Is a Universal Regularity Theory an Adequate Account of Laws of Nature?
We all seem to have, as Hempel points out, an instinctive idea of what a 'law' is - we seem to understand the forms they take, and what use they can be put to - but as with any concept where our intuitions predate a clear definition of the concept, we will have a somewhat difficult time pinning down exactly what qualifies as a law. I shall begin by attempting to show that a regularity theory of laws fits well with our intuitions and will be more useful than a denial of laws altogether. I shall then discuss the problem of 'accidental truths', and look at theories that attempt to avoid the difficulties posed by these; then I will briefly look at 'gruesome' predicates, and try to show why, whilst they may be a problem for induction, they pose less difficulty for the formulation of laws. Finally I shall put forward my rather stricter conception of laws, which denies definitional truths as laws.
The classic example of a law of nature (presumably because it is now so commonly accepted as holding) is that of gravity. We believe that it is not simply coincidence that the bodies in the universe seem to move in a particular pattern, but tend to argue that there is a rule that means that every massive object in the universe exerts a force attracting every other massive object in the universe, with a strength related to how far they are apart. Regardless of the specifics of the case, of the actual quantities involved, and even the question of whether this law actually holds, there is already (for the philosopher) a great leap to suggest that any such correlation might hold 'out there' in the universe, rather than just being our conception of how we see things occurring.
Hume, in denying that external events fitted together in the ways we tend to believe, undercut much of what might have been considered obvious about laws - and we must consider the possibility that, rather than there being any connection between events, however many times we see them happening, it may just be a psychological habit of humans to presume there is some underlying law. According to Hume, therefore, a law could be defined very weakly, as something that makes a claim about multiple similar events, and happens to have been true in the instances we have observed. Such a definition would make no claims about the universality of laws and would not comment on whether there was any reason for the events to happen, other than mere coincidence.
Luckily for the philosopher of science, this seems to be unsatisfactory as a conception of law, and we would prefer an idea which would embrace the possibility that laws might be universal concepts, able to predict as well as accurately describe previous occurrences, and also separate those things which just happen to exhibit the same sort of features by a fluke from those which were in some way always going to be similar due to an overriding rule. It would seem, therefore, that whilst this might be a more difficult conception of laws than that supported by sceptics who would deny that there are any underlying rules whatsoever, it will be far more useful a conception - and will of course be more accurate if there are actually rules of this form.
We have a problem, however, if we are simply attempting to define a law in terms that will satisfy our previous observations of the world, and hopefully future observations as well - namely that we risk creating any number of laws involving 'gruesome' predicates, that are constructed specifically to address previous occurrences yet make wildly different claims about the future. A 'gruesome' predicate is one that, like 'grue' which describes items that are green before time t and blue after time t, map exactly to another predicate for instances we have previously observed (in this case it describes just the same set of items, before time t as the predicate 'green' would) but differs significantly from that predicate for some or all future examples. I would argue, however, that 'gruesome' predicates may make prediction a problem (and thus cause real difficulties for a systematic formulation of an inductive principle) but that they do not pose difficulties for an actual definition of a law, since we will quite clearly be able to say that a law saying emeralds are green is correct whereas one claiming that emeralds are 'grue' is not if we observe that after time t they still appear to be green. I have no problem with accepting 'All emeralds are 'grue'' as a valid law (we know, for example, what it would mean for this law to hold, and understand all the words involved), but simply one that isn't true. The question of how we might go about deciding which laws are correct and which are not is a very interesting one but one that I would hope is not going to have to be written into the definition of what a law itself is.
So a conception of a law might claim that a law is simply a universally quantified sentence which is true, yet this does not seem satisfactory, because we can come up with examples of true universally quantified sentences which we would not want to call laws (and hopefully we believe that the intuitional belief that these instances are not laws reflects something more than just psychology). Consider the fairly trivial example that at a red-head convention we might be able to claim correctly that 'Everyone in this room has red hair', but we would surely not want to claim that this was due to a general law. How might we separate sensible laws such as gravity from ones which seem to be entirely coincidental, if our definition only requires universal coincidence for something to be a law?
