This one should be obvious, but I can’t quite see how to answer it.
The following principle is, I think, beyond serious dispute: If being F is an intrinsic property then having a part that is F is intrinsic.[1]
At first glance I would have this principle was just as safe: If being F is extrinsic then having a part that is F is also extrinsic. But I can’t find a quick proof of this that is completely universal. Here’s my initial thoughts.
Take an object that has an F part. Since F is extrinsic, it has a duplicate that is not F. Match that duplicate with duplicates of all the other parts of the object and voila a duplicate of the original object that has no parts that are F. Well obviously this won’t work because (a) we don’t know that the duplicate part could be not F while surrounded by all those other duplicate parts. (Remember F is extrinsic – surrounding it with other things is potentially relevant.) And of course now one of the other parts could be F. So that won’t work at all.
My second thought was to start with an object that is itself F although none of its parts are, and consider non-F duplicates of it. But this might be impossible, e.g. if F is the property of being a gunky object in a world that contains a G. So I’m a bit stuck for a proof here.
The principle sure looks sound, but I suppose that’s what Frege said about Axiom V.
Here’s an interesting specific instance of the principle to close with. Many people say that being negatively charged is extrinsic, because it’s defined in terms of its relational properties or some such. Is the property of having a part that is negatively charged extrinsic? Is it an extrinsic property of the universe we live in?
[1] I don’t mean it is beyond dispute – I just reserve the right to treat the disputants as less than serious.
UPDATE: No wonder I couldn’t prove this – it isn’t true. See David Chalmers’s comment below, or if you’d prefer the exposition version read on. An object is accompanied, in Langton and Lewis’s sense, if there is something in the world it does not overlap. Let F be “accompanied or simple”. This is pretty clearly extrinisic. (Our universe is not F but a duplicate of it in a larger world is F, for example.) But having a part that is F is a necessary property. Every simple has it in virtue of being itself simple and having itself as a part. And every non-simple can be divided into two parts, and each of the parts are F, so it has the property too. And the necessary property is intrinsic. So that settles that. Much thanks to David Chalmers for the example.