The Subtraction Argument in an Infinite World

1 Argument

The subtraction argument was formulated initially in Baldwin (1996). The argument takes off from three premises:

1. There might be a world with a finite domain of ‘concrete’ objects.

2. These concrete objects are, each of them, things that might not exist.

3. The non-existence of any one of these things does not necessitate the existence of any other such thing.

If we render these premises in more formal terms, one can prove that there exists a possible world with no concrete objects. The rough idea of the proof is that one starts with the world referenced by premise (A1) and then picks up one concrete object mentioned there. Using both (A2) and (A3), one arrives at another world that differs from the former one just by that object, i.e., the former world has the same concrete objects as the latter except for the object. Then one needs to pick up a concrete object from the latter world and repeat the whole procedure until one arrives at a world with no concrete objects.

It is worth noticing that the argument assumes that the relation of accessibility between possible worlds is transitive. However, since the mainstream of the subsequent discussions of the argument disregarded this aspect, I will gloss over it as well – this assumption should not undermine any of the claims I propound below while it will significantly simplify the narrative.

One of the objections raised against Baldwin’s argument concerns premise (A1). For instance, Rodriguez-Pereyra (1997) argues that a world with a finite number of concrete objects must not contain extended concrete objects since extended objects have an infinite number of parts, each of which is a concrete object. In order to meet this objection, both Rodriguez-Pereyra and others, e.g., Efird and Stoneham (2005), struggled to define a notion of a concrete object so that there can be a finite number of concrete, extended objects. As it usually happens, any such refinement may be confronted with a counterexample: see, for instance, the arguments of Hoffmann (2011), against using the notion of concrete^∗ objects defined in Rodriguez-Pereyra (1997). He defines concrete objects^∗ as follows: “x [is] concrete* if and only if x is concrete, memberless and a maximal occupant of a connected region.” (Rodriguez-Pereyra 1997, p. 164). Hoffmann (2011) pointed out that since there are possible worlds where there are no concrete* objects but where some objects are concrete, this refinement of Baldwin’s argument distorts the original idea. To address this objection, Rodriguez-Pereyra (2014) amends his earlier version of (A1): “There might be a world with a finite domain of concrete* objects.” becomes “There is a possible world with a finite domain of concrete^∗ objects and in which every concrete object is a (proper or improper) part of a concrete* object.”

One may wonder whether we can spare ourselves the fuss and modify this argument in such a way that (i) we do not need premise (A1), (ii) the modified argument is still conclusive, and (iii) its ontological assumptions are comparable to the those of the original argument. In other words, one may wonder whether we could have a subtraction argument when all proper possible worlds are infinite?

2 Infinitication

It should be evident that premise (A1) is essential for the argument as it stands. If we drop it, then we should either drop (A2), because the latter references back to the former, or we should at least re-interpret (A2), e.g., we may postulate that all concrete objects in any finite world are, each of them, things which might not exist – this may be seen as its stronger version. Then if we drop premise (A1) and/or premise (A2), there are models, e.g., the model defined in appendix 4.1 below, that shows (A3) may be satisfied, but all possible worlds contain some concrete objects – in fact each of them contains an infinite number of them. So, it looks as if the subtraction argument might require a deeper modification if we were to employ it in a scenario where no proper possible world contains a finite number of concrete objects.

First, let me slightly simplify the standard discourse about possible worlds. Namely, in what follows, I will identify possible worlds with sets of concrete objects, i.e., I assume that:

1. A concrete object x exists in a possible world w iff x∈w.

Actually, this assumption is false in most, if not all, cases because a set of concrete objects may usually be configured by different spatiotemporal and other relations, and each such configuration represents a different possible world. Nonetheless, this “subtlety” is of no relevance for what follows, since the number of possible worlds and their actual ontological status or structure are conceptually orthogonal to the case of metaphysical nihilism.

Let us now consider the following argument:

• (G1_informal) There is a possible world where each concrete object might not exist.

• (G2_informal) If for every concrete object (from a non-empty set of concrete objects) there is a possible world in which it does not exist, then there is a possible world in which all concrete objects (from this set) do not exist.^[1]

Both premises, particularly (G2_informal), require slightly more precise phrasing. So let W be the set of all possible worlds – all possible worlds mentioned below are supposed to belong to this set. (G1_informal) means that.

