One of the distinctive features of the Abrahamic faiths is the belief in a God who acts in human history. In both the Bible and the Koran, God is described as communicating with and through prophets and as performing signs and wonders. Christians believe, additionally, that God became incarnate, lived a human life, died, and was raised from the dead. In recent centuries, however, liberal and modernist movements, principally in Christian theology, have sought to demythologize their religion by rejecting the supernatural elements in the Bible as incompatible with modern science, and reconstructing a theology around the moral teachings of Jesus and the Hebrew scriptures. The most radical of such reconstructions rejects a supernatural God entirely. Less radical versions allow for a deistic God who created the world and established its initial conditions and natural laws, but does not intervene in creation.

What about modern science would lead theologians to undertake such a wholesale rethinking of their religion's assumptions? The line of reasoning one finds most frequently goes something like this. Modern science has revealed that the universe is governed by natural laws. Therefore, if we accept the laws proclaimed by the scientists, we must reject any kind of divine action in history that would violate those laws. Miracles, according to David Hume's influential definition, are events that violate natural laws (“On Miracles” in Hume [] 1902). And so, if we are committed to natural laws, we must reject miracles.

Historical Preliminaries and the Problem for Miracles

The foregoing is, of course, a very informal development of the argument. To assess its validity and soundness, we will need to develop it more precisely. But before we begin that task, it is worth noting that neither the father of the modern notion of natural law (Descartes) nor the man who developed the first successful formulations of laws unifying the explanation of motions near the Earth and the motions of the planets (Newton) would have found the foregoing line of reasoning congenial. Both believed in miracles. For Newton, miracles and providence were almost an obsession (Dobbs and Jacob ). Descartes held to so strong a view of God's free will that he believed that God could arbitrarily change not only natural laws, but even the canons of logic (Descartes , 291; , 22–55). Somewhere between Newton and Hume, something important must have changed in the conception of laws of nature.

When Descartes introduced the notion of “natural laws,” he really conceived of the regularities of nature on the model of prescriptive laws laid down by God for matter to obey—or at least (as matter cannot think and hence cannot truly obey) to conform to. In a slightly different vein, Newton and his circle, proponents of natural theology that they were, viewed natural philosophy (what we would today call “science”) as a clue to God's intentions; and hence they also viewed natural laws as expressing the will of the Divine Sovereign. For Hume and later writers, however, the notion of laws of nature had slipped free from its original theological moorings. Hume's own official analysis of laws and causation is arguably psychological and epistemic rather than metaphysical. But other writers, including many who have styled themselves “Humean,” have understood laws as regularities in nature itself.

In the past century, the most important heir to this broadly “Humean” tradition is the account (or accounts) of laws favored by the logical positivists and logical empiricists—really a number of accounts that differ in detail, but share a lineage and some central features. I shall thus refer at times to “the empiricist account” of laws, and at others to “empiricist accounts,” depending on whether the emphasis is on the shared core features or the distinctive variations upon them. What these empiricist accounts share at their core—and hence “the account” in the more generic sense—is the assumption that laws are (true) universally quantified claims about the real‐world behavior of objects. There are several variants of this account within the empiricist tradition. The simplest views the universally quantified conditionals expressing laws as materially adequate—that is, true of all actual events. Most versions add some sort of modal augmentation so that the laws hold true over counterfactual situations as well as actual ones, albeit not logically necessary. At a minimum, then, the empiricist account holds that laws express materially adequate, universally quantified claims about how objects actually behave. Or, less formally, it holds that laws express true universal claims about the real‐world behavior of objects. The principal difference between twentieth century empiricism and older forms of Human empiricism lies in the application of a type of logical regimentation that was unavailable in Hume's day, that of quantified logics.

If we combine this account of laws with Hume's definition of a miracle as an event that violates a natural law, the problem with miracles becomes apparent. If a law‐claim is a universal claim—a claim about what is true in every case—then a “violation” or exception is enough to falsify a law‐claim. Conversely, true law‐claims—real laws—cannot have exceptions. And thus, if we combine Hume's definition of ‘miracle’ with an empiricist account of laws, the notion of a miracle is incoherent. The problem is not so much that miracles have empirical evidence against them, but rather that there is a contradiction inherent in the very notion of an event that is an exception to an exceptionless law.

I take it that Hume's argument is valid. If laws are true universal claims and miracles are events that violate laws, then there can be no miracles. Indeed, if one takes Hume's characterizations of “law” and “miracle” as stipulative definitions, his argument is also sound, albeit at the risk of triviality. In order for the argument to be sound and nontrivial, two additional conditions must be met: Hume's characterization of “miracle” must correspond to the relevant usages of the term in theology, and the empiricist account of laws must be acceptable as an analysis of the laws that actually appear in the sciences.

