Explanation, control and generality

Robert L. Frazier

Christ Church


I would like to start by quoting a passage from Carl Hempel's `Explanation in science and in history' ([Hempel, 1962]).

Among the divers factors that have encouraged and sustained scientific inquiry through its long history are two pervasive human concerns which provide, I think, the basic motivation for all scientific research. One of these is man's persistent desire to improve his strategic position in the world by means of dependable methods for predicting and, whenever possible, controlling the events that occur in it. The extent to which science has been able to satisfy this urge is reflected impressively in the vast and steadily widening range of its technological applications. But besides this practical concern, there is a second basic motivation for the scientific quest, namely, man's insatiable intellectual curiosity, his deep concern to know the world he lives in, and to explain, and thus to understand, the unending flow of phenomena it presents to him.

These goals, prediction, control and explanation, will be the topic of this paper. I shall proceed by discussing the dominant philosophical view concerning explanation and show some consequences of adopting that view. During my discussion of explanation, I will argue that general laws or principles are not as relevant as might be thought to the explanation of particular phenomena. Finally I will discuss some of the consequences of accepting this conclusion for the relationship between theoretic and applied science.


When we ask why some physical phenomenon has occurred, we seem to be asking for an explanation of it. We think that we have made scientific progress when we can provide a scientific explanation: we have come to understand it. We also think that when we can give a scientific explanation of some phenomenon, we have a better chance of being able to predict it, change it, control it and/or replicate it. So explanation seems to have a central role in both theoretical and applied science. Given this, it is no surprise that philosophers of science have been concerned with the role of explanation in science, the nature of scientific explanation, and the criteria for such an explanation to be a good one.

In philosophical circles, the dominant view of explanation in science is `the covering law model of explanation'. Although traces of the model can be found in the works of J. S. Mill, its modern appeal is largely due to the careful statement of it by Carl Hempel, whom I previously quoted. Intuitively, this model proposes that an adequate scientific explanation of some particular concrete phenomenon shows how its occurrence follows from true general laws and accurate statements of relevant facts. Let me now give a bit more careful statement of the model.

Standardly, `explanandum' is used to refer to the statement of the occurrence of the phenomenon that we want to explain, and `explanans' is used to refer to the set of statements that figure in the explanation of the phenomenon. There are some conditions on what the statements in the explanans can be. Each explanans has two statements. One statement must be a general lawlike statement, or conjunction of such lawlike statements. A lawlike statement is one, that, if true, is a law of nature. The other statement must give contingent conditions that obtain prior to, or at the same time as, the phenomenon to be explained, or be a conjunction of such statements. Also, to prevent there being trivial explanations, such as there would be if an explanandum were also a statement in the explanans, the explanandum cannot follow from any single statement in the explanans. According to the covering law model, then, for there to be an explanation of some phenomenon, the statement that says it occurs, the explanandum, must be strictly deducible from the statements that give the explanation, the explanans.

Let me use a very simple example to demonstrate this structure. Say that we have a sample of water that is boiling. We might want an explanation of why it is. The covering law model says that we need an explanans expressing laws and contingent conditions that jointly entail the explanandum, that the water is boiling. The explanation might be something like this: (i) any water heated to 100C will boil; (ii) this sample of water is heated to 100C; therefore, (iii) this sample of water is boiling. The phenomenon of a sample of water boiling is explained because its occurrence follows from a general law, (i), and a particular contingent condition, (ii).

Of course, someone might object the explanation of why water boils at 100C is not a scientific explanation, because the boiling of water is not the right sort of phenomenon, and that there certainly are not any truly scientific lawlike statements about boiling water. I grant all of this. It is the structure that interests me. I could rephrase the whole thing in terms of water being in a gaseous state when its mean kinetic energy reaches a certain level. However, this would unnecessarily complicate things, at least for me.

