Ideal Science: David Lewis's Account of Laws 1
The account of laws David Lewis offers us is the least metaphysical of all those we shall examine. It is nearest to the straight empiricist regularity view, and attempts from the outset to put laws in touch with science. Lewis is well known to be a realist with respect to alternative possible worlds, but we shall see that this realism is not crucial here. The only metaphysics crucially involved is anti-nominalism: that is, a realist construal of the difference between natural and merely arbitrary classifications. 2 In addition, we shall see that Lewis's account has prima facie considerable success in meeting the criteria listed in the preceding chapter. But I shall argue that the successes are, in the end, only apparent. 3
The Definition of Law
Lewis first presented his account in Counterfactuals. There he refers to F. P. Ramsey's 1928 account of laws as consequences of those propositions which we should take as axioms if we knew everything and organized it as simply as possible in a deductive system. 4 John Earman points out that Ramsey was perhaps following John Stuart Mill's System of Logic which says about the expression Laws of Nature:
Scientifically speaking, [it] is employed . . . to designate the uniformities when reduced to their most simple expression. . . . According to one mode of expression, the question, What are laws of nature? may be stated thus: What are the fewest and simplest assumptions, which being granted, the whole existing order of nature would result? Another mode of stating the question would be thus: What are the fewest general propositions from which all uniformities which exist in the universe might be deductively inferred? 5
We shall look at Lewis's refinements in a moment, but note first that this sort of approach gives no indication of dependence on realism about possible worlds, or modal realism. While Lewis may present certain aspects of the account in terms of possible worlds, he can be followed there very well by someone who regards those as the theoretical fictions of semantics. Mill offers us here an account which seems really a sort of pattern for accounts of law, that could be elaborated by almost anyone, regardless of his views concerning the nature of necessity or of science.
The first difficulty to be faced is that there are innumerable true theories about the world, all of which entail the uniformities there actually are in nature. Mill suggests that we must pick out the theory which can be axiomatized by the fewest and simplest or (equivalently?) the fewest general propositions. This rather vague response to the difficulty leaves a number of open questions. Will there be a unique such theory? And is simplicity the only thing that matters? And is entailing the uniformities the single factual desideratum besides truth? And what is a uniformity anyway, or simplicity?
Lewis's refinement meets this difficulty as follows: There are innumerably many true theories (in the sense of: deductively closed sets of true sentences). Some of these are simpler than others, some are stronger (i.e. more informative) than others. What we value in science is both simplicity and strength, so we wish for a properly balanced combination. The laws are those sentences describing regularities which are common to all those true theories that achieve a best combination of simplicity and strength. (If there can be better and better combinations ad infinitum, this definition needs a technical adjustment, which I leave aside.)
I have written here as if simplicity, strength, and balance are as straightforward as a person's weight or height. Of course they are not, and the literature contains no account of them which it would be fruitful to discuss here. Strength must have something to do with information; perhaps they are the same. Simplicity must be a quite different notion. Note also that we have here three standards of comparison: simplicity, strength, and balance. The third is needed because there is some tension between the first and second, which cannot be jointly maximized. Sometimes a simple theory is also more informative. But if we have a simple theory, and just add a bit of information to it, so as to make it stronger, we will almost always reduce its simplicity. These are intuitive considerations that
surely everyone shares. As soon as we reflect on balance, however, the certainty of our intuitions dwindles fast. A person who weighs 170 pounds is overweight if he is five foot and underweight if he is six foot threethere we have a notion of balance. But how shall we gauge when a gain in simplicity is well bought for a loss of information? When does gain equal loss for two such disparate virtues?
So simplicity, strength, and balance are not straightforward. To utilize these motions uncritically, as if they dealt with such well-understood triads as under five foot five, over 200 pounds, overweight may be unwarranted. Still I shall leave the legitimacy of these notions unchallenged, in as far and for as long as I can. 6
As Lewis himself notes, there is a much more serious difficulty. 7 This difficulty appears exactly after we have agreed to the use of simplicity as a criterion. In what language(s) are these theories formulated? Suppose that we form a new language, in which simple predicates correspond to long and cumbersome constructions in our own. If we then translate two of our theories into this new language, the verdict of simplicity between them may be reversed. This is not merely an academic problem. When Poincar asserted that, despite the logical possibility of doing otherwise, physics would always remain wedded to Euclidean geometry, he based this assertion on a verdict of simplicity. But he spoke at a time before the exploitation of differential geometry. Thereafter, these very considerations of simplicity told against the retention of Euclidean geometry, just because geometric language had become so much more general and rich in its descriptive powers. Philosophers of course have discussed more academic examples, with novel predicates like: grue, meaning green if examined before the year 2000 and blue if not.
Should we count as best theories those whose translations win the competition in every language we could construct? Then we cannot expect to find any best theories at all. Even if one true theory in our language has, among theories in that language, the optimum combination of simplicity and strength, its translation into other language will not, in general, preserve this virtue. If on the other hand we ask for those theories which each have that pre-eminent status in some language, we must expect to obtain such a large class that they may have little more than tautologies in common.
The only remedy open to Lewis, as he himself explains, is to restrict the language(s) in which the candidate theories are allowed to be formulated. On what basis could this restriction be made? Lewis assumes that the correct language has the simple extensional structure studied in standard logic, with predicates as only non-logical terms. Each predicate has an extension: the class of things to which this predicate applies. Assert now that some classesfor example, the class of starsare marked by a real distinction, and other classesfor example, that of people whose names begin with Mare merely an arbitrary grouping. Then a good way to select a correct language appears: the basic predicates must each have as extension such a natural class. Other predicates may be introduced by definition only. Simplicity must be judged before any definitions are introduced.