Hume's definition of “miracle,” of course, was original and theologically contentious. The modern notion of natural laws, pioneered by Descartes, was little more than a century old when Hume wrote his Treatise, and so theologians prior to Descartes could not have had it in mind. (Indeed, some quite explicitly embraced very different definitions.) Hume's definitions, moreover, clearly imply the impossibility of miracles, and it is difficult to imagine that careful thinkers like Descartes and Newton would have missed this had they understood the terms ‘law’ and ‘miracle’ in the same way. And so, we might reasonably be suspicious that Hume made his task too easy by defining terms in a fashion that rendered the word ‘miracle' self‐contradictory by definition. A more general theological notion of ‘miracle’ might go something like this: a miracle is an event caused by special Divine action that would not or could not have occurred merely through the regular course of events in nature. Such a definition—or indeed any definition that does not define miracles as exceptions to exceptionless laws—would be sufficient to avoid Hume's argument against miracles.

But by Hume's day, another line of thought had also begun to develop, though it would not receive its most influential formulation until Laplace. This was the idea that a commitment to natural laws entails a commitment to determinism. And if this were correct, natural laws would present a serious problem for miracles even on a much broader interpretation of the word ‘miracle' If, as Laplace claimed, the laws of nature, combined with a complete description of the state of the universe at a time t, completely determine the state of the universe at any subsequent time t+∂, then nothing happens (at least after creation is complete) that could count as a miracle, even on this very broad definition.

The formulations of this view in the eighteenth and nineteenth centuries are, by contemporary philosophical standards, rather informal, and admit of a number of alternative interpretations. When I return to this matter in the “Empiricist Accounts of Laws” section below, I shall concentrate upon versions of empiricism that emerged out of the Vienna circle in the twentieth century, whose logical formulation will make it easier to assess the supposed tension between natural laws and miracles. In the “Miracles and Causal Powers” section below, I shall present a rival account of the nature of laws, which is much more compatible with miracles.

Laws and Their Philosophical Interpretations

What is a “natural law” or a “scientific law”? From the standpoint of philosophy of science, we must at least begin by dealing with the things called “laws” in the sciences themselves: things like Newton's inverse square law, the Boyle‐Charles law, Coulomb's law, and the laws for chemical reactions. Often, these take the form of mathematical equations. Of course, the equations themselves are meaningless until the variable letters are interpreted as representing quantitative phenomena like mass, charge, and velocity; and they receive these interpretations collectively in the form of a theory or model. The success of applying particular laws, and the theories and models in which they occur, gives us reason to believe that those laws make true claims (or approximately true claims, or an apt framework for making more particular true claims) about the relationships between variables.

Exactly what claim a law makes, however, is often underdetermined by the success of the law and the experimental evidence substantiating its truth (or approximate truth or aptness). For example, some favor the Humean regularist view that laws express mere empirical regularities among events, while others hold that they express unseen forces, dispositions, capacities, or powers. Within the regularist camp, there have been those who interpret laws as expressing only material generalizations and those who take them to express modal claims. Realists and instrumentalists may agree that a particular law is a good one, but disagree as to whether one of its variables expresses a real entity or property, or is simply instrumentally useful in understanding a system or predicting its kinematics. These are all philosophical accounts of the nature of laws, and the issues at stake between rival views are often distinct from those that are of interest to the scientist. Philosophers can agree on the merits of, say, general relativity or quantum mechanics, while disagreeing on exactly what those theories and the laws they employ should be interpreted as expressing. In some cases, however, it is possible to show that a particular philosophical account of laws does not do justice to laws as they are used in the sciences. This, I shall argue, is the case with the empiricist account of laws.

Empiricist Accounts of Laws

There are, in fact, a number of accounts of the nature of laws that have received the label “empiricist.” I am concerned in particular with those that have arisen out of the logical empiricist movement, which attempted to regiment a broadly Humean regularist account of laws by recasting laws in the rigorous logical formats of sentential and predicate calculi. The use of such logical resources opened the door to a number of alternative formulations of laws, such as the conditional form of, ceteris paribus, laws and the addition of modal operators to some more contemporary interpretations. What distinguishes the class of interpretations I am calling “empiricist” are two features:

E1. Universal quantification. Law‐claims are understood to be universally quantified claims, and a genuine law must be at least materially true.

E2. A domain of objects and events. The domain of the law, and the properties expressed by the predicate letters, consist of objects, their properties, and the events in which they take part. (In particular, neither the variables nor the predicate letters express such things as forces, dispositions, powers, or capacities.)

