Is it true to say (A) “Aragorn was at one point king of Gondor and Arnor” and false to say (-A) “Aragorn was at no point king of Gondor and Arnor?” My instinct is to say Yes, but if these are statements about the real world, the opposite is true, since Aragorn never existed.
We could say: For a culturally competent English speaker, the question assumes that we’re asking about some definite possible world w. In w, A is true and -A is false, while in the actual world -A is true and A is false.
But is there some possible world w in which Aragorn is king of Gondor and Arnor? My first instinct is to say Yes, the world portrayed in the Lord of the Rings. Call it Middle-Earth.
But is Middle Earth a single world? Tolkien is famous for his many, many versions of Middle-Earth, in many of which Aragorn becomes King of Gondor and Arnor. So maybe Middle-Earth is a set of possible worlds.
But are those possible worlds even possible? For one thing, some (or all?) versions of Tolkien’s mythology have internal inconsistencies, so they are not possible worlds. And even the ones that don’t (if any) might be impossible for some logical or metaphysical reason that we don’t now understand: they might only appear possible to us.
Let’s bite the bullet and say that none of the possible worlds in the set “Middle Earth” could ever have existed- they’re just not logically possible. Suppose further that any possible world in which A is true, if there is one, would be contained in the set “Middle Earth.” It follows that there are no possible worlds in which A is true.
-The statement “Aragorn was at one point king of Gondor and Arnor” is still true, but it’s true of an impossible world.
-When I say, “Aragorn was at one point king of Gondor and Arnor,” I’m not just saying that it’s possible for someone named Aragorn to be king of two kingdoms called Gondor and Arnor. That’s either true but boring or false for reasons I can’t understand. I’m successfully referring to an impossible individual (and when I say, “Frodo was never king of Gondor and Arnor,” I’m referring to another impossible individual and distinguishing him from the first).
-The set “Middle Earth” seems to be a set of impossible worlds, but not therefore an empty set of worlds, because I can state true facts about those worlds (“Aragorn was at one point king of Gondor and Arnor”), and I can distinguish between them (for example, I can distinguish the possible world in which the Ringbearer is named Frodo and the one in which he is named Bingo).
-All of this only works if the principle of explosion is false.