By adopting this remedy, Lewis makes it once more plausibleat least prima faciethat the class of laws, as defined, will be an appreciably informative set of sentences. The remedy certainly has historical precedent in metaphysics. Indeed, this insistence on the distinction between real and verbal or arbitrary classifications has often enough been taken as a defining difference between late medieval realists and nominalists. 8 So I shall call the position here adopted anti-nominalism.
The account so far already has notable virtues. A law is a statement; but since theories are logically closed, any logical equivalent of a law is also a law. A law is true; but true statements, however general or otherwise syntactically or semantically privileged, are not always laws. What is a law is an objective question, whose answer is independent of what science is actually developed in history, and indeed, independent of any other merely historical, psychological or otherwise anthropocentric fact. 9 And surely science attempts to find for us true, strong, simple theories; so a fortiori, by definition even, it seems that science is in pursuit of laws as defined by Lewis. Whether that is really so, we must now investigate.
2. The Definition of Necessity
So far Lewis's account answers the question of what the laws arein our world or, mutatis mutandis, in any world. But if something is a law, then it is not only true, but necessary. How does Lewis
honour this criterion? Since he is well known for his realism about possible worlds, we expect to see this come into play now. And so it does; but it does not play a crucial role for the account of laws as such. For the upshot is simplyas we shall seethat It is necessary that A is said to be true if A is implied by the laws of nature. This is, in effect, the definition of necessity. Since the notion of law has been defined previously, without reference to what is true in any other world, this way of honouring the criterion is open to even the most anti-metaphysical empiricist as well.
I shall be brief here about the connection between necessity and possible worlds, because I shall be discussing it at greater length in the next chapter (where it will play a crucial role). The logical warrant for the ideas which I shall now briefly describe will also be discussed at greater length there.
The logic of the word necessary is such that any definition of the form It is necessary that A is true in a world x if and only if A is true in every world which is possible relative to x, meets the logical criteria. There are distinct senses of necessary which in this way correspond to different relations of relative possibility. It suffices therefore for Lewis to define a particular such relation, relative physical possibility. For this he offers:
world y is physically possible relative to world x exactly if the laws of x are all true in y.
Note that he does not require that the laws of x also be laws of y; they need merely be true statements in that other world. The relevant sense of necessary is introduced by
It is physically necessary that A is true in world x if and only if A is true in every world which is physically possible relative to x.
And now we can deduce from these definitions:
It is physically necessary that A is true in world x if and only if A is implied by the laws of x.
Here implies means semantically implies; that is, certain premisses imply A exactly if A is true in every possible world in which those premisses are true.
This is a wonderful result. We may harbour a little doubt, due to its strength. Could not some individual matter of fact be
physically necessary without being entailed by the laws? But that question aside, a major desideratum has apparently been met.
Is this part of the account metaphysical, in the sense abhorrent to empiricists? I think not. We have here only a semantic analysis, a proposal for truth conditions which, in effect, define the clause It is necessary that. If one is not a realist about possible worlds, one may tacitly read world as model of our language. The definitional approach Lewis uses here is available to all. Anyone who feels he has an adequate notion of law, can go on to equate physical necessity with the status of being entailed by the laws. The use of this move by someone who believes all possible worlds to be real, neither weakens nor strengthens its benefits. Whether the benefits of this move are real, and not merely apparent, for the general account of laws, is an independent question.
This completes Lewis's account of laws and necessity. Its assumptions seem almost entirely acceptable even to empiricists (perhaps entirely, to many) and its prima-facie successes are remarkable. So why not be content to accept this reconstruction as conclusive?
3. Laws Related to Necessity
As we have found, on Lewis's account, the assertion that it is a law that A entails that it is physically necessary that A. This meets one of our main criteria. As we also saw, Lewis shows us here by example how the criterion can be met, through a stipulative definition, by anyone who feels he has already an adequate notion of law.
It is hard, however, to escape the feeling that if the criterion can be met satisfactorily in this way, then it must be devoid of all probative force. Doesn't Lewis meet the criterion by robbing it of significance?
The intuition behind the criterion is that the existence of a law that A makes it physically necessary that A (and, a fortiori, makes it true that A). I do not quite know what to make of this notion of making something necessary or true. Of course I know well enough the traditional terminology of the ground of necessity, and the recently sprung-up terminology of truth-makers. I also know the Aristotelian tradition of real necessities (as opposed to verbal necessities) grounded in real, substantial natures. So I, and you, gentle reader, have much circumstantial evidence to suspect that if laws are merely definitionally connected with necessity, then they won't do the right job. But none of this needs bother Lewis, if it only points to possible unhappiness with his account among eventual advocates of archaic or anachronistic ideas.
To some extent, the complaint may be formulated in terms of explanation: that something is a law should explain why something is physically necessaryand no fact can explain anything to which it is definitionally equivalent. I would like to postpone this form of the objection to another section below.
What if Lewis replied that we should not reproach him for honouring by means of a definition any equivalence which we independently accept? Then we may still fault him for not having tried to account for this equivalence, if there is one. Why should anyone accept it, let alone take it to be so basic that it might as well be built into our very language? This is not an idle question, I think. Let me give an example.
