Laws of nature 29 Ga Gb Gc Gd Accidental regularity Fa Fb Fc Fd that all Fs are Gs Ha Jb Ke Ld Gm Gp Gr Gr Accidental regularity Em Ep Er Er that all Es are Gs Hm Jp K Lr Gn Gq Gr Accidental regularity Dn q s Dr that all Ds are Gs Hn J Ks Ls Law that Ls are Gs Law that Ks are Gs Law that Js are Gs Law that Hs are Gs Figure 1.2 a few pages back.It has the same laws,but the additional facts mean that the accidental regularities now predominate over the genuine laws-the optimal systematization would include the regularity that Fs are Gs but exclude the law that Hs are Gs.If such a world is possible,and I think our intuitions suggest it is,then the systematic regularity theory gets it wrong about which the laws are.In this way the systematic theory fails to be materially adequate. Laws,regularities,and explanation I have mentioned that there are various things we want laws to do,which they are not up to doing if they are mere regularities.We have already looked at a related issue.One thing laws are supposed to do is support counterfactuals.Some opponents of the regularity theory think that this is sufficient to show that minimalism is mistaken,but I think that this is a difficult line to pursue (even if ultimately correct).More promising is to argue that laws cannot do the job of explaining their instances if they are merely regularities.Another issue is whether we can make sense of induction if laws are just regularities. The key to understanding why a mere regularity cannot explain its instances is the principle that something cannot explain itself.The notion of explanation is one we will explore in detail in the next chapter.However,for the time being.I hope it can be agreed
Figure 1.2 a few pages back. It has the same laws, but the additional facts mean that the accidental regularities now predominate over the genuine laws—the optimal systematization would include the regularity that Fs are Gs but exclude the law that Hs are Gs. If such a world is possible, and I think our intuitions suggest it is, then the systematic regularity theory gets it wrong about which the laws are. In this way the systematic theory fails to be materially adequate. Laws, regularities, and explanation I have mentioned that there are various things we want laws to do, which they are not up to doing if they are mere regularities. We have already looked at a related issue. One thing laws are supposed to do is support counterfactuals. Some opponents of the regularity theory think that this is sufficient to show that minimalism is mistaken, but I think that this is a difficult line to pursue (even if ultimately correct). More promising is to argue that laws cannot do the job of explaining their instances if they are merely regularities. Another issue is whether we can make sense of induction if laws are just regularities. The key to understanding why a mere regularity cannot explain its instances is the principle that something cannot explain itself. The notion of explanation is one we will explore in detail in the next chapter. However, for the time being, I hope it can be agreed Laws of nature 29
Philosophy of science 30 that for any fact F,F does not explain F-the fact that it rained may explain the fact that the grass is damp,but it does not explain why it rained. As a law explains its instances,it explains all of them and,therefore,it explains the fact that is the uniformity consisting of all of them.But a regularity cannot do this because,according to the above-stated principle,it cannot explain itself.To spell this out in more detail,let us imagine that there is a law that Fs are Gs.Let there in fact be only four Fs in the world,a,b,c,and d.So the generalization "all Fs are Gs"is equivalent to (A)(Fa Ga)(Fb Gb)&(Fc Gc)(Fd Gd)(nothing is F other than a,b.c. or d). The conjunction of all the instances is (B)(Fa Ga)&(Fb Gb)&(Fc&Gc)&(Fd Gd) (A)and (B)are identical except for the last part of (A),which says that nothing is F other than a.b,c.or d.Let us call this bit (C) (C)nothing is F other than a,b.c.or d S0(A)=(B)&(C). Thus,to say (A)explains (B)is the same as saying (B)&(C)explains (B).But how can that be?For clearly (B)does not explain(B),as already discussed.So if(B)&(C) explains (B)it is by virtue of the fact that(C).However,(C)says that nothing is F other than a,b,c,or d.That other things are not F cannot contribute to explaining why these things are G.Therefore,nothing referred to in (A)can explain why it is the case that (B). That we have considered a law with just four instances is immaterial.The argument can be extended to any finite number of instances.Many laws do have only a finite number of actual instances (biological laws for example).Nor does moving to an infinite number of instances change the nature of the argument.Even if it were thought that the infinite case might be different,then we could give essentially the same argument but, instead of considering the conjunction of instances,we could focus on just one instance. Why is a,which is an F,also a G?Could being told that all Fs are G explain this?To say that all Fs are Gs is to say if a is an F then a is a G,and all other Fs are Gs.The question of why a,which is an F,is also a G is not explained by saying that if a is an F then a is a G,because that is