The first reply might be that accidental truths fail because they're just that - accidental. If they were rule-bound they might be a law, but they are not, so they can't be. This seems to be no response at all, since, firstly, to say things have to be rule-bound to be a law is just to repeat what we're trying to work out, namely what it is for something to be a rule, rather than a simple repeatable occurrence, and secondly because clearly what are frequently called 'accidental truths' on the whole are not - there is a reason (a very good reason) why so many people with red hair are in one room at the same time - there is a convention going on - so we can't really claim that there is not explanation, even if we are going to say it is not one built into the workings of the universe.
Another response to this might be to take a counterfactual view of laws - and say that it's not just each person in a room having red hair that makes it a law but that every person that might have been in the room having red hair. Clearly in this case it isn't true that every person that might have been at the convention would have had red hair - people could have accidentally stumbled in or maybe the convention allowed (as most do) people interested in red-headedness, rather than just those exhibiting the quality themselves. If someone did walk in they wouldn't suddenly have red hair (in the way that if they walked in they'd automatically be at a red-head convention, or in a building, etc.), so this would not satisfy a counterfactual account of laws. The problem, of course, with counterfactuals is that they're very hard to test since, by definition, they test only things that haven't happened, and you can never go out and see simply what did occur.
A better response seems to be that suggested by one of the requirements of what Armstrong called the 'na´ve regularity theory', originally proposed by Molnar - that a law may contain only logical connectives, quantifiers and non-empirical predicates. This seems to quickly and easily remove the offending parts of our 'accidental truths' - ruling out 'laws' which include predicates such as 'any person in this room' which require specific knowledge of the world and which restrict the universality that has been protected by the requirement that the law must involve a universal quantification.
The 'na´ve regularity theory' also requires that any law should be a contingent fact, but it is not clear whether this claim, along with the idea that no empirical predicate may be used in the statement of the law is reasonable; I would argue that it is. A law, it seems to me, is a statement of the empirical connection between two concepts that we have an idea of without reference to experience. What do I mean by this? We presumably all know what massive objects are, and if someone did not it could be explained to them without any reference to any experience, only using concepts we can think of a priori, yet there are certain facts that could be true of such objects (and may, indeed, contingently be so), which we will not know a priori, but will have to observe through experience to become aware of.
To address first the requirement that no empirical predicates should be present in a law (and coming in a minute to the question of whether they must be restricted in the other direction, and must be contingent), it seems to me that this is more than just a handy way of removing certain awkward cases, but is in fact a required part of what a law would have to be. It does not seem to be terribly conducive to the truth to discuss particular arrangements of things which we know have occurred purely through observation and try to make claims about the connections between these things (or between these things and other more stable a priori concepts) which we attempt to universalise. Instead we should discuss concepts we already have a good grip of, without reference to the world around us, and only then consider the relationship between these things. We already seem to be making assumptions about how things are (and thus presumably about what laws hold) if we talk about a particular room in the same way as we talk about every room or the typical room.
Secondly, we must consider whether all laws must be contingent. It seems to me that they must, and that nothing that is necessarily could be considered to satisfy our instincts as to what we would usually term laws. Whilst some might claim that 'All water is H2O' or 'All bachelors are never-married men' are laws, I wouldn't agree that this was so. They seem to be simple truths, and do not tell us anything about the way that the universe actually works, or the way that things are held together. A law must connect two concepts that are known apart from experience by a connection that is only one known through experience, since laws seem to describe things about observations, not simply about the way we speak. Presuming (perhaps a little controversially, if we are going to do so without further discussion) that the a priori / necessary connection holds, and that if something is one it must be the other, we should presumably deny that nothing but the contingent will be able to tell us anything about our experience, or our universe 'out there' and thus will fail to fall under the boundaries of how I would like to define laws.
Overall, therefore, I think the na´ve regularity theory is an acceptable theory of laws, even if it is not as simple (and certainly not the simplest theory) as Armstrong's name seems to suggest it should be, being as it is a somewhat refined theory. Laws seem to describe certain brute facts about the universe and how it is constructed, and connect empirically two or more concepts that we know wholly without reference to experience, and can be used to predict accurately, so long as we know what laws actually hold, though knowing exactly how to choose which we believe and which we do not may well be another story entirely.