Consider the view that spatio-temporal relations among events are not sui generis but derive from physical relations such as connection by signals, physical contact, and identity through time. (We need not assume that the reduction is definitional, nor any particular version of the relational theory of space-time.) On such a view, one might wish to assert that it is physically impossible for any signal to connect events E and F. One would wish to assert this exactly if one held that E and F are simultaneous in some frame of reference. But this assertion cannot follow from any independent facts about E and F via general laws. For the only relevant facts concern their space-like separation, which derives from facts about signal connectabilitythe very subject of our statement.
Perhaps some will find this example fanciful, because they consider relational theories of space-time absurd. So let us delve into our uneasiness about the definitional link between law and necessity in yet another way.
To say that we have the concept of a law of nature must imply at least that we can mobilize intuitions to decide on proffered individual examples. Let us then consider a possible world in which all the best true theories, written in an appropriate sort of language, include the statement that all and only spheres are gold. To be
concrete, let it be a world whose regularities are correctly described by Newton's mechanics plus law of gravitation, in which there are golden spheres moving in stable orbits around one another, and much smaller iron cubes lying on their surface, and nothing else. If I am now asked whether in that world, all golden objects are spherical because they must be spherical, I answer No. First of all it seems to me that there could have been little gold cubes among the iron ones, and secondly, that several of the golden spheres could (given slightly different initial conditions) have collided with each other and thus altered each other's shapes. If my intuitions are a bit strong for your taste, perhaps you will at least grant that you feel no intuitive inclination to say Yes or to assert that the generalization is a law. But on Lewis's view it is a law in this world that all golden objects are spherical, and also physically necessary. I say this on the basis of intuitive judgements about simplicity and strength and their balance; but for such a simple world it does not seem difficult to find the best true theory.
Could it be argued that some presumption of laws was involved in my use of the terms gold and iron (along the lines perhaps of Wilfrid Sellars's Concepts as Involving Laws and Inconceivable Without Them)? I think the response is not open to Lewis, because his account of laws requires that the truth-values of all non-modal sentences be settled beforehand, before the laws can be identified. Could it be argued instead that the world I have described is indeed possible, but that I am wrong to balk at Lewis's conclusions about what its laws are? This would mean presumably that my intuitions are warped by my knowledge of gold and iron in our own world. But suppose I said the large spheres of that world were made not of gold, but of some substance I do not know, call it S. I am then willing to say that, as far as I can see, the simplest true descriptions of this world all contain the statements All spheres are S and All S-things are spheres. But do I feel I have the warrant to say that, in such a world, S-things must be spherical? I do not. Truth and simplicity just do not add up to necessity, as far as my intuitive reactions are concerned. No, I think that the consequences of Lewis's view for this sort of example can be swallowed only if one downgrades radically the connection between law and necessity.
4. Do Lewis's Laws Explain?
At first sight, it seems obviously true that laws of nature, in Lewis's sense, explain the phenomena. But after a second and third look, the mystery is rather that this should have seemed so at all. What could have led us to think so?
We may imagine the following train of thought (though Lewis definitely does not present it to us). Science explains; more generally, scientific theories explain. The best theories give the best explanations. But laws are the common part of all best theories. Therefore they are the ingredients present in any best, overall explanation of what the world is like. Surely that earns them the right to be called explanatory?
There are several assumptions here whose examination I wish to leave aside for now. 10 Even granting the assumptions, the therefore and surely hide a great deal. Consider: tautologies are a common part of all theories; but they are not explanatory. Also some very uninformative, near-tautologies are common to all the best theories, but would not be called explanatory, even if they mark the beginning of any explanation.
This is to the point for we do not know about the set of laws in Lewis's sensei.e. the common part of all the best theorieshow informative it is. That is because we do not know the diversity of best theories. So what could make us say at once: the common part of all the best theories must have the pre-eminent explanatoriness, which has always been claimed for laws of nature?
To give an analogy, suppose that the three material goods are money, houses, and land. I have as much of these as all rich men havedoes it follow that I am rich? No, for one has little money but much land, one has only a small garden but many houses, etc. What they all have is at least a little money, at least a little garden, and at least a little house. So do I, and am not rich at all.
This analogy is most troubling if one thinks of explanation
as requiring, as a minimum, the provision of sufficiently much relevant
information (sub specie whatever criteria of sufficiency and relevance
you like). The difficulty remains, I think, even if we connect explanation
essentially with unification rather than informationthe unification of loads
of little theories and bits of factual information, which are subsumed and no
longer isolated and separate. Thus to show why
achievement, we explain that Galileo's law for falling bodies
and Kepler's laws follow from
Let us turn to a second problem. We must still raise the independent question whether the best theories themselves really are the most explanatory. In a first, perhaps trivial form, this is the question whether simplicity and strength, properly balanced, make for explanatory power. (If so, the above point that the common elementi.e. the set of lawscannot be expected to have those virtues, poses a real problem.) To this, Lewis could respond that if he could be convinced they did not, he could revise the standard of comparison. Then he would say that the laws are what is common to the most explanatory, true theories, whatever explanatory means. But that is not such an easy way out as it looks. For there was a reason why Lewis chose simplicity and strength to begin: the evidence from reports about science, that something like those virtues are actually pursued there. Similar reports of course reach us from philosophers of science about the pursuit of explanation. But if this were indeed a third pursuit, might there not be a further trade-off, with scientists sometimes forgoing explanation for the sake of the other two desiderata? And if so, might it not be that what is common to the best theories is not only weaker than they, but drastically less explanatory, because the trade-off between explanation and the other virtues differs from best theory to best theory? That is exactly what is to be expected in such a caseand so the apparent way out is not a good way at all.
