What is Wrong with the Divers-Melia Argument: An Overview

Now, this argument generates considerable suspicion. First of all, Lewis is the first to recognise that Recombination does not suffice to generate alien worlds; but he does not

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DM note that S* being a subset of S is not enough to make S* alien-incomplete, since there might be indiscernible worlds. It is the fact that S* by definition omits some of the properties instantiated in S that makes S* incomplete. (DM 2002: 31)

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seem the slightest bit disconcerted by this. He simply comments that even though “...recombination will not generate alien worlds out of the parts of this world, it

nevertheless applies to alien worlds. ... If there are some, there are many more.” (Lewis 1986a: 92) In that sense, the Divers-Melia argument establishes nothing new – and certainly nothing that Lewis thought objectionable – for GR. There is no talk in Lewis (1986a) of Recombination (or the rest of the theory), for instance, having to provide some sort of ‘guarantee’ that the space of worlds is complete with respect to all basic alien possibilities. So, it is hard to see why one should take the DM argument to constitute a serious objection to GR. Nonetheless, let us set this issue aside for a moment. What the DM-requirement for GR’s ontological principles to ‘underwrite’ its analysis of possibility may amount to, and why DM think GR should do so, will be examined shortly (sections

2.6.3-2.6.6).

However that may be, the particulars of the argument themselves generate suspicion. For one, the principle (AC) that Divers and Melia first put forth as an alien-completeness principle for GR and then strike down, simply fails to even intuitively capture the thought that all aliens exist in logical space. For (AC) talks about a numbern of aliens and, surely, talk of number is irrelevant to stating universality: saying that any number of them are there, intuitively doesn’t answer the question of whether all of them are there. All in all, one is thus left with the feeling that even if (AC), which in any case is of DM’s own invention, cannot establish alien-completeness, then so much the worse for (AC) and for the argument which proceeds from it.

Especially disconcerting is the fact that the DM-argument to incompleteness crucially turns on the inessential numerical formulation of (AC). For, instead of focusing on features particular to GR, their argument simply exploits the general fact that infinite sets stand in a one-one correspondence with their infinite subsets. What Divers and Melia essentially establish is that, given an infinite number of Fs, any theory trying to guarantee the existence (or instantiation or inclusion) of all the Fs with a principle which states that for any number n of Fs, n Fs exist, is going to fail. But this is hardly surprising. The attempt is analogous to some putative realist about numbers trying to capture the fact that

every natural number exists by making it an axiom that that for any n, there are n distinct natural numbers. Or, it is analogous to a reductionist about the mental trying to ensure that the space of occurrences of C-fibres firing is complete with respect to the (let us suppose, infinite number of) occurrences of pain, by postulating that for any n there are n

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without a corresponding C-fibres firing occurrence. (Indeed, the only principle that ensures this is one that states that the truth-conditions for the analysis are satisfied.) Note, moreover, that the argument does not highlight a shortcoming particular to reductive

theories. The fact that (AC) is satisfied by both S and S* has nothing to do with whether (AC) employs primitive modality. For instance, (ACM) ‘For any possible n, npossible

objects exist across the GR-worlds and npossible distinct alien natural properties are instantiated among these objects’ is equally well satisfied by S and by S*, for the case where n is infinite, hence also fails.

As we saw, Divers and Melia anticipate dissatisfaction with (AC) by dismissing alternatives. Of particular interest, however, is the following principle, dismissed as “no stronger than [AC]” (DM 2002: 33):

(AC*) For every alien property P, there is a world w and an individual x in w such that x instantiates P. (DM 2002: 33)

(AC*) is of course trivial, for it says merely that each (non-empty) set of individuals has some individual as a member, and so is satisfied whether there are ten such sets, seventeen, or infinitely many. However if GR was allowed to espouse (AC*), which is stated in terms of universality, an argument that (AC*) cannot ensure that the GR-space of

possibilities is alien-complete because (AC*) is satisfied by S* which ex hypothesi is not alien-complete, would be question-begging. Of course (AC*) is trivial. But, between (AC) and (AC*) the latter is clearly a far better principle. For while (AC*) might fail to distinguish between different totalities, at least it constitutes an intuitive expression of plenitude, whereas (AC) fails on both counts.

