal flies in low temperature making up R₁. These differences are adequately explained by the mutation C₁. (c) Differences between a and flies having ~C₁ and ~C₂ the elements of R₁₂. These differences are adequately explained only by the combination of C₁ and C₂. In our example c₁ , c₂ and c ₁₂ are 5, 30 and 55 respectively. Since R₁, R₂ and R₁₂ are mutually exclusive and exhaust R, the total number of differences t must be the sum of

the differences involving these three sets:⁹

(4) t = c ₁ + c₂ + c_l2

It is important in this context that all differences are adequately explained. Neither C₁ nor C₂ by themselves adequately explain the differences in group (c), only the of C ₁ and C₂ does. Note also, that if we utilize the symmetry commented on in section 3 above and ask about the long wings of the flies in R using one of the short-winged flies as an object of comparison, then only one sufficient condition (viz. ~C₂) for long wings occurs in group (a), only one (~C₁) in (b), but two sufficient conditions (~C₁ and ~C₂) occur in group (c). Thus, whether we take these objects in (c) as explanandum objects or as objects of comparison, there is no longer any justification for separating the contributions of the two conditions. The differences in (c) cannot therefore be ’divided up’ between C₁ and C₂ but must be regarded as a separate category.

There is, of course, a sense in which we could say that C ₁ is involved in, or a part of, the explanations of the differences in group (c). The number of differences in which C₁ and C₂ will thus be involved, will be c ₁ + c₁₂ and c₂ + c ₁₂ respectively, where the sum exceeds t.

Although (4) leads to a natural additive measure of explanatory power, where the power of C₁ for instance, would be the fraction c₁/t, we will here consider only the following qualitative competitor of’ D 4:

D 5 C₁a has greater explanatory power than C₁a with respect to <a, E, R>, iff 

            C₁a adequately explains a greater number of constituents of <A, E, R> than doess C ₂a

that is, iff c₁ > c₂. By looking at the diagram above we see immediately that

(5) c₁ > c₂ iff P(C ₁x) < P (C₂x)

From (3) and (5) we get

(6) c₁ > c₂ iff P( Ex/C₁x) < P(Ex/C₁x)

Thus, under the assumption that C ₁ and C₂ are jointly sufficient and individually necessary for E, D 4 and D 5 are equivalent.

This equivalence holds in spite of the fact that D 4 does not treat the combined effect of C₁ and C₂ as a separate category. C₁ will raise the probability of E more thanwill, when c₁ + c ₁₂ > c₂ + c ₁₂. Since this will be the case precisely when c ₁ > c₂, the two definitions give the same result.

We must now consider the consequences of relaxing the assumptions of sufficiency and necessity. Suppose first that C₁ and C ₂ are not individually necessary for E, so that we get short-winged flies among the genetically normal and/or those raised in high temperature. We might for instance have some short-winged flies with C₂ and some with ~C₂, i.e. the ones in the lower left cell. If we wish to explain the short wings of a fly with C₂ , then C₂ will have some explanatory power, but if we pick a fly with ~C₂ the explanatory power of C ₂ will of course be zero. This is also what D 5 implies. Since short-winged flies will not be members of the reference class, their number will be irrelevant except, of course, in the sense that there may be fewer objects of comparison left. For instance, if 25 of the 30 flies in the lower left cell were short-winged, there would, for each short-winged fly in the upper left cell, be only 5 unique differences explained by low temperature. This, I think, is intuitively acceptable.

For D 4 there will be difficulties. The explanatory power of a cause will depend on which one of the short-winged flies occurs in the explanandum. But a probability statement like P(Ex/C₁x) will be insensitive to the difference.¹⁰ There will also be problems of the kind exemplified by the thrombosis case. Anything, which is statistically independent of thrombosis, will raise the probability of thrombosis more than will contraceptive pills. In view of these difficulties I think that a conclusion similar to that reached concerning D 2 and D 3 is justified, i.e. although it might be possible to modify D 4 in an acceptable way, D 5 is preferable. Furthermore, even in those cases where the definitions coincide, there is reason to regard D 5 as more fundamental than D 4. It was the analysis of the explanandum on which D 5 was directly based, which provided the rationale for D 4 and enabled us to answer the question about the relevance of the population for the individual case.

Suppose now that C₁ and C₂ are not sufficient for E. For D 4 this only means that we would have a smaller number of short-winged flies in the upper left cell of the diagram and a smaller number of events to explain. As long as the distribution of the objects of comparison remains the same, the relative importance of C₁ and C₂ will not be altered.

For D 5, however, things will be more complicated. If C₁ and C₂ are not sufficient for E, there must be some other condition (or conditions) C₃ (say a nutritional factor), such that only the combination of all three conditions is sufficient for E. But this means that for instance C₁ will not adequately explain all the differences between a and those objects which have ~C₁ and C₂, but only those differences between a and objects which have ~C₁, C₂ and C ₃. In terms of our example, the mutation will not adequately explain all the differences between 5 short-winged fly and those long-winged flies, which are genetically normal and have been raised at room temperature, because some of these long-winged flies, namely those lacking the nutritional factor, would not have been short-winged even if they had been mutations. Introducing a third condition thus gives rise to a new partition of the reference class and the corresponding class of differences, and the explanatory power must now be divided up as follows:

(7) t = c₁+c₂+ c ₃ + c₁₂ + c ₁₃ + c₂₃ + c₁₂₃

Here the difference between D 4 and D 5 shows up, D 4 would give C ₁ a greater explanatory power than C₂ , if

