Subtraction arguments aim to show that there are possible worlds in which there exist no concrete objects at all: they are arguments for the possibility of metaphysical nihilism. The argument must show not only that no concrete objects necessarily exist, but also that no disjunction, (object o exists) v (object o’ exists) v (object o” exists) v …. v (object o”’ exists), is necessarily true. The argument must show it is false, in other words, that every world includes some concrete object or other. It seems obvious that no theist can accept the conclusion of subtraction arguments, since theists are committed to the thesis that God is concrete and exists in every world. But there are views nearby to metaphysical nihilism that theists might find congenial.
I find subtraction arguments really intriguing. Here’s the earliest version of the argument due to Baldwin (1996, Analysis). But see Rodriguez-Pereyra’s recent paper, ‘The Subtraction Arguments for Metaphysical Nihilism: compared and defended’.
(A1) There is a possible world with a finite domain of concrete objects.
(A2) These objects are, each of them, things which might not exist.
(A3) The non-existence of any one of these things does not necessitate the existence of any other such things.
All three premises have been questioned. (A1) is false, sone have argued, since some spatiotemporal concrete object in each world (that includes such objects) has infinitely many concrete parts. Or, perhaps each concrete object is an ur-element of a set that is itself concrete. Here again we have infinitely many concrete objects (most of which are concrete sets). But let’s concede (A1), and let the world in question be w.
Premise (A2) looks like the contingency claim that, for each object o in w, there is some world w’ in which o fails to exist. But it could be satisfied if we discover that, for each object o in w, there is some time t in each world at which o fails to exist. Even if o exists in every world–for instance, even if God exists in every world–we might wonder whether o exists at all times in every world.
We might wonder as well whether the argument for God’s necessary existence also an argument for His omnitemporal existence (existence at all times in every world) or even, more weakly, permanentism (His existence at each time in every world). Concerning the latter two, God might exist at each time in every world, but not at all times; it might be true that he will exist at every time, but does not now).
(A2) might also be satisfied if we learn that, though each o in w exists in every world, o is not concrete in every world in which it exists. Views more or less like this have been defended (see Linsky, Zalta, and Williamson). Could God exist in every world but be concrete only in some worlds? Or, could he exist in every world, at all times, but be concrete only at some times in some worlds? The question here again is whether the argument for God’s necessary existence is also an argument for his necessary concrete existence.I’m not sure how to answer these. Let’s concede (A2).
The main problem with the subtraction argument is premise (A3). In order for the argument to be valid, (A3) must be much stronger than it appears. It is not enough that, for each object o in w, and for every concrete object x, it is not the case that o’s non-existence does not entail the existence of x. That is, it is not enough that (A3′) is true (letting x range over all concrete objects and o range over the concrete objects in w, and including an existence predicate in addition to quantifiers).
(A3′) (∀o)(∀x)~☐(~(o exists) ⊃ (x exists))
This is equivalent to the simpler,
(A3”) (∀o)(∀x)◊(~(o exists) & ~(x exists))
What (A3”) tells you is that it’s possible for each o in w to co-non-exist with each concrete object x. It is a version of the Humean doctrine of the denial of necessary connections. What we need is that it is possible for each o in w to co-non-exist with all other concrete objects. So, to make the argument valid, we need something like (A3*).
(A3*) (∀o)~☐(~(o exists) ⊃ (∃x)(x exists))
And (A3*) is just equivalent to the simpler (A3*’)
(A3*’) (∀o)◊(~(o exists) & ~(∃x)(x exists))
What (A3*’) tells us is that it’s possible for each o in w to co-non-exist with all concrete objects.
It is true that (A1) – (A3*’) entails metaphysical nihilism: that is, there is some world containing no concrete objects at all. But (A3*’) already commits non-nihilists to a view they cannot accept. No one who is non-nihilist would accept the premise that each of the objects o in w can co-non-exist with all concrete objects! That’s exactly what non-nihilists are anxious to deny.
But there are weaker versions of (A3) that non-nihilists might find congenial.
(A4) (∀o)◊(~(o exists) & ~(∃x)(x is concrete))
It might be acceptable that each object o in w is such that there is a world w’ in which o fails to exist and nothing is concrete. In this case, it can be true in w’ that concrete objects other than o exist, they are simply not concrete in w’. This does commit you to the view that concrete objects are not essentially concrete. Or, even weaker, (A5)
(A5) (∀o)◊(~(o is concrete) & ~(∃x)(x is concrete))
According to (A5) each object o in w is such that there is a world in which o fails to be concrete and nothing at all is concrete. This allows that concrete objects, including o, exist in that world, but they are not concrete there. (A6) is even weaker.
(A6) (∀o)(∃t)◊(~(o is concrete at t) & ~(∃x)(x is concrete at t))
According to (A6), for each object o in w there is some time in some world w’ at which o is not concrete and neither is anything else. Each object o in w might be concrete at some time in w’, and so might other objects. But there is some time t in w’ at which nothing is concrete. What is true about concrete objects in w’ before and after t, we just don’t know.
I think each of these will get you versions of metaphysical nihilism that, perhaps, theists might be more receptive.