It was a comfortable enough flight over that I spent more time sleeping than doing things worthy of note. Surprisingly enough, it was The Iliad that kept making me drowsy. The various battle scenes were fine to stay awake through – though I hadn’t realised just how horribly detailed they could be. The problem was old King Nestor. Nestor’s role, for those who aren’t familiar, is largely to try and calm the tensions in the Achean camp, and his main weapon is the long-winded speech. It didn’t seem to help much with Agammemnon and Achilles, but it inevitably worked with me. By the middle of the story, all I had to hear was, “Then good King Nestor rose” and I was sound asleep.

Maybe if I hadn’t slept so much I would have figured out more about stars. But maybe not, for I think I was a little stuck just where I was. Here’s the basics. (For background on stars, see Ted Sider’s papers here and here. Be warned though, this is possibly the most esoteric philosophical question I’ve ever thought about, and that’s not a trivial comparison class.)

Ideally, we’d like to define F* as being F minus maximality. But that won’t do for two reasons.

First, it suggests that when F is not maximal, then F* = F. And that isn’t always right. Let F be the property of being human or weighing more than sixteen stone. This is not maximal – it’s not always the case that the large part of something that weighs more than sixteen stone does not weigh more than sixteen stone. But nor is it the case that F* = F. A large part of me is F*, but it is not F.

Second, this kind of conceptual subtraction in general is not defined. (I think Lloyd Humberstone has a paper on this somewhere, but I don’t quite know where. Wiggins makes quite a bit of this point in his response to Parfit in the 3rd edition of Sameness and Substance. That was the best part of the new edition I thought.) If F can be analysed as G and H, then F minus G is just H. But where F cannot be so analysed, F minus G is not clearly defined. The problem is that there’s nothing remotely like the unique factorisation theorem for concepts or for properties. What we’d like is that F minus G is the property H such that H and G is equivalent to F. But there are too many such properties H. There’s a few ways we might try to discriminate amongst them, mostly using strong appeals to naturalness at crucial points, but as far as I can tell the general problem is hopeless. And I have a suspicion this territory has been worked over in the literature, so I won’t go through it all here.

Let’s try getting to starring more directly. First hypothesis: An F* is something that massively overlaps an F. This gets the right result in most cases, but it doesn’t work in general. In fact, massively overlapping an F is neither necessary nor sufficient for being an F*.

Against necessity: imagine a ball with a small lump on one side. The lump is not massive, but it is big enough to make the ball something other than a sphere. Consider the part of the ball apart from the lump. It is a sphere*, for it has everything necessary for being a sphere other than being maximal, but it does not massively overlap a sphere.

Against sufficiency: Cusack is the heaviest man in Ireland. But not by much. He is only a few ounces heavier than Lenehan. If Cusack’s right hand were suddenly to fall off, Lenehan would be heavier. Let F = is the heaviest man in Ireland, and let a be the mereological difference between Cusack and his right hand. Is a an F*? It seems to be not. It does not have what it takes to be the heaviest man in Ireland, for it is less heavy than Lenehan. But it does massively overlap an F.

An F* is not just a duplicate of an (actual or possible) F. This is I think a necessary condition for being an F*, but it is not sufficient. The counterexamples to sufficiency are easy. I’m a duplicate of a possible uncle, but I am not an uncle*. Still, we do seem to have a necessary condition here, and that may be worth something.

What we intuitively want for a definition of star is something like the following. A thing a is F* iff if a is the right kind of thing to have maximal properties, it has F. The last conditional is not a material conditional, so we can’t easily use it in an analysis. But we can do something.

The kind of thing that’s apt to have maximal properties is just a thing that does have some or other natural maximal property. (I’ll come back to why there has to be a restriction to natural maximal properties here in a bit.) Roughly, then, an F* is something that if it has any natural maximal properties, it is F. Say an object is pretty iff it has any natural maximal properties. Here’s a first pass at trying to define F*, at least for cases where F is reasonably natural.

Another little definition that will be helpful. Say F is intrinsic to the Gs iff being F entails being G and the following holds. Any bijection between the Gs in w1 and the Gs in w2 that maps objects onto duplicates always maps Fs onto Fs and non-Fs onto non-Fs. (That’s actually a little rough. For some purposes we need to also say that for any collection of objects the fusion of their images under the bijection is a duplicate of their fusion. I’ll assume that where necessary.) A lot of extrinsic properties are nonetheless intrinsic to the Gs for suitable G. (Every property, I think, is intrinsic to the things – that’s sort of a weak version of the truthmaker principle.) For instance, the property of being the heaviest man in Ireland is intrinsic to the men in Ireland.

Here’s my attempt then at getting F*. Let G be any natural maximal property such that F is intrinsic to the Gs. Let a be some object in a world w that massively overlaps a pretty object. If a is pretty, then a is F* iff a is F. If not, let b be the pretty object. Let P be the set of pretty objects apart from b in w. Let w’ be a world in which a duplicate of a, call it a’, is pretty. Consider any bijection from the Gs plus a in w onto the Gs in w’. If a is F*, then a’, the image of a under the bijection, should be F. The reason is that a’ is just like a in all respects necessary for being F, it is an intrinsic duplicate and the world is just the right way for a’ to be F, and since a’ is G, and G is a natural maximal property, a’ is pretty so it is apt to have maximal properties. That much all seems relatively uncontroversial, I think.

Let me now make a bold conjecture. If for all such G all such bijections map a onto an F, then a is an F*. The little argument above was that this is a necessary condition for being F*. The hypothesis is that it’s sufficient. I don’t really have an argument that this is sufficient, which is why it is a particularly bold hypothesis. I do, however, have something that may be a counterexample. In fact I may have two. (An extremely bold hypothesis in that case.)

Let F be the property of being the best hitter in baseball. Right now, I presume, Barry Bonds has that property. Let a be a large part of Barry Bonds, say all of him less one hair. I take it that a is F*, and as far as I can tell, my theory delivers that result. But what of poor c, which is the mereological difference between Barry and both of his hands. I think c is not F* – it is not at all the right kind of thing to be the best hitter in baseball, for it has no hands. But I can’t immediately see a G such that being the best hitter in baseball is intrinsic to the Gs, and any suitable bijection does not map c onto the best hitter in baseball. The worry is that being the best hitter in baseball might not be intrinsic to any group more coarse-grained than the things, so there’ll be no bijections of the type I described, so on all such bijections c will be mapped onto the best hitter in baseball. Maybe I’m wrong about that, so the bold conjecture might be right. And maybe c really is F*, the intuitions here are not particularly clear.

A different kind of problem arises with properties like being the mereological difference between a human and its longest hair. Note this is maximal, but we don’t want to say an object with this property is pretty. The difference between me and my longest hair, call it d, has this property, call it F, but it is not pretty. That’s why I restricted the definition of prettiness to those things with natural maximal properties. But now consider d minus its longest hair – call that e. Surely e is F*. But there’s no way at all for my definition of starring to work in that case, for it is only defined for cases where the things that are F are pretty, or at least where they could be pretty. I’m actually not too worried about that. Maybe I don’t have a definition of starring, but necessary and conditions for being an F* for cases where F is reasonably natural. That would still be progress I think, though maybe not much progress.