Hempel and Oppenheim’s essay “Studies in the Logic of Explanation,” published in volume 15 of the journal Philosophy of Science, gave an account of the deductive-nomological explanation. A scientific explanation of a fact is a deduction of a statement (called the explanandum) that describes the fact we want to explain; the premises (called the explanans) are scientific laws and suitable initial conditions. For an explanation to be acceptable, the explanans must be true.
According to the deductive-nomological model, the explanation of a fact is thus reduced to a logical relationship between statements: the explanandum is a consequence of the explanans. This is a common method in the philosophy of logical positivism. Pragmatic aspects of explanation are not taken into consideration. Another feature is that an explanation requires scientific laws; facts are explained when they are subsumed under laws. So the question arises about the nature of a scientific law. According to Hempel and Oppenheim, a fundamental theory is defined as a true statement whose quantifiers are not removable (that is, a fundamental theory is not equivalent to a statement without quantifiers), and which do not contain individual constants. Every generalized statement which is a logical consequence of a fundamental theory is a derived theory. The underlying idea for this definition is that a scientific theory deals with general properties expressed by universal statements. References to specific space-time regions or to individual things are not allowed. For example, Newton’s laws are true for all bodies in every time and in every space. But there are laws (e.g., the original Kepler laws) that are valid under limited conditions and refer to specific objects, like the Sun and its planets. Therefore, there is a distinction between a fundamental theory, which is universal without restrictions, and a derived theory that can contain a reference to individual objects. Note that it is required that theories are true; implicitly, this means that scientific laws are not tools to make predictions, but they are genuine statements that describe the world—a realistic point of view.
There is another intriguing characteristic of the Hempel-Oppenheim model, which is that explanation and prediction have exactly the same logical structure: an explanation can be used to forecast and a forecast is a valid explanation. Finally, the deductive-nomological model accounts also for the explanation of laws; in that case, the explanandum is a scientific law and can be proved with the help of other scientific laws.
Aspects of Scientific Explanation, published in 1965, faces the problem of inductive explanation, in which the explanans include statistical laws. According to Hempel, in such kind of explanation the explanansgive only a high degree of probability to the explanandum, which is not a logical consequence of the premises. The following is a very simple example.
The relative frequency of P with respect to Q is r
The object a belongs to P
Thus, a belongs to Q
The conclusion “a belongs to Q” is not certain, for it is not a logical consequence of the two premises. According to Hempel, this explanation gives a degree of probability r to the conclusion. Note that the inductive explanation requires a covering law: the fact is explained by means of scientific laws. But now the laws are not deterministic; statistical laws are admissible. However, in many respects the inductive explanation is similar to the deductive explanation.
– Both deductive and inductive explanation are nomological ones (that is, they require universal laws).
– The relevant fact is the logical relation between explanans and explanandum: in deductive explanation, the latter is a logical consequence of the former, whereas in inductive explanation, the relationship is an inductive one. But in either model, only logical aspects are relevant; pragmatic features are not taken in account.
– The symmetry between explanation and prediction is preserved.
– The explanans must be true.