what we want explained.On the other hand,facts about other Fs,even all of them,simply do not impinge on this F. It should be noted that this objection covers,in effect,all the forms of the regularity theory,not just the simple regularity theory.For more sophisticated versions operate by paring down the range of regularities eligible as laws and exclude those that fail some sort of test.But they still maintain that the essence of lawhood is to be a regularity,and the relation of explanation between law and instance is still,according to the regularity theorist,the relation between a regularity and the instance of the regularity.However sophisticated a regularity theory is,it cannot then escape this criticism.For instance,the deductive integratibility required by Lewis and Ramsey does not serve to provide any more unity to a law than is provided by a generalization.That the generalization is an axiom or consequence of the optimal axiomatic system does nothing to change the fact that it or the regularity it describes cannot explain its instances
that for any fact F, F does not explain F—the fact that it rained may explain the fact that the grass is damp, but it does not explain why it rained. As a law explains its instances, it explains all of them and, therefore, it explains the fact that is the uniformity consisting of all of them. But a regularity cannot do this because, according to the above-stated principle, it cannot explain itself. To spell this out in more detail, let us imagine that there is a law that Fs are Gs. Let there in fact be only four Fs in the world, a, b, c, and d. So the generalization “all Fs are Gs” is equivalent to (A) (Fa & Ga) & (Fb & Gb) & (Fc & Gc) & (Fd & Gd) & (nothing is F other than a, b, c, or d). The conjunction of all the instances is (B) (Fa & Ga) & (Fb & Gb) & (Fc & Gc) & (Fd & Gd) (A) and (B) are identical except for the last part of (A), which says that nothing is F other than a, b, c, or d. Let us call this bit (C) (C) nothing is F other than a, b, c, or d So (A)=(B) & (C). Thus, to say (A) explains (B) is the same as saying (B) & (C) explains (B). But how can that be? For clearly (B) does not explain (B), as already discussed. So if (B) & (C) explains (B) it is by virtue of the fact that (C). However, (C) says that nothing is F other than a, b, c, or d. That other things are not F cannot contribute to explaining why these things are G. Therefore, nothing referred to in (A) can explain why it is the case that (B). That we have considered a law with just four instances is immaterial. The argument can be extended to any finite number of instances. Many laws do have only a finite number of actual instances (biological laws for example). Nor does moving to an infinite number of instances change the nature of the argument. Even if it were thought that the infinite case might be different, then we could give essentially the same argument but, instead of considering the conjunction of instances, we could focus on just one instance. Why is a, which is an F, also a G? Could being told that all Fs are G explain this? To say that all Fs are Gs is to say if a is an F then a is a G, and all other Fs are Gs. The question of why a, which is an F, is also a G is not explained by saying that if a is an F then a is a G, because that is what we want explained. On the other hand, facts about other Fs, even all of them, simply do not impinge on this F. It should be noted that this objection covers, in effect, all the forms of the regularity theory, not just the simple regularity theory. For more sophisticated versions operate by paring down the range of regularities eligible as laws and exclude those that fail some sort of test. But they still maintain that the essence of lawhood is to be a regularity, and the relation of explanation between law and instance is still, according to the regularity theorist, the relation between a regularity and the instance of the regularity. However sophisticated a regularity theory is, it cannot then escape this criticism. For instance, the deductive integratibility required by Lewis and Ramsey does not serve to provide any more unity to a law than is provided by a generalization. That the generalization is an axiom or consequence of the optimal axiomatic system does nothing to change the fact that it or the regularity it describes cannot explain its instances. Philosophy of science 30
Laws of nature 31 A case that seems to go against this may in fact be seen to prove to rule.I have an electric toaster that seems to have a fault.I take it back to the shop where I bought it, where I am told"They all do that,sir".This seems to explain my problem.Whether or not I am happy,at least I have had it explained why my toaster behaves the way it does. However,I think that this is an illusion.Being told that Emily's toaster does this,and Ned's and Ian's too does not really explain why my toaster does this.After all,if Emily wants to know why her toaster behaves that way,she is going to be told that Ned's does that too and Ian's and so does mine.So part of the explanation of why mine does this is that Emily's does and part of the explanation of why Emily's does this is that mine does. This looks slightly circular,and when we consider everyone asking why their