Laws of nature 19 criticism is also made that it is not even necessary for there to be a regularity for the corresponding law to exist.That is,there are laws without the appropriate regularities. Regularities that are not laws I will start with the objection that being a regularity is not sufficient for being a law. Consider the following regularities: (a)All persisting lumps of pure gold-195 have a mass less than 1,000 kg. (b)All persisting lumps of pure uranium-235 have a mass of less than 1,000 kg.3 Both (a)and (b)state true generalizations.But (a)is accidental and (b)is law-like.It is no law that there are no lumps of the pure isotope of gold-we could make one if we thought it worth our while.However,it is a law that there are no such lumps of uranium- 235,because 1,000 kg exceeds the critical mass of that isotope (something less than a kilogram)and so any such lump would cause its own chain reaction and self-destruct. What this shows is that there can be two very similar looking regularities,one of which is a law and the other not. This is not an isolated example.There are an indefinite number of regularities that are not laws.Take the generalization:all planets with intelligent life forms have a single moon.For the sake of argument,let us imagine that the Earth is the only planet in the universe with intelligent life and that there could exist intelligent life on a planet with no moons or more than one.(For all I know,these propositions are quite likely to be true.) Under these circumstances,the above generalization would be true,even though there is only one instance of it.But it would not be a law;it is just a coincidence.The point here is that the SRT does not distinguish between genuine laws and mere coincidences.What we have done is to find a property to take the place of F which has just one instance and then we take any other property of that instance for G.Then "All Fs are Gs"will be true. And,with a little thought,we can find any number of such spurious coincidental regularities.For most things that there are we could list enough of their general properties to distinguish one thing from everything else.So,for instance,with a person,call her Alice,we just list her hair colour,eye colour.height,weight,age,sex,other distinguishing features,and so on in enough detail that only Alice has precisely those qualities.These qualities we bundle together as a single property F.So only Alice is F. Then choose some other property of Alice (not necessarily unique to her),say the fact that she plays the oboe.Then we will have a true generalization that all people who are F (i.e.have fair hair,green eyes,are 1.63 m tall,weigh 59.8 kg,have a retrousse nose,etc.) play the oboe.But we do not want to regard this as a law,since the detail listed under F may have nothing whatsoever to do with an interest in and talent for playing the oboe. The minimalist who wants to defend the SRT might say that these examples look rather contrived.First,is it right to bundle a lot of properties together as one property F? Secondly,can just one instance be regarded even as a regularity?(If it is not a regularity then it will not be a counter-instance.)However,I do not think that the minimalist can make much headway with these defences. To the general remark that the examples look rather contrived,the critic of the SRT has two responses.First,not all the cases are contrived,as we can see from the gold and
criticism is also made that it is not even necessary for there to be a regularity for the corresponding law to exist. That is, there are laws without the appropriate regularities. Regularities that are not laws I will start with the objection that being a regularity is not sufficient for being a law. Consider the following regularities: (a) All persisting lumps of pure gold-195 have a mass less than 1,000 kg. (b) All persisting lumps of pure uranium-235 have a mass of less than 1,000 kg.9 Both (a) and (b) state true generalizations. But (a) is accidental and (b) is law-like. It is no law that there are no lumps of the pure isotope of gold—we could make one if we thought it worth our while. However, it is a law that there are no such lumps of uranium- 235, because 1,000 kg exceeds the critical mass of that isotope (something less than a kilogram) and so any such lump would cause its own chain reaction and self-destruct. What this shows is that there can be two very similar looking regularities, one of which is a law and the other not. This is not an isolated example. There are an indefinite number of regularities that are not laws. Take the generalization: all planets with intelligent life forms have a single moon. For the sake of argument, let us imagine that the Earth is the only planet in the universe with intelligent life and that there could exist intelligent life on a planet with no moons or more than one. (For all I know, these propositions are quite likely to be true.) Under these circumstances, the above generalization would be true, even though there is only one instance of it. But it would not be a law; it is just a coincidence. The point here is