What Are Laws of Nature

What Are Laws of Nature?

Bas C. van Fraassen

This question has a presupposition, namely that there are laws of nature. But such a presupposition can be cancelled or suspended or, to use Husserl's apt phrase, bracketed. Let us set aside this question of reality, to begin, and ask what it means for there to be a law of nature. There are a good half-dozen theories that answer this question today, but, to proceed cautiously, I propose to examine briefly the apparent motives for writing such theories, and two recent examples (Peirce, Davidson) of how philosophers write about laws of nature. Then I shall collect from the literature a number of criteria of adequacy that an account of such laws is meant to satisfy. These criteria point to two major problems to be faced by any account of laws.

1. The Importance of Laws

What motives could lead a philosopher today to construct a theory about laws of nature? We can find three. The first comes from certain traditional arguments, which go back at least to the realist-nominalist controversy of the fourteenth century. The second concerns science. And the last comes from a reflection on philosophical practice itself; for while in the seventeenth century it was scientific treatises that relied on the notion of law, today it is philosophical writings that do so.

The motive provided by the traditional arguments I shall spell out in the next section, drawing on the lectures of Charles Sanders Peirce.

The second and much more fashionable motive lies in the assertion that laws of nature are what science aims to discover. If that is so, philosophers must clearly occupy themselves with this subject. Thus Armstrong's What Is a Law of Nature? indicates in its first section, the nature of a law of nature must be a central ontological concern for the philosophy of science.

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This does indeed follow from the conception of science found among seventeenth-century thinkers, notably Descartes. Armstrong elaborates it as follows. Natural science traditionally has three tasks: first, to discover the topography and history of the actual universe; second, to discover what sorts of thing and sorts of property there are in the universe; and third, to state the laws which the things in the universe obey. The three tasks are interconnected in various ways. David Lewis expresses his own view of science in such similar comments as these:

Physics is relevant because it aspires to give an inventory of natural properties. . . . Thus the business of physics is not just to discover laws and causal explanations. In putting forward as comprehensive theories that recognize only a limited range of natural properties, physics proposes inventories of the natural properties instantiated in our world. . . . Of course, the discovery of natural properties is inseparable from the discovery of laws. ¹

But what status shall we grant this view of science? Must an account of what the laws of nature are vindicate this viewor conversely, is our view of what science is to be bound to this conception? We know whence it derives: the ideal of a metaphysics in which the sciences are unified, as parts of an explanatory, all-embracing, and coherent world-picture (recall Descartes's philosophy as a whole is like a tree whose roots are metaphysics, whose trunk is physics, and whose branches, . . . are all the other sciences). But this ideal is not shared throughout Western philosophy, nor ever was.

By its fruits, of course, shall we know this tree. If, starting with this conception, philosophers succeed in illuminating the structure of science and its activities, we shall have much reason to respect it. I do not share this conception of science, and do not see prima facie reason to hold it.

On the other hand, if metaphysics ought to be developed in such a way that the sciences can be among its parts, that does indeed place a constraint on metaphysics. It will require at least a constant series of plausibility argumentsto assure us that the introduction of universals, natural properties, laws, and physical necessities do not preclude such development. But this observation yields, in itself, only a motive for metaphysicians to study science, and not a motive for philosophers of science to study metaphysics.

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The third and final motive, I said, lies in our reflection on philosophical practice itself. Even in areas far removed from philosophy of science, we find arguments and positions which rely for their very intelligibility on there being a significant distinction between laws and mere facts of nature. I can do no better than to give an example, in section 3 below, of one such philosophical discussion, by Donald Davidson, about whose influence and importance everyone is agreed.

2. Peirce on Scholastic Realism

The traditional arguments are two-fold: to the conclusion that there must be laws of nature, and quite independently, to the conclusion that we must believe that there are such laws. The first argues from the premiss that there are pervasive, stable regularities in nature (sometimes itself backed up by noting the success of science). But no regularity will persist by chancethere must be a reason. That reason is the existence of a law of nature.

The second argues that if the preceding be denied, we are reduced to scepticism. If you say that there is no reason for a regularitysuch as that sodium salts always burn yellowthen you imply that there is no reason for the regularity to persist. But if you say there is no reason, then you can't have any reason to expect it to persist. So then you have no basis for rational expectation of the future.

Charles Sanders Peirce asserted, correctly, that the general form of such arguments appeared well before the idea of laws of nature appeared in its modern sense. ² Arguments of this form were given by the scholastic realists of the late Middle Ages against the nominalists. Peirce himself devoted the first section of his fourth lecture, The Reality of Thirdness, in his 1903 lecture series at Harvard, to his own variant of these arguments. ³ The lecture starts with the assertion that something quite beyond what nominalists acknowledge, is operative in nature. Dramatically opening his hand to the audience, Peirce displayed a stone (piece of writing chalk?):

Suppose we attack the question experimentally. Here is a stone. Now I place that stone where there will be no obstacle between it and the floor, and I will predict with confidence that as soon as I let go my hold upon the stone it will fall to the floor. I will prove that I can make a correct

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prediction by actual trial if you like. But I see by your faces that you all think it will be a very silly experiment.