toaster does this strange thing.we can see that the explanations we are all being given are completely circular. So why does being told "They all do that"look like an explanation?The answer is that,although it is not itself an explanation,it points to an explanation."They all do that" rules out as unlikely the possibilities that it is the way I have mistreated the toaster,or that it is a one-off fault.In the context of all the other toasters behaving this way,the best explanation of why my toaster does so is that some feature or by-product of the design or manufacturing process causes the toaster to do this.This is a genuine explanation,and it is because this is clearly suggested by the shopkeeper's remark that the remark appears to be explanatory.What this serves to show is precisely that regularities are not explanations of their instances.What explains the instances is something that explains the regularity, although the fact of the regularity may provide evidence that suggests what that explanation is. Laws,regularities,and induction If regularities do not explain their instances,then a question is raised about inductive arguments from observed instances to generalizations.The critic of minimalism,whom I shall call the full-blooded theorist says that laws explain their instances and that inferring a law from the observation of its instances is a case of inference to the best explanation e.g.we infer that there is a law of gravitation,because such a law is the best explanation of the observed behaviour of bodies(such as the orbits of the planets and the acceleration of falling objects).(Inference to the best explanation is discussed in Chapters 2 and 4). Because,as we have seen,the minimalist is unable to make sense of the idea of a law explaining its instances,the minimalist is also unable to employ this inference-to-the- best-explanation view of inductive inference.4 For the minimalist,induction will in essence be a matter of finding observed regularities and extending them into the unobserved.So,while the minimalist's induction is of the form all observed Fs are Gs therefore all Fs are Gs,the full-blooded theorist's induction has an intermediate step:all observed Fs are Gs,the best explanation of which is that there is a law that Fs are Gs, and therefore all Fs are Gs. Now recall the problem of spurious (accidental,contrived,single case,and trivial) regularities that faced the simple regularity theory.The systematic regularity theory solves this problem by showing that these do not play a part in our optimal systematization.Note that this solution does not depend on saying that these regularities
A case that seems to go against this may in fact be seen to prove to rule. I have an electric toaster that seems to have a fault. I take it back to the shop where I bought it, where I am told “They all do that, sir”. This seems to explain my problem. Whether or not I am happy, at least I have had it explained why my toaster behaves the way it does. However, I think that this is an illusion. Being told that Emily’s toaster does this, and Ned’s and Ian’s too does not really explain why my toaster does this. After all, if Emily wants to know why her toaster behaves that way, she is going to be told that Ned’s does that too and Ian’s and so does mine. So part of the explanation of why mine does this is that Emily’s does and part of the explanation of why Emily’s does this is that mine does. This looks slightly circular, and when we consider everyone asking why their toaster does this strange thing, we can see that the explanations we are all being given are completely circular. So why does being told “They all do that” look like an explanation? The answer is that, although it is not itself an explanation, it points to an explanation. “They all do that” rules out as unlikely the possibilities that it is the way I have mistreated the toaster, or that it is a one-off fault. In the context of all the other toasters behaving this way, the best explanation of why my toaster does so is that some feature or by-product of the design or manufacturing process causes the toaster to do this. This is a genuine explanation, and it is because this is clearly suggested by the shopkeeper’s remark that the remark appears to be explanatory. What this serves to show is precisely that regularities are not explanations of their instances. What explains the instances is something that explains the regularity, although the fact of the regularity may provide evidence that suggests what that explanation is. Laws, regularities, and induction If regularities do not explain their instances, then a question is raised about inductive arguments from observed instances to generalizations. The critic of minimalism, whom I shall call the full-blooded theorist says that laws explain their instances and that inferring a law from the observation of its instances is a case of inference to the best explanation— e.g. we infer that there is a law of gravitation, because such a law is the best explanation of the observed behaviour of bodies (such as the orbits of the planets and the acceleration of falling objects). (Inference to the best explanation is discussed in Chapters 2 and 4). Because, as we have seen, the minimalist is unable to make sense