that the SRT does not distinguish between genuine laws and mere coincidences. What we have done is to find a property to take the place of F which has just one instance and then we take any other property of that instance for G. Then “All Fs are Gs” will be true. And, with a little thought, we can find any number of such spurious coincidental regularities. For most things that there are we could list enough of their general properties to distinguish one thing from everything else. So, for instance, with a person, call her Alice, we just list her hair colour, eye colour, height, weight, age, sex, other distinguishing features, and so on in enough detail that only Alice has precisely those qualities. These qualities we bundle together as a single property F. So only Alice is F. Then choose some other property of Alice (not necessarily unique to her), say the fact that she plays the oboe. Then we will have a true generalization that all people who are F (i.e. have fair hair, green eyes, are 1.63 m tall, weigh 59.8 kg, have a retroussé nose, etc.) play the oboe. But we do not want to regard this as a law, since the detail listed under F may have nothing whatsoever to do with an interest in and talent for playing the oboe. The minimalist who wants to defend the SRT might say that these examples look rather contrived. First, is it right to bundle a lot of properties together as one property F? Secondly, can just one instance be regarded even as a regularity? (If it is not a regularity then it will not be a counter-instance.) However, I do not think that the minimalist can make much headway with these defences. To the general remark that the examples look rather contrived, the critic of the SRT has two responses. First, not all the cases are contrived, as we can see from the gold and Laws of nature 19
Philosophy of science 20 uranium example.We can find others.One famous case is that of Bode's "law"of planetary orbits.In 1772.J.E.Bode showed that the radii of known planetary orbits fit the following formula:0.4+0.3x2(measured in astronomical units)where n=0 for Venus,1 for the Earth,2 for Mars,and so on,including the minor planets.(Mercury could be included by ignoring the second term,its orbital radius being 0.4 astronomical units.) Remarkably,Bode's law was confirmed by the later discovery of Uranus in 1781.Some commentators argued that the hypothesis was so well confirmed that it achieved the status of a law,and consequently ruled out speculation concerning the existence of a possible asteroid between the Earth and Mars.Bode's "law"was eventually refuted by the observation of such an asteroid,and later by the discovery of the planet Neptune, which did not fit the pattern.What Bode's non-law shows is that there can be remarkable uniformities in nature that are purely coincidental.In this case the accidental nature was shown by the existence of planets not conforming to the proposed law.But Neptune,and indeed Pluto too,might well have had orbits fitting Bode's formula.Such a coincidence would still have been just that,a coincidence,and not sufficient to raise its status to that of a law. Secondly,the critic may respond that the fact that we can contrive regularities is just the point.The SR'T is so simple that it allows in all sorts of made-up regularities that are patently not laws.At the very least the SR'T will have to be amended and sharpened up to exclude them.For instance,taking the first specific point,as I have stated it the SRT does not specify what may or may not be substituted for F and G.Certainly it is an important question whether compounds of properties are themselves also properties.In the Alice example I stuck a whole lot of properties together and called them F.But perhaps sticking properties together in this way does always yield a new property.In which case we might want to say that only uncompounded properties may be substituted for F and G in the schema for the SRT. However,amending the SRT to exclude Fs that are compound will not help matters anyway,for two reasons.First,there is no reason why there should not be uncompounded properties with unique instances.Secondly,some laws do involve compounds-the gas laws relate the pressure of a gas to the compound of its volume and temperature.To exclude regularities with compounds of properties would be to exclude a regularity for which there is a corresponding law.To the second point,that the regularities con-structed have only one instance,one rejoinder must be this:why cannot a law have just one instance?It is conceivable that there are laws the only instance of which is the Big Bang. Indeed,a law might have no instances at all.Most of the transuranium elements do not exist in nature and must be produced artificially in laboratories or nuclear explosions. Given the difficulty and expense of producing these isotopes and because of their short half-lives it is not surprising that many tests and experiments that might have been carried out have not been.Their electrical conductivity has not been examined,nor has their chemical behaviour.There must be laws governing the chemical and physical behaviour of these elements under circumstances which have never and never will arise for them. There must be facts about whether nobelium-254,which is produced only in the laboratory,burns in oxygen and,if so,what the colour of its flame is,what its oxide is like,and so forth;these facts will be determined by laws of nature,laws which in this case have no instances