Would it matter if the laws by themselves do not form a very explanatory theory? Well, it matters if you take the conceptual link with explanation to be crucial to the idea of law. Here I have my third problem. If I write down the law of radioactive decay, it is simply a sentence that might, as far as looks and content are concerned, be a mere truth (so Lewis would say). Could the fact that this sentence is a theorem of all the best theories be cited as
the explanation of the present behaviour of the Geiger counter in the presence of radium?
Let us turn the question around. 11 Why would it not be regarded as the right sort of fact about the world? Well, if this fact explains why radium behaves in that fashion, should we not be able to say that in the absence of this fact, ceteris paribus, radium need not display this regularity? Suppose the contrary, therefore; suppose there is a best theory in which this sentence is not a theorem. That means presumably that some ultimate science treats the equation describing radioactive decay as an ancillary fact, theoretically isolated, which may be used in conjunction with deep principles of a totally different sort to explain the behaviour of Geiger counters, in a footnote. Suppose also that only one, or at least a small minority, of the best theories are like this; so if we could see them, we would regard them as admirable logical trickery, achieving the aims of science by far-fetched logical devices. How would this look to someone who takes the idea of law seriously, someone who is strongly inclined to insist that not mere facts but only laws can explain the phenomena? Could he see the existence of such a theory as showing that the putative law of radioactivity decay is not a law? He would instead say, I think, that the appearance of explanation can be produced by the logically outr, but not real explanation.
To sum up: I have four serious doubts about whether the laws of this world, in the sense of Lewis, explain what happens. The first is that explanation is crucially dependent on information, and that what is common to all the best explanations may not be informative enough to be explanatory itself. The second is that the best theories, by the criteria of simplicity and strength, may not be the best explanations. (They might not be, namely if explanation is crucially dependent on some other feature, which requires sacrifice of simplicity, strength, or the balance between them.) The third doubt is that the fact that something is a lawin the sense of Lewisis perhaps not the sort of fact that gives us an explanation at all, at least of the type laws were meant to give. Finally, I noted one additional doubt in passing: the initial impression here that of course Lewis's laws explain must largely come from their location at the ideal end of science, so to say; and I distrust that. This last doubt is the most weighty to me; but before turning to its examination, we must still stop to look at the properly metaphysical ingredient of Lewis's view.
5. Lewis's Anti-Nominalism 12
Anti-nominalism is the view that some classes correspond to real distinctions, and others do not. This view has appeared in many varieties, since Plato introduced it in his theory of Forms.
Thus some philosophers say that a class corresponds to a real distinction if its members have some property or universal in common, which nothing else shares. For example, they might say, all green objects have a property in common, namely the colour green. But the objects which are gruewhich means, examined before 2000 ad and green, or else not examined before 2000 ad and bluedo not have a special property in common. The predicate green stands for a real property and the predicate grue does not. This is one possible account of the idea, and it involves, besides anti-nominalism, a definite further idea about the the existence of a certain kind of abstract entities and their relation to ordinary objects. Other philosophers speak instead of natural kinds and say that mice do, and humans do, constitute natural kinds (mouse-kind and humankind) but their sum does not (there is no mouse-or-humankind).
Lewis pointed out quite correctly in his paper New Work for Universals that his account of laws could be saved from a serious problem by the addition of some such anti-nominalism. (See section 1 above.) The laws are to be taken as the theorems of all the best theories formulated in a correct language. A correct language is one whose predicates all correspond to real distinctions. If we call a class natural exactly if it marks such a natural classification, corresponding to a real distinction, then the requirement is: each predicate of a correct language applies to all and only the members of a certain natural class.
As Lewis also saw, it does not matter in the present context what form of anti-nominalism is embraced. Any of them will save him. So I shall also limit myself in this discussion to the minimum tenets of anti-nominalism. The division between natural classes and merely arbitrary or artificial classifications is just assumed to be drawn somehow.
Now if laws are to be what science hopes to provide in the end, then science had better hope to formulate its theories in a correct language. And the guardians of this correctness can only be the scientists themselves. What basis could there be for this hope? Is there anything in the process of scientific theorizing, theory choice, or theory evaluation which would tend to lead it to correct language?
A priori only two types of affirmative answers could be given here. We could suggest that humans have a special insight into the difference between natural and unnatural classes, and that this insight is one of the guiding factors in science. On the other hand, we could suggest that, without any such insight, scientists will tend to end up with natural predicates due to the ruthless weeding out of theories by empirical and/or theoretical success and failure. Let us look at each alternative in turn.
Is it plausible to think that we humans are naturally fit to distinguish real distinctions among all the ones we can describe? Recall that whatever we say must be combined with the following assertion: the most basic predicates of science will in the long run tend to correspond to real classes. But the distinctions which we use so easilygreen vs. blue, hard vs. soft, mouse vs. catdo not at all belong to the basic categories of physical science. Nor are they likely to do so in the future. Indeed, science has progressively undermined the primacy of those categories which have priority for us. Colours have had second-class citizenship for centuries, and the biological speciesparadigm for Aristotle's formshave lost their theoretical status with the advent of evolution.