1. There is a possible world w such that for each x∈w, there is a possible world v such that x∉v.

Now premise (G2_informal) is understood to be satisfied in any possible world, so it is to mean:

1. If for every concrete object x∈A there is a possible world v such that x∉v, then there is a possible world u such that for every concrete object x∈A, x ∉u.

Now if one identifies possible worlds with sets of objects, as I do in this paper, premise (G2) may be expressed as follows:

1. If ∅≠U⊆W, then there is a possible world v such that v⊆∩U.^[2]

Now I will show that (G3) is equivalent to (G2). Suppose that (G3) holds. Let A be a non-empty set of concrete objects. Suppose that for every object in A there is a possible world in which this object does not exist. Let U be a set of such worlds, i.e., U = {w∈W: ∃x∈A x∉w}. Since A is not empty, U is non-empty either. (G3) implies then that there is a possible world v∈W such that v⊆∩U. Since U is defined in such a way that no object from A belongs to ∩U, therefore no object from A exists in v as well, i.e., v is a possible world where all objects from A do not exist.

Suppose now that (G2) holds. Let U be a non-empty subset of W. Consider now set − (∩U), i.e., the complement of ∩U.^[3] Since possible worlds are rendered here as sets of objects, −(∩U) is a set of possible objects. If −(∩U) is empty, then ∩U=∪W. Since for any possible world v, it is the case that v⊆∪W, therefore v⊆∩U. Otherwise, i.e., when −(∩U) is not empty, for every x, if x∈−(∩U) then there is a possible world u, which belongs to U, such that x ∉u, i.e., x does not exist in u. (G2) implies now that there is a possible world, say v, in which all concrete objects from −(∩U) do not exist, i.e., −(∩U)⊆ −v. Therefore, v⊆∩U.

Now it is easy to show that (G1) and (G3) entail that ∅∈W. To see this, let w ₀ be a possible world whose existence is postulated by (G1). Then consider set UG such that.

w∈UG ≡ w∈W ∧ ∃x (x∉w ∧ x∈w ₀).

UG is not empty given premise (G1). Therefore premise (G3) implies that there is a possible world v such that v⊆∩UG. I will show that ∩UG=∅ on pain of inconsistency. Suppose otherwise and let a∈∩UG. If ∅∈UG, then ∩UG =∅. Thus, let a∈w_a for some proper possible world w_a∈UG. (In fact, a∈w for all w∈UG.) Using the definition of UG, we get that a∈w ₀. Then (G1) implies that there is a possible world in W, let’s call it w _−a, such that a∉w _−a.^[4] Since a∉w−a and a∈w ₀, therefore w _−a ∈UG. Since a∈w for all w∈UG, we know that a∈w _−a. As a result, both a∈w _−a and a∉w _−a. So ∩UG=∅ and therefore v=∅.

It should be obvious that both premises (G1) and (G2/G3) are needed to conclude that there exists an empty possible world. If it is not obvious, consider:

1. the model described in appendix 4.2 below, which exemplifies that it can be the case that all possible worlds are proper even if (G2/G3) is satisfied;

2. the model from appendix 4.1, which exemplifies that it can be the case that all possible worlds are proper even if (G1) is satisfied.

Is this piece of reasoning suasive, i.e., can we reasonably expect that if one is open-minded about metaphysical nihilism, then showing to him or her that (G1) and (G2/G3) imply that there is an empty possible world makes this view more acceptable?^[5] Instead of addressing this question directly, I will compare its premises to the original premises (A1)–(A3). If they turn out to be sufficiently similar and if they happen to make comparable ontological assumptions, then my modification may be considered as a viable alternative to the original.

Perhaps we should start by explicating the latter premises in the set-theoretical vernacular used in this paper.^[6] (A1) says that there is a possible world w such that |w|=n for some natural number n>0. Then premise (A2) seems to have it that for each x in (any) world w that is referred to by (A1), there exists v∈W such that x∉v. Since this interpretation makes (A2) dependent on (A1), we might consider a stronger, “independent” interpretation thereof: “In every finite world all concrete objects are, each of them, things which might not exist”, i.e., for each object x in any world w such that |w|=n for some natural number n, there exists v∈W such that x∉v – I will refer to this version as (A2+). (A3) says that for each w∈W and for each x∈w, there exists v∈W such that v= w\{x}.^[7]

From the functional point of view, premise (G1) replaces Baldwin’s premises (A1) and (A2) in the sense that both have it that all concrete objects might not exist. The obvious difference is that Baldwin’s argument assumes that the number of these contingent objects is finite, and my version thereof does not.