uranium example. We can find others. One famous case is that of Bode’s “law” of planetary orbits. In 1772, J.E.Bode showed that the radii of known planetary orbits fit the following formula: 0.4+0.3×2n (measured in astronomical units) where n=0 for Venus, 1 for the Earth, 2 for Mars, and so on, including the minor planets. (Mercury could be included by ignoring the second term, its orbital radius being 0.4 astronomical units.) Remarkably, Bode’s law was confirmed by the later discovery of Uranus in 1781. Some commentators argued that the hypothesis was so well confirmed that it achieved the status of a law, and consequently ruled out speculation concerning the existence of a possible asteroid between the Earth and Mars. Bode’s “law” was eventually refuted by the observation of such an asteroid, and later by the discovery of the planet Neptune, which did not fit the pattern. What Bode’s non-law shows is that there can be remarkable uniformities in nature that are purely coincidental. In this case the accidental nature was shown by the existence of planets not conforming to the proposed law. But Neptune, and indeed Pluto too, might well have had orbits fitting Bode’s formula. Such a coincidence would still have been just that, a coincidence, and not sufficient to raise its status to that of a law. Secondly, the critic may respond that the fact that we can contrive regularities is just the point. The SRT is so simple that it allows in all sorts of made-up regularities that are patently not laws. At the very least the SRT will have to be amended and sharpened up to exclude them. For instance, taking the first specific point, as I have stated it the SRT does not specify what may or may not be substituted for F and G. Certainly it is an important question whether compounds of properties are themselves also properties. In the Alice example I stuck a whole lot of properties together and called them F. But perhaps sticking properties together in this way does always yield a new property. In which case we might want to say that only uncompounded properties may be substituted for F and G in the schema for the SRT. However, amending the SRT to exclude Fs that are compound will not help matters anyway, for two reasons. First, there is no reason why there should not be uncompounded properties with unique instances. Secondly, some laws do involve compounds—the gas laws relate the pressure of a gas to the compound of its volume and temperature. To exclude regularities with compounds of properties would be to exclude a regularity for which there is a corresponding law. To the second point, that the regularities con-structed have only one instance, one rejoinder must be this: why cannot a law have just one instance? It is conceivable that there are laws the only instance of which is the Big Bang. Indeed, a law might have no instances at all. Most of the transuranium elements do not exist in nature and must be produced artificially in laboratories or nuclear explosions. Given the difficulty and expense of producing these isotopes and because of their short half-lives it is not surprising that many tests and experiments that might have been carried out have not been. Their electrical conductivity has not been examined, nor has their chemical behaviour. There must be laws governing the chemical and physical behaviour of these elements under circumstances which have never and never will arise for them. There must be facts about whether nobelium-254, which is produced only in the laboratory, burns in oxygen and, if so, what the colour of its flame is, what its oxide is like, and so forth; these facts will be determined by laws of nature, laws which in this case have no instances. Philosophy of science 20
Laws of nature 21 So we do not want to exclude something from being a law just because it has few instances,even just one instance,or none at all.At the same time the possibility of instance less laws raises another problem for the SRT similar to the first.According to the SRT empty laws will be empty regularities-cases of"all Fs are Gs"where there are no Fs.There is no problem with this;it is standard practice in logic to regard all empty generalizations as trivially true.What is a problem is how to distinguish those empty regularities that are laws from all the other empty regularities.After all,a trivial empty regularity exhibits precisely as much regularity as an empty law. Let us look at a different problem for the SRT.This concerns functional laws.The gas law is an example of a functional law.It says that one magnitude-the pressure of a body of gas-is a function of other magnitudes,often expressed in a formula such as: T P=k where Pis pressure,Tis temperature,Vis volume,and k