Why silly? Because we all know what will happen.

But how can we know that? In words to be echoed later by Einstein, Podolsky, and Rosen, he answers If I truly know anything, that which I know must be real. The fact that we know that this stone will fall if released, is the proof that the formula, or uniformity [which] furnish[es] a safe basis for prediction, is, or if you like it better, corresponds to, a reality. A few sentences later he names that reality as a law of nature (though for him that is not the end of the story).

Do we have here the first or the second argument, or both? We very definitely have the second, for Peirce clearly implies you have no right to believe that the phenomena will continue the same in the future, unless you believe in the reality in question. But the reality cannot be a mere regularity, a fact about the future ungrounded in the present and past, for that could not be known. Peirce did recognize chance, and agreed that anything at all could come about spontaneously, by chance, without such underlying reasons. Therefore he does not subscribe to the validity of There is a regularity, therefore there must be a reason for it, since no regularity could come about without a reason. However he does not allow that we can know the premiss of that argument to be true, unless we also know the conclusionnor to believe the premiss unless we believe the conclusion. This is a subtle point but important.

He gives the example of a man observed to wind his watch daily over a period of months, and says we have a choice: (a) suppose that some principle or cause is really operative to make him wind his watch daily and predict that he will continue to do so; or else (b) suppose it is mere chance that his actions have hitherto been regular; and in that case regularity in the past affords you not the slightest reason for expecting its continuance in the future. It is the same with the operations of nature, Peirce goes on to say, and the observed regularity of falling stones leaves us only two choices. We can suppose the regularity to be a matter of chance only, and declare ourselves in no position to predict future casesor else insist that we can predict because we regard the uniformity with which stones have been falling as due to some active general principle.

There is a glaring equivocation in this reasoning, obscured by a judicious choice of examples. Sometimes by chance is made to mean due to no reason, and sometimes no more likely to happen than its contraries. Of course, I cannot logically say that certain events were a matter of chance in the second sense, and predict their continuation with any degree of certainty. That would be a logical mistake. Nor do I think that a person winds his watch for no reason at all, unless he does it absent-mindedly; and absent-mindedness is full of chance fluctuations. But I can quite consistently say that all bodies maintain their velocities unless acted upon, and add that this is just the way things are. That is consistent; it asserts a regularity and denies that there is some deeper reason to be found. It would be strange and misleading to express this opinion by saying that this is the way things are by chance. But that just shows that the phrase by chance is tortured if we equate it to for no reason.

Perhaps we should not accuse Peirce of this equivocation, but attribute to him instead the tacit premiss that whatever happens either does so for a reason or else happens no more often than its contraries. But that would mean that a universe without lawsif those are the reasons for regularitieswould be totally irregular, chaotic. That assertion was exactly the conclusion of the first argument. Hence if this is how we reconstruct Peirce's reasoning, we have him subscribing to the first argument as well. His indeterminism would then consist in the view that individual events may indeed come about for no reason, but not regularities. ⁴

Peirce knew well the contrary tradition variously labelled nominalist and empiricist, which allows as rational also simple extrapolation from regularities in past experience to the future. He saw this represented most eminently by John Stuart Mill, and attacked it vigorously. The following argument appears in Peirce's entry Uniformity in Baldwin's Dictionary (1902). ⁵ Of Mill, Peirce says that he was apt to be greatly influenced by Ockham's razor in forming theories which he defended with great logical acumen; but he differed from other men of that way of thinking in that his natural candour led to his making many admissions without perceiving how fatal they were to his negative theories (ibid. 76).

Mill had indeed mentioned the characterization of the general uniformity of nature as the fact that the universe is governed by general laws. ⁶ (He did not necessarily endorse that form of language

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as the most apt, though he does again use it in the next paragraph.) Any particular uniformity may be arrived at by induction from observations. The peculiar difficulty of this view lies in the impression that the rule of induction gives, of presupposing some prior belief in the uniformity of nature itself. Mill offered a heroic solution:

the proposition that the course of nature is uniform is the fundamental principle, or general axiom, of Induction. It would yet be a great error to offer this large generalization as any explanation of the inductive process. On the contrary I hold it to be itself an instance of induction, and induction by no means of the most obvious kind. (Collected Works, 392)

According to Peirce, Mill used the term uniformity in his discussions of induction, to avoid the use of law, because that signifies an element of reality no nominalist can admit. But if his uniformity meant merely regularity, and implied no real connection between the events covered, it would destroy his argument. Thus Peirce writes:

It is, surely, not difficult to see that this theory of uniformities, far from helping to establish the validity of induction, would be, if consistently admitted, an insuperable objection to such validity. For if two facts, A and B, are entirely independent in their real nature, then the truth of B cannot follow, either necessarily or probably, from the truth of A. (Collected Papers, 77)

But this statement asserts exactly the point at issue: why should A, though bearing in itself no special relation to B, not be invariably or for the most part be followed by B? It is true that there would be no logical necessity about it, nor any probability logically derivable from descriptions of A and B in and by themselves. But why should all that is true, or even all that is true and important to us, be logically derivable from some internal connection or prior circumstance?