From the formal point of view, it is evident that the conjunction of (A1) and (A2) imply premise (G1) and that the latter does not imply the former – see, for example, the model described in appendix 4.1 below. Nevertheless, premise (A2+) is logically weaker than (G1). First, the model from 4.2 shows that the former does not entail the latter. Secondly, it can be shown that the latter implies the former. If there is a possible world, say w0, such that every concrete object in this world may not exist, as (G1) stipulates, then there is no concrete object that exists in all worlds – since if there was, it would belong to w0 and, consequently, it would not exist in some world (other than w0). Thus, for each object, both from w0 and from any other world, there is a world where this object does not exist. As a result, the consequent of premise (A2+) is satisfied, even if its antecedent is not. This, incidentally, shows that (G1) implies not only (A2+), but also (A2++): for each object x in any possible world, finite or infinite, there exists v∈W such that x∉v. In fact, (G1) is equivalent to (A2++) – provided that W is not empty.

Therefore, I find premise (G1) at least comparable to premises (A1) and (A2) with respect to its ontological assumptions. Actually, the fact that (G1) does not require that there be any finite worlds makes it ontologically less demanding than (A1).

From the functional point of view, premise (G2/G3) replaces Baldwin’s premise (A3) (in the interpretation of Efird and Stoneham (2005)) – both have it that, so to speak, contingencies of concrete objects are summative: if any object from a set of concrete objects may not exist, then all of them may not exist. Visibly, premise (A3) also assumes that all objects in any finite world are contingent while (G2/G3) does not. A more significant difference is that Baldwin’s argument requires, metaphorically speaking, that this summation of contingencies may be realised in stages (“one by one”) while my version allows that it can be done in one step. The ontological ramifications of this difference are far from being clear. First, given Baldwin’s proof, one can argue that there is no difference between the “one-by-one” summation, which is described by (A3) (in the interpretation of Efird and Stoneham (2005)), and a finite “bulk” summation:

1. For each possible world w and for any finite subset z of w, there exists v∈W such that v=w\z.

In other words, one can prove that (A3) is logically equivalent to (A3bulk) – in fact Baldwin’s argument shows the gist of this proof.

Thus, exploiting the metaphor of subtractions, one may compare premises (A3) and (G2/G3) as follows:

1. premise (A3) has it that

   1. all concrete objects in all possible worlds are contingent and

   2. any finite set of concrete objects can be subtracted (from any world where they all exist);

2. premise (G2/G3) has it that

   1. if a set contains only contingent concrete objects, then this set can be subtracted (from any world).

Therefore, the difference in question is essential only if we deal with infinite sets of contingent objects. I cannot think about any ontological view that would explicitly allow us to distinguish between such finite and infinite summation of contingencies. I can imagine that someone might maintain that the notion of ontological contingency is intrinsically related to agency:

1. A concrete object is contingent iff there is an agent who is capable of bringing it about that this object ceases to exist.

Then the difference between the finite and infinite summation amounts to whether there exists an agent that can destroy an infinite number of objects. The latter issue is, by all means, ontologically significant because, among other things, any system of theistic metaphysics should imply that there are such agents, and its atheistic counterpart might deny it. Nonetheless, the existence of such agent is relevant for the difference in question only in the presence of the above definition, which definition may be seen objectionable as an established account of contingency.

From the formal point of view, (G2/G3) is logically independent of (A3). In the model defined in appendix 4.3, the former, but not the latter, is satisfied, and the model from 4.1 shows that the latter, but not the former, may be satisfied.

For the reasons mentioned above, I find that (G2/G3) is comparable to premise (A3) with respect to its ontological assumptions. One may then consider the compound of (G1) and (G2/G3) as an alternative version of the original argument for metaphysical nihilism.

3 Conclusions

In summary let me claim that the original subtraction argument for metaphysical nihilism can be modified so that it delivers its conclusion even if it is not possible that the number of concrete objects is finite. The modification sketched in this paper is logically independent of the original, so that it cannot be interpreted as being logically weaker or stronger than the original. It is just a different form thereof. Nonetheless, although not identical, both share insights into the nature of ontological contingency: (a) some objects are contingent, (b) objects’ contingencies are summative.