These two features are, of course, dissociable. One could interpret laws as quantified claims, at least some of which are about forces, dispositions, powers, or capacities. Conversely, one could interpret laws as expressing forces, dispositions, powers, or capacities without employing a quantifier, as we shall see in the “Miracles and Causal Powers” section. Defining “empiricist” accounts in a fashion that excludes quantified accounts ranging over forces is to some extent a stipulation on my part. My reason for restricting the scope of “empiricism” in this way for the purposes of this article is that it is the combination of E1 and E2 that results in a problem for miracles, and addressing this problem is the main target of this article. If the reader would prefer to draw the boundaries of empiricism more broadly, the same point could be made by distinguishing “kinematic” empiricist accounts (those whose domains exclude forces, dispositions, powers, and capacities) from “dynamic” empiricist accounts (which allow or even favor them).

Empiricist Laws, Determinism, and Miracles

The empiricist account (in my narrow “kinematic” sense) raises problems for miracles (and likewise for libertarian free will) because empiricism can be strengthened to imply determinism by the simple addition of one more claim: namely, that every physical event falls under at least one natural law of the right type. Suppose that a law‐claim L* is the assertion some law L obtains in nature. If L* implies that event E1 entails the subsequent occurrence of event E2, then E1 occurring without E2 following would, on the empiricist interpretation, imply that L was not a genuine law, as a single counterexample materially falsifies a universally quantified claim. (And, as a modally strengthened version of such a claim would imply its material truth as well, a counterexample would falsify the modal version too.) In short, even God could not make it the case that L* was a true universally quantified claim that had a genuine counterexample, because that would be self‐contradictory. It would not be self‐contradictory for God to do things at one time that did not conform to the regularities that take place at every other time; but in such a case, there would not be a genuine empiricist law corresponding to those regularities.

Of course, there are several different sorts of laws, and the implications of the empiricist interpretation are different for each of them. The most important for our concerns are what might be called causal laws—that is, law‐claims that say something about the relationship between antecedent and consequent conditions, like the laws of gravitation or electromagnetism. These laws apply to all bodies that have mass or charge, respectively. And, if interpreted as claims about how bodies with these properties actually behave in their interaction with one another, their application to a particular antecedent state of affairs (the cause) yields a particular consequent state of affairs (the effect). All macroscopic bodies have mass; and so, if the gravitation law makes true, materially‐adequate, and nonprobabilistic claims about the resulting state of an antecedent event described in terms of mass and position, it would seem that the behavior of bodies with mass would have to be deterministic, on pain of the laws turning out to be materially false.

Not all types of scientific laws have this consequence. Laws of statics are about the state of a system at a single time, and probabilistic laws do not specify a unique resultant state. But even if the gravitation law were the only law that had this feature, the implications would be considerable. It would leave open the possibility that objects that do not possess mass might behave indeterministically, which, in turn, would open the doors for the possibility of such things as indeterminacy in quantum electrodynamics and free thought by God or Cartesian souls. It might also allow for revelations through prophecy and visions. But it would preclude a great number of biblical miracles, such as the parting of the Red Sea or the ascent of Elijah into the sky, as those events do involve bodies with mass behaving in a fashion contrary to what the gravitation law would predict. So, if we interpret the gravitation law as a universal claim about antecedent and consequent conditions, either such miracles do not happen, or else they falsify the law‐claim.

Of course, some versions of empiricism deny that there is actual causation in nature at all, treating “causation” as either constant conjunction or a projection of an associative linkage from the mind onto the world. But even quantifying only over actual events and treating “cause” and “effect” as mere names of antecedent and subsequent events, we would have to conclude that miracles do not occur. Moreover, most forms of late twentieth‐century empiricism strengthened the notion of a law beyond material adequacy to cover counterfactual cases and rule out accidental regularities. And if the scope of such conditionals covers counterfactual cases, and the conditionals are true and universally quantified, then miracles could not take place without rendering the laws false.

The empiricist interpretation of laws thus supplies a short road to at least a limited form of determinism—a form that would seem to be strong enough to create a problem for many of the biblical miracles. If a nonprobabilistic causal law‐claim L* expresses a nonprobabilistic law L, and L* makes claims about the condition that always follows if the antecedent conditions are met, then either (a) any object O that falls under the scope of L behaves as described by L*, or else (b) L* is false and L is not a genuine law. And there is at least one nonprobabilistic causal law that would seem to apply to all macroscopic events, because all macroscopic bodies have mass. Formulations of the gravitation law, at least, are nonprobabilistic in form and are interpreted by empiricists as universal claims about antecedent and consequent conditions such as position and motion. And most of the biblical miracles involve bodies with mass behaving in ways that do not correspond to the gravitational conditionals. And so the empiricist account, while not directly asserting determinism, provides a basis for implying determinism with respect to a class of events into which many of the biblical miracles fall. Similar cases could no doubt be made, on the basis of different laws, for biblical miracles such as the multiplication of loaves and fishes, Jesus changing water into wine at Cana, the resurrection of Lazarus, or Philip's sudden change of geographic location after his conversation with the Ethiopian eunuch.