If, as looks likely, the universe is irreducibly chancy, that is, involves non-deterministic events and essentially probabilistic laws, the previous version of the covering law model would not work for all explanations. This is because probabilistic laws or statements will not support deduction. For example, from the facts that smoking increases the probability of my getting cancer and that I smoke, we cannot deduce that I will get cancer. At most we can conclude that I have increased the chances of my getting cancer. In an acknowledgment of this feature of probabilistic statements, Hempel introduced a complementary version of the covering law model that does not require the explanandum to follow deductively from the explanans. Instead, when the laws are probabilistic, the explanans has to make the explanandum highly likely. So we have two versions of the covering law model of explanation, the deductive-nomological model (DN), which requires strict laws and deduction, and the inductive-statistical model (IS), which requires probabilistic laws and inductive support.

Although, as I mentioned, the covering law model is the dominant model of explanation, neither the DN model nor the IS model is without its difficulties. I will quickly explain a couple of difficulties with these views both to let you see better how they are supposed to work, and to give you some idea of how at least some philosophers evaluate such views.

One of the main technical difficulties with the DN model is that it may not get the direction of explanation correct. Suppose that we have as part of an explanans a lawlike statement that says a barometer falls if and only if a storm is coming. If we also have as part of the explanans an empirical statement saying that the barometer is falling, we can derive the statement that the storm is coming. Yet we would not want to say that the falling of the barometer explains the coming of the storm, it is merely evidence for it. Now suppose that the explanandum is that the barometer is falling and the statement giving empirical conditions in the explanans is that a storm is coming. This looks like the right direction for explanation. It is the coming of the storm (with the appropriate law) that explains why the barometer is falling, not the other way around. Yet both patterns seem to satisfy the DN model.

One problem with the probabilistic version of the covering law model, the IS model, is that it seems to preclude the possibility of explaining unlikely events. Remember the IS model says that the explanans is supposed to support the conclusion that the explanandum is highly likely. Let us assume that there is some probabilistic law that says if a certain amount of chemical, call it C, is applied to a kind of plant, call it P, there is .9 chance that the plant to which C is applied will be dead after 24 hours. We apply the appropriate amount of C to some P plant and go away for 24 hours. When we come back, the plant is still alive. Is not the plant's being alive explained by the small chance the law gives to this? However, according to the IS model, it cannot explain why the plant is alive because the law and statement of empirical conditions will not make it highly likely that it is alive.

Of course efforts have been made to overcome these difficulties with the covering law model, with varying degrees of success. The difficulty I want to discuss is not so technical, but rather concerns the requirement that one of the statements in the explanans be a law, or lawlike statement. The problem is as follows. For an explanation to be a good one, it is required that the explanans be true, or well confirmed. So the laws must be true, or well confirmed. However, laws that are like this, true or well confirmed, almost invariably involve descriptions of what happens in idealized situations, or situations involving a high degree of abstraction. So it becomes unclear whether or how what is true in these idealized situations actually explains what happens in situations that are non-ideal, or do not involve significant abstractions. More importantly, even if there is some account of how the idealized explains the non-idealized, it is difficult to see how this kind of explanation is the sort that could underlie successful attempts to control the the world, as we want to do in applied science.

It is easiest to see why this is so if we consider an example. Let me return to a previous example: explaining why a sample of water is boiling. Of course the law, or regularity, that I gave earlier is much too crude. Let me be a bit more careful. We need to start with a variable for the temperature of the water, "t". In addition, we need a constant, K, which has a value of 100C. The simplest version of the law would be this: t = K, in effect turning the variable t into a constant. To be adequate, this proposed physical law would have to have universal applicability. It fails in this respect. Water at sea level might boil at 100C, but water at higher altitudes does not boil at that temperature. At 10,000 feet above sea level, for instance, water is supposed to boil at 90C.

It is certainly easy enough to make the proposed law concerning boiling water more general by describing a relationship between altitude and temperature. Such a law would require only a bit more machinery: a variable that takes a measurements of altitude (or pressure) as its value, say "a", and a constant that expresses the relationship between altitude and the boiling temperature of water, let us again use "K". A more general version of the law might go something like this: t = 100 - a(K), where t is the boiling temperature of water in degrees Celsius, a is the altitude deviation from sea level in hundreds of meters and K=1/3. If a is zero then t is 100C, and for each unit change of a, t changes by -(a(K)). For example: if a decreases by one, t increases by 1/3 degree; if a increases by 3, t decreases by 1 degree. Upon reflection, we can see, I hope, that this proposed law correctly takes into account variations in altitude.