is a constant.In regarding this as a law,we believe that Tand Vcan take any of a continuous range of values,and that P will correspondingly take the value given by the formula.Actual gases will never take all of the infinite range of values allowed for by this formula.In this respect the formula goes beyond the regularity of what actually occurs in the history of the universe.But the SRT is committed to saying that a law is just a summary of its instances and does not seek to go beyond them.If the simple regularity theorist sticks to this commitment,the function ought to be a partial or gappy one,leaving out values that are not actually instantiated.Would such a gappy "law"really be a law?One's intuition is that a law should cover the uninstantiated values too.If the SRT is to be modified to allow filling in of the gaps,then this needs to be justified.Furthermore,the filling in of the gaps in one way rather than another needs justification.Of course there may well be a perfectly natural and obvious way of doing this,such as fitting a simple curve to the given points. The critic of the SRT will argue that this is justified because this total (non-gappy) function is the best explanation of the instantiated values.But the simple regularity theorist cannot argue in this way,because this argument accepts that a law is something else other than the totality of its instances.As far as what actually occurs is concerned, one function which fits the points is as good as any other.From the SRT point of view. the facts cannot decide between two such possible functions.But,if the facts do not decide,we cannot make an arbitrary choice,say choosing the simplest for the sake of convenience,as this would introduce an arbitrary element into the notion of lawhood. Nor can we allow all the functions that fit the facts to be laws.The reason for this is the same as the reason why we cannot choose one arbitrarily and also the same as the reason why we cannot have all empty generalizations as laws.This reason is that we expect laws to give us determinate answers to questions of what would have happened in counterfactua/situations-that is situations that did not occur,but might have. Laws and counterfactuals Freddie's car is black and when he left it in the sun it got hot very quickly.The statement "Had Freddie's car been white,it would have got hot less quickly"is an example of a
So we do not want to exclude something from being a law just because it has few instances, even just one instance, or none at all. At the same time the possibility of instance less laws raises another problem for the SRT similar to the first. According to the SRT empty laws will be empty regularities—cases of “all Fs are Gs” where there are no Fs. There is no problem with this; it is standard practice in logic to regard all empty generalizations as trivially true.10 What is a problem is how to distinguish those empty regularities that are laws from all the other empty regularities. After all, a trivial empty regularity exhibits precisely as much regularity as an empty law. Let us look at a different problem for the SRT. This concerns functional laws. The gas law is an example of a functional law. It says that one magnitude—the pressure of a body of gas—is a function of other magnitudes, often expressed in a formula such as: where P is pressure, T is temperature, V is volume, and k is a constant. In regarding this as a law, we believe that T and V can take any of a continuous range of values, and that P will correspondingly take the value given by the formula. Actual gases will never take all of the infinite range of values allowed for by this formula. In this respect the formula goes beyond the regularity of what actually occurs in the history of the universe. But the SRT is committed to saying that a law is just a summary of its instances and does not seek to go beyond them. If the simple regularity theorist sticks to this commitment, the function ought to be a partial or gappy one, leaving out values that are not actually instantiated. Would such a gappy “law” really be a law? One’s intuition is that a law should cover the uninstantiated values too. If the SRT is to be modified to allow filling in of the gaps, then this needs to be justified. Furthermore, the filling in of the gaps in one way rather than another needs justification. Of course there may well be a perfectly natural and obvious way of doing this, such as fitting a simple curve to the given points. The critic of the SRT will argue that this is justified because this total (non-gappy) function is the best explanation of the instantiated values. But the simple regularity theorist cannot argue in this way, because this argument accepts that a law is something else other than the totality of its instances. As far as what actually occurs is concerned, one function which fits the points is as good as any other. From the SRT point of view, the facts cannot decide between two such possible functions. But, if the facts do not decide, we cannot make an arbitrary choice, say choosing the simplest for the sake of convenience, as this would introduce an arbitrary element into the notion of lawhood. Nor can we allow all the functions that fit the facts to be laws. The reason for this is the same as the reason why we cannot choose one arbitrarily and also the same as the reason why we cannot have all empty generalizations as laws. This reason is that we expect laws to give us determinate answers to questions of what would have happened in counterfactual situations—that is situations that did not occur, but might have. Laws and counterfactuals Freddie’s car is black and when he left it in the sun it got hot very quickly. The statement “Had Freddie’s car been white, it would have got hot less quickly” is an example of a Laws of nature 21
