Strong and Weak Humeanism
Humeans distinguish between how things are in themselves and how they are related to other things. The latter, they say, is always a contingent matter: Even though this cup of tea is about 20m away from a book and stands on a table, it could very well not be 20m away from the book and not stand on the table. In slogan form, there are no necessary connections between distinct entities.
Understood literally, this leads to a position one might call strong humeanism:
For any distinct possible things x and y, it is possible that x and y both exist, and it is also possible that x but not y exists.
This does not yet capture the Humean intuition that even though the cup and the book are 20m away from each other, they could well have been 30m from each other. We could try to strengthen strong humeanism to account for these intuitions, e.g. thus:
For any possible thing x and any extrinsic property P, it is possible that x has P, and it is also possible that x lacks P.
But this is far too strong. One problem is that for the second formulation to imply the first, 'property' must not be understood in a particularly restricted way. It must be a property of the cup that it exists in a world where there also exists a book. But once we allow Cambridge properties, nothing prevents us from considering necessary or impossible properties such as existing in a world where it is true that there is a/no largest prime. That looks like an extrinsic property, hence by the strong principle it should be contingent whether the cup has it. But it is not. That's not a very serious problem though because it is controversial whether necessary and impossible properties should ever be counted as extrinsic.
Another problem is that among extrinsic properties we might find properties like 'being a large hippopotamus in a French zoo', and it is very doubtful whether the cup of tea could possibly have this property. To get rid of such counterexamples, we must somehow restrict 'extrinsic property' to purely extrinsic properties, since being a large Hippopotamus presumably implies having certain intrinsic properties.
Another problem is that strong humeanism implies that nothing can have any extrinsic property essentially. But if extrinsic properties matter at all for the counterpart relation -- as they surely do -- it is hard to see how this could be so. Whether some cup in some world is (a counterpart of) the cup over here depends at least in part on where it is located, by whom it is used, where and how it was manifactured, and so on. Even if, pace Kripke, none of these properties in isolation is essential to the cup, at least the disjunction of them seems to be: If some cup is located in an entirely different surrounding, used by entirely different people, was manifactured in an entirely different way etc., then what makes it a counterpart of the cup over here? I don't think it strictly follows from strong humeanism that the counterpart relation is entirely intrinsic, but it certainly follows that it is more intrinsic than we ordinarily take it to be.
(One could try to restrict 'property' in the humean principle to exclude disjunctive properties. On the one hand, I'm not sure if that wouldn't undermine the principle; on the other hand, I don't think it would help much. It is still implausible to rule out non-disjunctive extrinsic essentials. Basically the same holds for the attempt to return to the first formulation of strong humeanism.)
Strong humeanism is not an attractive position. Yet somehow the problems it faces do not really refute the basic idea that this thing over there, the cup, could exist in all kinds of different circumstances, whether or not we still call it a cup in such circumstances or regard it as a counterpart of the original thing. Remember the humean distinction between how things are in themselves and how they are related to other things. The basic idea is that it is possible for some thing to be in itself just like this cup while very differently related to other things. Let's call this kind of position Weak Humeanism. Again we can try more precise statements like
For any distinct possible things x and y, it is possible that an intrinsic duplicate of x exists together with an intrinsic duplicate of y, and it is also possible that an intrinsic duplicate of x exists without an intrinsic duplicate of y.
or
For any possible thing x and any (purely) extrinsic property P, it is possible that an intrinsic duplicate of x has P, and it is also possible that an intrinsic duplicate of x lacks P.
These look much more plausible. Indeed, they could be taken as analytic: If P is shared by all intrinsic duplicates of x, this just means that P is not (purely) extrinsic.
(This raises two interesting questions in Lewis scholarship: 1. To what extent is Lewis' Principle of Recombination, that looks a lot like Weak Humeanism, analytic? 2. To what extent do which of Lewis' definitions of 'intrinsic' presuppose the Principle of Recombination?)
Unfortunately, now that we have a defensible formulation of Humeanism, Anti-Humeanism has become a very strange position: Do Anti-Humeans reject analytic truths?
Maybe the best way to state the difference between Humeanism and Anti-Humeanism is that the former believe in intrinsic properties and the latter do not.