A Note on Substraction Arguments
December 28, 2014 — 23:42

Author: Michael Almeida  Category: Uncategorized  Tags: , ,   Comments: 5

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.

Comments:
  • David Efird

    Dear Professor Almeida,

    I find your post fascinating, and, as someone who has worked a long time on the subtraction argument(s), I’m glad to see that you’re interested in this topic. The debate on the soundness of these arguments hasn’t moved forward much in the last few years, and I’m sure you will have many good things to say about it. As an initial thought about your post, I wonder why you say:

    ‘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.’

    Tom Stoneham and I have long defended a version of the subtraction argument and a conception of concreteness on which the subtraction argument is sound and it doesn’t threaten the (necessary) existence of God. On our account,

    ‘A concrete object is one which exists at a location in spacetime, has some intrinsic quality, and is such that if it has a boundary, it has a natural boundary.’ (From ‘Justifying Metaphysical Nihilism: A Response to Cameron’, published in PPQ in 2009. (This definition of concreteness is a slightly revised version definition from our original definition in ‘The Subtraction Argument for Metaphysical Nihilism’ published in JPhil in 2005.

    Briefly stated, the reason, we proposed this conception of what it is to be concrete is that it seemed to describe the kind of object we wonder about when we wonder why there is anything at all, and also the kind of object we have pre-theoretical intuitions about.

    Because God has no location in spacetime, at least in how I conceive of God, God doesn’t count as a concrete object. So, as a theist, I’m happy to accept both the soundness of the subtraction argument, as Tom and I have defended it along with conception of concreteness we have defended, and also the necessary existence of God. Do you see any contradiction in this combination of views?

    Best wishes,
    David (Efird)

    December 29, 2014 — 14:52
  • Michael Almeida

    Hi David (if I may),

    Call me Mike, please. Thanks for this! Yes, I thought about how to phrase what I called a problem for theists. I actually hedged in the phrasing, saying it “seems obvious that no theist can accept the conclusion of subtraction arguments”. But since I take all causally efficacious things to be concrete, I take God to be concrete. It is also difficult for me not to see God as temporal (omnitemporal), and so (assuming they’re too close to distinguish) spatiotemporal. In any case, I don’t want it to turn out that God could not occupy a Lewisian world.

    On the other hand, I think you’re right that this not the notion concrete being that typically figures in subtraction arguments. The argument is typically about material, spatiotemporal objects, sometimes about sets of such objects. But with or without that assumption, it looks to me like something like (A3*’) must be an assumption in the argument. It must be assumed that the non-existence of any concrete object o in w must fail to entail the existence of all other concrete objects x. I don’t think non-nihilists believe (A3*’).

    (A3*’) (∀o)◊(~(o exists) & ~(∃x)(x exists))

    I think both your principle (B) (interpreting A3) and Rodriguez-Pereyra’s principle (ϒ) (also interpreting A3) entail (A3*’). But since non-nihilists would reject (A3*’), I’m guessing they’d reject (B) and (ϒ).

    December 29, 2014 — 16:50
  • Michael Almeida

    Does David Efird’s principle (B) (interpreting A3) and Gonzalo Rodriguez-Pereyra’s principle (ϒ) (also interpreting A3) entail (A3*’)? If so, then since non-nihilists would certainly reject (A3*’), they’d also reject (B) and (ϒ). Here’s a small proof that they do, under the assumption of (A2) (I don’t offer a proof here, but you can see that they’re actually provably equivalent under (A2)).

    Here is Rodriguez-Pereyra’s principle (ϒ):

    (γ) ∀w∀x[x exists in w ⟶ {∃w*(~(x exists in w*)) ⟶ ∃w**(w** and w differ only in that in w** neither x nor its parts exist)}].

    Here is David Efird’s principle (B):

    (B) ∀w∀x[x exists in w ⟶ ∃w*{~(x exists in w*) ⟶ ∀y(y exists in w* & y exists in w)}].

    And of course (A3*’):

    (A3*’) (∀o)◊(~(o exists) & ~(∃x)(x exists))

    For the case of (B), assume that (A3*’) is false. In that case there is some o in w such that, for no world w does o fail to exist and no other concrete object exist. In short, if (A3*’) is false, then for some o in w, o’s non-existence entails that some concrete object or other exists. But then (B) is false. For some w’ that includes exclusively some o in w, there is no w* such that o does not exist in w* and everything that exists in w* exists in w’. And that’s because o’s non-existence entails that some concrete object or other exists in w’ (this follows from the rejection of (A3*’).

    A similar argument shows that (γ) entails (A3*’) under the assumption of (A2). The argument here is slightly complicated by the fact that Rodriguez-Pereyra restricts attention to concrete* objects that are a subset of concrete objects. We could remove the assumption of (A2) by modifying (A3*’) in ways that do not make it any more palatable.

    (A3**) (∀o)(~☐(o exists) ⟶ ◊(~(o exists) & ~(∃x)(x exists))

    No non-nihilist would accept this version of (A3) either. But it is entailed directly by (B) and (γ).

    December 31, 2014 — 11:10
  • Mark Rogers

    Hey Dr. Almeida! 
    You say “For the case of (B), assume that (A3*’) is false. In that case there is some o in w such that, for no world w does o fail to exist and no other concrete object exist. In short, if (A3*’) is false, then for some o in w, o’s non-existence entails that some concrete object or other exists.” But could it not be specifically correct to say that 

    [“For the case of (B), assume that (A3*’) is false. In that case there is (at least one o in w ) such that, for no world w does o fail to exist and no other concrete object exist. In short, if (A3*’) is false, then for some o in w, o’s non-existence entails that some concrete object or other exists.”]? This seems to suggest that God could co-exist with other necessary beings.

    January 4, 2015 — 8:55
    • Michael Almeida

      I don’t see a logical difference: ‘some’ means ‘at least one’. But I agree that the argument is generally presented as an argument concerning concrete objects other than God.

      January 4, 2015 — 10:44
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