Hempel's "Covering-Law" model
Let's say that an explanation is a relationship between some explanatory information (the explanans) and a description of an event to be explained (the explanandum). The covering-law model, associated with Carl Hempel, includes two sorts of explanation, Deductive-Nomological or D-N explanation, and Inductive-Statistical or I-S explanation.
A number of conditions are said to be required for an adequate explanation (see C&C p. 770 for the first four, p. 711ff for the fifth -- note especially the informal explanation at the bottom of p. 714):
The explanation must be an argument with the explanans as premises and the explanandum as conclusion. The premises of the argument must make the conclusion highly probable (certain, in the case of D-N explanation). (e.g. C&C p. 685)
The explanans must contain at least one general law (a universal law for D-N explanation; a statistical law for I-S explanation).
The explanans must have empirical content. [Why include this? Hempel himself noted that this actually follows from the first condition, since you can't derive an empirical conclusion from premises that have no empirical content. He includes it simply to make this requirement specific. What he wants to rule out is explanations, such as the explanation of properties of living organisms by means of an "entelechy" or "vital force," which he regards as completely untestable and hence without empirical content. See Hempel, "Studies in the Logic of Explanation," in Hempel, Aspects of Scientific Explanation (New York: Macmillan, 1965), p. 248: "the point deserves special mention because . . . certain arguments which have been offered as explanations in the natural and in the social sciences violate this requirement." His specific examples include the "entelechy" example just mentioned (p. 257; cf. Carnap in C&C, pp. 679-682), and certain kinds of teleological explanations (p. 256).]
The explanans must be true.
(I-S explanation only) The explanans must have maximal specificity. [What does this mean, and what is it designed to rule out? The problem involves explanations which lose their force when additional facts are made known. For example, I might attempt to explain why someone survived a bout with pneumonia by appealing to the particular facts C1 that the person had pneumonia (you can't recover from a disease you haven't had!) and C2 that the person took penicillin, together with the general statistical law that 90% of people with pneumonia who take penicillin recover. (Confession: I just made up this so-called statistical law; I have no idea how close it is to being accurate.) However, if we now learn that our subject was 98 years old and had a serious heart condition, we might doubt that our explanation suffices after all, for surely the percentage of 98-year-olds with heart conditions who recover from pneumonia after being administered penicillin will be much lower. So the requirement is that the particular conditions include all the relevant facts about the individual.]
Observations about the Covering Law Model
Hempel makes a number of interesting observations about his model, including these:
On his model, explanation and prediction are "structurally identical" (C&C pp. 695ff). That is, the logical structure of an explanation and of a prediction are the same; the difference concerns only what we know at a particular point in time. If we already know that the explanandum has occurred, then an argument conforming to the model is an explanation; if we don't yet know whether the explanandum has occurred (or will occur), then an argument conforming to the model is a prediction.
The model is a kind of ideal case, and particular arguments offered as explanations may fail to live up to it in different ways, including (a) elliptical explanations, in which some initial conditions or (more likely) laws are merely implicit; and (b) partial explanations, in which we have an explanation of an event falling under a broader description than that in the explanandum, but not an explanation of why it falls under the more specific description offered by the explanandum.
Problems for the Covering-Law Model
1. Are the conditions jointly sufficient? (That is, must anything that satisfies all the conditions count as a genuine explanation?) (If not, then we should try to think what additional conditions are required.)
Some counterexamples seem to show that an "explanation" could satisfy all of the criteria listed even though the explanatory information was completely irrelevant to the explanandum.
one famous example of this kind, which we might call a case of complete irrelevance: why didn't Mr. X become pregnant? We offer the following explanation: Mr. X took the pill regularly; anyone who takes the pill regularly will not become pregnant; therefore Mr. X did not become pregnant. The explanans contains a general law, it has empirical content, the explanation is a deductively valid argument, the explanans is true -- all the conditions seem to be met, and yet the proposed explanation is clearly bogus. [example due to Wesley Salmon; included in Ruben, p. 725] This sort of case in some ways is the flip side of the examples that led Hempel to the requirement of maximal specificity. Those examples involved leaving out information that was highly relevant. These examples involve, instead, including information that is not relevant.
Here is a related but slightly different case (also discussed by Ruben, pp. 721-722). This one is a case of causal preemption (that is, an event occurs which would cause the explanandum if given a chance, but something else causes the explanandum first, "preempting" the would-be cause). Why did Jones die when he did? He ate a pound of arsenic five minutes earlier, and anyone who eats a pound of arsenic dies within 24 hours. However, what the explanation does not mention is that Jones was run over by a bus immediately before his death. The arsenic would have caused his death if the bus hadn't, but in fact the arsenic never had a chance; the bus got him first.
