Is There An Analytic Limit of Genuine Modal Realism?
John Divers and Joseph Melia argue in their paper "The Analytic Limit of Genuine Modal Realism" (Divers and Melia 2002) for two claims:
a) several objections referring to a supposed circularity in the explanation of modality in genuine modal realism (GMR) can be met (including Lycan`s objection that the notion of individual to play its role in GMR has to be understood as possible individual);
b) there remains the problem of a non-circular specification of the completeness of the postulated set of possible words.
I agree with their first claim. Their answer to Lycan, however, seems to provide arguments that there is no completeness problem for GMR, i.e. that their claim (b) is false.
Divers and Melia draw a helpful analogy between systems of logic and axiomatic characterisations of a set of possible words. A characterisation –and so the set of possible worlds that fits it– has to be sound and it has to be complete. The characterisation is sound iff the set of worlds which fits it contains no impossibilities. The characterisation is complete iff for any possibility –to be explained shortly– there is a corresponding possible world.
Lycan`s objection is discussed in connection with soundness. Lycan (1991) argues that GMR should admit no world which has an impossible individual, since a corresponding world would be an impossibility, the set of possible worlds would not be consistent. This, Lycan argues, implies that the notion of individual employed by GMR has to be taken as possible individual in the first place. Divers and Melia (cf. p. 23) agree with Lycan about excluding impossible individuals, but disagree that this entails that ‘individual’ means possible individual. They argue for the soundness of local possibility. Local possibilities are possibilities with respect to individuals of the actual world or recombinations of parts of the actual world. They ‘believe that the principle of recombination keeps us with the realm of the possible, we claim that this principle rules out impossible worlds.’(pp.23-4) So they believe the actual world to be consistent. There are no impossible objects in the actual world. (Intuitively speaking: what else should ‘impossible object’ mean than that something impossible is not actual.) So local possibility is sound.
A related argument, however, can be applied outside of the realm of local possibility, since for GMR there is nothing special about the actual world – a part from the fact that we are living in the actual world (cf. Lewis 1986, p.92). If there is nothing logically special about the actual world, if it is consistent so should be all the other words. So there are no impossible individuals in these worlds and once more the principle of recombination, which applies to them as well (cf. Lewis 1986, p.92), keeps us with the realm of the possible. So the set of all possible worlds is sound.
Divers and Melia discuss the completeness problem in the light of alien properties. An alien property is ‘a natural property that is not instantiated by any individual in [the actual world], and is not analysable as a conjunctive or structural property built up from constituent that are all instantiated by parts of this world’(p.27) Let us consider as broad possibility possibilities that also deal with alien properties. Divers and Melia consider it plausible that there are alien properties, and if there are any, there are infinitely many. The completeness problem can be posed as the question whether GMR can supply an axiom securing that for any possibility –however alien– there is a corresponding possible world. Diver and Melia propose (p.30)
(OAN) For any n there are n objects that, between them, instantiate n distinct alien natural properties.
They then argue that given a set S of possible worlds with denumerably many alien natural properties, there is a set S* failing to contain denumerably many of them (as does the set of even numbers with respect to the set of natural numbers) which still fulfils (OAN), since for any n there are n alien natural properties left in S*, although there are possibilities –those involving the missing alien natural properties in S*– to which no possible world in S* corresponds. So (OAN) does not guarantee completeness. As a possible remedy they consider universal quantification (p.33)
For every alien property Pi, there is a world wi and there
is an individual xi such that
(xi is in wi and xi instantiates P2i+1).
As an axiom this is, of course, once again satisfied by S* with respect to its alien properties (counting them P2, P4 …). From a naïve point of view, however, one would argue: S* fails to meet (iii), since P3 is missing in S*. This naïve reasoning assumes an independently given set of natural alien properties P regarding which S* fails to meet (iii). Is there such a set? In GMR ontology properties are not basic. Divers and Melia reject the idea of supplementing the ontology, since, they claim, their argument could be repeated (cf. p.34). Indeed properties are not needed. Lewis himself, although also speaking about alien properties, speaks about alien individuals first:
"Among all the possible individuals there are, some are parts of this world; some are not, but are duplicates of parts of this world; some, taken whole, are not duplicates of any part of this world, but are divisible into parts each of which is a duplicate of some part of this world. Still other possible individuals are not thus divisible: they have parts, no part of which is a duplicate of any part of this world. These I call alien individuals." (Lewis 1986, p.91)
Given alien individuals alien properties can be defined. Individuals are basic to GMR. Like the worlds are simply there (in logical space), individuals are there, all of them are there. So the set of all individuals is there. That the principle of recombination cannot give us all these individuals is irrelevant to them being there. The set of individuals is given independently of a model set S’ of possible worlds, whereas properties are given only after a model set S’ has been given. So we can lay down a condition of models in GMR referring to the set of all individuals. And –as we have seen in the discussion of soundness– there are no impossible individuals in GMR. The set of all individuals, including the alien individuals, is consistent. We can try again to give a completeness condition:
(OAI) For every individual there is a world which contains that individual.
Since the set of individuals contains even the alien individuals there are, given (OAI), worlds containing them, therefore all alien properties, derived from these alien individuals, are there. Diver and Melia´s S* violates (OAI) since all individuals instantiating alien natural property P3 are missing in S*.
Adding (OAI) to the ontological axioms adds completeness to soundness of GMR.
What do we learn then? Given that you meet Lycan´s objection in the way Divers and Melia do, claiming the absence of impossibilia, there seems to be no analytic limit of GMR.
Divers, J. and J. Melia 2002: ‘The Analytic Limit of Genuine Modal Realism’. Mind, 111, pp. 15-36.
Lewis, D. 1986: On the Plurality of Worlds. Oxford: Blackwell.
Lycan, W. 1991: ‘Pot Bites Kettle’. Australasian Journal of Philosophy, 69, pp.212-3.
Priest, G. 1987: In Contradiction. Dordrecht: Martinus Nijhoff Publishers.
Some advocates of paraconsistency
(cf. Priest 1987) claim that our world is not consistent, since it contains
inconsistent objects (say the Russell-set or sentences being both true and
false), but that is a completely different debate that should not concern us
 I`m not decided whether we should answer Lycan´s objection that way and whether there might be more serious, say epistemological, problems for genuine modal realism. That is another discussion.