The convictions expressed by Peirce are strong, and have pervaded a good half of all Western philosophy. Obviously we shall be returning to these convictions, in their many guises, in subsequent chapters. A law must be conceived as the reason which accounts for uniformity in nature, not the mere uniformity or regularity itself. And the law must be conceived as something real, some element or aspect of reality quite independent of our thinking or theorizing

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not merely a principle in our preferred science or humanly imposed taxonomy.

3. A Twentieth-Century Example: Davidson

Concepts developed or analysed in one part of philosophy tend to migrate to others, where they are then mobilized in arguments supporting one position or another. From the roles they are expected to play in such auxiliary deployment, we should be able to cull some criteria for their explication. A good example is found in recent philosophy of mind.

Is there mind distinct from matter? Peter felt a sudden fear for his safety, and said I know him not. The first was a mental event, the second at least in part a physical one. But materialists say that the mental event too consisted solely in Peter's having a certain neurological and physiological stateso that it too was (really) physical. Donald Davidson brought a new classification to this subject, by focusing on the question whether there are psychophysical laws. Such a law, if there is one, might go like this: every human being in a certain initial physiological state, if placed in certain circumstances, will feel a sudden fear for his or her safety. Davidson denies that there are such laws, yet asserts that all mental events are physical.

It may make the situation clearer to give a fourfold classification of theories of the relation between mental and physical events that emphasizes the independence of claims about laws and claims of identity. On the one hand there are those who assert, and those who deny, the existence of psychophysical laws; on the other hand there are those who say mental events are identical with physical and those who deny this. Theories are thus divided into four sorts: nomological monism, which affirms that there are correlating laws and that the events correlated are one (materialists belong in this category); nomological dualism, which comprises various forms of parallelism, interactionism, and epiphenomenalism; anomalous dualism, which combines ontological dualism with the general failure of laws correlating the mental and the physical (Cartesianism). And finally there is anomalous monism, which classifies the position I wish to occupy. ⁷

This last position is that every strict law is a physical law, and most if not all events fall under some such lawwhich they can obviously do only if they admit of some true physical description.

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Therefore most if not all events are physical. This is consistent provided that, although every individual mental event has some physical description, we do not assert that a class of events picked out by some mental descriptionsuch as a sudden feeling of fearmust admit some physical description which appears in some strict law.

This point of consistency is easy enough to see once made. It does not at all depend on what laws are. But whether the position described even could be, at once, non-trivial and truethat does depend on the notion of law. If, for example, there were no distinction between laws and true statements in general, then there obviously are psychophysical laws, even if no interesting ones. Imagine an omniscient being, such as Laplace envisaged in his discussion of determinism, but capable also of using mental descriptions. Whatever class of events we describe to It, this being can list all the actual members of this class, and hence all the states of the universe in which these members appear. It can pick out precisely, for example, the set of conditions of the universe under which at least one of these states is realized within the next four years. Davidson must object that what It arrives at in such a case is in general not a law, although it is a true general statement. ⁸

The form of objection could be anthropomorphic: although It could know that, we humans could not. Then the cogency of the objection would hinge on the notion of law involving somehow this distinction between what is and is not accessible (knowable, confirmable, . . . ) to humans. The position of anomalous monism would no longer have the corollary Therefore most if not all events are physical, but rather something like: every event which we humans could cover in some description that occurs in a humanly accessible (knowable, or confirmable, or . . . ) general regularity, is physical. In that case the position would seem to have no bearing at all on the usual mindbody problems, such as whether the mental supervenes on the physical (which means, whether our mental life being otherwise would have required the physical facts to be otherwise).

Davidson's objection to the story about this omniscient genie would therefore need to be non-anthropomorphic. It would have to insist on a distinction between what the laws are and truths in general, independent of human limitations. The reason this being would not automatically arrive at a law, by reflection on just any

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class of events we mentioned to It, would have to be due to a law being a special sort of fact about the universe.

Davidson himself notes this presupposition of his argument, and places the burden of significance squarely on the notion of law. What he then goes on to say about laws is unfortunately in part predicated on the logical positivists' very unsuccessful approach to the subject, and in part deliberately non-committal: There is (in my view) no non-question begging criterion of the lawlike, which is not to say that there are no reasons in particular cases for a judgment (Essays, 217). This statement, which begins his discussion of laws, itself presupposes the positivists' idea that laws are simply the truths among a class of statements (the lawlike ones) singled out by some common element of form or meaning, rather than by what the world is like. (Davidson comments nomologicality is much like analyticity, as one might expect since both are linked to meaning (p. 218). This presumption was later strongly criticized, for example by Dretske; at this point we should note only that it is dubitable, and not innocuous. I do not mean to go further into how Davidson discusses laws here; the point I wanted to make should now be clear.