of the idea of a law explaining its instances, the minimalist is also unable to employ this inference-to-thebest-explanation view of inductive inference.14 For the minimalist, induction will in essence be a matter of finding observed regularities and extending them into the unobserved. So, while the minimalist’s induction is of the form all observed Fs are Gs therefore all Fs are Gs, the full-blooded theorist’s induction has an intermediate step: all observed Fs are Gs, the best explanation of which is that there is a law that Fs are Gs, and therefore all Fs are Gs. Now recall the problem of spurious (accidental, contrived, single case, and trivial) regularities that faced the simple regularity theory. The systematic regularity theory solves this problem by showing that these do not play a part in our optimal systematization. Note that this solution does not depend on saying that these regularities Laws of nature 31
Philosophy of science 32 are in themselves different from genuine laws-e.g.it does not say that the relationship between an accidental regularity and its instances is any different from the relationship between a law and its instances.What,according to the minimalist,distinguishes a spurious regularity from a law is only its relations with other regularities.What this means is that a minimalist's law possesses no more intrinsic unity than does a spurious regularity. Recall the definition of "grue"(see p.18).Now define "emerire"thus: X is an emerire=either X is an emerald and observed before midnight on 31 December 2000 or X is a sapphire and not observed before midnight on 31 December 2000. On the assumption that,due to the laws of nature,all emeralds are green and all sapphires are blue,it follows that all emerires are grue.The idea here is not of a false generalization (such as emeralds are grue),but of a mongrel true generalization formed by splicing two halves of distinct true generalizations together.Now consider someone who has observed many emeralds up until midday on 31 December 2000.If their past experience makes them think that all emeralds are green,then they will induce that an emerald first observed tomorrow will be green;but if they hit upon the contrived (but real)regularity that all emerires are grue,then they will believe that a sapphire first observed tomorrow will be blue.As both generalizations are true,neither will lead this person astray.The issue here is not like Hume's or Goodman's problems-how we know which of many possible generalizations is true.Instead we have two true generalizations,one of which we think is appropriate for induction and another which is not-even though it is true that the sapphire is blue,we cannot know this just by looking at emeralds.What makes the difference?Whatever it is,it will have to be something that forges a link between past emeralds and tomorrow's emerald,a link that is lacking between the past emeralds and tomorrow's sapphire. I argued,a couple of paragraphs back,that in the minimalist's view there is no intrinsic difference between a spurious regularity and a law in terms of their relations with their instances.And so the minimalist is unable to give a satisfactory answer to the question:What makes the difference between inducing with the emerald law and inducing with the emerire regularity?Being intrinsically only a regularity the law does not supply the required link between past emeralds and future emeralds;any link it does provide is just the same as the one provided by the emerire regularity that links emeralds and sapphires.(Incidentally,the case may be even worse for the systematic regularity theorist,as it does seem as if the emerire regularity should turn out to be a derived law because it is derivable from two other laws.) The full-blooded view appears to have the advantage here.For,if laws are distinct from the regularities that they explain,then we can say what makes the difference between the emerald law and the emerire regularity.In the former case we have something that explains why we have seen only green emeralds and so is relevant to the next emerald,while the emerire regularity has no explanatory power.It is the explanatory role of laws that provides unity to its instances-they are all explained by the same thing. The contrast between the minimalist and full-blooded views might be illustrated by this analogy:siblings may look very similar(the regularity),but the tie that binds them is not this,but rather their being born of the same parents,which explains the regularity of their similar appearance
