Subtraction arguments begin with the modest assumption that there are worlds that do not include more than a finite number of concrete (or concrete*) objects. Each of these concrete objects has the essential property of not existing necessarily. Each of these concrete objects also has the essential property of being such that their non-existence entails the existence of no other concrete objects: each has the essential property of possibly co-non-existing with everything concrete. If any one of the finite objects in these worlds fails to have the essential property of possibly co-non-existing with every concrete object, the subtraction argument fails.
We might find that, in the spirit of modal ecumenism, this is a tolerable view. Why not believe that some finite number of concrete objects, all of which exist in some odd world w, have this interesting modal property of possibly co-non-existing with all other concrete objects?
Well, first, it occurred to me that the assumption is overkill. Why not assume that just one of the objects in w has this interesting modal property. This assumption alone ensures that there is some concrete-empty world somewhere in metaphysical space. So , then, why not just assume that object x1 in w, but no other objects x1, x3….xn in w, has the property of possibly co-non-existing with all other concrete objects?
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.
Like several other contributions to The Puzzle of Existence, the essay by the late E. J. Lowe is devoted to the question whether there might have been nothing. Lowe calls the view that there might have been nothing ‘metaphysical nihilism,’ and he offers an argument against a certain version of it.
Lowe’s paper begins with some very helpful context-setting. In 1996, Peter van Inwagen had argued that there is a possible world which was ’empty’ in the sense of containing only abstract objects, and no concrete objects. However, according to van Inwagen, out of the infinitely many possible worlds, only one is the empty world, hence the probability that the empty world is actual is 0.
Van Inwagen’s argument was published together with a response by Lowe. In his article in Puzzle, Lowe summarizes his earlier argument as follows:
Some abstract objects exist necessarily, and so exist in every possible world. But all abstract objects depend on there being concrete objects – although not necessarily the same concrete objects in every possible world. Hence, concrete objects exist in every possible world, even if there is no necessary concrete being (182-183).
Lowe’s key examples of necessarily existing abstract objects were numbers, which he then thought were required to ground arithmetic truths. One important difficulty of Lowe’s understanding of the grounding of abstracta in concreta was that it required him to deny the existence of the null set and the number 0.
After 1996, two things happened: a number of objections to Lowe’s argument were raised, and (for reasons independent of those objections) Lowe stopped believing in numbers. The current paper reformulates the argument in a way that relies only on universals and ‘impure’ sets. Lowe’s argument is, essentially, that universals depend ontologically on their instances, and sets depend ontologically on their members, but there can be no cycles or regresses of ontological dependence, hence, if there are abstracta, there must be concreta. Lowe continues to reject the existence of the null set, and consequently of all ‘pure’ sets, on grounds that the null set cannot be properly grounded.
My key worry as I was reading the paper concerns a shift in Lowe’s characterization of van Inwagen’s position, against which he is supposed to be arguing. At the beginning of the paper, he describes van Inwagen as arguing for the existence of a world empty of concreta, but conceding that that world still contains abstracta (182). But later in the paper, Lowe characterizes van Inwagen as arguing that “there is an ’empty’ world in which there exist no concrete objects but abstract objects do exist” (187). At this point, Lowe makes it clear that he is no longer arguing that there must be concrete objects, but only that there can’t be abstracta without concreta. In other words, what was earlier a concession of van Inwagen’s has become part of van Inwagen’s thesis.
To his credit, Lowe explicitly addresses this worry in the very last paragraph of the paper. However, his response is concessive: he is indeed no longer arguing that there must be concreta. “Doesn’t that significantly reduce the metaphysical significance of the argument?” he asks rhetorically (194). I certainly think so. However, Lowe is certainly right that the fact that this argument is of lesser metaphysical significance than the argument he once tried to offer does not mean that the present argument is not interesting or significant. If Lowe is right about the ontological dependence of abstracta on concreta and the well-foundedness constraints on ontological dependence, there are wide-ranging metaphysical consequences.
(Cross-posted at blog.kennypearce.net.)