The assumptions involved are that there is a significant concept of natural law, that the distinction between laws and truths in general is non-anthropomorphic and concerns what the world is like, and that the correct account of laws must do justice to all this. These are indispensable to Davidson's classification of philosophical positions on mind and matter, to the arguments for his position, and for the significance of that position. ⁹ This is a striking illustration of how general philosophy had, by our century, learned to rely on this notion of law.

4. Criteria of Adequacy for Accounts of Laws

If we do have the concept of a law of nature, this must mean at least that we have some clear intuitions about putative examples and counterexamples. These would be intuitions, for example, about what is and what is not, or what could be and what could not be, a law of nature, if some sufficiently detailed description of the world is supposed true. It does not follow that we have intuitions of a more general sort about what laws are like. But when we are offered ideas of this more general sort, we can test them against our intuitions about specific examples.

The use of such examples and our intuitive reactions to them serves at least to rule out overly simplistic or nave accounts of laws of nature. Their use has also led to a number of points on which, according to the literature, all accounts of laws must agree. None of these points is entirely undisputed, but all are generally respected.

Disagreements about the criteria should not dismay us at the outset. As Wittgenstein taught, many of our concepts are cluster conceptsthey have an associated cluster of criteria, of which only most need be satisfied by any instance. The more of the criteria are met, the more nearly we have a clear case. This vagueness does not render our concepts useless or emptyour happiness here as elsewhere depends on a properly healthy tolerance of ambiguity.

In what follows I shall discuss about a dozen criteria found in the literature. Some are less important, or more controversial than others. We can use them to dismiss some nave ideas, especially cherished by empiricistsand in subsequent chapters bring them to bear on the main remaining accounts of law. Nowhere should we require that all the criteria be met; but any account should respect this cluster as a whole.

Universality

The laws of nature are universal laws, and universality is a mark of lawhood. This criterion has been a great favourite, especially with empiricists, who tend to be wary of nearly all the criteria we shall discuss subsequently. There is indeed nothing in the idea of universality that should make philosophical hackles rise, nor would there be in the idea of law if a law stated merely what happens always and everywhere. The hope that this may be so must surely account for the curiously uncritical attitude toward this notion to be found in even the most acute sceptics:

Whitehead has described the eighteenth century as an age of reason based upon faiththe faith in question being a confidence in the stability and regularity of the universal frame of Nature. Nothing can better illustrate Hume's adherence to this faith, and its separation in his mind from his philosophical scepticism, than his celebrated Essay Of Miracles. The very man who proves that, for all we can tell, anything may be the cause of

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anything, was also the man who disproved the possibility of miracles because they violated the invariable laws of Nature. ¹⁰

That does not make Hume inconsistent. If what a law is concerns only what is universal and invariable, the faith in question could hardly impugn Hume's scepticism about mysterious connections in nature beyond or behind the phenomena. For in that case it would merely be a faith in matters of fact, which anyone might have, and which would notunlike the monkish virtuesbar one from polite society (the standard Hume himself so steadfastly holds out to us).

Unfortunately this mark of universality has lately fallen on hard times, and that for many reasons. Let us begin with the point that universality is not enough to make a truth or law of nature. No rivers past, present, or future, are rivers of Coca-Cola, or of milk. I think that this is true; and it is about the whole world and its history. But we have no inclination to call it a law. ¹¹ Of course we can cavil at the terms river, Coca-Cola, or milk. Perhaps they are of earthly particularity. But we have no inclination to call this general fact a law because we regard it as a merely incidental or accidental truth. Therefore we will have the same intuition, regardless of the terms employed. This is brought out most strikingly by parallel examples, which employ exactly the same categories of terms, and share exactly the same logical form, yet evoke different responses when we think about what could be a law. The following have been discussed in various forms by Reichenbach and Hempel: ¹²

I guess that both are true. The first I'm willing to accept as putatively a matter of law, for the critical mass of uranium will prevent the existence of such a sphere. The second is an accidental or incidental factthe earth does not have that much gold, and perhaps no planet does, but the science I accept does not rule out such golden spheres. Let us leave the reasons for our agreement to one sidethe point is that, if I could be law, if only a little law, and 2 definitely could not, it cannot be due to a difference in universality.

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Another moral that is very clear now is that laws cannot be simply the true statements in a certain class characterized in terms of syntax and semantics. There is no general syntactic or semantic feature in which the two parallel examples differ. So we would go wrong from the start to follow such writers as Goodman, Hempel, and Davidson in thinking of the laws as the true lawlike statements.

