Is There a Null World?

A while back–roughly 18 yrs ago to be precise–Brian Leftow had articulated what might be a plausible account for how necessary abstract entities (the Platonic horde! as Richard Davis calls it) depend on God.  Leibniz had insisted that the ‘essences or possibles’ absolutely depend on God such that not only would they not have existed if God did not exist, but that they would not have even been possible.  Leftow takes inspiration from this to construct a semantics of ‘impossible worlds’ as he calls it.

In Religious Studies 42 (2006): 371–391, Richard Davis puts this view under scrutiny and finds it wanting.  I intend here to take a breif look at one point that Davis finds puzzling in Leftow’s account.

Davis points out that Leftow professes to be an actualist.  On this view, all that exists, exists in this world.  The entire set of concrete and abstract entities exist in this world, which we will call C.  Among C’s members are worlds or sets of atomic propositions.  There are non-null worlds–maximal sets of atomic propositions such that for every proposition p, either p or –p.  A possible world is a non-null world who’s members are capeable of being true conjointly or individually.

Now according to Leftow, C contains not only possible worlds, but also impossible worlds.  There are non-null impossible worlds–worlds where there is at least one proposition p such that both p and –p are included in that world.  But then, Leftow says, there are also impossible worlds that are null worlds, or worlds with the null set of propositions–a set with exactly 0 members.  In fact, there is exactly one null-world, the world in which God does not exist.  God’s non-existence occurs if and only if the null world obtains.  In his Liebnizian Cosmological Argument, there is a burning question here however that Leftow tries to anticipate:

That God’s non-existence occurs in the null world does not entail that the proposition ‘God does not exist’ exists in the null world.  It does not exist there.  In the null world, no propositions exist, and so none are true (or false).  God’s nonexistence is a logical ‘black-hole’, sucking all the propositions of a world into itself.

But is Leftow convincing here?  One would think that if there is a world in which God does not exist, then doesn’t the proposition “God does not exist”, exist in that world, right?  Leftow tries to solve the problem using Adams’ in/at distinction according to which there a propositions about possible worlds that are true not in those worlds, but at or of them.  But I confess that I do not understand what it means (sorry John) for a proposition to be true of the null world and that proposition not be true in it.  Davis offers an argument that seems to me intuitively true:

(1) Necessarily, if God did not exist, then the null world would have obtained.

(2) Necessarily, the null world obtains if and only if nothing exists.

(3) Necessarily, nothing exists if and only if it is true that nothing exists.

And now from (1)–(3) we can infer:

(4) Necessarily, if God did not exist, then it would have been true that nothing exists.

(5) Necessarily, if God did not exist, then the proposition ‘Nothing exists’ would have been true. 

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