Also in favour of (AC*) is that its cost is not really comparable to the cost of a far broader trivial principle put forth by Manuel Bremer, who, noting that GR-worlds and individuals simply are there, “all of them are there” (Bremer 2003: 81), contends that, given that properties are sets of individuals, the following will thus suffice for alien- plenitude:

(AIC) “For every [world-bound] individual there is a world which contains that individual.” (Bremer 2003: 81; c.f. DM 2003: 83 for the qualification in brackets)

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Naturally, Divers and Melia reject Bremer’s proposal as trivial. (DM 2003: 84) For one, as they correctly note, while it is true that all the individuals ‘are simply there’, by the same token, so are all sets of individuals, so the distinction does little work here. (DM 2003: 85) They conclude that if GR were allowed to go with such a trivial proposal, it could, by the same token, take the even shorter route and say:

(AWC) “All the worlds exist.” (DM 2003: 84)

DM rightly reject (AWC) and (AIC) on the basis that they don’t do any work to capture the notion that there are ‘no gaps in logical space.’ (DM 2003: 85) But (AWC) and (AIC) are not comparable with (AC*). Bremer’s proposal throws the baby out with the bath-water. For Lewis’ Recombinationalready allows GR to non-trivially state that all

non-alien individuals are there, by recombining actual individuals. Moreover, Recombination would be sufficient to ensure that the total space of worlds is alien- complete, given some base set of aliens to recombine. So it is only at the level of alien

individuals, and indeed only basic aliens, that the question of triviality even arises. Bremer’s simply grants too much. (AC*), although trivial, is not comparable in cost to (AIC), (AWC) or any other such trivial principle that constitutes an empty affirmation of the entire GR-ontology.

And besides, nothing prevents GR from adding a further principle, to the effect that the number of alien properties is infinite (call it (INF)), to supplement (AC*). (And while, just as with (AC), (AC*)+(INF) cannot rule out that logical space is as per S* rather than

S, they are nonetheless sufficient to block an argument proceeding in the manner of Divers and Melia: For, by definition(AC*) can distinguish between any two sets of worlds, when one of these is ex hypothesi alien-incomplete. Or else, what right do DM have to quantify over properties beyond those in the range of (AC*)?)

I think the best strategy for GR is to let (AC*) trivially capture the fact that all basic alien properties are there, and allow Recombinationto recursively work from that basis to ensure alien-completeness. This is, after all, what Lewis seems to be doing when he says that “if there are some [alien worlds], there are many more.” (Lewis 1986a: 92) DM might reply as they do to Bremer: “To see what a postulate adds to the theory – do not guess: consider which metaphysical scenarios its truth rules out and which it permits.” (DM 2003: 84) In reply, I say two things: One, while (AC*) may be compatible with ‘different metaphysical scenarios’ as to the number of alien possibilities (which can be

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taken care of by a different principle like INF), it is not compatible with a metaphysical scenario that there are aliens that fail to be instantiated. Two, as I will argue shortly, the Divers-Melia idea that GR must ‘rule out’ certain metaphysical scenarios and ‘permit’ others in order to be accepted as a theory of modality that gives the right results, seems ill- fitted for an informal metaphysical theory such as GR.

In sum, what Divers and Melia establish is that GR is unable to establish non-trivially the truth of a universal statement of infinite instances, especially not by talking about the

number of instances. But so what? Any realist theory would fail in the attempt to ensure all of an infinite range of entities are accounted for, non-trivially, and especially by

enumerating them. So their argument does not highlight limitations particular to GR. Moreover, Lewis is well aware that Recombination does not ‘guarantee’ that logical space is complete with respect to all (basic, natural) alien possibilities, yet does not perceive this state of affairs worthy of extended discussion. I say, as Lewis does, let the basic alien- plenitude principle go trivial (supplementing it, if need be, with an additional principle to the effect that the number of aliens is infinite) and let Recombinationtake it from there. There is enough content in the rest of the theory, I contend, to make up for this.

Why do Divers and Melia disagree? I examine their reasons in the next two sections. Sections 2.6.3 and 2.6.4 elucidate and reject the Divers-Melia requirement that GR must in some sense ‘guarantee’ that logical space is complete with respect to the (alien)

possibilities. Sections 2.6.5 and 2.6.6 examine and reject possible reasons why DM might think that GR has to provide such a guarantee.