Philosophy of science 22 counterfactual statement.It is not about what did happen,but what would have happened in a possible but not actual (a counter-to-fact)situation (i.e.Freddie's car being white rather than black).The counter-factual in question is true.And it is true because it is a law that white things absorb heat less rapidly than black things.Laws support counterfactuals. We saw above that every empty regularity will be true and hence will be a law, according to the SRT.This is an undesirable conclusion.Counterfactuals help us see why.Take some property with no instances,F.If we allowed all empty regularities to be laws we would have both law 1 "it is a law that Fs are Gs"and law 2 "it is a law that Fs are not-Gs".What would have happened if a,which is not F,had been F?According to law 1,a would have been G,while law 2 says a would have been not-G.So they cannot both really be laws.Similarly,we cannot have both of two distinct functions governing the same magnitudes being laws,even if they agree in their values for actual instances and diverge only for non-actual values.For the two functional laws will contradict one another in the conclusion of the counterfactuals they support when we ask what values would Phave taken had T'and Vtaken such-and-such (non-actual)values. Counterfactuals also underline the difference between accidental and nomic regularities.Recall the regularities concerning very large lumps of gold and uranium isotopes.There are no such lumps of either.In the case of uranium-235,there could not be such lumps,there being a law that there are no such lumps.On the other hand,there is no law concerning large lumps of gold,and so there could have been persisting 2,000 kg lumps of gold-195.In this way counterfactuals distinguish between laws and accidents. Some philosophers think that the very fact that laws support counterfactuals is enough to show the minimalist to be wrong (and the SRT supporter in particular).The reasoning is that counter-factuals go beyond the actual instances of a law,as they tell us what would have happened in possible but non-actual circumstances.And so the minimalist must be mistaken in regarding laws merely as some sort of summary of their actual instances. This argument seems powerful,but I think it is not a good line for the anti-minimalist to pursue.The problem is that counterfactuals are not any better understood than laws,and one can argue that our understanding of counterfactuals is dependent on our notion of law or something like it,in which case corresponding to the minimalist account of laws will be a minimalist account of counterfactuals.You can see that this response is plausible by considering that counterfactuals are read as if there is a hidden clause,for instance "Freddie's car would have got hot less quickly had it been white and everything else and been the same as far as possible".(Which is why one cannot reject the counterfactual by saying that had Freddie's car been white,the Sun might not have been shining.)The clause which says that everything should be the same as far as possible requires among other things,like the weather being the same,that the laws of nature be the same.So one can say that laws support counterfactuals only because counterfactuals implicitly refer to laws.Counterfactuals therefore have nothing to tell us about the analysis of laws. Consider the fact that laws do not support all counterfactuals-particularly those counterfactuals relating to the way things would be with different laws.For instance one could ask how quickly would two things have accelerated towards one another had the gravitational constant G been twice what it is.The actual law of gravitation clearly does not support the correct answer to this counterfactual question
