Nomological possibility, chancy laws and zero-fit
Let's say that something X is nomologically possible if it is true at some world where the actual laws of nature are true. The actual laws may or may not be laws at this world. All we require is that they are true there.
Now consider a chancy law according to which a coin tossed in some standard way has a 50 percent chance of landing heads. For this to be a law at some world w means that it is part of the best theory of w, or that it represents the actual propensities in w, or something like that. What does it mean for it to be merely true at a world?
It can't mean that 50% of all coin tosses really land heads. Otherwise "it is a law that p" would not entail "p". Could it mean that coin tosses have a certain propensity to come up heads, or that the statement is part of the best theory of w? Then "it is a law that p" and "p" would coincide for chancy p. That's also odd: non-chancy laws appear to be limit cases of chancy laws; but for non-chancy laws, truth doesn't coincide with lawhood. Moreover, the proposal could turn nomological possibilities into impossibilities: if the chance of heads is 0.5, then it is unlikely, but not (intuitively) impossible that almost all coins end up tails. But the fact that almost all coins really do end up tails might be incompatible with it being a law that the chance of heads is 0.5. It will be so on the best-system account.
I think we should rather say that X is nomologically possible relative to chancy laws L iff X is not assigned probability 0 by L; and therefore that a chancy law is true at a world w (as opposed to: is a law at w) iff it does not assign probability 0 to any event in w.
(Chances can vary with time. If the actual chance of X is 0.5 at t1, but 0 at t2, is an X-world nomologically possible or impossible relative to the actual world? Or is it possible at t1 and impossible at t2? None of that. Nomological possibility does not mean having probability greater than 0: in a non-chancy world, all and only actual events have probability greater than 0; but lots of alternative events are nomologically possible. Chancy laws give us history-to-chance conditionals: if the history up to time t is H, then the chance of E at t is x. A world w is nomologically impossible iff for some H and E both obtaining at w, our laws say that the chance of E given H is 0.)
But here is a problem: some philosophers, for example Adam Elga in his "Infinitesimal chances and laws of nature" (2004), claim that the actual world history has probability 0 by our actual laws: consider some radium atom A that decayed at some time t today. It could just as well have decayed at continuum many other times; so the probability that it would decay at t is 0. I'm not yet convinced by that, because I'm not yet convinced that chances have to be real numbers. But if Elga is right about chances, and I am right about nomological possibility, then the actual world is nomologically impossible. Not good.
Hi Wo,
Suppose you didn't go with Lewis's very particular theory of chance, whereby chances arise out of distributions of non-chancy facts via helping to systematize those facts in best theory.
You may then be up for objective propensities. Roughly, "point particle x has 50 pc chance of decay at r", or somesuch, would have standing comparable to "point particle x being spin up at r".
Now suppose you said that for a world to be nomologically possible relative to laws that attributed 50pc chance of decay to things of type F, was for all Fs in that world to have a suitable propensity-property. It wouldn't follow, I think that it'd follow that in such a world, Fs having that propensity was a matter of law: it might be in such a world that the best systematization depicted that as a coincidence (suppose that the best system represented chances of decay for particles in general as declining over history, but that was only a single F, half-way through, which happened to have a 50 pc propensity for decay.)
Anyway, I'm not sure what to say about the issue in the Lewisian setting (or Armstrong's treatment of chance, for that matter.)