Two Notions of Necessity (and the Theistic Arguments that Conflate Them)

There are two notions of necessity floating around that easily get conflated: (i) exists in all possible worlds, and (ii) can't not exist. But (ii) can't be captured by (i); (ii) is more fine-grained than (i).  Indeed, it's epistemically possible that a being is necessary in sense (i), but not in sense (ii). 

To see this, say that a world stub is some initial temporal segment of a possible world (whether beginningless or not). Now consider that it it's epistemically possible for a god (an uncreated, metaphysically independent being) G to exist in the world stub of every possible world, and yet go out of existence at some time downstream of the world stub of at least one -- but perhaps many, and perhaps even every -- possible world (say it commits suicide due to eternal boredom, or it's annihilated by some other being downstream of one or more world stubs). It's therefore epistemically possible for G to be necessary in sense (i), but not in sense (ii).

This has non-trivial implications for some theistic arguments. Some contemporary theistic arguments --  "minimal modal ontological arguments" (as van Inwagen defines them), certain Leibnizian cosmological arguments, etc. --  deploy S5 modal logic to show that an Anselmian being currently exists. In particular, they aim to show that

1. A necessary being exists in at least one possible world.

and then infer from (1) and Axiom S5 of S5 modal logic to infer that

2. A necessary being exists in every possible world.

And finally, from (2) they infer that

3. A necessary being exists.

Now of course many -- myself included -- have raised doubts about (1). But the preceding discussion raises a problem for the inference from (2) to (3). For as we've seen above, (3) doesn't follow from (2). Therefore, even if one establishes that there is a necessary being in the sense captured by sense (i) above -- viz., the necessity operator of modal logic --, one has not thereby established that such a being currently exists. And because of this, theistic arguments of the sort mentioned above that rely on an inference from (2) to (3) to establish God's existence are bound to fail.