Indeed, evolution suggests a status for the distinctions we naturally make, that removes them far from the role of fundamental categories in scientific description. Classification by colour, or currently stable animal-mating groups is crucial to our survival amidst the dangers of poison and fang. This story suggests that the ability to track directly certain classes and divisions in the world is not a factor that guides scientists in theory choice. For there is no such close connection between the jungle and the blackboard. The evolutionary story clearly entails that such abilities of discrimination were selected for, by a filtering process that has nothing to do with successful theory choice in general. Indeed, no faculty of spontaneous discrimination can plausibly be attributed a different status within the scientific account of our evolution. Even if successful theory choice will in the future
aid survival of the human race, it cannot be a trait selected for already in our biological history. 13
Perhaps the distinctions we are able to track directly, and those which science honours in its basic terminology, are both natural. Perhaps so; but the question was whether theory choice could tend to favour correct language because the scientists can tell directly what is a natural predicate and what is not. We can't say Yes to this on the basis that scientists have a good eye for natural classes of a sort which do not correspond to the basic scientific predicates. To be fit for one task (avoidance of the common poisons, snakes, etc.) does not make one automatically fit for another.
So let us look at the second alternative. Is it possible that the selection of the more successful theoriesvis--vis experimental data and theoretical criteriawill tend to favour formulation in a correct language? We must do a thought experiment here first. Suppose that at a certain point in history, all the primitive scientific predicates are natural ones. Now suppose that one scientist devises a theory which is simpler and more informative than any to be had so farbut only by the use of new theoretical terms which do not stand for natural classes. Why should we think that his theory should be judged inferior? New theoretical terms are typically not definable in terms of the old, and on the other hand, are typically required for radical theoretical innovation. No, I expect that this would be the end of the natural classes' winning streakthe incorrect language would take over.
How could we designate this as an evil day for science? Should we predict that scientific progress will be held up? But it is quite conceivable that this errant new theory is part of one of the best theories, formulated in its language. And it is conceivable that this new theory is simpler when formulated in its language, than any of its translations into correct languages. After all, that was the reflection that set David Lewis on this round to begin!
The suspicion I have at this point is this: if there really is an objective distinction between natural classes and others, and if laws in the sense of Lewis are what science hopes to formulate in the long run, then the only possible evidence for a predicate being natural is that it appears in a successful theory. If that is so, then science can never be guided even in part by a selection of natural over unnatural predicates. For the judgement of inferiority of any terminology on such a basis can then be made only in retrospect,
on the basis of some other lack of success. But in the absence of any selection for natural predicates, in independent fashion, at the time of theory choice or evaluation, we can have no reason to expect that science will tend to develop such a correct language.
But even worse follows. To be precise: if the only link we have is that a predicate is more likely to be natural if it occurs in a successful theory, then we shall never have warrant to think that any predicate is natural. This sounds paradoxical, but consider the following example. Suppose
1 per cent of all available predicates have feature F
2 per cent of all predicates which appear in successful theories have feature F
feature F is not correlated with any independently checkable characteristic.
Then it is clearly true that a predicate is more likely to be natural if it occurs in a successful theoryindeed, twice as likely. Yet we shall never have reason to have any but an extremely low opinion of any predicate's claim to naturalness.
I submit that there is no plausible way to improve on this dismal picture. To think that our opinion of such a claim could cumulatively improve would require something like this: every time a predicate survives theory change, we must raise our opinion of its claim to naturalness. But that is exactly what would be plausible if independent selection in favour of naturalness were going on in theory changethe opposite of our present hypothesis.
Could we stand the problem on its head and identify the natural predicates, in Peircean fashion, as just those which will in the long run remain part of the evolving scientific account of the world? In that case, what the natural predicates are comes to depend on the actual history of our science, and perhaps on counterfactual judgements about how it would continue to evolve in the absence of nuclear holocausts, Armageddon, and the like. A major desideratum for the account of lawsthat it makes them independent of any historical, psychological, or other anthropocentric factorsappears to be abandoned. In any case, we cannot make any such suggestion to Lewis, who, quite properly, (a) wants an account of what the laws are of any world, inhabited by scientists or not, and (b) would evaluate counterfactuals in terms of what the laws are and not define laws by means of counterfactuals.
We see therefore that the anti-nominalist manuvre, by saving Lewis's account from one peril, has precipitated it into another. For it has produced unchartable distances between Lewis's best theoriesand hence lawsand the theories we could reasonably hope for at the ideal end of science. The two ideals have been radically separated. We turn now to a very different line of thought that will point to the same separation.
6. Laws Related to the Pursuit of Science
One of the features of [Lewis's] account is that, on the assumption that scientific theorizing is an attempt to achieve the best overall deductive system, it explains why we are normally justified in believing that the axioms and theorems of the best available scientific theories are (or approximate) laws.
John Earman, The Universality of Law 14
The last, but one of the most important criteria for an account of laws is this: the account should make it plausible that laws of nature are the truths which science aims to discover. My phrasing should not be too strictly or prejudicially construed. If the account makes it plausible that the laws, as defined, are part of the theoretical description of the world provided by science in the long run, if all goes ideally wellthat is enough. At first sight, Lewis's account has this very important virtue. For prima facie, our hope for science, and our expectation of its achievement if all should go ideally well, is that science reach one of the best true theories in our world. And by definition, all the ways in which this hope could be realized will lead to the laws of nature, in Lewis's sense.
That this prima-facie virtue should be a real one, we found to be a crucial concern for the other desideratum of explanation. Perhaps laws in the sense of Lewis can be expected to explain only in the sense that anyone tends to grant at once that scientific theories explain. (We contrast this with the idea that laws should explain why the phenomena are necessarily what they are, in some more substantial sense, which certain authors refuse to grant for laws in the sense of Lewis.) But that expectation too requires a previous conviction, that laws are identifiable as just the sort of truths we may ideally expect to find in our scientific theories.
