The Deductive-Nomological model (D-N model for short), also called the Covering Law model, was a product of logical positivism. This fact points to one reason, as noted in the previous chapter, that logical positivists are not instrumentalists. Both positivists and instrumentalists recognize the predictive function of theories. Positivists recognize also an explanatory function, largely in terms of the D-N model, while instrumentalists do not recognize the explanatory function of scientific theories. This model is called “deductive” because explanation in this model is conceived as a form of deductive argumentation. It is called “nomological” because of the central position that laws play in the model.¹ Before we begin in detail on the model, then, let us take a look at the narrower concept of scientific law.

LAWS

The Meaning of “Scientific Law”

The term law brings a sense of universality, indubitability, of the absolute. To call a claim a scientific law seems to settle the question. There is nothing further to ask. Yet, such rhetorical power holds little weight among most philosophers. A deeper look at “law” in a scientific context is in order. Let us take as paradigmatic (not in the Kuhnian sense) Isaac Newton’s first law of motion, the law of inertia: an object in motion remains in motion until acted on by an external force; an object at rest remains at rest until acted on by an external force. This law is well known among scientists, nonscientists, philosophers, and nonphilosophers. Also, among nonphilosophers and nonscientists, it seems to be conceived in the absolutist sense alluded to earlier—it is a principle of reality to be found everywhere without exception—but also, at least anymore, without critical reflection.

The statement of a law is most clearly expressed in the form of what logicians call a universal affirmative statement (all A’s are B’s) or a conditional statement (if … then). For example, the law of inertia can be expressed thus: “All objects at rest/in motion are objects that will remain at rest/in motion until acted upon by an external force”; or, “If an object is at rest/in motion, then it will remain at rest/in motion until acted upon by an external force.” In either form it predicates a common quality on some subject in question. Here, that subject would be material objects. The universal affirmative form indicates clearly the generalized aspect of a law: that the predicate applies to all material objects, which points to generalization as a necessary condition for a law. Though a necessary condition, generalization is not a sufficient condition for a law, in fact, there are many generalizations that are clearly not laws. Hempel (1966) presents this instructive example: “All solid spherical masses of pure gold weigh less than 100,000 kilograms.” This is a true generalization as far as we can tell. That is, anywhere you look you are not going to find a mass of gold weighing more than 100,000 kilograms. Although true and a generalization, this is not a law. It is what is called an accidental generalization. Here, “accidental” is used in the philosophical sense to mean not an unintended error or mistake but merely a contingent truth. A contingent truth is true due to specific circumstances, which may have been different themselves, thereby not leading to the situation held to be contingently true. In this case, the fact that it is true that there are no 100,000+-kilogram boulders of gold could be true—given different contingencies. It may even be possible to construct such a boulder if we could collect enough gold. Compare this accidental generalization to this second example: “All solid spherical masses of pure plutonium weigh less than 100,000 kilograms.” Though similar in form, this is a different type of statement. It is different because of why it is true versus why the “gold” statement is true. As noted, the “gold” statement is an accidental generalization, which is only contingently true because of the way in which gold has been distributed around the universe. The “plutonium” statement is true because any mass of plutonium (due to the nature of plutonium) would become unstable and explode before reaching 100,000 kilograms. So there could not be 100,000+ kilograms of plutonium. This indicates that laws are something more than contingent truths. The same seems to be so with our inertia example. It seems like more than an accident or contingency to say that all material objects at rest remain so. To call it a “law” suggests something deeper.

