the impossibility of the reduction of the modal to the non-modal

Scott Shalkowski's "The Ontological Ground of Alethic Modality" presents an interesting thesis, part of which can be summarized by the claim that a Lewisian possible worlds style reduction of modal truths to non-modal truths cannot be non-circular without being arbitrary. The reason is that in order for this sort of reduction to go through the class of possible worlds (which are supposed to underwrite our claim that, for example, 'it's possible that an oak tree is 1000m tall' iff in some accessible, "nearby" possible world there is a 1000m tall oak tree) must be of exactly the right size. That is, first, the class of possible worlds (C) must not include impossible worlds or entities (like a round square) because including such entities spoils our getting at what's possible when we quantify over C. And, second, C must not exclude any genuinely possible world.

I'm assuming the idea is that if a world in C included a round square as one of its inhabitants, then the program of reducing "it's possible that..." to a question of quantification over C is thwarted -- it's not possible that there is a round square, and so we don't want to affirm the truth of 'There is c ε C such that there is a round square that's included in c', but this sentence may very well hold on an impossible worlds view. Of course, one might hold that we shouldn't restrict C to just the set of possible worlds, but should include impossibilia as well on the basis that the restriction smacks of circularity. But then, as the possibility of a round sqaure illustrates, we've lost the intuitive attraction that the world theory held.

On the other hand, C must include every possible world; no such world can be left out. If one were to be left out, then an entity or certain configuration (call it 'e') would be possible, but the sentence 'It's possible that e' would come out false on the possible worlds analysis of modality. We see that if the possible worlds analysis is to get off the ground, then the class C is bounded above by impossibility and below by what's possible.

The observation is that modal concerns (what's possible and impossible) factor into the setting of both bounds, and so a possible worlds account as a reductive story about modality is either circular or arbitrary.

Being interested in a deflationary account of modality in general (and more specifically semantics for the locution 'it is necessary that S'), I was particularly struck by something in the concluding remarks. Shalkowski claims on page 686:

That an expression means what it does involves not merely the fact that the expression has been or is being used in certain ways, but also the fact that it is permissible to use it in novel circumstances in some limited ways. That meaning is projectible, but restricted, is just the fact that it is possible to use the expression in certain ways but not others and still accord with the conventions of a given language. Expressions with the same previous usage but with different projections onto novel cases differ in meaning. Thus the story with meaning is, in the final analysis, a modal story and not the proper basis for the foundation of modality.

Where does this leave us with regard to the deflationary account in which 'it is necessary that S' is analyzed in terms of 'it is analytic that S'? Well, if modal notions are involved generally in meaning (in that 'it's possible that α is F' can be explained in terms of when it's appropriate to make the predication 'is F' of a certain individual 'α', and these appropriateness or assertability conditions are grounded in the meanings or conceptual content of the predicates and names involved) and specifically in claims of analyticity, then it seems that we haven't deflated modal claims, rather we've made a claim that modal claims involving a certain locution should be analyzed in terms of the application conditions of predicates and reference axioms for names. If we hold the view that one grasps concept A iff one can rightly apply the predicate 'is A' to all and only things that fall under A, then we've characterized modal claims involving a certain locution in terms of concepts and conceivability.