Best-systems accounts of laws and second-order laws

On best-systems accounts of laws, p is a law provided that p is a theorem of the best system, where the best system is the one that optimizes informativeness and simplicity (and maybe fit, if we want probabilistic laws). But we also seem to have second-order laws, such as symmetry principles like Lorentz-invariance. Can we fit such second-order laws into a best-systems account?

Say that a symmetry S is a permutation of the collection of worlds. Given a proposition p, let p^S be a proposition that is true at a world w if and only if p is true at S(w). Say that a propositon is invariant under a class U of symmetries provided that p is true at w if and only if p^S true at w for all S in U. A nice way to formulate invariance principles is then to say that the conjunction of the laws is invariant under U, for some appropriate class of symmetries U.

On a best systems analysis, then, U-invariance holds in a world w if and only if the conjunction of the axioms of w's best system is U-invariant. But that's not enough for us to have a second-order invariance law. To have a second-order invariance law, we need that it be a theorem of the best system that the conjunction of the axioms of the best system is U-invariant.

Suppose w is some world with deterministic first-order Newtonian laws, an absolute distinction between rest and motion, and a second-order laws that says that the laws are Galileian-invariant. Is this likely to work out on a best-systems analysis? Let the first-order Newtonian axioms be N₁,...,N_n. These will be all part of the best-system. Now, N₁,...,N_n do not entail that the laws are Galileian-invariant. For there will be worlds where N₁,...,N_n are laws, but on a best-systems analysis there is a further law, L₀, where L₀ is not Galilean-invariant (perhaps all of the particles initially are at rest, and it that they are all at rest might then make it in as an axiom of the best system). Since the second-order claim that the laws are Galileian-invariant does not follow from N₁,...,N_n, it follows that for the second-order claim to be a law, the best system needs something more than N₁,...,N_n. But it seems possible that (a) it is a second-order law that the laws are Galileian-invariant and yet (b) the best system is just N₁,...,N_n as adding stuff that entails the second-order law to the axioms does not add enough predictive value beyond that N₁,...,N_n already have to justify the loss of simplicity.