(8) c₁ + c₁₂ + c ₁₃ + c₁₂₃ > c₂ + c ₁₂ + c₂₃ + c₁₂₃

whereas D 5 requires that

(9) c₁ > c₂

Now (8) and (9) will be equivalent when c₁₃ = c ₂₃ and this will occur only when C₃ is evenly distributed between C₁ and C₂. For instance, if the number of genetically normal flies were equal to the number of high temperature flies, and the nutritional factor were present in all and only mutated flies, then D 4 would give mutation and low temperature equal weight, and D 5 would give low temperature priority. It may not be self-evident that D 5 comes closer to our intuitions, which are probably not very clear anyway, but I do think that it has a slight advantage over D 4. Under the above assumptions, if we compare with a fly in the high temperature group, low temperature would be sufficient for short wings (since all flies in this group have the nutritional factor). A genetically normal fly, however, would not have had short wings if it had been a mutation (since no flies in this group has the nutritional factor). I think it is reasonable that this is reflected in a greater explanatory power of low temperature. Let me end this section with a methodological remark. When it is said, for instance, that genes are more important for a certain phenotypic property than the environment, this cannot mean that the genes play a greater role in determining that property or that genes are somehow more powerful. The truth of a statement about the relative importance of causal conditions will depend on the actual frequencies of these conditions. Thus, the relative importance of the environment to short wings could easily be changed by lowering the temperature in which the fruit flies are raised. The relative importance of genes and environment is in an important sense relative to genes and environment. This conclusion agrees well with the views of population geneticists (e.g., Lewontin (1974)) but does not seem to be recognised in other fields, although it should hold for any pair of causes in any area of science.

7. A note on complex conditions

There is a final question in this context which is important in itself and which will further elucidate the relationship between D 4 and D 5. Reality can be categorised in many different ways, and so can the causal conditions for an effect. It may be asked, then, what the effect of a different slicing of the causal pie would be on the relative importance of two conditions. In most cases, of course, nothing definite could be said about this, but there is at lest one case which merits attention. Suppose that one condition C₁ has greater explanatory power with respect to some explanandum than another condition C and that C can be construed as a conjunction of the more ’elementary’ conditions C₂ and C₃ It would then seem reasonable to expect that C₁ has greater explanatory power than any of these conditions taken singly. Intuitively, a large explanatory power of C₁ should not be reduced by a finer slicing of other explanatory conditions. This is equivalent to the following ’invariance requirement’:

IR The explanatory power of (C₁ & C₂) can never be less than the explanatory power of C₁.

It is easy to see that D 5 satisfies this requirement and that D 4 violates it. Suppose that C adequately explains <a, E, R> (where R has c elements) and that C is actually a conjunction of C₁ and C₂. As we have seen before, we can separate the explanatory contributions (c₁ , c₂ and c₁₂) of these conditions and their interaction by a partition of the reference class and the corresponding explananda <a, E, R ₁>, <a, E, R₂> and <a, E, R ₁₂>. Since R₁, is a subset of R, c₁ cannot exceed c.

D 4 could satisfy IR only if it were always the case that

(10) P(E/C₁C₂)  > P(E /C₁)

which is plainly false, even if it is assumed that C₁ and C₁C₂ both raise the probability of E. The case of disjunctive properties does not offer any additional problems. If Ca adequately explains <a, E, R> and C is a disjunction of C₁ and C₂, then the reference objects must all lack both properties, and the reference class cannot be partitioned. There may, of course, be different sorts of explanandum objects, some with both C₁ and C₂ and some with only one of these conditions. In the latter case, only the condition that is present can have any explanatory power. The situation becomes different if we ask why E is absent among the members of R, but this case will just be the mirror image of that discussed in the previous paragraph, because the explanatory condition ~(C₁ v C₂) is now a conjunction. I think that this agrees with intuition. If low temperature and a certain genetic mutation were each sufficient for short wings, so that the disjunction of these conditions explained a certain set of differences, the explanatory power of one condition taken singly would depend on which fly we were discussing and which condition(s) was present in this particular fly.

Received on February 7, 1983.

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Notes

I am indebted to Bengt Hansson for valuable suggestions and detailed criticism of earlier versions of this paper.

1 This term was introduced by Dray (1957) to refer to the classical theory of Hempel & Oppenheim. See Hempel (1965).

2 The main source here is Salmon (1970). Other variations on the same theme are Hansson (1975) and Gärdenfors (1980). Van Fraassen (1980) also contains a discussion.

3 Other selection criteria are discussed in Nagel (1961) and Martin (1972). Van Fraassen, who notes the incompatibility of these criteria, suggests that selections are determined by context.

5 The reference class should not be equated with what van Fraassen (1980) calls a ’contrast class’. The latter consists variously of ’alternatives to the event’ (p. 129) and ’a set of propositions’ (p. 142), while the former consists of objects. It might be possible, though, to generalise these concepts so that they cover each other.

6 Gärdenfors (1980) has developed a theory in which ’surprise reduction’ is taken as a feature of all explanations.

7 See the references in note 2.

8 A discussion of this and other problems in connection with the probabilistic theory of causality is given in Salmon (1980).

9 Although the variables on which (4) is based are qualitative, there is a striking similarity with the models resulting from the statistical technique known as ’analysis of variance’. An example is the so-called ’linear model’ of quantitative genetics, where phenotypic variance is broken down into additive components of genetic, environmental and interactive variance:

s² _p = s² _g + s² _e + s² _ge

Cf., Falconer (1960). It is characteristic of the fields which utilise this technique that variance, which is a measure of differences, is assumed to be explained by another variance. This view is becoming increasingly important in many sciences but is difficult to handle with hitherto existing theories of explanation.