The reader should note that the ease with which the empiricist can move from an account of laws to determinism is a consequence of the two assumptions about laws cited earlier as E1 and E2: that is, that laws are understood as universal and kinematic claims about how objects always actually behave. Universal claims about forces, causal powers, or probabilities do not similarly lead to determinism; nor do weaker sorts of generalizations about actual events that are not framed in something like the form of quantified classical logic. The combination of these two assumptions thus comes very close to sneaking determinism into the picture through the back door of logical quantification, with the small loophole that it does so only for those objects and events to which at least one causal law applies.

It is difficult to assess exactly how ideas from the sciences, and philosophical interpretations thereof, came to have an influence even in academic theology, much less in popular religious consciousness. My guess is that the notion that a world with natural laws must also be a deterministic world that has no room for miracles is one that took hold by way of a mixture of explicit argumentation and more intuitive assumptions about the implications of modern science. The importance of particular characterizations of laws in philosophy of science, however, stems not so much from their being the source of assumptions that are relevant to miracles, as from the fact that they allow us to assess those assumptions more carefully. There really does seem to be at least a prima facie problem for miracles, given the empiricist account of laws. And, as I shall argue later in the article, the same problem does not arise on an important alternative account.

Familiar Theological Objections

Beginning in “More Fundamental Problems with the Empiricist Account,” I shall argue that the empiricist account, in all of its “kinematic” formulations, is deeply flawed in ways that have already been explored ably, and perhaps decisively, in philosophy of science. Before turning to this main critique, however, I shall briefly discuss several familiar objections that do not so fully reject the empiricist account.

More Fundamental Problems with the Empiricist Account

The objection from QM claimed that the empiricist account of laws was not a viable account of particular laws endorsed by the scientists. One can, however, make a more fundamental case that empiricism misunderstands the nature of causal laws in general, including nonstatistical and classical laws. Empiricism's principal error is this: it takes causal laws to be claims about what actually happens (that is, about particular events like motions of bodies). Taken in this way, many laws—for example, the law of gravitation—would turn out to be false. Indeed, it is doubtful that they would ever describe the actual unfolding of events—the kinematics, if you will—correctly. But the problem here is not a problem for the laws themselves, but for the empiricist account of them, because such laws are really claims of a different kind: namely, claims about things like forces such as gravitational attraction. Or, more generally, they make claims about factors that make a regular contribution to how particular sequences of events unfold. They are dynamic rather than kinematic claims.

The basic point can be made very simply. Consider Newton's gravitation law, F = g(m₁ * m₂)/r². This equation can be expressed in English by saying that the gravitational attraction between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Suppose now that we were to interpret this according to empiricist lights as a claim about how bodies with mass actually behave—that is, how they move. Interpreted in this way, the law would imply that two bodies of equal mass, dropped from the same height above the Earth, should fall with equal acceleration and reach the ground at the same time. Now imagine taking two sheets of paper of equal mass, crushing one into a tight ball, and folding the other into a paper airplane, and then dropping them. Try it yourself if you are in doubt, but they will not fall at the same rate.

Now if the gravitation law actually implied that the two pieces of paper must reach the ground at the same time, this example would demonstrate that the law‐claim is false, and I could expect a Nobel prize in due order. But, of course, that is not at all what the example shows. What it shows, and what even nonphysicists like myself already know, is that gravity is not the only principle at work. In this case, aerodynamics plays a dominant role; but we could easily come up with other examples where principles of magnetism (substitute a metal object for one piece of paper and add a sufficiently strong magnet to the experimental setup) or psychology (make one of the objects a hundred dollar bill and the experimental location a crowded room) would play roles of similar importance. These are all possibilities about which the gravitation law tells us absolutely nothing. It speaks only to one of the factors contributing to the motions of objects. And the same is true, mutatis mutandis, for laws of aerodynamics, magnetism, and so on. In short, each law, taken individually, does not make or license any claims about the actual motions of objects. Rather, each expresses a single regular factor that can contribute to the actual motions.

This, moreover, is the interpretation that most naturally recommends itself when one reads Newton or subsequent scientists without trying to force their claims into the garb of quantified logic. The inverse square law F = g(m₁ * m₂)/r² does not look like an equation about motion, but about force. Newton did also derive from it kinematic equations describing ideal planetary orbits at equilibrium. But even this required the addition of another Newtonian innovation: the summation of forces as the bridge between theory and prediction, in this case gravitation and inertia. More generally, the actual motions of objects are determined by the combination of all of the forces acting upon them—something that goes beyond the scope of any single law. Indeed, it is only in the case of dynamic laws that we can make sense of the idea of composition of forces. If each law individually determined how objects would behave, there would be no need for a summation of forces; and, indeed, such a summation would produce a prediction of resultant motion different from those based on the individual laws. (And, of course, if those laws were interpreted according to empiricist lights, this would falsify the laws.)

