I think the reason behind this background assumption in Divers-Melia that GR should ensure, by means of a series of ontological axioms, that its space of worlds is complete, and so that (P) is a materially adequate analysis, can be gathered from an earlier paper by Divers (1997) drawing from Shalkowski (1994). Shalkowski’s argument (in section 2.4) was that for (P) to be accurate, it must be the case that for all and only ways a world can be, some world is; and consequently that (P) is an accurate analysis only if in some objectionable sense modal. We saw, with Sider (2003), that the latter simply doesn’t hold, or else, no adequate reductive analysis of anything would ever be possible. Now, Divers also grants this point, saying “there can be no appropriate material conception of
circularity – circularity in extension – since to require non-circularity in that sense is simply to require that the condition of material adequacy should not be satisfied” (Divers 1997: 146). But, as we also saw, Shalkowski seems to run together the question of whether (P) is materially adequate, which is merely a factual matter, with the question of whether (P) is arbitrary, which is a matter of judgement, thus requiring demonstration. And while Divers (1997) recognises the illegitimacy of the former, he fails to recognise the irrelevance of the latter, instead drawing special attention to its importance as a condition “on the success (qua demonstrable non-arbitrariness)” (Divers 1997: 157 my italics) of the proposed reduction.
So, Divers (1997) takes the challenge for justification inherent in Shalkowski’s (1994) arbitrariness charge to constitute a legitimate request for GR to present, as part of its non- modal theoretical tenets, a defence ofthe adequacy of (P), in order for us to be able to
judge the analysis put forth in (P), if only by GR-lights, as adequate. (Divers 1997: 155- 157) In particular, it looks like Divers (1997) argues for the following triad:
(I) For (P) to be deemedadequate, GR must defend (P) as non-arbitrary.
(II) To defend (P) as non-arbitrary, GR must show that its ontology supports the truth of (P).
(III) If GR is to remain non-circular, the defence of the non-arbitrariness of (P) must be presented in non-modal terms.
But whether this triad is acceptable or not depends entirely on what it means for GR to show that its ontology supports (P). If the relevant notion of support is taken to be the Divers-Melia strong notion of axiomatic entailment, then the conjunction of (I)-(III)
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effectively amounts to the unreasonable demand that, to defend (P) as acceptable, GR must non-modally and non-trivially entail that the (inherently modal) satisfaction conditions of (P) are met in the theory. It seems to me that it is this illegitimately strong demand, which matures in Divers-Melia (2002) into the argument that unless objectionably modal, GR cannot put (P) forth as an adequate analysis of possibility.
Notably, even if we accept the reasonable claim in (II) that GR ought to be able to
justify (P) as non-arbitrary, it simply doesn’t follow that such a defence can only consist in an non-trivial entailment of (P)’s left-to-right accuracy condition by the theory. The non- arbitrariness challenge can more gently be met by any argument that gives (P) a
presumption of truth, given the background ontology of the theory. Here is a nice one by Sider:
“Suppose reality is just the way Lewis thinks it is. [...] There is then the question of whether there is room in this reality for modality. Within this multiverse, is there a candidate property we can identify with the property of being a possible proposition? The answer seems to be yes—it is the property of being a proposition that is true at some Lewis-world. As shown, this property can be defined in entirely non-modal terms (in terms of spatiotemporal notions and the restriction of quantifiers). Thus, an adequate non-modal definition of ‘possible’ can be given, if Lewis’s ontology is indeed correct.” (Sider 2003: 197) 108
DM might object that this argument simply side-steps the problem by packing all
contentious questions in the first sentence in the quote, asking us to ‘suppose reality is just the way Lewis thinks it is’. They might say, this is precisely the point, we do not have a non-modal ontological principle that establishes the Lewisian ontological picture that is supposed to accompany (P). But, the point is that we do not need any independent such principle in order to know what picture of reality GR puts forth; GR’s commitment to the truth of (P) (alongside, presumably, a commitment to modal truths) gives us that. For, GR is simply committed to the truth of (P), and, thereby, it is committed to reality being such that (P) is true. We can accept that much without also making the unreasonable demand that GR articulate (P)’s truth-condition non-modally.
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See relatedly Wright (2007: 163). Wright’s overall aims in his (2007) are strictly unrelated to the present endeavour. However Wright, too, suggests that in the case of a possible-world-semantical analysis of modal claims, “a theoretical explanation of content is achieved via an equivalence in truth-conditions under the hypothesis of the ontology of the explicating theory.” (Wright 2007: 163, my italics)
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Now, we have a further question: Can we judge (P) as adequate for the job for which it has been put forward? The answer, as Sider points out, is yes. For the assumption of (P)’s truth allows us to proceed with the proposed reduction. In other words, upon
assuming Lewis’ pluriverse, an analysis of possibility can be had. But it is very doubtful that the same can be said, upon assuming, say, the existence of pencils in one’s drawer. For while there is no reason to believe that there are not enough worlds in Lewis’ pluriverse for (P) to offer an adequate analysis, there is reason to believe there are not enough pencils in one’s drawer for an analogous analysis involving pencils. So, while the assumption of the truth of (P) allows us to proceed with the reduction, on the contrary, the assumption that for every possibility there is a (distinct) pencil in one’s drawer will most likely lead to its own refutation.109 In that sense, the GR-ontology supports the reduction in (P) in a way that pencils do not support a similar reduction of possibility to pencils.
Hence (P), unlike its pencil-analogue, can be judged an adequate analysis of modality.110 More than that, GR offers yet further support for (P) by putting forth Recombination, which strengthens the presumption of (P)’s truth, by giving us an idea as to what it would take for (P) to be true, thus strengthening our licence to judge (P) as adequate. Finally, as Sider remarks, there is the additional “...question of whether it is reasonable to believe that Lewis’s ontology is correct”. But as he continues “...here Lewis has his Quinean answer—we ought to believe in his ontology because of its theoretical utility.” (Sider 2003: 197) Notice that this, too, allows us to increase our credence in (P).
In sum, putting emphasis on a need by GR to defend or justify the adequacy of its proposed analysis need not result in anything as strong as the DM (2002) demand that the ontological component of GR axiomatically entail the accuracy of (P). So the DM- demand on GR can safely be ignored as too strong and their incompleteness result as irrelevant to the question of the reductive success of the theory.
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Unless after all someone came up with an analysis of possibility via pencils, which did not require there to be so many of them. Indeed, if a reduction of possibility to pencils, bottle-caps or any other item were shown to be serviceable, then it is a good question on what basis we ought to reject it. For, what such analysis would have shown is that possibility can indeed be reduced to the properties of any small finite collection of objects and their interrelationships.
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Even Divers (1997) remains partly unconvinced: “Why should the genuine modal realist not be thought to have discharged his theoretical obligations by expressing either belief or disbelief when confronted with any given ontological claim, rather than requiring-as it were-an informative characterisation of his intended domain of quantification as a whole? Are actualists in possession of such a characterisation, or are their existential beliefs best and adequately expressed in terms of belief in these and disbelief in those?” (Divers 1997: 154 fn 7) Indeed.
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