Other counterexamples apparently show that an "explanation" could satisfy all the criteria, and be relevant, and yet fail to constitute a genuine explanation. This is because of what is sometimes called the symmetry of D-N explanation.
the flagpole (cf. Salmon et al., Introduction to the Philosophy of Science): on a flat, level piece of ground we have a 12' tall flagpole. The sun is at an elevation of 53.13 degrees. We can deduce that the length of the flagpole's shadow is 9'. But also we can deduce the height of the flagpole from the length of the shadow and the elevation of the sun, or the elevation of the sun from the height of the flagpole and the length of its shadow! Only the first of these deductions looks explanatory. (Problem: we can use effects to "predict" causes, but not to explain them.)
the barometer. We can predict the coming of a storm by a drop in the barometer, but this does not seem explanatory. (Drop in barometer and coming of storm both explained by a common cause, the drop in atmospheric pressure.)
From the positions of the earth, moon, and sun, together with the laws of celestial mechanics, we can deduce when the next total eclipse of the sun will occur. This deduction seems like a reasonable explanation of the eclipse (after it occurs). However, we can also use those present positions to retrodict an earlier eclipse. This doesn't seem explanatory.
2. Are the conditions necessary? (That is, must a genuine explanation meet all the listed conditions?) If not, we should abandon the ones that aren't necessary.
Must a successful explanation meet the first condition? That is, must the explanans make the explanandum likely?
paresis. Famous example due to Michael Scriven. (See C&C p. 775) Apparently (at least at the time Scriven was writing) nothing was known about the cause of paresis except that only people who have had syphilis get it. However, only a small percentage of people with syphilis contract paresis. Why did S get paresis? Scriven suggests that it is an explanation to say that S had syphilis. (In Hempel's view, this is only a partial explanation.)
throwing a die. Why did the die come up a 4? Assuming that the
outcome of a roll of the die is genuinely random, it seems that all that
we can say is that the die will come up a 4, on average, 1/6 of the
time, and this was just one of those times. This seems the only
explanation available, but it does not make the explanandum likely.
Must a successful explanation meet the second condition? That is, must the explanans contain general laws?
(Notice that these two questions are very closely related to Hempel's thesis of structural identity between explanation and prediction. If a prediction could fail to be an explanation, then, since Hempel's conditions do guarantee that we could predict the explanandum, the conditions are not sufficient. If an explanation could fail to be a potential prediction, then the conditions are not necessary.)Causal Relevance Model
One suggestion about what a more adequate model might look like is that to explain an event is to provide causal information relevant to the event's occurrence. Such views have been offered by Wesley Salmon, David Lewis, and others.
One version of this account is offered by Ruben (in C&C pp. 730-745). Ruben notes that the problems of irrelevant "explanations" and of symmetries leading to "explanations" that seem backward seem to have to do with causation: the reason irrelevant factors are irrelevant is that they have no causal effect on the explanandum, and the reason symmetry leads to explanations that seem backward is that we are explaining causes in terms of their effects.
One proposed solution is to add some sort of causal condition to Hempel's other conditions -- for example, the requirement that the explanans include causal information. However, the obvious ways to add such a condition still leave it possible that the added causal information is irrelevant! Ruben suggests that we should hold that an explanation gives the cause of the explanandum -- not just an argument from which we can deduce the explanandum (or make it probable) and that includes causal information of some sort, but events that actually caused the explanandum.
If we agree, how should we add this requirement? We could add as a condition of adequacy that C1, . . ., Cn must include (or be) the causes of the explanandum. But Ruben's suggestion is more radical. Once we add something like that condition, he suggests, the need for the other conditions becomes questionable.
Ruben suggests, in fact, that to explain an event e we need only cite its cause c. "c caused e" is an explanation of e all by itself, without the need for (a) an argument of any sort, or (b) the inclusion of general laws of any sort. (In many cases there will be general laws that are relevant, and citing these may be interesting and relevant, but it is not necessary in order to have an explanation.)Pragmatics of Explanation
Causal relevance cannot be the whole story, however. How much causal information is necessary? It is too much to require that an explanation include all the causal factors that led to the explanandum; in that case no actual explanations would satisfy the criteria. On the other hand, it is not enough to require that some causal information be offered. Van Fraassen (in The Scientific Image) suggests that the correct account will appeal to pragmatic factors involving the situation in which an explanation is requested. He mentions two such factors in particular.
First, we want information which is salient given our interests. Our interests determine which factors we hold constant and which we envision varying. The civil engineer may ask, "Why did the accident occur, given that the braking distance was such-and-such?" and conclude that the explanation is the shrubbery too close to the intersection. The auto manufacturer, on the other hand, may ask, "Why did the accident occur, given that the shrubbery was there?" and decide that the answer is that the braking distance was too short.
The second factor is the contrast class: we want to know why the explanandum occurred rather than one of the other possibilities we are interested in. A thief may want to know why Willie Sutton robbed banks rather than convenience stores or Laundromats; in that case his answer, "That's where the money is," provides a good explanation. On the other hand, a police officer may want to know why he robbed banks rather than getting a job; this requires a different explanation. The contrast class may explain why the same question may seem answerable in some contexts but not in others. Why did Frank get paresis? If this means why did Frank, in contrast to other members of the population at large, get paresis, then a good answer is "because he had syphilis." But if the question means "Why did Frank, in contrast to other people with syphilis, get paresis" then there is no known answer.
(More on van Fraassen on the pragmatics of explanation here.)