Unfortunately, there is also an important problem with this version of the proposed law: it is false. Water as we find it in our environment, for example in streams and when we turn on the tap, is not a very homogeneous substance. Different samples usually contain different varieties and concentrations of particulates and gases. In addition, most samples have parts at varying altitudes or pressures. Each of these features may affect the boiling temperature. Again, it may be objected that even this proposed law is too simple, and an even more complicated law would have the required universality. I suggest that it is doubtful that we could ever formulate a law that took into account all of the various solutions that can constitute water. This is where idealization and abstraction come in to play. If we want to be accurate, we could, instead of giving a law concerning water, propose a law having applicability to H$_{2}$O and in very highly controlled situations do experiments establishing the relationship between H$_{2}$O being in a gaseous state, temperature and pressure. The resulting law might well be true. Unfortunately it also fails to apply to very many actual phenomena. There are very few actual samples of H$_{2}$O around. A way of putting a roughly similar idea is that other things being equal, water will boil at 100C, but that other things are seldom equal.

It may be argued that my example misses an important point. The law concerning the H$_{2}$O does explain what happens when we heat water because the samples of water that we have do not deviate from H$_{2}$O enough to matter. My response is twofold. First, according to the covering law model, if the occurrence of boiling water does not follow from the explanans, then there is no explanation. Some might want to say that it is a partial explanation or an explanation that is near the truth. Perhaps this is true. An explanation of why some sample of water is in a gaseous state will have to appeal to the fact that the sample is largely comprised of H$_{2}$O, so in this sense it is part of the explanation.

Second, whether or not it matters that the water sample deviates only slightly from H$_{2}$O depends on the context of the question. If we are trying to control some aspect of the environment, it depends on how accurate our information has to be and what our purposes are in trying to control the environment. If we are conducting an experiment in a laboratory environment, it may be that a very small deviation from H$_{2}$O prevents us from getting the results that we want.

At least as important, especially when it comes to practical questions, such impurities could make a huge difference when considering a very large number of events that depend on the entire group of features that the substance has. Here is an hypothetical example to illustrate this point. Let us say that each day there are millions of people who boil water, whether from taps, wells, or rivers, to make tea and coffee. What fuel resources are required to allow them to do this? If we knew that all of the water was H$_{2}$O (and the amount used at any one time and the altitudes at which it was used and the exact efficiency of the method of heating it, etc.), then we would know fairly well how much fuel was needed. However, we do not. Indeed we know that there are small variations in the amount of fuel that it takes to bring each sample to the boil. Some water takes a bit more fuel, some a bit less. These millions of very slight differences could add up to a huge deviation from the amount of fuel that would be needed if all of the water were H$_{2}$O. So if the question is how much fuel is needed to provide for coffee and tea, the deviation does matter and the general principle concerning the boiling temperature of H$_{2}$O is of little use.

The difficulty in applying general laws to particular events in order to explain them or control them is even more evident when it comes to phenomena that depend on a larger number of features. For example, in Agriculture and the Environment ([Briggs and Courtney, 1985]) there is a discussion of various formal models concerning yield increases as a result of using fertilizers (the Misterlick equation, and two equations suggested by the FAO). I take the statements of these models to fill the role of lawlike statements in the covering law model of explanation. They are supposed to be used to explain, predict and control. For my purposes, I do not need to present them, because the point I want to make rests on the comments made by the authors after the discussion of them. They say that there is a `parabolic relationship between crop yield and fertilizer application' (p. 111), and then go on to say the following:

It should also be possible, in theory, to make use of these equations to find the optimum rate of fertiliser application. A range of factors, however, complicate the calculations. First is the fact that the constants in the equations vary from one crop to another; the experimental data on one crop cannot be used to predict responses of other crops. Secondly, environmental variables such as weather, soil type and water availability all influence the relationships. Thirdly, management factors play an important part and variables such as sowing date, previous fertiliser practice and cropping sequence may all affect crop responses. Finally, the relationships are also sensitive to nutrient interactions, so that responses depend upon the degree of limitations imposed by other nutrients. (p. 111)

As one more example, in a discussion of the productivity of arable lands, the authors say the following of a particular formal model:

