Just a few thoughts I thought I'd air here.
Is GMR contingently true? So far I've seen nothing that would indicate that if there are a plurality of concrete worlds there are necessarily a plurality of concrete worlds. Consider a Lewisian universe in which all worlds with green rabbits aren't there. That seems just as metaphysically possible as the universe in which all Lewisian worlds are there. If that's true, any given Lewisian world is a contingent entity (the fact that he lacks the vocabulary to describe what that would mean irregardless). In other words, most actualist possible worlds exist in virtue of the world possibly being a certain way (and thus all possibilities are necessarily represented). Lewisian worlds don't have this to fall back on.
utterly contingent.... disgusting!
Also, I had a couple chats with the reductionists of the class about the epistemology of modality. The question had come up, "how can we know about possibilities". This could be directed to either realist theory.
Well, I'll attempt an empiricist approach to this problem. We can observe actual properties and relations and conceptually divide them from their instances. We can also notice scientific trends, or natural laws. If we can conceive of a few properties and relations combined, and notice that nothing in our natural laws prohibits this combination, voila! We've conceived of a possibility. Of course hard determinists will have to be more strict about the laws they employ, and give a story about a causal history. But that's ok, we don't need to be hard determanists (at least no present natural law entails HD).
If we grant Lewis that all possibilities are realized in a concrete world, then voila! We know stuff about other concrete worlds. However, if all these worlds are contingent, we have less cause to grant then to Lewis.
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You say that Lewis lacks the resources to ensure that all possibilities are necessarily represented. Divers' O12 is supposed to guarantee that. It might be unclear how it does this; Lewis discusses this issue in On the Plurality of Worlds. He says he explicitly endorses the following:
(1) absolutely every way that a world could possibly be is a way that some world is, and
(2) absolutely every way that a part of a world could possibly be is a way that some part of some world is. (pp. 86)
Lewis wants (1) and (2) to guarantee that worlds are abundant; logical space is in some sense complete, and lacks gaps. But he worries that (1) and (2) have trivial readings. So what to do to characterize them non-trivially?
To get what he wants, Lewis appeals to a principle that has become fairly famous in metaphysics: Humean recombination. Says Lewis:
"I suggest we look to the Humean denial of necessary connections between distinct existences. To express the plenitude of possible worlds, I require a principle of recombination according to which patchng together parts of different possible worlds yields another possible world. Roughly speaking, the principle is that anything can coexist with anything else, at least provided they occupy distinct spatiotemporal positions. Likewise, anything can fail to coexist with anything else. Thus if there could be a dragon, and there could be a unicorn, but there couldn't be a dragon and a unicorn side by side, that would be an unacceptable gap in logical space, a failure of plenitude. And if there could be a talking head contiguous to the rest of a living human body, but there couldn't be a talking head separate from the rest of a human body, that too would be a failure of plenitude. . ."
"I cannot altogether accept the formulation: anything can coexist with anything. for I think the worlds do not overlap, hence each thing is part of only one of them. A dragon from one world and a unicorn from a second world do not themselves coexist either in the dragon's world, or in the unicorn's world, or in a third world. . ."
"Ordinarily I would replace trans-world identity by counterpart relations, but not here. I cannot accept the principle: a counterpart of anything can coexist with a counterpart of anything else. . ."
"It is right to formulate our principle of recombination in terms of similarity. It should say, for instance, that there is a world where something like the dragon coexists with something like the unicorn. But extrinsic similarity is irrelevant here, so I should not speak of coexisting counterparts. Instead, I should say that a duplicate of the dragon and a duplicate of the unicorn coexist at some world, and that the attached talking head has at some world a separated duplicate. . ."
"Not only two possible individuals, but any number should admit of combination by means of coexisting duplicates. Indeed, the number might be infinite. . ."
"But now there is trouble. Only a limited number of distinct things can coexist in a spacetime continuum. It cannot exceed the infinite cardial number of the points in a continuum. So if we have more than continuum many possible individuals to be copied, or if we want more than continuum copies of any single individual, then a continuum will be too small to hold all the coexisting things that our principle seems to require. . ."