counterfactual statement. It is not about what did happen, but what would have happened in a possible but not actual (a counter-to-fact) situation (i.e. Freddie’s car being white rather than black). The counter-factual in question is true. And it is true because it is a law that white things absorb heat less rapidly than black things. Laws support counterfactuals. We saw above that every empty regularity will be true and hence will be a law, according to the SRT. This is an undesirable conclusion. Counterfactuals help us see why. Take some property with no instances, F. If we allowed all empty regularities to be laws we would have both law 1 “it is a law that Fs are Gs” and law 2 “it is a law that Fs are not-Gs”. What would have happened if a, which is not F, had been F? According to law 1, a would have been G, while law 2 says a would have been not-G. So they cannot both really be laws. Similarly, we cannot have both of two distinct functions governing the same magnitudes being laws, even if they agree in their values for actual instances and diverge only for non-actual values. For the two functional laws will contradict one another in the conclusion of the counterfactuals they support when we ask what values would P have taken had T and V taken such-and-such (non-actual) values. Counterfactuals also underline the difference between accidental and nomic regularities. Recall the regularities concerning very large lumps of gold and uranium isotopes. There are no such lumps of either. In the case of uranium-235, there could not be such lumps, there being a law that there are no such lumps. On the other hand, there is no law concerning large lumps of gold, and so there could have been persisting 2,000 kg lumps of gold-195. In this way counterfactuals distinguish between laws and accidents. Some philosophers think that the very fact that laws support counterfactuals is enough to show the minimalist to be wrong (and the SRT supporter in particular). The reasoning is that counter-factuals go beyond the actual instances of a law, as they tell us what would have happened in possible but non-actual circumstances. And so the minimalist must be mistaken in regarding laws merely as some sort of summary of their actual instances. This argument seems powerful, but I think it is not a good line for the anti-minimalist to pursue. The problem is that counterfactuals are not any better understood than laws, and one can argue that our understanding of counterfactuals is dependent on our notion of law or something like it, in which case corresponding to the minimalist account of laws will be a minimalist account of counterfactuals.11 You can see that this response is plausible by considering that counterfactuals are read as if there is a hidden clause, for instance “Freddie’s car would have got hot less quickly had it been white and everything else and been the same as far as possible”. (Which is why one cannot reject the counterfactual by saying that had Freddie’s car been white, the Sun might not have been shining.) The clause which says that everything should be the same as far as possible requires among other things, like the weather being the same, that the laws of nature be the same. So one can say that laws support counterfactuals only because counterfactuals implicitly refer to laws. Counterfactuals therefore have nothing to tell us about the analysis of laws. Consider the fact that laws do not support all counterfactuals—particularly those counterfactuals relating to the way things would be with different laws. For instance one could ask how quickly would two things have accelerated towards one another had the gravitational constant G been twice what it is. The actual law of gravitation clearly does not support the correct answer to this counterfactual question. Philosophy of science 22
Laws of nature 23 Laws that are not regularities-probabilistic laws So far there is mounting evidence that being a simple regularity is not sufficient for being a law.There are many regularities that do not constitute laws of nature: (a)accidental regularities (b)contrived regularities (c)uninstantiated trivial regularities (d)competing functional regularities. The natural response on the part of the minimalist is to try to amend the SRT by adding further conditions that will reduce the range of regularities which qualify as laws.So the original idea is maintained,that a law is a regularity,and with an appropriate amendment it will now be that laws are a certain sort of regularity,not any old regularity. This is the line I will examine shortly.But before doing so I want to consider an argument which suggests that being a regularity is not even sufficient for being a law. That is,there are laws that are not simple regularities.If this line of thinking were correct, then it would be no good trying to improve the SRT by adding extra conditions to cut down the regularities to the ones we want,as this would still leave out some laws that do not have a corresponding regularity. The problem concerns probabilistic laws.Probabilistic laws are common in nuclear physics.Atomic nuclei as well as individual particles are prone to decay.This tendency to decay can be quantified as the probability that a nucleus will decay within a certain period.(When the probability is one-half,the corresponding period of time is called the halflife.)So a law of nuclear physics may say that nuclei of a certain kind have a probability p of decaying within time t What is the regularity here?The SRT,as it stands,has no answer to this question.But an answer,which satisfies the minimalist's aim of portraying laws as summaries of the individual facts,is this.The law just mentioned will be equivalent to the fact that,of all the relevant particles taken together,a proportion p will have decayed within t.