Philosophers of science sometimes have to do the hard work of explaining how scientists explain things. The 20th century German-American philosophers of science Carl G. Hempel and Paul Oppenheim applied themselves, in 1948, to such an effort in their jointly written paper “Studies in the Logic of Explanation.” In this detailed and logically labyrinthine text, Hempel and Oppenheim collaborated, as they did often, to strenuously define what is commonly called the “deductive-nomological model” of scientific explanation. It is not clear when the actual moniker “deductive-nomological” (hereafter the D-N model) came about, since a close examination of the 1948 paper seems to make no mention of the model with these precise words. Nevertheless, a quick glance through an extensive bibliography of Hempel’s published work shows that Hempel was making professional use of the term “deductive-nomological” in the early 1960s (Hempel 399). Hempel also calls the D-N model the “covering-law model” in various essays on the topic (69). In this paper we will set out to examine and define the model as it relates to scientific explanation and prediction. Then we’ll turn to another philosopher of science who was also greatly interested in the topic of scientific explanation Wesley C. Salmon. Salmon’s 1985 paper “Scientific Explanation: How We Got from There to Here” was instrumental in pointing out several problems with the “D-N model”. Salmon’s criticism dislodged Hempel and Oppenheim’s model enough to cause its descent from its once authoritative place as the generally accepted method by which to scientifically explain things. In all fairness, Hempel did anticipate problems with the model and in subsequent years (after 1948 and especially in the 1960s) and he tried to ameliorate some of them. As mentioned, we’ll be briefly looking at Salmon’s critique and to bits of Hempel’s own reevaluation to see what can be concluded from there and if we should reject the D-N model outright.
First, let’s endeavor to define the D-N model using Hempel and Oppenheim’s terms. In a straightforward way, Hempel and Oppenheim open their paper “Studies in the Logic of Explanation” with a general statement about scientific explanation itself “[t]o explain the phenomena in the world of our experience, to answer the question ‘why?’ rather than only the question of ‘what?’ is one of the foremost objectives of empirical science” (206). Right away this sets the tone for the extent their philosophical project. In other words, they will want to show how the ‘why?’ questions of scientific inquiry can be empirically and logically legitimized in a basic formulaic way. Hempel and Oppenheim going about this by looking at bare-bones pattern of scientific explanation starting with the Latin terms “explanandum” and the “explanans” (Hempel, Oppenheim 207). To clarify this pair of terms will mean that the explanans is more or less the set/s of premises that ‘explain’ the deduction of the ‘explanation’ which they accordingly dub the explanandum. This is just another way of saying that the explanans explain the explanandum. A footnote next to the two terms indicates that these words are derived from the Latin “explanare” (Hempel, Oppenheim 222). To be sure, these are the logical components of a scientific explanation.
The explanans (the explanation) has two important subcategories “[…] one of these contain certain sentences C1, C2, …, Ck which state certain antecedent conditions; the other is a set of sentences L1, L2, …, Lr which represent general laws” (Hempel, Oppenheim 207). Put another way, we have to have a finite number of laws (natural or otherwise) and a finite number of conditions that are to be used to legitimately give a scientific explanation for a given phenomena. And so, the explanandum (the explained) must “be a logical consequence of the explanans” (208). The following logical schema illustrates the essential logical structure as Hempel and Oppenheim formally presented it in their paper.
Logical schema of the deductive-nomological model (Hempel, Oppenheim 209).
If it’s not already apparent, the deductive part of the explanation requires that the explanans as a whole set must be true and must be able to be verified with empirical “experiment or observation” (Hempel, Oppenheim 208). As for the nomological (re: lawful, having to do with laws, etc.) part of the D-N model, we’ll have to include the minimum of one general law in our set of explanans. Hempel and Oppenheim give a short example of how this explaining works, for instance, why a rowboat’s (normally straight) oar will appear to be bent in water. The explanation for the bent oar follows from two general laws (L1, L2, …, Lr), “—mainly the law of refraction and the law that water is an optically denser medium than air…” (207). The other ‘antecedent conditions’ (C1, C2, …, Ck) are obvious practical things, such as the oar being put in the water, the oar is normally straight and not actually distorted to begin with, the water is clear enough to see the illusion of the distorted oar, etc. Importantly, scientific predictions can also be contained within the D-N model with an easy reversal of the explanans and explanandum, whereby the explanans is given prior to the explanandum—i.e. an explanation is offered to predict upcoming phenomena. This order can get easily confused when we are trying to situate explanation with respect to prediction, but it should remembered that the explanation comes after the phenomena to be explained.