Empiricist Variations: Ceteris Paribus Laws and Ideal Worlds

The main criticism I have offered of the empiricist account—that most laws would, if interpreted as universally quantified claims about real‐world events, turn out to be false—is in some ways a familiar one. It has been offered at a similar level of generality (and with equally critical intentions) by Nancy Cartwright (, , ), and I have developed it at greater length in my Laws, Mind and Free Will (Horst ). And, in a weaker form, it was recognized to exist for some laws by the logical empiricists themselves, who attempted to handle “exceptions” to laws through the use of boundary conditions and ceteris paribus clauses. These embed the claim actually stated in the law equation within a specification of the conditions under which it is true. (Or by specifying the conditions under which it is not true.) In sentential logic, C ⊃ L, where C specifies the conditions under which L is true. In predicate logic, the schematic form would be ∀[quantified variable(s)](C ⊃ L), where C and L are shorthand for formulas containing bound variables.

There are indeed scientific laws that are understood to hold only under particular conditions—for example, only for matter in the gaseous phase, or for fluids in laminar flow. But such ceteris paribus laws were generally assumed to be different in form from fundamental laws like gravitation, which were viewed as “strict and exceptionless.” Perhaps, however, concerns about “exceptions” to all laws can be handled by treating all laws as ceteris paribus laws? This interpretation is suggested by Cartwright in one of her early essays, where she suggests the following interpretation of the gravitation law: “If there are no forces other than gravitational forces at work, then two bodies exert a force between each other which varies inversely as the square of the distance between them, and varies directly as the product of their masses” (, p. 58).

This interpretation, however, has two fatal flaws. First, there are no such cases in the real world, so the law would be empirically vacuous. Second, the law, thus interpreted, would say nothing at all about cases where there are more than two bodies and/or nongravitational forces. But the gravitation law does tell us something about such cases—indeed, it tells us exactly the same thing it tells us about any two‐body cases: namely, the contribution of pairwise gravitational interaction to the overall causal mix. In other words, this interpretation of the law would seriously misinterpret its scope of application.

A slightly more sophisticated and more recent approach is to treat laws as expressing true universal claims about the behavior of objects, not in the real world, but in an “ideal world” (Giere ). This gambit, however, suffers from much the same problems. Because the laws, thus interpreted, do not make any claims about the real world, they would be empirically vacuous by definition; and by the same token, they would get the scope of real scientific laws wrong, as these do say something true about real‐world interactions. Moreover, consider how different a world would have to be for it to, say, lack any nongravitational forces. Since most physical particles have properties in addition to mass, they would not exist in such a world, and likewise, mutatis mutandis for other laws and the forces they express. Indeed, there would be no such worlds in which more than one force was present, and hence no particles with, say, both mass and charge.

So, neither the original version of the empiricist account, nor those modified to include ceteris paribus clauses or to refer to ideal worlds, is a viable account of the nature of the laws employed by scientists. Moreover, the arguments marshaled toward this conclusion are based solely on philosophy of science, and in no way depend on any assumptions about miracles or supernatural agents. As a result, the argument against miracles based on the empiricist account of laws cannot be sound. This, in itself, is a significant conclusion. Suspicion of miracles, even among Christian theologians, has certainly grown significantly since the time of Hume's arguments against miracles. And to the extent that the implications of empiricist accounts of laws, whether Humean or those of the twentieth century, fueled this suspicion, rejection of the empiricist accounts of laws provides a powerful defeater for those arguments. It may be true that if empiricist accounts of laws were correct, miracles would be impossible. But that need not trouble the advocate of miracles if the antecedent of that conditional is, as I have argued, false.

Moreover, the source of the problem with the empiricist account of laws is precisely the feature that generated problems for miracles: the “kinematic” interpretation of laws as claims about what always actually happens. This suggests that the supposed problem for miracles may be an artifact of a particular philosophical interpretation of laws. To explore this possibility further, let us turn to a major rival account of laws, sometimes called the “causal account.”