Here is where the conditional statement of law may be helpful. Let us restate the “plutonium” claim as a conditional statement: “If x is a mass of plutonium, then x will weigh less than 100,000 kilograms.” The conditional expression of a law brings out the logical connection involved. For the sake of analysis, let us introduce some terminology. Any conditional statement can be broken down into two simpler statements. The first is the statement that is placed between the “if” and the “then.” In this case, that would be “x is a mass of plutonium.” That statement in the context of the conditional statement is called the “antecedent.” The second is the statement following the “then.” In this case that would be “x will weigh less than 100,000 kilograms.” In the context of the conditional statement, that is called the “consequent.” To assert the truth of a conditional statement is not to assert the truth of either the antecedent or the consequent. Rather, it is to assert a logical relationship between the two. Take for example the conditional statement, “If I am in Philadelphia, then I am in Pennsylvania.” This is true (assuming that we are referring to the Philadelphia that indeed is in Pennsylvania; not to allow that assumption would be sophistry), regardless of whether you happen to be in Philadelphia or not. This statement is true if you are in Pittsburgh, Paris, or Philadelphia. Consider in addition the statement, “If it’s night time, then the sun is down.” Again, this statement is true, whether it happens to be day or night. For what a conditional statement asserts is a logical connection between the antecedent and the consequent. The connection asserted is that any time the antecedent is true the consequent will be true also. Conditional statements are also called hypothetical statements: given the hypothesis that it is night time, then the sun must be down. Given the hypothesis that I am in Philadelphia, then I must be in Pennsylvania. Therefore, regarding the “plutonium” claim, what is asserted is a logical connection between the statement that some given thing x is a mass of plutonium and the statement that x is no more than 100,000 kilograms. Any time it is true that some x is a mass of plutonium, it will also be true that said mass will weigh no more than 100,000 kilograms. That is a true assertion, given what was noted earlier about the instability of plutonium. Compare this amended claim though: “If the moon had been composed of pure plutonium, it would weigh less than 100,000 kilos.” This too is a true conditional statement. Even though the moon that we know (the large orb in the sky that is 3,474 km in diameter) would certainly weigh much more were it composed of pure plutonium, it could not have reached that size and mass if it were composed of pure plutonium. Thus, it would have to be under the given mass limit. But if we make the same type of claim about gold, we get a different result: “If the moon had been composed of pure gold, it would weigh less than 100,000 kilos.” This statement is false. There is nothing about gold to prevent the moon (hypothetically) to be composed purely of it and retain the mass we identify with the moon in actuality. Thus, the assertion of the logical connection between antecedent and consequent for this statement is false. There is then some necessity in the logical connection between the antecedent and the consequent in the conditional expression of a scientific law.

And here is where things become difficult. What is the nature of this necessity? When discussing necessary truths in contrast to contingent truths, the most common meaning is logically necessary truth (or statement). But as we have learned earlier, logically necessary truths (analytic truths) tell us nothing new about the world. They merely assert relations between concepts. And certainly we expect scientific laws to tell us something about the world. Second, logically necessary truths are typically obvious and knowable through rational contemplation, whereas we typically recognize scientific laws as in need of empirical study and support to be asserted. Third, a sure test of any proposed logical truth is that its denial should be a contradiction. For example, refuting the logically necessary truth “All bachelors are unmarried” would result in a contradiction because denying that each and every bachelor is unmarried is inconsistent with the definition of “bachelor.” Or even more simply, negating “The red truck is red.” would also result in a contradiction. Denying a scientific or natural law does not result in a logical contradiction. Denying that a pure plutonium moon would have a mass less than 100,000 kilograms is not in itself a contradiction. Denying that a particle of matter will exhibit the properties of inertia is also not a logical contradiction. It is perfectly conceivable that some piece of matter would not exhibit the properties of inertia or that a hunk of plutonium might have a mass greater than 100,000 kilograms whereas it is not conceivable that a bachelor could be married and still be a bachelor or that a red truck could not be red.

These questions of necessity fall within the branch of logic known as modal logic. Modal logicians refer to different forms of necessity. We might say that a natural law may not be logically necessary but may instead be physically necessary. To be logically necessary is to be necessitated by the laws and structure of logic. To be physically necessary is to be necessitated by the laws and structure of the physical world. The perspicuous reader should see the problem with this answer. Such physical necessity is precisely what we are trying to establish with scientific laws. Thus, to base the necessity of natural laws on the concept of physical necessity is to beg the question: What seems to be part of the concept of a scientific law is that it is grounded in reality in some metaphysical sense. In descrying laws of nature, scientists are plumbing the deep structure of the universe. Once these studies turn metaphysical, empirical science—its methods, its justifications, its authority—begins to fall away.