In addition, the model is designed to be linked to related socio-economic models which take account of government policies, public demand, agrarian structure and so on. Whether our understanding will ever be sufficient to pull together all these complex and conflicting factors into a single, realistic model of arable farming systems is debatable, but one function of such work is that it illustrates the range of factors that are involved in determining productivity, and provides a framework for future research. (pp. 245-246)

There is a general point that I want to make based on these quotations and my, perhaps overlong, discussion of the boiling point of water. I grant that we may be able to isolate conceptually certain features or factors of complicated situations and to give very general accounts of what difference changes to the factors would make to outcomes in very idealized situations. However, and this is my main point, that we can develop these very general accounts does not mean that they will be very useful or accurate in explaining, predicting or controlling phenomena that will occur in non-idealized situations. Think of it in the following way. Each actual phenomenon that we want to explain, etc., involves a multidimensional problem. General idealized accounts tell us what happens in one dimensional situations. The move from general idealized accounts to solutions of multidimensional problems is made difficult because there is no clear function relating them.

Indeed, sometimes I wonder whether there could be such functions. Assume that there really are chaotic systems, e.g., the weather system. Although the principles underlying a chaotic system are simple and general, the occurrence of phenomena satisfying these principles are especially sensitive to variations in initial conditions. If there are a number of dimensions to a multidimensional problem that involve chaotic systems, then it is very difficult to see how we could formulate anything approaching a function from the general principles governing the various individual dimensions to an overall solution.

Let me sum up my argument so far. In the philosophy of science, the dominant model of explanation is the covering law model. This model says that we explain particular phenomena by showing how their occurrence follows from general laws and particular conditions, or are made highly likely by probabilistic laws and particular conditions. Explanation is a precursor to prediction and control. We use the understanding of phenomena, our explanations of them, to predict, to create or to control them. The problem with this model, I have argued, is that underlying it is the idea that we can usefully apply principles or laws that are essentially about idealized situations to situations that are not idealized. My conclusion is that, at best, this method gives us a very incomplete explanation of actual phenomena, or a complete explanation of phenomena that do not actually occur, ones that are idealized.

If general laws aren't useful for explanation or prediction or control, what good are they?

From what I have said so far, it might be concluded that I am in some sense anti-science, or at least anti-theoretical science. More importantly, someone might conclude that I would deny that coming to grips with very general laws has practical benefits. However, nothing could be further from the truth. I believe that comprehending the most general laws, principles or regularities greatly deepens our understanding of the world and has practical benefits. My point is more limited. These principles do not directly figure into explanations of actual phenomena that we encounter, nor do they allow us to control the world as we find it.

The most general laws or principles are true of, or applicable to, idealized situations. However, they give us information about the sorts of forces or capacities that occur in the actual world. They give us information about the deep patterns of the world. They tell us what sorts of factors our explanations of actual phenomena are going to have to take into account. In a sense, what they do is to organize and systematize empirical laws, laws or generalizations that take more of a particular context into account. Let me give an example.

Say that I take a sample of water from my tap each day at noon and heat it until it boils. After a time I tally the results and find that the water always boils at between 100.5C and 100.8C. Assume that this gives me some justification for believing that it will do so in the future. Say that we conduct similar experiments with samples from other sources and find that most have a a fairly narrow range of boiling temperatures. We then formulate empirical laws for them. E.g., any sample from this source will boil at .... We might even formulate a more general empirical law concerning the range for all the samples. Now we do an analysis of the water, find that it is mostly H$_{2}$O, but contains various minerals and gases. Samples from different sources have slightly different compositions. Coming to comprehend that H$_{2}$O has a boiling temperature of 100C helps us to understand what is going on here, but is not enough to explain it. It even gives us a strategy for getting the boiling temperature closer to 100C. We might think that the purer we make the water, the closer the boiling temperature will be to 100C. (Of course the strategy is not guaranteed to succeed. In purifying the water we may remove a substance that lowers the boiling temperature, while leaving one that raises it. When both substances were in the water, the one lowering the boiling temperature partially cancelled out the effect of the other. Since the one lowering the boiling temperature is removed, the water is purer, but now has a higher boiling temperature.)