The Causal Account

Over the past 40 years, a number of philosophers have suggested an alternative type of account of the nature of laws, which we may label causal accounts. On these accounts, laws are understood to express what have variously been described as “causal powers,” “causal capacities,” “causal dispositions,” or “potential partial causal contributors.” Gravitational force would be a good paradigm example for explicating causal accounts. What the gravitation law expresses is a “power” or “capacity” objects with mass have to influence each other in a particular way: namely, by exerting an attractive force described by the inverse square law. This force is a contributor to the real‐world motions that ensue, but is generally only a partial contributor, as other forces are usually at work as well. And in the case of some laws, the contribution is only “potential” or conditional, as there may be additional ancillary conditions that must be met for the potential to be activated (e.g., some chemical reactions only take place in the presence of a catalyst, the law of speciation through variation and selection will cease to be in play if planetary conditions cease to support life, and so on).

The observation leveled critically at the empiricist account—that each law speaks only to its own domain—is thus a central feature of causal accounts. And a commitment to the truth of a given law leaves us uncommitted on the further question of what other causal contributors there might be. The discovery of universal gravitation neither presaged nor excluded the possibility of electromagnetic force. Moreover, commitment to a given law leaves us uncommitted, not only on the question of what additional laws there may be, but also on the question of whether there might be additional causal contributors that are anomic.

Another way of putting the same point (and stressing continuities with Newton) is to say that laws like the gravitation law express dynamic rather than kinematic claims. Of course, there are laws that are kinematic rather than dynamic. To illustrate this distinction more clearly, compare Newton's inverse square law with an important precursor: Galileo's claim that projectile motion is parabolic, the product of combining a horizontal component that covers equal distances in equal times with a vertical component that is accelerated linearly over equal times (Galilei [1974], Theorem 1, Proposition 1). Galileo's “theorem” is a mathematically formulated claim about the geometry of a particular type of motion. The principles from which it is derived—the “component motions”—are neither dynamic principles (they are not separate states that cause the actual motion) nor real motions, but abstractions (“formal causes” in the language still used in Galileo's day). The inverse square law, by contrast, expresses the force two bodies exert on one another as a function of their masses and the distance between them. It can be used to calculate approximate real‐world motions through summation of forces using vector algebra, often with the inclusion of nongravitational contributions such as inertia. And it yields ideal models of motion in cases of equilibrium, such as the elliptical form of a stable planetary orbit.

In light of this distinction, we might rephrase the criticism of the empiricist account by saying that the error was to mistake dynamic laws (like universal gravitation) for kinematic laws. However, this is potentially misleading: kinematic laws like Galileo's theorem of parabolic motion are by no means exceptionless exact descriptions of real‐world motion either—a fact to which anyone who has launched a projectile in a strong wind can attest. The parabolic motion of projectiles is an idealized model of a special case of the summation of gravitational and inertial forces, in which other real‐world factors like aerodynamics have been bracketed. It is the dynamic laws that are more fundamental here, and the kinematic model's fit with real‐world behavior is a function of the strength of the real‐world contributions of the factors bracketed from the model.

On the causal account, laws—and the theories and models associated with them—provide insight into the dynamic principles that contribute to real‐world events, and they do so in a piecemeal fashion. Each law attempts to capture causal regularities of a particular sort, and says nothing about what other causal powers there might be. In the terminology I developed in Laws, Mind and Free Will (Horst ), each law presents a model of a domain of phenomena, a model that is idealized in that it “brackets” indefinitely many other factors that may be at work in the real world. Of course, there are also questions about how well each law models even its own domain: the values of the constants may be incorrect or approximate, the law may employ a logarithmic equation while the phenomenon may be better described by a power law, and so on. But no matter how well a law succeeds at describing the dynamic regularities of its own domain, its usefulness for prediction is further affected by what other causal powers are “in play” in a given situation. When we use a model to predict real‐world kinematics, we treat the world as though only that law were in play, even though we know full well that this is generally not the case. Thus, when we are testing a law, theory, or model, we build experimental setups that screen off other causal factors as far as it is possible to do so. And, assuming we do not forget that we are dealing with an idealized model, we understand that the predictions generated by the model will approximate real‐world events to the extent that other potentially relevant causal factors are not in play. What kind of approximation is “good enough” will depend on the context. And, again to borrow terminology developed in Horst (), a law, model, or theory may be called “apt” for a given context when it yields predictions that are good enough for the pragmatic constraints of that context.

The causal account lends itself naturally to explaining summation of forces, but also highlights the limitations of that technique. It is precisely because the laws pick out distinct causal contributors that we need an additional technique of calculating how those contributions combine to produce actual resultant phenomena. In classical mechanics, vector algebra was a suitable technique for this, as inertia, gravitation, the mechanical force of collisions, and electromagnetism were taken to be fundamental, independent, and nonprobabilistic, and each yields a contribution to resultant motion expressible by a vector. What such vectors express are not component motions, but component forces that together result in a single actual motion. And, given these assumptions, the application of vector algebra should yield increasingly good approximations of real‐world behavior to the extent that (a) all of the contributions of these types are brought into the calculation and (b) there are no additional causal contributors of different types.