Does Lewis's account fare well with this desideratum, upon due reflection? To reach the set of laws, on Lewis's account, we successively narrow down the sets of true statements by means of criteria of selection conceived of as purely syntactic and semantic (as opposed to pragmatic, which would admit as relevant also historical, psychological, or other contextual factors). Everything will be well, therefore, only if we can maintain either that actual theory choice in the history of science is by such criteria, or that it should be, or that it would be under ideal conditions. If not, we cannot plausibly expect science to reach one of Lewis's best theories, even if all goes ideally well. But can we maintain some such thesis about the history of science?
I have three reasons for saying No. The first is that the criteria for theory choice in science are not Lewis's criteria of selection, and do not have the same general character. The second is that even if Lewis's selection criteria were among those guiding scientific theory choice, his purpose would be defeated by the presence of additional criteria. The third is that even if Lewis's selection criteria were the actual and sole actual criteria utilized in theory choice, reflection on our starting-point will make it impossible to conclude that science tends toward one of Lewis's best theories as end point, even ceteris paribus.
Lewis's selection criteria, to separate out the best theories, are four: truth, simplicity, strength, balance. There is a fifth, or perhaps I should say zeroth, criterion: the selection is made from theories formulated in a correct language (languages with natural predicates). Now actual science begins with theories not known to be true, but in any case, not very simple, not very strong, with regrettable sacrifices of simplicity for strength or vice versa, and formulated with predicates for which we claim no virtue beyond familiarity. The progress of science will not choose among these; it will modify them. We envisage therefore two processes: one a logical subdivision of the whole class of theories, and the other a trajectory through that class, starting from a specific point. Question: should we expect the trajectory to land in the target area which the selection marks out?
First of all we suspect that when theories are in competition, and one has the advantage of simplicity, that this advantageous simplicity is a human, historically conditioned one. At this point
in time, the language used is not entirely correct, but it is familiar to the contesting scientists. David Lewis's criterion of simplicity would be applied (by someone outside history) as follows: first translate both theories into the correct language, then compare those two new formulations as to simplicity. Even if the general criterion of simplicity is the same, the verdict may well be reversed by the translation. But that real simplicity, which becomes apparent only upon formulation in a language which the contesting scientists neither have nor know, cannot affect the outcome of the contest! That outcome will only be affected by the simplicity felt and appreciated by the contestants.
This problem arises even if we think that the general notion of simplicity is the same for the actual scientists as for someone not historically conditioned in the same way. Of course, the problem is much worse if scientists' peculiar education or aesthetic sensibilities enter their judgements of simplicity. In that caseand I fear it may be sothe pious sound of the word simplicity may be the only link between the two sorts of evaluation.
We also suspect that if two theories are in competition, and one has the advantage of strength (that is, informativeness), the strength is peculiarly historical. Forinformation about what? Any sort of information? Information has a generally agreed upon measure in the simple context of communication engineering. But if we laud a theory for informativeness, that measure is not intended, I am sure. For in practice we call a theory more informative if it answers more of our questionsand we are highly selective in what questions we pose. I think all scientists agree on the value of accurate prediction of empirical phenomena. But even there Thomas Kuhn has charted historical variations in what empirical information scientific theories have been required to give. 15 In addition, theories may be more or less informative about what goes on behind the phenomena. The putative information it gives there is evaluated quite differently by different scientists, at least until it issues in new empirical predictions. This is amply illustrated by the differing nineteenth-century views on the value of atomic and molecular underpinnings for thermodynamics.
The general point is this: even if the measure of information is objective, and is just what Lewis thinks it is, the operative principle of theory-evaluation will be in terms of valuable information. When the scientific community apparently judges that one theory is more
informative than its competitors, and should therefore be favoured, that favoured theory may well be one that is really less informative all told. The reason is that historically conditioned values are modifying the judgement tacitly. The perceived superiority with respect to desired and valued information, will issue in that apparent judgement of greater informativeness pure and simple.
So far my first reason. Now I turn to the second reason:
even if the actual evaluation of simplicity and strength in the history of
science coincide with the historically unconditioned evaluation, things may
still not work out. The reason is that there may be additional criteria
operative in the historical evaluationand I think there are. I would
especially mention the advantage a theory may have if it is more easily
combined with theories outside the context. For example, Lord Kelvin objected
Perhaps there are still further criteria at work in the history of science, beyond those considered by Lewis. All we need is some suspicion of this sort. For then we have immediate reason to suppose that the process of theory choice will go awry, from Lewis's point of view. All it needs is some extra criteria. Consider this parallel. One child says: the best objects in this room are the largest. A second child says nothing but begins to compare the objects it finds two by two. If it always discards the smaller, we may reasonably expect thatif it is not interrupted or deceived or whateverit will end up holding one of those items which the first child considered best. But if the second child has an additional criterionif, for example, it regretfully puts aside any object, however large, if it is redwe no longer have that expectation. The largest objects may be a very different class from the set of largest non-red objects. We can no longer expect their selections to be the same, if the second child displays any decided preference guided by colourfor example, if it always puts aside the red object unless it is at least twice as large as the other. As long as any other proclivity is at work, the outcome will depend a great deal on the actual composition of the room's contents, and we have no logical way to speculate about that.