Wesley Salmon in his (above mentioned) paper writes that Hempel and Oppenheim were already anticipating potential problems with the D-N model, such that not all scientific explanations will fall under the D-N model, “…some are probabilistic or statistical” (243). This means that the general laws (L1, L2, …, Lr) used in the D-N model will then be replaced by a finite set of statistical laws (minimum of one), thereby changing the model to an “inductive-statistical (I-S)” model (Salmon 242). Salmon writes that Hempel readily recognized that statistical probability was by definition somewhat ambiguous and therefore problematic “[t]he source of the problem of ambiguity is a simple and fundamental difference between universal laws and statistical laws” (244). As we already know, there are often (nay always) exceptions to the rules, probabilistic arguments usually account for what will ‘probably’ happen with a certain percentage of not happening. Salmon cites the famous black swan example that refuted the previously accepted inductive argument that all swans were white (244). There is always going to be a fractional percentage of a statistical explanation that is not successful because statistical law simply cannot account for an absolute (100%) deductive assurance, therefore it had to be dubbed inductive. Hempel tried to resolve this problem with his “requirement of maximal specificity (RMS)” (Salmon 245). This required that all appropriately relevant details be included in the explanation. But, determining what is appropriate can be a dilemma, and this is why Hempel came up with (and later rejected) his so-called “principle of essential epistemic relativity of I-S explanation” (246). In other words, the maximum amount of specificity works fine with the D-N model, but doesn’t work well with the I-S model because the knowledge base (episteme) needed is still only inductive, so we are restricted by the relative knowledge base of any given explanation which can easily slide into over-generalized explanations.
Salmon plainly states that “[t]he hegemony of logical empiricism regarding scientific explanation did not endure for long” (249). Among the already stated problems with the D-N model anticipated by Hempel and Oppenheim, there were other problems that caused it to lose traction as a way to explain things scientifically. In his paper, Salmon concludes that he wishes to bring causality back into scientific explanation. As he admits, causes will serve to explain things. A cause can explain certain effects. Another consideration Salmon addresses is time or better said “temporal asymmetries” (257). This is similar to his causal condition, where we can explain things by earlier events but not the other way around. Sylvan Bromberger’s flagpole example is given by Salmon to illustrate problems with Hempel and Oppenheim’s D-N model. On a sunny day at a certain time a flagpole, given any number of conditions, casts a shadow of a certain length that can be deduced from the conditions, re: the height of the flagpole, the time of day, etc. We can then use the similar conditions to determine the height of the flagpole “…[y]et hardly anyone would allow that the length of the shadow explains the height of the flagpole” (Salmon 249). The point here has to do with the way Hempel and Oppenheim’s D-N model seems to be devoid of causal ways of explaining things, because even if we can try to show that the shadow explains the height of the flagpole it just doesn’t causally work out that way since, as Salmon explains, causality is asymmetrical, the effects don’t always explain the causes.
In closing, it would make sense to avoid entirely rejecting Hempel and Oppenheim’s D-N model. And the reason is simple. Given the above arguments that show distinct problems with the model, we can still accept it, not as the sine qua non of scientific explanation, but as one, out of a choice of many, by which to explain something. We often fall into the binary illusion that we need to choose things on a ‘one or the other’ basis without a full consideration of the options. After all, there is more than one way to explain things.
Hempel, Carl G. The Philosophy of Carl G. Hempel. Ed. James H. Fetzer. New York, NY: Oxford University Press, 2001. Print.
Hempel, Carl G., Paul Oppenheim. “Studies in the Logic of Explanation.” Introductory Readings in the Philosophy of Science. Eds. E.D. Klemke, Robert Hollinger, David Wÿss Rudge and A. David Kline. Amherst, NY: Prometheus Books, 1998. pp. 206-224. Print.
Salmon, Wesley. “Scientific Explanation: How We Got from There to Here.” Eds. E.D. Klemke, Robert Hollinger, David Wÿss Rudge and A. David Kline. 241-263.
 Incidentally, as far as we can see, the I-S model is not mentioned by Hempel till the 1960s, and is not covered in the 1948 paper.