What, if anything, explains the difference between a law of nature and an accidentally true generalization? 

The modern debate over this question appears to be taking place between two schools of thought.  There is the "Humean" school of the Mill-Ramsey-Lewis view of Laws of Nature as regularities in nature.  And there is the "anti-Humean" school of the Dretske-Tooley-Armstrong view of Laws of Nature as relations of necessity between Universals.  At the metaphysical level, the two sides to the debate have vastly different conceptions of the relation between Nature and the Laws.  The Humeans maintain that the Laws are descriptions of matters of fact -- the facts determine the Laws.  The anti-Humeans maintain that it is the Laws that determine matters of fact. 

What the Humean school undoubtedly has right is that most of the things we honour with the label "Law of Nature" are nothing more than recognized patterns in the mass of experience we have with Nature.  Scientists search for and discover larger patterns in our experience and generate sweeping generalizations, frequently applicable only in "ideal" circumstances, and often understood to hold only "approximately".  The more basic of these generalizations, the more of science that they support, the more likely they are to be graced with the honorific of "Law of Nature".  The patterns involved need not be clear and crisp patterns of unquestionable regularity.  Because we are such consummate pattern recognition devices, they can be rather muddy partial patterns, that hold only "for the most part".  Sometimes, of course, we are fallible and the patterns are not really there.  As Nancy Cartwright has pointed out these generalizations must be viewed through the noise and fog of "non-existent idealized circumstances", "ceteris paribus", "mutatis mutandis", and a multitude of similar caveats. 

So for the Humean, what differentiates an accidentally true generalization from a law of nature is, in the first place, that the pattern we have noticed must be expected to continue into the future.  We must anticipate that our future experiences of nature will reveal that the pattern continues unchanged.  If not, then the generalization is not the identification of a persisting regularity of nature, and is not a candidate for a law of nature.  In the second place, the generalization must play some role in underpinning the rest of our understanding of nature.  If the candidate generalization can be shown to be the foundation of enough of such other lesser laws and rules, then it is honoured with the "law of nature" label.  In other words, to a Humean, the Laws of Nature systematize our descriptions of our experiences of Nature.  The Laws of Nature are the basic theorems or axioms of a model/theory (or consistent collection of models/theories) that most simply accounts for the widest range of experiences. 

What the anti-Humean school undoubtedly has right, on the other hand, is that the Laws of Nature have an unmistakable element of necessity about them.  According to the anti-Humeans, the Humeans ignore this intuitive element.  The anti-Humeans therefore rely specifically upon it to mark the distinction between laws of nature and accidentally true generalizations.  But anti-Humeans face the serious, as yet unresolved, metaphysical difficulty of providing an explanation of just what this necessity amounts to.  Just how it is explained varies according to the anti-Humean philosopher.  But claiming the necessity to be brute is to ignore the problem, and labelling it as the relation of Necessitation is to merely label the problem. 

But consider the analysis of causation by Messrs. Fair, Dowe, and Salmon.  If their approach is correct, causation is best understood in terms of the conservation of some small number of key quantities (like energy, momentum, and so forth).  And if that is the case, then the conservation restrictions governing causation, would be laws of nature.  In fact, on this basis, they would be the only "fundamental" laws of nature.  All the rest of what we commonly refer to as laws of nature would be just higher level "common" laws of nature, deriving their stature from the causation defined in the "fundamental" laws, and subject to all the problems highlighted by Cartwright.  But the still unanswered question is, just what ensures that energy, say, is conserved across causal interactions?  How is the relation of necessitation cashed out?  While it is obvious that there is some kind of necessity at work here, it is entirely unclear just how to understand that necessity.  But consider that it is part of what it means to be the surface of a sphere that dictates that the surface is finite, unbounded and equal to 4πr2.  Possibly it is part of what it means to be a Universe that those conserved quantities are conserved. 

This is just one example of a description of the relation of necessitation that anti-Humeans rely upon to distinguish laws of nature from accidentally true generalizations.  If an inductive generalization has none of the required necessity about it (in the case of the causation example, has none of the required causal relations about it), then it is classed as an accidentally true generalization.  If it does have the required necessitation (or causal properties), then it qualifies as a "common" laws of nature.

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