Problems and Limitations of Scientific Laws

Let us return to the empirical method. Scientific laws are established largely through empirical observation and inductive inference—they usually make reference to other laws also. Our studies have already established the limitations inherent in these. The only way to tell that matter has the property of inertia is to test it. The only way to know with absolute certainty that all matter does is to test every piece of matter. Not only that, we would have to test every piece of matter for all time, because it is at least conceivable that some bit of matter might gain and lose the property of inertia. We are epistemically left then with the limitations of induction, as well as the limitations of perception that we studied in Chapter 9. “Law” in that deep sense of scientific law that is often suggested in the use of that word can never be established. This leaves many scientists and philosophers of science more modestly conceiving of scientific laws as mere observable patterns. Recognizing them as established patterns allows us to use them as elements of theories without making unfounded metaphysical presumptions. But that still seems to strip away what seems to be a cherished sense of what scientific law means.

Another problem with scientific laws is that even if we could establish them with certainty, they are often in an important sense not really true. Yet, they are still accepted as true. This is because scientific laws are often conceived and structured in terms of ideal conditions. Take our law of inertia exemplar again. Has anyone ever observed an object truly (cosmically?) at rest? Certainly, we can observe objects at rest relative to the Earth. That is, practically, how Newton established and tested his laws. Yet, conceptually, they refer as well to rest relative to the universe. It is not even clear whether there is such a thing. They are also conceived as in a friction-free environment. Take the third law of motion (for every action there is an equal and opposite reaction) for a clearer example here. Because friction operates on matter (friction of surfaces or friction of air) most experiments testing this law will be slightly imperfect. The reaction will usually not be precisely equal and opposite. The more general point here is that laws (again conceived ideally) leave out many small, difficult to control for variables. Largely, these variables are of little significance. As a practical matter the laws turn out to be “close enough.” They work. They allow us to predict and explain phenomena. Yet, in a technical sense, they are not “true.” They do not reflect the world we have, the phenomena we observe, but only approximations of those phenomena. The problem is that to precisely describe such phenomena (taking into account all variables big and small) would be impractically complex and not generalizable. And it is the generalizing feature of laws that makes them useful and helps theoretically bring together disparate phenomena into a comprehensible statement about the world we know.

One way to conceive of these “approximate” laws of nature is as elliptical generalizations in which variables are elided; however, if included, they would accurately reflect the real structure of the world. These are sometimes called ceteris paribus generalizations. Ceteris paribus means “all things being equal” in Latin. All things being equal, an object at rest will remain at rest until acted on by an outside force. All things being equal, there will be an equal and opposite reaction for every action. The point here is to attempt to retain the force of the law (as an absolute, true statement), while recognizing that it is not really true. Yet, some critics still find this too speculative, too metaphysical (Cartwright, 1980/2000). But here we get to the underlying problem of this analysis for the last few paragraphs. Remember the focus here is supposed to be on explanation. However, we have been conflating this discussion of explanation and a discussion of description. In order to accept laws as successful in constructing explanations, we need not accept them as true descriptions about reality (Cartwright, 1980/2000). The two goals are distinct and should not be confused. This way of thinking leads toward a more pragmatic conception of laws and explanation.

BACK TO THE D-N MODEL

First some basics: It is customary in philosophical discussions of explanation to employ a little Latin because English does not provide the same simple and accurate terminology. We use the term explanandum to refer to the phenomenon to be explained. “Explanandum statement” is the sentence that refers to that phenomenon. The term explanans statements refers to the specific explanatory sentences, those sentences that do the explaining. And the term explanans refers to the collection of explanans statements within a given explanation.

The D-N model, as described by Hempel (1966), includes two basic requirements for a good explanation. First, “the explanatory information adduced affords good grounds for believing that the phenomenon to be explained did, or does, indeed occur” (Hempel, 1966, p. 48). This requirement is referred to as explanatory relevance and refers both to some basic intuitions about what an explanation is and the deductive form by which the D-N model is understood. The second requirement is that the explanans must be empirically testable, which of course conforms to the empirical standard held by logical empiricists like Hempel as well as most scientists and philosophers of science.

According to the D-N model, the explanans of an explanation composes a set of premises that deductively imply the explanandum statement, which of course refers to the explanandum itself. Let us attempt an explanation of the operation of a mercury thermometer.

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