Summation of forces becomes more problematic in cases where assumptions of fundamentalness and independence cannot be made, which would include any special science laws but particularly those involving feedback systems (such as the brain) where the behavior of the different “units” is mutually dependent. Moreover, many scientific laws do not lend themselves to combination through vector algebra, whether because they are not quantitative, or because two quantified laws do not deal in commensurable measures (e.g., chemical properties like pH are quantified, but are not spatial like particle descriptions in classical mechanics), or because the ways two laws are idealized prevent their recombination. (Cartwright gives the example of two laws of chemical combination for a substance X, one for what happens when it is mixed with an acid, the other for what happens when it is mixed with a base, pointing out that what happens when you mix X with both acid and base cannot be obtained by adding the results of the two separate equations.)

While we should allow for methods of “composition of forces” other than vector algebra, it is not clear (a) that these are available for all cases, (b) that, when available, they produce exact results even in ideal cases, or indeed (c) whether their unavailability should be viewed as pointing to a “dappled world” (Cartwright ) or an artifact of the cognitive nature of science (Horst ). However, if one starts, as causal theorists do, with individual laws, theories, and models as expressions of individual potential regular causal contributors, and regards techniques for summing forces as things that we then seek in order to apply dynamic laws to produce kinematic predictions, one may be left with a very different set of expectations than if one begins with such assumptions as that laws are universally quantified claims or that scientific laws do (or should, or could) yield a comprehensive and self‐consistent framework for explaining everything that happens.

Early modern thinkers hit upon the idea of the world as a great mechanism operating in accordance with a few comprehensible and nonprobabilistic laws. While many of them (notably Isaac Newton) held that God can and does tinker with the machine from time to time, the idea of the nomic world‐machine pulled their successors down the path through a ‘God of the gaps’ to deism or atheism. But if we start by looking at the sciences as we actually find them, we find a number of separate laws, theories, and models of different regular patterns in nature, some of them suggesting particular types of causal powers. We find ways of combining these that are apt in some, but not all, contexts, and partial (but seldom complete) reductions between them. The methodological principle implied is two‐sided. On the one hand, when we are applying known laws and models for prediction and explanation, we treat them as if only the causal powers they express are in play. On the other hand, we are well aware that this “as‐if” assumption is unlikely to be true, and are open to the possibility of the discovery of additional causal powers and additional insights into how they work in tandem.

It is crucial to keep both sides of the methodology in mind. Treating a phenomenon as a closed (and perhaps deterministic) system is something we do for purposes of comprehension and calculation. It is a good methodology for testing theories and applying them to problems. But it should not be mistaken for a kind of metaphysical principle, either that these laws we are using are the only factors really at work, or even that the universe is a closed (and perhaps deterministic) system under some complete set of laws. Methodologically, the following should be regarded as open questions: (1) what other nomic causal powers there may be, (2) whether there are anomic causal powers, (3) how they interact to produce real‐world phenomena, and (4) whether the kinds of laws we can comprehend can be made into a system that is at once comprehensive and self‐consistent, or whether there are features of them (such as the idealized character they may need to be comprehensible to minds like ours) that may prevent this.

Miracles and Causal Powers

On the causal account, a commitment to any set of laws is quite compatible with a commitment to miracles. Of course, nothing in the scientific laws implies that miracles or other anomic events do occur. But neither are they precluded or even rendered unlikely. This tends to bear out the suspicion, raised at the end of the “Empiricist Variations: Ceteris Paribus Laws and Ideal Worlds” section above, that the apparent conflict between natural laws and miracles may be an artifact of the very feature of the empiricist account that made it unacceptable on other grounds. But more importantly, it shows that on a major account of laws that is more viable than the empiricist account, there is no conflict between the existence of laws and the occurrence of miracles.

Let us try to spell this out more specifically. On a causal account, what actually happens is a function of all of the causal powers that are in play. We know a number of lawful causal powers that explain a very great deal about the natural world. But there is no reason to assume (except in an “as‐if” way for purposes of experiment or prediction) that these are all of the causal powers that are ever in play. Perhaps least controversially, there are very likely laws—perhaps even very fundamental laws—that we do not yet know about. But the fact that we understand the world of nature through the discovery of its lawful dynamic regularities by no means implies that there are not also causal powers that are not lawful or regular. Even if the laws (or at least some of the laws) tell us about causal factors that are always at work, this says nothing about whether there may be anomic causal factors as well, such as free will or God's miraculous interventions. To the extent that science is in the business of finding regularities, the irregular, the spontaneous, the anomic is systematically screened out of scientific laws and theories, and experimental methodologies aimed at finding the regularities will overlook it. Nomic Divine action could be tested through experimental intervention. (Though, of course, the experiment would not prove that the cause was Divine, but only that it was lawful.) Anomic Divine actions—miracles—cannot become the object of controlled experiment, because by definition, there is not a nomic relationship between variables.

