Restricted Identity III: Too Many Solutions?
So there are several ways to make sense of restricted identities. Which is the right one? Maybe there is no fact of the matter.
The difference depends on which contexts are regarded as referentially transparent and which as opaque. And that in turn depends on how the referents are individuated. For instance, (de re) ascriptions of modal properties will be transparent iff the referents of singular terms are such that they determine the truth value of all such ascriptions, perhaps because they (the referents) are fusions of world-bound individuals with their counterparts, or because they are Carnapian individual concepts, or because they simply contain some hidden tag that determinately settles all their modal properties. At any rate, for de re modal contexts to be referentially transparent, the referents have to provide us with a function from worlds to world-bound individuals, as that's what we need to determine the the truth value of those ascriptions. Alternatively, if we hold that those contexts are referentially opaque, we decide that the referents do not contain that information. Instead, we put the information into another aspect of meaning, which we call the terms' intension. Is the difference really more than just a relabeling of semantic vocabulary?
One might object that it's not open to us to choose any objects we please and call them the referents: the referent of "the amazon" is definitely not a trans-world-monstrosity or a Carnapian obscurity or a set-theoretic construction -- it is the amazon. So if I say that we can just as well take some of these intensional thingies as referent, I either have to deny that the referent of "the amazon" is the amazon, or claim that the amazon is such an intensional thingie, which clearly it is not.
Well, I'm really not sure whether it is. Consider the hidden-tag view, aka haecceitism. On this view, there is some hidden property of the amazon that determines which things in other worlds are its counterparts. So on this view, the amazon is an intensional thingie. Even more so on Kripke's view on which that it is simply a primitive fact which things at which worlds are counterparts of (as Kripke would say, identical with) the amazon. But even if the amazon is not an intensional thingie, I'm not very keen on satisfying the folk theory of reference. If the folk says that "the amazon" refers to some entity that turns out to be useless for compositional semantics, I think we should rather ignore that intuition than clutter semantics with the useless semantic value.
Here is something that looks at least like a prima facie reason to prefer less intensional construals of reference, and correspeondingly to say that at least some of the restricted identities are real identities. Consider again the proposal that A1 and A2 are not identical because they differ in modal properties, even though for some reason or other "At our world, A1 = A2" is true. But we counterpart theorists have learned that every ordinary sentence about ordinary things should be regarded as implicitly prefixed by "At our world". Hence the ordinary identity statement "A1 = A2" (with the modifier left tacit) is also true. Hence by semantic descent A1 and A2 are identical.