## Models of laws

In The Metaphysics within Physics, Tim Maudlin raises a puzzling objection to Humean accounts of laws. (Possibly the same objection is raised by John Halpin in several earlier papers such as "Scientific law: A perspectival account".)

Scientists often consider very different models of putative laws. Such models can be understood as miniature worlds or scenarios in which the relevant laws obtain. On Humean accounts, the laws at a world are determined by the occurrent events at that world. The problem is that rival systems of laws often have models with the very same occurrent events. Whether this is a problem depends on what we mean by "the relevant laws obtain". Maudlin:

Let's suppose (and how can one deny it) that every model of a set of laws is a possible way for a world governed by those laws to be. (Maudlin, p.67)

To see the problem this creates, assume H is a history of occurrent events which fits two incompatible systems L1 and L2. For Humeans, H is a complete model, so presumably it is a model of both L1 and L2. By Maudlin's supposition, H is then governed by both L1 and L2, which is impossible.

But why should we follow Maudlin's "supposition"?

Imagine it is a law that all ravens are black, and let L be this law. We must distinguish two propositions: (a) that all ravens are black, and (b) that it is a law that all ravens are black. Correspondingly, we must distinguish two sets of worlds, or models: those where (a) is true and those where (b) is true. (This is not controversial; nobody thinks that lawhood is the same as truth.) The content of our law L is (a), the proposition that all ravens are black. So what is a model of L? The most natural proposal, I would have thought, is to say that a model of L is a situation in which the content of L is true. Any situation in which all ravens are black is therefore a model of L. Maudlin's "undeniable" supposition is instead that only those situations should count as models of L in which L is a law: the situation must be "governed by those laws".

Of course it doesn't really matter how we use the word 'model'. We could call situations in which L is true models of L in the weak sense, and situations in which it is a law models of L in the strong sense. As I said, I think the weak sense is more natural. Maudlin himself slips into it on the very next page, where he reminds the reader of the crucial point that "different laws share the same models".

More important is whether we really have a conflict between Humeanism and scientific practice. When physicists consider models of, say, Newtonian mechanics, they consider hypothetical situations in which the laws of Newtonian mechanics are true. According to Maudlin, they further restrict their attention to situations in which the Newtonian laws are laws. But do we really need to assume this in order to make sense of scientific practice? I don't see why. On the contrary, I think it is plausible that physicists only care about what kinds of hypothetical situations satisfy the relevant equations: the content of the relevant laws.

(Perhaps Maudlin's argument is based on a simplistic conception of the logic of nomic possibility. It is plausible to say that a model of a physical theory should be a situation that is nomically possible on the assumption that the theory is true. I suspect Maudlin takes nomic necessity to be an S5 modality. It then follows that if w is nomically possible relative to L, then L must be a law at w. It also follows that if some proposition p (say, that I had salad for lunch) is not nomically necessary, then it is nomically necessary that p is not nomically necessary. But these assumptions are highly controversial. On Humeans accounts of lawhood, they are false, and I don't think this can be regarded as a cost. If it is a law that all ravens are black, does it really follow that it is also a law that it is a law that all ravens are black?)