We can agree in the intuitions invoked above, before any detailed analysis of universality. But we have also already discerned some reason to think that the analysis would not be easy. In fact, it is extremely difficult to make the notion precise without trivializing it. The mere linguistic form All . . . are . . .  is not a good guide, because it does not remain invariant under logical transformations. For example, Peter is honest is in standard logic equivalent to the universal statement Everyone who is identical with Peter, is honest. To define generality of content turns out to be surprisingly difficult. In semantics, and philosophy of science, these difficulties have appeared quite poignantly. ¹³ Opinions in the literature are now divided on whether laws must indeed be universal to be laws. Michael Tooley has constructed putative counterexamples. ¹⁴ David Armstrong's account requires universality, but he confesses himself willing to contemplate amendment. ¹⁵ David Lewis's account does not require it. ¹⁶ In Part III we shall find an explication of generality allied to concepts of symmetry and invariance. While I regard this as important to the understanding of science, the generality we shall find there is theory-relative.

The criterion of universality, while still present in discussion of laws, is thus no longer paramount.

Relations to Necessity

In our society, one must do what the laws demand, and may do only what they do not forbid. This is an important part of the positive analogy in the term laws of nature.

Wood burns when heated, because wood must burn when heated. And it must burn because of the laws which govern the behaviour of the chemical elements of which wood and the surrounding air are composed. Bodies do not fall by chance; they must fall because of the law of gravity. In such examples as these we see a close connection between law and must, which we should stop to analyse.

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Inference.

The most innocuous link between law and necessity lies in two points that are merely logical or linguistic. The first is that if we say that something is a law, we endorse it as being true. The inference

is warranted by the meaning of the words. This point may seem too banal to mentionbut it turns out, surprisingly, to be a criterion which some accounts of law have difficulty meeting. One observes of course that the inference is not valid if of nature is left out, since society's laws are not always obeyed. Nor does it remain valid if we replace law of nature by conjecture or even well-confirmed and universally accepted theory. Hence the validity must come from the special character of laws of nature. In Chapter 5, the problem of meeting this criterion will be called the problem of inference.

Intensionality.

The second merely logical point is that the locution It is a law that is intensional. Notice that the above inference pattern (1) does remain valid if we replace a law of nature by true. But something important has changed when we do, for consider the following argument:

This is certainly correct, but loses its validity if we now replace true again by a law of nature. Another example would be this: suppose that it is a law that diamonds have a refraction index > 2, and that as a matter of fact all mankind's most precious stones are diamonds. It still does not follow that it is a law that all mankind's most precious stones have a refraction index > 2. Here we see the distinction between law and mere truth or matter of fact at work.

Our first two criteria are therefore merely points of logic, and I take them to be entirely uncontroversial.

Necessity Bestowed.

The moon orbits the earth and must continue to do so, because of the law of gravity. This illustrates the inference

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from It is a law that A to It is necessary that A; but this must be properly understood.

The medievals distinguished the necessity of the consequence from the necessity of the consequent. In the former sense it is quite proper to say If all mammals have hair then whales must have hair, because whales are mammals. The must indicates only that a certain consequence follows from the supposition. For law this point was therefore already covered above. The criterion of necessity bestowed is that there is more to it: if It is a law that A is true then also, rightly understood, It is necessary that A is true. This necessity is then called physical necessity or nomological necessity (and is now often generalized to physical probability).

Empiricists and nominalists have always either rejected this criterion or tried to finesse it. For they believe that necessity lies in connections of words or ideas only, so ultimately the only necessity there can be lies in the necessity of the consequence. This is not altogether easy to maintain, while acknowledging the preceding points of logic. Yet their persistent attempts to reconstrue the criterion of necessity bestowed, so that it is fulfilled if properly understood, show the strength of the intuition behind it. ¹⁷

Necessity Inherited.

There is a minority opinion that what the laws are is itself necessary. ¹⁸ This point definitely goes beyond the preceding, for logic does not require what is necessary to be necessarily necessary. More familiar is the idea that there are many different ways the world could have been, including differences in its laws governing nature. If gravity had obeyed an inverse cube law, we say, there would have been no stable solar systemand we don't think we are contemplating an absolute impossibility. But we could be wrong in this.

Of course, if laws are themselves necessary truths, their consequences would inherit this necessity. Therefore the strong criterion of necessity inherited entails that of necessity bestowed. And since what is necessary must be actual, the criterion of necessity bestowed entails that of inference. The entailments do not go in the opposite direction. So three of the criteria we have formulated here form a logical chain of increasing strength.

Explanation

Such writers as Armstrong insist that laws are needed to explain the phenomena, and indeed, that there are no explanations without laws. This is not in accord with all philosophical theories of explanation. ¹⁹ A more moderate requirement would be that laws must be conceived as playing an indispensable role in some important or even pre-eminent pattern of explanation.

There does indeed appear to be such a pattern, if there is an intimate connection between laws and necessity (and objective probability). It may even be the pre-eminent pattern involved in all our spontaneous confrontations with the world. Witness that Aristotle made it the key to narrative and dramatic structure in tragedy:

And these developments must grow out of the very structure of the plot itself, in such a way that on the basis of what has happened previously this particular outcome follows either by necessity or in accordance with probability; for there is a great difference in whether these events happen because of those or merely after them. (Poetics, 52^a1722)

This account of tragedy bears a striking resemblance to Aristotle's account of how science must depict the world, in his Physics. ²⁰ The parallel is no accident, though one must admit that Aristotle's demands upon our understanding of nature persisted longer than those he made upon our appreciation of literature.