Finally I turn to my third reason. Even if the criteria of historical
theory choice were all and only those described by Lewis in his theory of laws, all would not be well. For the evolution of science as a whole is historically conditioned by its starting-point, and by the schooled imaginations of its practitioners. Again, an analogy. Let one extraterrestrial visitor to earth judge that the most beautiful animals here are the largest, most active ones. He, she, or it must have some criterion of balance in mind; but obviously the class he thus selects contains elephants and perhaps a few others. Now let a second such alien begin to breed mice, always selecting from each generation the largest, most active ones. He will eventually have large active mice, but not large, active animals. For a large mouse is still a small animal.
This ends my critique of a law-oriented eschatology of science. Should we now add that if I am right, so much the worse for science? Are all reasons to think that science would not even ideally arrive in Lewis's target area ipso facto reasons to expect that science will fail in its proper task?
By no means. As I see it, science aims to give us theories which are empirically adequate. The practitioners commit themselves to one theoretical framework rather than another, if they judge, to the best of their cognitive ability, that this is more likely to serve the aim of empirical success. They are acting in good faith if their selection criteria do indeed, by their own lights, help rather than hinder, and at least do not sabotage, the pursuit of this aim. That they should always act as best as they can, by their own lights, is their ethic and their conscience. They have also been very successful in this pursuit, and have as much reason as anyone to believe in their enterprise. What I have been arguing is that this positive trust in the actual process of science, establishes no link between its eschatology and Lewis's laws. That the process of science leads to greater empirical success always gained through more beautiful intellectual constructions, if all goes well, is implied (modulo the meaning of all goes well). But that it leads to laws in Lewis's sense, is not implied. Thus there is no reason to equate Lewis's laws with what science pursues.
7. A Parable
High in the mountains by the eastern sea, the magicians have their own kingdom. It is small, compared to ours, not much larger than our largest city, but rich with the gifts of magic and nature. High
up it lies, on a still plateau where the rising sun brings warmth early every morning before it turns to us. The magicians who live there seek to draw us with their subtle powers, but are hindered by the frailties of our own intellect and flesh.
In our kingdom, all manner of weakness of the eyes is hereditary. Our kings, whom few have seen, were always the most far- and clear-sighted creatures on earth. In the sky they sawso it is toldstars of fire to which they gave many wonderful names, likening them to warriors, beasts, and jewel-studded girdles. Our soldiers too were always far-sighted, andthen as nowstrike fear in the heart of all that lives and moves beneath the sun. They detect enemies before they come within stone-throwing distance, and signal each other with mirrors glinting in the sun. We ordinary people of lesser stock, the craftsmen, fishers, and scholars, see as much as we need; though compared to them we live as if in mist and haze.
This story is told of long ago. The magicians sent a dream to three kings, three soldiers, and three scholars. The dream revealed the magical kingdom in all its glory, with such felt hope and grace as to be at once infinitely desirable. Each dreamer resolved to seek the kingdom. But our minds are clouded in proportion to our eyesight, so the kings, soldiers, and scholars did not learn equally much. The kings saw clearly the magicians' houses and castles, the high mountains, and a brilliant star which they recognized, at its zenith. The soldiers saw only a mountainside, and green meadows in the dawn; by the shadows they judged that the place must lie due east. They could not discern houses from rocks, nor see any star. But such was the longing this dream inspired, that they knew it held a prize beyond what any campaign could bring. Lastly, the scholars, as captivated as the others, received no inkling of whether the place was high or low, though they too saw how the rising sun cast the shadows. Each group began its journey east, quite unbeknownst to the others.
Many obstacles lay in the kings' path: rivers and ravines, hunger-maddened goblins and wolves, cliffs too steep to climb and lakes too wide to swim. Almost every day they were diverted, now left then right, out of their way. But each night the kings saw their guiding star and each dawn set out towards it. After three years and a day, the kings ascended the eastern mountains, and were welcomed into the magicians' city.
The soldiers, trained to find their way across difficult terrain, and to judge direction accurately from shadows cast by sun and moon, struck east. Coming upon the hills, they ascended. But the hills proved low, judging by their memories of old campaigns, and they knew they had not come to the place they sought. Eventually, climbing almost unscalable cliffs, they came to the top of a mountain. As far as they could see, there were no heights comparable to this. The high meadows were green and berries abounded, a lake held trout. In the earth they found silver and gold, the bees gave up their honey, the trees gave them wood for building. In their dream they had not seen the great magical castles, nor did they have the kings' grasp of how high the eastern mountains are. So there they stayed, still a year's journey from the east, in bounty undreamed of in their old soldiers' lifebut still in poverty and want compared to the kings.
The journeying scholars did not have the kings' eyesight, nor the soldiers' fieldcraft. They did not know the place they sought was high in the mountains. The magicians' kingdom could after all have been as glorious and rich if it had been in a valley, and the east would still have been east if the land had run everywhere level to the sea. So they sought only the east and indeed, if they had journeyed due east they would have arrived. To guide them they had a lodestone compass, fashioned by our finest craftsmen. They attached a small light to the lodestone, which they sighted through narrow slits in a screen, so as to draw a line with true direction. Thus their determination of the compass points was exceedingly fine by night and day. Always after an obstacle they used a small sand-clock to gauge the time they had needed, departing from true; set up their compass again, and adjusted their path. Yet at every turn, some minute angle was lost, whether to south or north. The proportion of deflections favoured, ever so slightly overall, the south. After five years of travel they came upon the sea, where they found a land of milk and honey, warmth and welcome among a friendly people. There were green fields and the sweet taste of dates ripened in the sun. To the north, across an arid desert, there lay soaring mountains, they were told. But they had come to the easternmost shore, and there they stayed. A half year's journey to the north, lay the incomparable intellectual splendours of the magicians' land, where scholarship had already bloomed for ten thousand years.
