covering law model the view of scientific explanation as a deductive argument which contains non-vacuously at least one universal law among its premises. The names of this view include ‘Hempel’s model’, ‘Hempel-Oppenheim (HO) model’, ‘Popper-Hempel model’, ‘deductivenomological (D-N) model’, and the ‘subsumption theory’ of explanation. The term ‘covering law model of explanation’ was proposed by William Dray. The theory of scientific explanation was first developed by Aristotle. He suggested that science proceeds from mere knowing that to deeper knowing why by giving understanding of different things by the four types of causes. Answers to why-questions are given by scientific syllogisms, i.e., by deductive arguments with premises that are necessarily true and causes of their consequences. Typical examples are the ‘subsumptive’ arguments that can be expressed by the Barbara syllogism: All ravens are black. Jack is a raven. Therefore, Jack is black. Plants containing chlorophyll are green. Grass contains chlorophyll. Therefore, grass is green. In modern logical notation, An explanatory argument was later called in Greek synthesis, in Latin compositio or demonstratio propter quid. After the seventeenth century, the terms ‘explication’ and ‘explanation’ became commonly used.
The nineteenth-century empiricists accepted Hume’s criticism of Aristotelian essences and necessities: a law of nature is an extensional statement that expresses a uniformity, i.e., a constant conjunction between properties (‘All swans are white’) or types of events (‘Lightning is always followed by thunder’). Still, they accepted the subsumption theory of explanation: ‘An individual fact is said to be explained by pointing out its cause, that is, by stating the law or laws of causation, of which its production is an instance,’ and ‘a law or uniformity in nature is said to be explained when another law or laws are pointed out, of which that law itself is but a case, and from which it could be deduced’ (J. S. Mill). A general model of probabilistic explanation, with deductive explanation as a specific case, was given by Peirce in 1883.
A modern formulation of the subsumption theory was given by Hempel and Paul Oppenheim in 1948 by the following schema of D-N explanation: Explanandum E is here a sentence that describes a known particular event or fact (singular explanation) or uniformity (explanation of laws). Explanation is an argument that answers an explanation-seeking why-question ‘Why E?’ by showing that E is nomically expectable on the basis of general laws (r M 1) and antecedent conditions. The relation between the explanans and the explanandum is logical deduction. Explanation is distinguished from other kinds of scientific systematization (prediction, postdiction) that share its logical characteristics – a view often called the symmetry thesis regarding explanation and prediction – by the presupposition that the phenomenon E is already known. This also separates explanations from reason-seeking arguments that answer questions of the form ‘What reasons are there for believing that E?’ Hempel and Oppenheim required that the explanans have empirical content, i.e., be testable by experiment or observation, and it must be true. If the strong condition of truth is dropped, we speak of potential explanation.
Dispositional explanations, for non-probabilistic dispositions, can be formulated in the D-N model. For example, let Hx % ‘x is hit by hammer’, Bx % ‘x breaks’, and Dx % ‘x is fragile’. Then the explanation why a piece of glass was broken may refer to its fragility and its being hit: It is easy to find examples of HO explanations that are not satisfactory: self-explanations (‘Grass is green, because grass is green’), explanations with too weak premises (‘John died, because he had a heart attack or his plane crashed’), and explanations with irrelevant information (‘This stuff dissolves in water, because it is sugar produced in Finland’). Attempts at finding necessary and sufficient conditions in syntactic and semantic terms for acceptable explanations have not led to any agreement. The HO model also needs the additional Aristotelian condition that causal explanation is directed from causes to effects. This is shown by Sylvain Bromberger’s flagpole example: the length of a flagpole explains the length of its shadow, but not vice versa. Michael Scriven has argued against Hempel that explanations of particular events should be given by singular causal statements ‘E because C’. However, a regularity theory (Humean or stronger than Humean) of causality implies that the truth of such a singular causal statement presupposes a universal law of the form ‘Events of type C are universally followed by events of type E’. The HO version of the covering law model can be generalized in several directions. The explanans may contain probabilistic or statistical laws. The explanans-explanandum relation may be inductive (in this case the explanation itself is inductive). This gives us four types of explanations: deductive-universal (i.e., D-N), deductiveprobabilistic, inductive-universal, and inductiveprobabilistic (I-P). Hempel’s 1962 model for I-P explanation contains a probabilistic covering law P(G/F) % r, where r is the statistical probability of G given F, and r in brackets is the inductive probability of the explanandum given the explanans: The explanation-seeking question may be weakened from ‘Why necessarily E?’ to ‘How possibly E?’. In a corrective explanation, the explanatory answer points out that the explanandum sentence E is not strictly true. This is the case in approximate explanation (e.g., Newton’s theory entails a corrected form of Galileo’s and Kepler’s laws). See also CAUSATION, EXPLANATION, GRUE PARADOX , PHILOSOPHY OF SCIENC. I.N.