What exactly is this criterion, that laws must explain the phenomena? When a philosopheras so many doraises explanation to pre-eminence among the virtues, the good pursued in science and all natural inquiry, he or she really owes us an account of why this should be so. What is this pearl of great price, and why is it so worth having? What makes laws so well suited to secure us this good? When laws give us satisfying explanations, in what does this warm feeling of satisfaction consist? There are indeed philosophical accounts of explanation, and some mention laws very prominently; but they disagree with each other, and in any case I have not found that they go very far toward answering these questions. ²¹

Hence we should not get very far with this criterion for accounts of laws, if its uses depended greatly on the philosophical opinions of what explanation is. Fortunately there are two factors which keep us from being incapacitated here. The first factor is the very large measure of agreement on what counts as explanation when we are confronted with specific, concrete examples. The other factor

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is the great degree of abstraction which characterizes many discussions of law. In Chapter 6, for example, we shall be able to take up Dretske's and Armstrong's arguments concerning what is for them the crucial argument form of Inference to the Best Explanationand its relation to lawswithout ever having to reproach them for the fact that they nowhere tell us what an explanation is.

We shall encounter a certain tension between the criteria regarding the connections of law with necessity on the one hand, and with science on the other. Here the concept of explanation could perhaps play an important mediating role: If explanation is what we look for in science, while necessity is crucial to explanation and law crucial to necessity, then that tension may perhaps be aufgehoben in a higher unity. We shall have to see.

Prediction and Confirmation

That there is a law of gravity is the reason why the moon continues to circle the earth. The premiss that there is such a law is therefore a good basis for prediction. The second traditional argument which I briefly sketched aboveand illustrated from Peirce's lecturegoes on: and if we deny there is such a reason, then we can also have no reason for making that prediction. We shall have no reason to expect the phenomenon to continue, and so be in no position to predict.

If there is a problem with this argument today, it is surely that we cannot be so ready to equate having reason to believe that A with believing that there is a reason for A. Linguistic analysis in philosophy makes us very wary of such pretty rhetoric. But the equation might perhaps hold for the special case of empirical regularities and laws. Certainly, a form of this second traditional argument is found very prominently in Armstrong's book. After canvassing some views on what laws are, he notes a possibility which he says was brought to his attention by Peter Forrest:

There is one truly eccentric view. . . . This is the view that, although there are regularities in the world, there are no laws of nature. . . . This Disappearance view of law can nevertheless maintain that inferences to the unobserved are reliable, because although the world is not law-governed, it is, by luck or for some other reason, regular. ²²

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Armstrong replies immediately that such a view cannot account for the fact that we can have good reasons to think that the world is regular. He gives an argument for this, which I shall discuss in Chapter 6.

A little of the recent history of confirmation appeared along the way in the article by Davidson which we discussed above. While Davidson attempts no definition or theory of what laws are, he says among other things that laws are general statements which are confirmed by their instanceswhile this is not always so for general statements. This makes sense if laws are the truths among lawlike statements, and if in addition we (who assess the evidence) can distinguish lawlike statements from other generalities. For else, how can instances count for greater confirmation?

But this idea receives rather a blow from the parallel gold and uranium examples we discussed above. These parallel examples are so parallel in syntactic form and semantic character that the independent prior ability to distinguish lawlike from other general statements is cast into serious doubt.

We should also observe that for writers on laws there isand perhaps must bea crucial connection between confirmation and explanation. For consider the following argument: that it is a law that P could be supported by claims either of successful explanation or of successful prediction (or at least, successful fitting of the data). But prediction cannot be enough, for the second sort of claim works equally well for the bare statement that A: It is a law that A entails or fits factual data only in so far as, and because, A does. Hence confirmation for the discriminating claim It is not only true but a law that A can only be on the basis of something in addition to conforming evidence. One traditional candidate for this something extra is successful explanation.

This observation gives us, I think, the best explanation of why advocates of laws of nature typically make Inference to the Best Explanation the cornerstone of their epistemology.

Counterfactuals and Objectivity

Philosophy, being a little other-worldly, has always been fascinated with the conditional form If (antecedent) then (consequent). When the antecedent is false (the conditional is contrary to fact or counterfactual) what speculative leaps and fancies are not open

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to us? If wishes were horses then beggars would ride; if gravity had been governed by an inverse cube law there would have been no stable solar system; if Caesar had not crossed the Rubicon, . . . . Being also a little prosaic, the philosopher sets out to find the bounds of fancy: when must such a conditional be true, when false?