Many generations have repeated this tale, which could only have
come to us from a returning king, still shining with the magicians' knowledge. The soldiers remained, happy, in the lower mountains, and the scholars, also happy, by the eastern shore. Are we right to describe our fellow scholars of so long ago, as in error? They truly travelled east, by the finest determination human hands and sight allowed them. Of course they realized that their instrumentation was not infinitely fine, and that such a journey could not have a single, pre-ordained end. But what they found, at the easternmost point by their reckoning, was paradise by their lightsthey would not have been content with less. Yet we sigh; their light seems dim and poor to us who, though of the same benighted kin, have pictured to ourselves magicians, kings, and stars. Some say the tale is not a history of long ago, but a vision of our far future. In these republican days, some even say that our kings never had their fabled power of sight, and no one ever will. Whatever be true, we pity those scholars, our brothers, who only found happiness, but never that true home with its true riches.
8. Conclusion: Deceptive Success
The reason why I liked Lewis's theory of laws must have been clear from the beginning. First of all, the account involves very little that could be associated specifically with (pre-Kantian) metaphysics. Lewis himself is a realist about possible worlds, but his account of laws could be accepted word for word by someone who regards possible worlds as (semantic) theoretical fictions. The second reason is that the account makes a real effort to establish a link with science. The laws, as defined, should be good candidates for what science will ideally arrive at, and the fundamental principles of science should be good candidates for laws. By defining laws in terms of good, better, and best theories about our world, Lewis makes a sincere effort to honour this desire.
So what are the difficulties that render the account inadequate? They are of two sortsthe ones I have taken up, and the ones that Lewis himself points out in later writings.
In his Philosophical Papers, vol. ii, David Lewis proposes an amendment to his account of laws. He introduces the notion of objective chance, in the first instance to broaden his account to cover the probabilistic theories of contemporary physics. This
generalization of the notion of law he concludes, ruefully, not to admit of the sort of treatment given to non-probabilistic laws. So he admits chance as a separate category. Then, in his definition of law, he replaces the set of true theories, by the set of those theories which never had any chance of being false. He writes The field of eligible competitors is thus cut down. But then the competition works as before. The best system is the one that achieves as much simplicity as is possible without excessive loss of strength, and as much strength as is possible without excessive loss of simplicity. A law is a regularity that is included . . . in the best system (p. 126).
I have chosen to concentrate on Lewis's original theory for three reasons. The first is that the difficulties I see for the original, more limited account seem to me to persist almost entirely intact for the recent, amended account. It is true that after cutting down the field of competitors, the criteria of simplicity, strength, and balance have less work to do. But we can't really tell how much less; so we cannot evaluate the import of this remark at all. Secondly, I wanted to make clear that difficulties faced by an account of laws are not brought on by its recourse to metaphysics. In Lewis's original account, there is an absolute minimum of metaphysics, and I did not need to raise an objection to this minimal presence as such, to find what I regard as debilitating difficulties. This will make clear to ametaphysical philosophers, I hope, that accounts of law turn to more metaphysics out of need, not idiosyncratic preference. With Lewis's amended account, this would not be nearly so clear, because with chance as a separate and irreducible notion, the reality of possible worlds does become crucial. Finally, I conjecture that new difficulties introduced by the ontological reification of chance will affect Lewis's new account as much as some others to be discussed. (These last two reasons will be clearer, I think, after the discussions of chance and its relation to opinion in the next three chapters.) A conjecture is not a firm reason, but it may incline.
To complete our overview let me summarize the problems discussed in this chapter which already affect the earlier version of the account.
It is true that the account does not presuppose modal realism. Unfortunately, the moderation with respect to metaphysics made the account vulnerable to charges that it does not respect real necessity, in several ways. 16 Secondly, the laws of this world, in Lewis's sense, are not at all guaranteed to be explanatory. If the
best theories are the best explanations, then those laws are part of every best explanation of the world as a whole. But the laws themselves may well lack those very features that make the best theories explanatory. And thirdly, the attempt to link up with science founders, in my opinion, inevitably. For the criteria for better and best theories utilized, must be such as to leave it an objective matter, independent of history and psychology, what truths are laws. That means that the equation we are tempted to trustthe best theories are those theories which science could or might reach, should all go ideally wellis simply divorced entirely from the equation that defines best theories for Lewis. I see no remedy for this.
Most of all, we see here the dilemma posed by the problems of inference and of identification, which I discussed at the end of the preceding chapter. Lewis formulated his definitions in such a way that there can be no question about the validity of It is a law that A; therefore, A. The inference problem is thus successfully handled. And to begin, it seemed that identification too was unproblematic. But that turned out not to be so, because the criteria for better and best theoriescrucial to the definition of lawwere not translation invariant. The consequent introduction of the notion of natural classes and predicates, led to an identification problem which I believe to be unsolvable. The attempts at identification examined put laws out of touch with science even if otherwise granted to be workable.
As we go on now to other accounts of laws, we shall find more and more pre-Kantian metaphysics, and at the same time, less and less contact with science. For Lewis's account there was still a point to serious discussion of the eschatology of sciencethere will be little point to it later in this part. The notion of necessity, and the idea of very strict criteria for explanation of what is as what has to bethese will be honoured all the more. I cannot hide my conviction that if Lewis's account had been more successful, it would have been foolish to look furtherbut there it is. The last hope for an empiricist account of law, that a little sacrifice to anti-nominalism would ward off peril, is gone.
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