As regular readers of my blogs know I am interested in epistemic and metaphysical versions of uncertainty in economics. A chance encounter with Jessica Wilson a few years ago made me realize that there is a live topic (in analytic metaphysics) on metaphysical indeterminacy. I am blown away by a paper by Elizabeth Barnes & Robert Williams (hereafter B/W), "Theory of Metaphysical Indeterminacy." I never expected to see an elegant account of metaphysical indeterminacy in terms of a "fully classical and bivalent logic and semantics." (2) [Recall this and this.]
To see how they respect such a 'classical' approach, here's one important feature of how they handle metaphysical indeterminacy: "When p is metaphysically indeterminate, there are two possible (exhaustive, exclusive) states of affairs—the state of affairs that p and the state of affairs that not-p—and it is simply unsettled which in fact obtains." (9) Now, to avoid misunderstanding: the unsettledness they speak of here is not epistemic. (Below I return to this.) Note, too, that they rely on modality ("possible"); this is not accidental: they rely on insights from work on modality. In fact, their approach opens up fascinating perspectives on modality. They spell out the modal moves as follows:
[P]ossible worlds are abstract objects which represent (classically complete) ways the world might be...there is a single world which is actualized. The actualized world is the abstract object which represents things exactly as they are in the reality consisting of us and our surroundings.
Once we have the set of all the abstract possible worlds in place, we can carve it up in various ways...What we want to do here is introduce another division amongst the worlds that will single out a new modality—the worlds which are precisificationally possible...There will be more than one world in the space of precisifications just in case there is indeterminacy in reality. (9-10)
Obviously, the above does not do full justice to the ingenuity of their technical approach. Now, one apparent down-side of their approach is that it might well become a bit cumbersome to represent vague objects. Most mountains, say, just have vague boundaries (even though we can make uncontroversial claims about the vast majority of things on what side of the boundary they are); this is a clear fact about our concrete reality (the actual world). [If you disagree, just accept it by stipulation for the sake of argument.] It's easy on B/W’s approach to represent that it is vague if something belongs to the mountain or not; but I suspect that the space of precisifications (the halo) will get cumbersome (or indefinitely large) in order to represent that a mountain just is a vague object.*
For, if I understand the argument correctly, in every world that is precisified, that is the actualized world, non-vague identity (A=A) necessarily holds. So, we need the space of precisifications to represent concrete vagueness in a kind of indirect way, that is, by way of incompatible, possible state of affairs. As an aside (and this is not a criticism!), if this much is right, then the precisifed world, this abstract object, represents in such a way such that it forgoes the idea that the representation (or language) is a kind of mirror to or map of concrete reality.
The B/W framework is compatible with the very controversial idea that there is vague identity in the actual world, if not de re than at least de dicto; this they exhibit in another very interesting paper. To oversimplify, one moral of their story is that one cannot appeal to one’s classical logic alone to motivate rejection of metaphysical indeterminateness. (One might say, indeed, that’s an empirical matter!)
Yet, I am left with a concern (or probably a confusion): recall that in every world that is precisified, that is the actualized world, non-vague identity (A=A) necessarily holds. So it seems that when we work in their formal system, we are forced into saying that [A] in a precisified world there can be no vague identity (even though we can indirectly characterize metaphysical indeterminateness just fine). But that means that the friend of metaphysical indeterminacy must say that [B] vague identity as such is 'possible' in the space of precifications, but not in the precisified world.+ The worry is that [A] & [B] together seem to re-raise the specter that the whole B/W proposed approach is somehow epistemic rather than metaphysical because a way to understand why something is possible in the space of precifications while impossible in the precisified world is that there are epistemic or representational barriers between the space of precifications and the precisified world (just as there are epistemic or representational barriers between the actualized world and the concrete world).
Obviously, the previous paragraph is not the way B/W understand their own project. And I don’t think one is forced into the reading I have proposed. So, I welcome suggestions on how to think my way out of this.**
*By contrast, Jessica Wilson's less formal approach gets this result easily by characterizing this mountain as a determinable.
+Oddly, it remains a live option in concrete reality.
**I thank my graduate students for lively discussion on this material—and they are not responsible for my mistakes.