In general terms, what the causal theorist should say about miracles is something like this. Miracles (if indeed they occur) involve the introduction, by special Divine action, of causal powers that are not grounded in natural laws. The miracle is a result of all of the causal powers involved, just like any other event in the world. The difference consists solely in the fact that there are causal factors at work that are not “routinely” in play simply by dint of the natural laws.

Of course, for something to be a miracle that happens in the natural world (as opposed to, say, someone being “caught up into heaven” in a vision), the event must begin with God willing something and setting supernatural forces into motion. This falls outside the realm of science, and quite possibly of human understanding. But equally, it must end with things happening in some particular time and place, and to particular bits of matter: say, Jesus’ physical body walking on the surface of the Sea of Galilee, or Elijah being lifted bodily from the surface of the Earth. In between, there will be a string of “secondary causes” through which God works: processes that are natural in their operation but supernatural in their origin.

I am pessimistic about the prospects of speculating about the particulars of how any particular miracle might come about. Its origin in the Divine will and the introduction of supernatural causal powers into the world seem to be things that lie, at best, in the realm of pious conjecture. It is, of course, possible to speculate about the added causal powers. Was Elijah lifted by a miraculous attractive power above him, pushed up by a repulsive power below him, or did God perhaps set into motion an actual whirlwind that lifted him up like Dorothy and Toto in The Wizard of Oz? Did the multiplication of the loaves and fishes involve the creation of new matter, or a miraculous reorganization of existing atoms? Again, such questions strike me as lying in the realm of pious conjecture and not of academic philosophy.

What philosophy can provide here is a compatibility proof: if, as causal accounts (to my mind rightly) claim, laws express dynamic principles that contribute to the real‐world behavior of objects rather than universal claims about how the objects actually behave, a commitment to such laws does not imply (or even suggest) a commitment to determinism or to the causal closure of the universe under natural laws. Of course, demonstrating the compatibility of miracles with natural laws does not imply (or even suggest) that miracles actually occur. The causal account is also compatible with the possibilities (a) that the universe is deterministic, or (b) that the universe is indeterministic, but that the indeterminism is not a result of special divine intervention. Any attempt to adjudicate such questions would require a much broader inquiry, one that would need to draw upon debates in contemporary physics and cosmology as well as in theology. My intention here, by contrast, has been much more focused: namely, to show (1) that one very influential view of laws that has called miracles into question is untenable on other grounds, and (2) that at least one alternative view of laws is available that is compatible with a belief in miracles. To the extent that a person—say, a Christian or other theist—has prima facie reason to believe that miracles occur, the compatibility proof serves as a defeater for a particular objection. And to the extent that said objection has been historically influential in casting doubt upon miracles, the existence of a defeater for that objection is important for those whose theological beliefs include a commitment to miracles.

Of course, such a compatibility proof holds far less interest for anyone who does not feel any prima facie reason to believe in miracles. And whatever evidence there may be for the occurrence of miracles must be of a different sort from the evidence for scientific laws. The occurrence of miracles can neither be tested through experiment nor understood in natural terms. Our experiments cannot force God to perform a miracle; and, while in some cases, there may be plausible physical evidence and testimony that a miracle has occurred, there is no way of ruling out the possibility that these may be products of confabulation, forgery, illusion, or artifacts of data collection. While theists hold that miracles can serve to inspire religious belief, even their own sacred texts make it clear that skepticism is always a possible response. Likewise, even if miracles are accomplished through secondary causes, a real miracle requires there to be some action of God upon the world that involves some other, nonnatural sort of causation. How such causation might come about is beyond the reach of the natural sciences, and may well be beyond human understanding altogether. We cannot fully specify a mechanism through which miracles occur, because insofar as we regard an event as miraculous, an important component of any explanation would have to take the form of a claim such as “and then God intervened.”

Summary

Since the late seventeenth century, both theists and atheists have suggested that the commitment of the Abrahamic religions to miracles is in tension, or is even outright incompatible, with a commitment to the existence of natural laws. I have argued that such an incompatibility arises, not from the laws themselves, but from a particular interpretation of the nature of laws, the empiricist account. This account is untenable for reasons arising from within philosophy of science, and thus no sound argument can be based upon it. The more plausible alternative, the causal account, is compatible with miracles, and the general framework of summation of forces can be extended to include the proximate causes of miracles, though, of course, the initial causal contribution of Divine action stands outside the scope of scientific inquiry, and perhaps of human understanding.