There is one potentially large class of cases where the answer is clear. If B follows from A with necessity, then If A then B is true and If A then not B is false. Thus if iron must melt at 2000C, it follows that this iron horse-shoe would melt if today it were heated to that temperature . . . this is clear even if the horseshoe remains at room temperature all day. At midnight we will be able to say, with exactly the same warrant, that it would have melted if it had been heated to 2000C. Many other such conditionals command our intuitive assent: Icarus' father too would have fallen if his wings had come loose, and so would I if I had stepped off the little platform when I went up the cathedral tower in Vienna. We observe that in all such cases we intuitively agree also to describe our warrant in terms of laws. These facts about iron and gravity are matters of law; if it is a law then it must be so; and if it must be so then it will or would be so if put to the test.

This large class of cases falls therefore very nicely under the previous criterion of necessity bestowed. But the requirement that laws be the sort of thing that warrant counterfactuals, has a much greater prominence in the literature. Is there more to it?

In the mid-1940s, Nelson Goodman and Roderick Chisholm made it clear that there are mysteries to the counterfactual conditional, which had escaped their logical treatment so far. This treatment did indeed fit necessary implications. Typical sanctioned argument patterns include

But can all conditionals derive from necessities in this way? Consider: if I had struck this match, then it would have lit. It does not follow that if I had struck this match, and it had been wet at the time, then it would have lit. Nor, if I agree that the latter is false, do I need to retract the former. I can just say: well, it wasn't wet. We see therefore that counterfactual conditionals violate

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the principles of reasoning which govern strict or necessary conditionals.

How are we to explain this mystery? Goodman did not explain it, but related it to laws. ²³ We can, he said, support a counterfactual claim by citing a law. We cannot similarly support it by merely factual considerations, however general. For example it is a fact (but not a matter of law) that all coins in Goodman's pocket were silver. We cannot infer from this that if this nickel had been in his pocket then, it would have been silver. On the other hand it is also a fact and a matter of law that silver melts at 960.5C. Therefore if this silver had been heated to that degree, it would have melted. This observation, Goodman thought, went some way toward clearing up the mystery of counterfactual conditionals. The mystery was not thereby solved, so the connection was inverted: giving warrant for counterfactual conditionals became the single most cited criterion for lawhood in the post-war literature.

But the mystery was solved in the mid-1960s by the semantic analysis due to Robert Stalnaker and extended by David Lewis. Unfortunately for laws, this analysis entails that the violations of those principles of inference that work perfectly well for strict conditionals are due to context-dependence. The interesting counterfactuals which do not behave logically like the strict ones do not derive from necessities alone, but also from some contextually fixed factual considerations. Hence (I have argued elsewhere) science by itself does not imply these more interesting counterfactuals; and if laws did then they would have to be context-dependent. ²⁴ Robert Stalnaker has recently replied to this that science does imply counterfactuals, in the same sense that it implies indexical statements. ²⁵ An example would be:

This is a context-dependent sense of implies (not of course the sense which I had in mind), because the referent of you depends on context. Stalnaker's point is quite correct. But it leads us to conclude at best that the speaker may believe that some law is the case, and holds its truth-value fixed in a tacit ceteris paribus clause (which gives the counterfactual sentence its semantic content in this context). This is certainly correct, but is equally correct for any other sort of statement, and cannot serve to distinguish laws from mere truths or regularities. I suspect that the real use of Goodman's requirement concerned counterfactuals considered true in cases where the corresponding physical necessity statement is also implied. If so, the requirement coincided in philosophical practice with the requirement of bestowed necessity.

Context-Independence.

In view of the above, however, it is important to isolate the sense in which law statements cannot be context-dependent. Stalnaker's sort of example leaves us with the requirement:

Related to the context-independence of the locution It is a law that, but not at all the same, is the point that laws are to be conceived of as objective.

Objectivity.

Whether or not something is a law is entirely independent of our knowledge, belief, state of opinion, interests, or any other sort of epistemological or pragmatic factor. There have definitely been accounts of law that deny this. But they have great difficulty with such intuitively acceptable statements as that there may well be laws of nature which not only have not been discovered and perhaps never will be, but of which we have not even yet conceived. ²⁶

Relation to Science

We come now to a final criterion which is of special importance. Laws of nature must, on any account, be the sort of thing that science discovers. This criterion is crucial, given the history of the concept and the professed motives of its exponents.

This criterion too is subject to a number of difficulties. First of all, there is no philosophically neutral account of what science discovers, or even what it aims to discover. Secondly, although the term law has its use in the scientific literature, that use is not without its idiosyncracies. We say: Newton's laws of motion, Kepler's laws of planetary motion, Boyle's law, Ohm's law, the law of gravity. But Schroedinger's equation, or Pauli's exclusion principle, which are immensely more important than, for example,

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Ohm's law, are never called laws. The epithet appears to be an honour, and often persists for obscure historical reasons.

Attempts to regiment scientific usage here have not been very successful. Margenau and Lindsay note disapprovingly that other writers speak of such propositions as copper conducts electricity as laws. ²⁷ They propose that the term should be used to denote any precise numerical equation describing phenomena of a certain kind. ²⁸ That would make Schroedinger's equation, but not Pauli's exclusion principle, a law. Even worse: it would be quite easy to make up a quantitative variant of the rivers of Coca-Cola example which would meet their criterion trivially.