(Note that we would find out what the value of p is by looking at the proportion that decays in observed samples.Another feature that might be included in the law is resiliency,i.e.that p is the proportion which decays in all appropriate subpopulations,and not just the population as a whole.) The problem arises when we consider each of the many particles individually.Each particle has a probability p of decaying in time t.This is perfectly consistent with the particle decaying well before t or well after t.So the law allows for any particle to decay only after time t*,which is later than t.And what goes for one particle goes for all.Thus the law is consistent with all the particles decaying only after t*,in which case the proportion decaying by t is zero.This means that we have a law the instances of which do not form the sort of regularity the minimalist requires.The minimalist requires the proportion p to decay by t We would certainly expect that.But this is by no means necessary.While it is extremely unlikely that no particle will decay until after t,it is not impossible.Another,less extreme,case would be this.Instead of the proportion p decaying within t,a very slightly smaller proportion than p might decay.The chance of this is not only greater than zero,i.e.a possibility,but may even be quite high. The minimalist's guiding intuition is that the existence and form of a law is determined by its instances.Here we have a case where our intuitions about laws allow
Laws that are not regularities—probabilistic laws So far there is mounting evidence that being a simple regularity is not sufficient for being a law. There are many regularities that do not constitute laws of nature: (a) accidental regularities (b) contrived regularities (c) uninstantiated trivial regularities (d) competing functional regularities. The natural response on the part of the minimalist is to try to amend the SRT by adding further conditions that will reduce the range of regularities which qualify as laws. So the original idea is maintained, that a law is a regularity, and with an appropriate amendment it will now be that laws are a certain sort of regularity, not any old regularity. This is the line I will examine shortly. But before doing so I want to consider an argument which suggests that being a regularity is not even sufficient for being a law. That is, there are laws that are not simple regularities. If this line of thinking were correct, then it would be no good trying to improve the SRT by adding extra conditions to cut down the regularities to the ones we want, as this would still leave out some laws that do not have a corresponding regularity. The problem concerns probabilistic laws. Probabilistic laws are common in nuclear physics. Atomic nuclei as well as individual particles are prone to decay. This tendency to decay can be quantified as the probability that a nucleus will decay within a certain period. (When the probability is one-half, the corresponding period of time is called the half-life.) So a law of nuclear physics may say that nuclei of a certain kind have a probability p of decaying within time t. What is the regularity here? The SRT, as it stands, has no answer to this question. But an answer, which satisfies the minimalist’s aim of portraying laws as summaries of the individual facts, is this. The law just mentioned will be equivalent to the fact that, of all the relevant particles taken together, a proportion p will have decayed within t. (Note that we would find out what the value of p is by looking at the proportion that decays in observed samples. Another feature that might be included in the law is resiliency, i.e. that p is the proportion which decays in all appropriate subpopulations, and not just the population as a whole.) The problem arises when we consider each of the many particles individually. Each particle has a probability p of decaying in time t. This is perfectly consistent with the particle decaying well before t or well after t. So the law allows for any particle to decay only after time t*, which is later than t. And what goes for one particle goes for all. Thus the law is consistent with all the particles decaying only after t*, in which case the proportion decaying by t is zero. This means that we have a law the instances of which do not form the sort of regularity the minimalist requires. The minimalist requires the proportion p to decay by t. We would certainly expect that. But this is by no means necessary. While it is extremely unlikely that no particle will decay until after t, it is not impossible. Another, less extreme, case would be this. Instead of the proportion p decaying within t, a very slightly smaller proportion than p might decay. The chance of this is not only greater than zero, i.e. a possibility, but may even be quite high. The minimalist’s guiding intuition is that the existence and form of a law is determined by its instances. Here we have a case where our intuitions about laws allow Laws of nature 23