Suppose ("towards a contradiction," as they say, except we will still have some caveats) that "necessarily" is a vague predicate. Then imagine representing a sorites problem for necessity, and take some presumably contingent being, like a mere quantum particle fluctuating in and out of existence. (This is not so clear as we would like, since one might style quantum flux to be itself a necessary process, such that when this flux infuses our world with a particle, it necessarily infuses our world with a particle, and then the particle exists by necessitation in that sense, even though it will be necessitated to cease to exist in the next Planck flicker of time.)
Now, say that this bit of substance has a degree of necessity that is minimal: it has some necessity to it, but too little to count as robustly or nontrivially necessary. Say that we can conceive of its degree of necessity increasing in the sense that it becomes actual in more and more possible worlds. Modulo David Lewis, say that this means that our lonely particle acquires more and more counterparts in other possible worlds.
If the above has any stable, approachable meaning (which is suspect), then go to the claim that there is a particle that exists or has counterparts in every possible world. Since necessity, here, is truth in all possible worlds, we seem to have strictly marked out a fully necessary particle; there was no vague transition point, then. So it seems that we could not construct this particle's necessity as a vague property; or then being necessary doesn't seem vague.
However, what is the meaning of "all possible worlds"? Or: what is the meaning of the word "all"? This is where the mystery of unrestricted quantification comes in. Storer[10] reads:
... The predicativist, however, is not willing to make the supposition of determinacy for the sets of natural numbers: instead she sees them as open-ended, as will be explained below. It is this open-endedness that means that quantification over the sets does not always have a determinate meaning, and that therefore makes the circularity of an impredicative set-specification into something viciously circular. ... One expression of the open-endedness thought is the distinction between generalizations made with ‘any’ and those made with ‘all’. This distinction was first explicitly drawn by Russell. (Russell in fact attributes the idea to Frege, but this attribution is dubious given Frege’s absolutism about the domain of quantification.) Intuitively, ‘any’ expresses schematic generality, whereas ‘all’ expresses the logical product of its instances. The meaning of ‘all’ is therefore dependent on a determinate range of instances. The use to which Russell put the distinction was to generalize over domains which are not determinate: indenitely extensible domains, such as the domains of propositions and properties.
If the attempt to determine "all" or "everything" is itself indeterminate (for us mortals, at least), then there is at least enduring subjective vagueness with respect to "all possible worlds," and so necessity becomes an example of higher-order vagueness, i.e. of the vagueness of the predicate "vague" itself. That is, since a distinction in orders can be numerically indexed, then we can speak of first-order, second-order, ... etcetera-order vagueness. What number do we stop at? Is there any reason to think that n for nth-order vagueness can ever be fixed at some finite value—or any specific infinite value, even? Here we appeal to the distinction between relative and absolute infinity, where the latter is absolutely all forms of infinity, except given what we just said about the word "all," then again we are referring to something open-ended and indeterminate for us, indeed something that seems necessarily indeterminate.
If it can be necessary that something is indeterminate, does not necessity participate in indeterminacy in the end, to some extent, after all?