Faced with this situation, some writers have reserved law for low-level, empirical regularities, thus classifying the law of conservation of energy rather than Boyle's law as terminological idiosyncrasy. To distinguish, these low-level laws are also called phenomenological laws, and contrasted with basic principles which are usually more theoretical. Science typically presents the phenomenological laws as only approximate, strictly speaking false, but useful. According to Nancy Cartwright's stimulating account of science, the phenomenological laws are applicable but always false; the basic principles accurate but never applicable. ²⁹ It is therefore not so easy to reconcile science as it is with the high ideals of those who see it as a search for the true and universal laws of nature.

The criterion of adequacy, that an account of laws must entail that laws are (among) what science aims to discover, is therefore not easy to apply. Certainly it cannot be met by reliance on a distinction embodied in what scientists do and do not call a law. Nor, because of serious philosophical differences, can it rely on an uncontroversial notion of what the sciences (aim to) discover. The criterion of objectivity we listed earlier, moreover, forbids identification of the notion of law with that of a basic principle or any other part of science, so identified. For if there are laws of nature, they would have been real, and just the same even if there had been no scientists and no sciences.

It appears therefore that in accounts of law, we must try to discern simultaneously a view of what science is and of what a law is, as well as of how the two are related. These views must then be evaluated both independently and in terms of this final criterion, that they should stand in a significant relationship.

Earlier in this century, the logical positivists and their heirs

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discussed laws of nature, and utilized that concept in their own explications and polemics. I shall not examine those discussions in any detail. If we look at their own efforts to analyse the notion of law, we find ourselves thoroughly frustrated. On the one hand we find their own variant of the sin of psychologism. For example, there is a good deal of mention of natural laws in Carnap's The Logical Syntax of Language. But no sooner has he started on the question of what it means to say that it is a law that all A are B, than he gets involved in the discussion about how we could possibly verify any universal statement. On the other hand there is the cavalier euphoria of being involved in a philosophical programme all of whose problems are conceived of as certain to be solved some time later on. Thus in Carnap's much later book Philosophical Foundation of Physics, ³⁰ we find him hardly nearer to an adequate analysis of laws or even of the involved notions of universality or necessitybut confident that the necessary and sufficient conditions for lawhood are sure to be formulated soon. The culmination of Carnap's, Reichenbach's, and Hempel's attempts, which is found in Ernest Nagel's The Structure of Science, was still strangely inconclusive. In retrospect it is clear that they were struggling with modalities which they could not reduce, saw no way to finesse, could not accept unreduced, and could not banish.

Having perceived these failures of logical empiricism, some philosophers have in recent years taken a more metaphysical turn in their accounts of laws. I shall focus my critique on those more recent theories.

5. Philosophical Accounts: The Two Main Problems

Of the above criteria, never uniformly accepted in the literature, five seem to me pre-eminent. They are those relating to necessity, universality, and objectivity and those requiring significant links to explanation and science.

But apart from the more or less piecemeal evaluation these allow, of all proffered philosophical accounts of laws, there will emerge two major problems. I shall call these the problem of inference and the problem of identification. As we shall see, an easy solution to either spells serious trouble from the other.

The problem of inference is simply this: that it is a law that A,

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should imply that A, on any acceptable account of laws. We noted this under the heading of necessity. One simple solution to this is to equate It is a law that A with It is necessary that A, and then appeal to the logical dictum that necessity implies actuality. But is necessary univocal? And what is the ground of the intended necessity, what is it that makes the proposition a necessary one? To answer these queries one must identify the relevant sort of fact about the world that gives law its sense; that is the problem of identification. If one refuses to answer these queriesby consistent insistence that necessity is itself a primitive factthe problem of identification is evaded. But then one cannot rest irenically on the dictum that necessity implies actuality. For necessity, now primitive and unexplained, is then a mere label given to certain facts, hence without logical forceBernice's Hair does not grow on anyone's head, whatever be the logic of hair.

The little dialectic just sketched is of course too elementary and nave to trip up any philosopher. But it illustrates in rudimentary fashion how the two problems can operate as dilemma. We shall encounter this dangerous duet in its most serious form with respect to objective chance (irreducibly probabilistic laws), but it will be found somehow in many places. In the end, almost every account of laws founders on it.

Besides this dialectic, the most serious recurring problem concerns the relation between laws and science. The writers on laws of nature by and large do not so much develop as presuppose a philosophy of science. Its mainstay is the tenet that laws of nature are the sciences' main topic of concern. Even if we do not require justification for that presupposition, it leaves them no rest. For they are still required to show that science aims to find out laws as construed on their account. This does not follow from the presupposition, even if it be sacrosanct.

While I cannot possibly examine all extant accounts of laws, and while new ones could spring up like toadstools and mushrooms every damp and gloomy night, these problems form the generic challenge to all philosophical accounts of laws of nature. In the succeeding three chapters, we shall see the three main extant sorts of account founder on them.

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