Understanding and applying very general laws can give practical results in a variety of ways. One is in creating new objects, ones whose properties we thoroughly understand. I think that there is an important distinction between creating objects and controlling the world around us. Unfortunately, the former often passes for the latter. When we create objects, such as polymers, new chemical compounds, computer chips, and, perhaps, species, we are able to do so in a much more controlled environment than we face in trying to control some part of the world, for example, as in when we try to improve the yield on some bit of land. In creation, we, to put it crudely, put the properties in that we want, while in controlling we have to deal with what we are presented with. Since in the creation of new objects the environment is much more controlled, it more closely approaches the idealized situations to which very general laws apply. Creating new objects, however, is not the same as controlling the world, because the idealized laws that allow us to create the objects do not tell us how to control the interaction of those objects and a larger context. For example, in a controlled environment we can create or isolate chemicals that will be poisonous to certain pests. What we do not know as a result of creating it is what effects using it in an uncontrolled environment will have, nor do we know how to control those effects. We are creating a substance whose properties we know, but we are not controlling the world because we do not control what effects it will have in a larger context.

Theoretical science and applied science

Science, as I have been exposed to it, is largely theoretical. I was taught that generality in a theory is an important virtue. Indeed, it sometimes seems that the holy grail of science is a unified theory of everything. There is something to be said for this picture of science. Many of the objects we use depend for their existence on highly theoretical science, for example, the computer on which I wrote this paper, the printer on which I printed it, the airplane on which I flew here, etc. It might be that the abundance and relative low cost of the food in the countries in which I have lived is a result of technology made possible by very theoretical science, although something should also be said about the temperate climate and the good soil. Of course this is only part of the story. As well as allowing us to create objects and substances that add to our lives, theoretical science has given us the ability to destroy all life. It is a truism that knowledge, scientific knowledge, can be used for good or ill.

In addition to the practical results of very general theories, there are the benefits that we get from simply discovering more about the world. It is theoretical science that tells us about the fundamental forces, capacities, structure and properties of the universe in which we live.

However, where theoretical science, at least in my view, has difficulties is in showing us how to control the world and how to explain the occurrence of particular phenomena. This is because it does not say enough, and cannot say enough, about the intersections where various general theories meet. General theories surely inform us about the world, but do not give us the whole story. These intersections are, I believe, the domain of applied science.

If general theories cannot supply the answers in these intersections, how do we arrive at them? This is not perfectly clear to me, but I have a few thoughts about it. The first is that there is generality and there is generality. Perhaps this comment needs a bit of explaining. Given particular contexts, or local conditions, there might be generalities at a certain level that can be applied to other situations where the same local conditions exist, but which wouldn't hold where the local conditions do not exist. These generalities, which are associated with local conditions, might include close enough approximations to the actual influences to allow for explanation. How are these not very general generalizations arrived at? By looking for the results of the forces, capacities and properties that more abstract theories say are there, seeing how they work out in the local conditions and making inductive generalizations based on these observations. In other words, developing empirical generalizations informed by abstract theory.


In the philosophy of science, the covering law model is the dominant model of explanation in the sciences. There are two versions of the model. One proposes that for a particular phenomenon to be explained, its occurrence must follow from general laws and particular conditions. The other version says that probabilistic laws and particular conditions must make the occurrence of the phenomenon highly likely. I argued that the application of general laws (along with particular conditions) cannot explain particular phenomena. This is because general laws are idealizations, which cannot be applied correctly to non-idealized situations. I then suggested that the purpose of such general laws is to inform us about the capacities, properties and forces that existed and to give us some information about their characteristics. Finally I suggested that the explanation of particular phenomena was the domain of applied science, informed by general laws and principles.


Briggs and Courtney, 1985
Briggs, D. and Courtney, F. (1985).
Agriculture and Environment.
Longman, London.

Hempel, 1962
Hempel, C. (1962).
Explanation in science and history.
In Colodny, R. G., editor, Frontiers of Science and Philosophy, pages 7-33. Allen & Unwin, London.
Also in [Ruben, 1993].

Ruben, 1993
Ruben, D.-H., editor (1993).
Oxford University Press, Oxford.

Robert L. Frazier 2009-10-27