"Our principle therefore requires a proviso: 'size and shape permitting'. The only limit on the extent to which a world can be filled with duplicates of possible individuals is that the parts of a world must be able to fit together within some possible size and shape of spacetime. Apart from that, anything can coexist with anything, and anything can fail to coexist with anything. . ."
"This leaves a residual problem of plenitude: what are the possible sizes and shapes of spacetime? Spacetimes have mathematical representations, and an appropriate way to state plenitude would be to say that for every representation in some salient class, there is a world whose spacetime is thus represented. It is up to mathematics to offer us candidates for the 'salient class'." (pp. 87-90)
So this is where Divers is getting O12 from. And hopefully it's now clearer just what work O12 is supposed to be doing.
Hopefully it's clear from the last comment that Lewis takes himself to be providing a metaphysical account of all of logical space--reality in the very broadest sense. This makes it obvious, to me, that Lewis intends his account to be necessarily true, if true. There are a couple of other hints that he holds this if more support is needed. One place is in a footnote to Plurality on pp. 224:
"I am inclined to agree with Unger that we have reason to reject hypotheses that involve gratuitous arbitrainess, and therefore suggest--unacceptably--that the geography of logical space is a contingent matter."
Elsewhere, he is clearer (pp. 112). Since he's also addressing the "knowledge" issue here, I'll quote at length:
"So we have (a) boundary between knowledge that does and that doesn't require causal contact with the subject matter. It is a principled boundary . . . Modal and mathematical knowledge together fall on the right side of the line. Our contingent knokwledge that there are donkeys at our world requires causal contact with donkeys, or at least with what produces them. Our necessary knowledge that there are donkeys at some worlds--even talkng donkeys, donkeys with dragons as worldmates, and what have you--does not require causal acquaintance either with the donkeys or with what produces them. It requires no observation of our surroundings, because it is no part of our knowledge of which possible world is ours and which possible individuals we are."
"If you think that all knowledge requires causal acquaintance with the subject matter, I think that is just hasty generalization. But if you concede that mathematical knowledge does not, and yet you insist that knowledge of other-worldly donkeys does, the I doubt that you really regard the latter as non-contingent modal knowledge. I suspect that you suspect that other worlds must really be parts of actuality, not alternative possibilities. We considered this as an objection in its own right . . . it is better to take it straight than to entangle it with issues about knowledge. If the other worlds would just be part of actuality, modal realism is kaput. If not, then the knowledge we have concerning donkeys at other possible worlds is not on a par with the knowledge we lack concerning donkeys in remote or hidden parts of this world. We should not be misled by a false analogy between the two. The former is part of our modal knowledge of what worlds there are. The latter would be part of our knowledge about which world is ours; we gain such knowledge by interacting causally with the world around us, and the problem is that we interact mainly with its nearby and unhidden parts . . ."
"Can you really not know that 2 + 2 = 4, or that there are no true contradictions, when you fully understand and accept the statement? I doubt it." (pp. 112-113)
Lewis devotes from 108-122 of Plurality to issues concerning knowledge of possibilities.
I think it is clear that we have knowledge of possibities. Forget about Lewis for a second. Now consider which things you could do, or things that could happen, in the next few seconds, or minutes, or days, or years. Lots of things come to mind. At least for some of these, you know they are alternatives for you. That's not to say you're infallible; you can be wrong about what the alternatives really are for you. Note that there's no real deep mystery here: people tend to get better at evaluating their alternatives as they live longer and assigning respective likelihoods to different outcomes. This seems to roughly stem from a capacity for counterfactual reasoning. We would be seriously screwed without knowledge of possibilities. (Maybe we could get by with well-justified beliefs about possibilities, but I'm assuming that the "modal skeptic" won't find this any less mysterious.)
Another example, from Kripke: Consider two fair six-sided dice. How many outcomes can result from rolling the dice? The answer is 36, and you know it is.
A similar example, from Williamson: Suppose you have 4 knife blades and 4 knife handles, and each blade will fit any handle. How many knives could you make? The natural answer is '16', not '4', even though you could only ever have at most 4 at once. Claim: your knowledge of non-actual possibilites best explains how you got the natural, correct answer.
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