are in themselves different from genuine laws—e.g. it does not say that the relationship between an accidental regularity and its instances is any different from the relationship between a law and its instances. What, according to the minimalist, distinguishes a spurious regularity from a law is only its relations with other regularities. What this means is that a minimalist’s law possesses no more intrinsic unity than does a spurious regularity. Recall the definition of “grue” (see p. 18). Now define “emerire” thus: X is an emerire= either X is an emerald and observed before midnight on 31 December 2000 or X is a sapphire and not observed before midnight on 31 December 2000. On the assumption that, due to the laws of nature, all emeralds are green and all sapphires are blue, it follows that all emerires are grue. The idea here is not of a false generalization (such as emeralds are grue), but of a mongrel true generalization formed by splicing two halves of distinct true generalizations together. Now consider someone who has observed many emeralds up until midday on 31 December 2000. If their past experience makes them think that all emeralds are green, then they will induce that an emerald first observed tomorrow will be green; but if they hit upon the contrived (but real) regularity that all emerires are grue, then they will believe that a sapphire first observed tomorrow will be blue. As both generalizations are true, neither will lead this person astray. The issue here is not like Hume’s or Goodman’s problems—how we know which of many possible generalizations is true. Instead we have two true generalizations, one of which we think is appropriate for induction and another which is not—even though it is true that the sapphire is blue, we cannot know this just by looking at emeralds. What makes the difference? Whatever it is, it will have to be something that forges a link between past emeralds and tomorrow’s emerald, a link that is lacking between the past emeralds and tomorrow’s sapphire. I argued, a couple of paragraphs back, that in the minimalist’s view there is no intrinsic difference between a spurious regularity and a law in terms of their relations with their instances. And so the minimalist is unable to give a satisfactory answer to the question: What makes the difference between inducing with the emerald law and inducing with the emerire regularity? Being intrinsically only a regularity the law does not supply the required link between past emeralds and future emeralds; any link it does provide is just the same as the one provided by the emerire regularity that links emeralds and sapphires. (Incidentally, the case may be even worse for the systematic regularity theorist, as it does seem as if the emerire regularity should turn out to be a derived law because it is derivable from two other laws.) The full-blooded view appears to have the advantage here. For, if laws are distinct from the regularities that they explain, then we can say what makes the difference between the emerald law and the emerire regularity. In the former case we have something that explains why we have seen only green emeralds and so is relevant to the next emerald, while the emerire regularity has no explanatory power. It is the explanatory role of laws that provides unity to its instances—they are all explained by the same thing. The contrast between the minimalist and full-blooded views might be illustrated by this analogy: siblings may look very similar (the regularity), but the tie that binds them is not this, but rather their being born of the same parents, which explains the regularity of their similar appearance. Philosophy of science 32
Laws of nature 33 A full-blooded view-nomic necessitation The conclusion reached is this.A regularity cannot explain its instances in the way a law of nature ought to.This rules out regularity theories of lawhood.The same view is achieved from the reverse perspective.We cannot infer a regularity from its instances unless there is something stronger than the regularity itself binding the instances together. The task now is to spell out what has hitherto been little more than a metaphor,i.e. there is something that binds the instances of a law together which is more than their being instances of a regularity,and a law provides a unity not provided by a regularity. The suggestion briefly canvassed above is that we must consider the law that Fs are Gs not as a regularity but as some sort of relation between the properties or universals Fness and Gness. The term universal refers to properties and relations that,unlike particular things,can apply to more than one object.A typical universal may belong to more than one thing,at different places but at the same time,thus greenness is a property that many different things may have,possessing it simultaneously in many places.A first-order universalis a property of or relation among particular things;so greenness is a first-order universal. Other first-order universals would be,for example:being (made of)magnesium,being a member of the species Homo sapiens,being combustible in air,and having a mass of 10 kg.A second-order universal is a property of or relation among first-order universals. The property of being a property of emeralds is a second-order universal since it is a property of a property of things.The first-order universal greenness has thus the second- order property being a colour.Being generous is a first-order property that people can have.Being a desirable trait is a property that properties can have-for instance,being generous has the property of being desirable;so the latter is a second-order universal. Consider the law that magnesium is combustible in air.According to the full-blooded view this law is a relation between the properties of being magnesium and being combustible in air.This is a relation of natura/necessity.It is not just that whenever the one property is instantiated the other is instantiated,which would be the regularity view. Rather,necessitation is supposed to be stronger.The presence of the one property brings about the presence of the other.Necessitation is therefore a property (more accurately a relation)of properties.This is illustrated in Figure 1.3,where the three levels of particular things,properties of those things(first order universals)and relations among properties (second order universals)are shown.The arrow,which represents the law that emeralds are green,is to be interpreted as the relation of necessitation holding between the property of being an emerald and the property of being green. The advantages of this view are that it bypasses many of the problems facing the minimalist.Many of those problems involved accidental regularities or spurious "cooked- up"regularities.What were problems for the minimalist now simply disappear- necessitation among universals is quite a different matter from a regularity among things. The existence of a regularity is a much weaker fact than necessitation between universals. Two universals can coexist in precisely the same objects without one necessitating the other.We would have something like the diagram in Figure 1.3,but without the top layer and so without the arrow.This is the case in a purely accidental regularity.In an accidental regularity every particular has both of two universals,but the universals are not themselves related.This explains why not every regularity is a law.It also explains
A full-blooded view—nomic necessitation The conclusion reached is this. A regularity cannot explain its instances in the way a law of nature ought to. This rules out regularity theories of lawhood. The same view is achieved from the reverse perspective. We cannot infer a regularity from its instances unless there is something stronger than the regularity itself binding the instances together. The task now is to spell out what has hitherto been little more than a metaphor, i.e. there is something that binds the instances of a law together which is more than their being instances of a regularity, and a law provides a unity not provided by a regularity. The suggestion briefly canvassed above is that we must consider the law that Fs are Gs not as a regularity but as some sort of relation between the properties or universals Fness and Gness. The term universal refers to properties and relations that, unlike particular things, can apply to more than one object. A typical universal may belong to more than one thing, at different places but at the same time, thus greenness is a property that many different things may have, possessing it simultaneously in many places. A first-order universal is a property of or relation among particular things; so greenness is a first-order universal. Other first-order universals would be, for example: being (made of) magnesium, being a member of the species Homo sapiens, being combustible in air, and having a mass of 10 kg. A second-order universal is a property of or relation among first-order universals. The property of being a property of emeralds is a second-order universal since it is a property of a property of things. The first-order universal greenness has thus the secondorder property being a colour. Being generous is a first-order property that people can have. Being a desirable trait is a property that properties can have—for instance, being generous has the property of being desirable; so the latter is a second-order universal. Consider the law that magnesium is combustible in air. According to the full-blooded view this law is a relation between the properties of being magnesium and being combustible in air. This is a relation of natural necessity. It is not just that whenever the one property is instantiated the other is instantiated, which would be the regularity view. Rather, necessitation is supposed to be stronger. The presence of the one property brings about the presence of the other. Necessitation is therefore a property (more accurately a relation) of properties. This is illustrated in Figure 1.3, where the three levels of particular things, properties of those things (first order universals) and relations among properties (second order universals) are shown. The arrow, which represents the law that emeralds are green, is to be interpreted as the relation of necessitation holding between the property of being an emerald and the property of being green. The advantages of this view are that it bypasses many of the problems facing the minimalist. Many of those problems involved accidental regularities or spurious “cookedup” regularities. What were problems for the minimalist now simply disappear— necessitation among universals is quite a different matter from a regularity among things. The existence of a regularity is a much weaker fact than necessitation between universals. Two universals can coexist in precisely the same objects without one necessitating the other. We would have something like the diagram in Figure 1.3, but without the top layer and so without the arrow. This is the case in a purely accidental regularity. In an accidental regularity every particular has both of two universals, but the universals are not themselves related. This explains why not every regularity is a law. It also explains Laws of nature 33