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?

This seems like a reasonable suggestion, but the reason why the subtraction argument cannot make this minimal assumption is pretty clear. It’s because x1 has the modal property of possibly co-non-existing with all other concrete objects, *if and only if* x2, x3, . . ., xn have that modal property, too. Indeed, even the assumption that a single concrete object in some remote possible world w has the property of possibly co-non-existing with all other concrete objects entails that* every concrete object* in *every possible world* has the property of possibly co-non-existing with all other concrete objects. Let R denote that property (strictly, relation). The first reason to doubt that any object o in w has R is the following.

P. ☐((∀o)(Ro ≣ ☐(∀y)Ry))

(P) states that, necessarily, a concrete object o in w has the modal property R of possibly co-non-existing with all other concrete objects if and only if every concrete object y in every possible world w’ has the property of possibly co-non-existing with all concrete objects. But the problem generalizes. The second reason to doubt is (P’).

P’. ☐(◊(∃x)Rx ≣ ☐(∀y)Ry)

(P’) states that, necessarily, some possible concrete object x has the modal property R of possibly co-non-existing with all other concrete objects if and only if every concrete object y in every possible world w’ has the property of possibly co-non-existing with all concrete objects.

Subtraction arguments fail unless (P) and (P’) are true (indeed, unless it is assumed that ☐(∀y)Ry.) So, allowing that some o in w has property R is equivalent to allowing that every concrete object, in every world, has property R. But why grant that it is *impossible* that the non-existence of any possible object entails the existence of some other possible object or other. How would we know that, in some very distant world, there aren’t *fissioners*: concrete objects that cease to exist only by fissioning into two distinct fissioners? In any world in which a fissioner x does not exist there are two other concrete objects (fissioners y and z). I don’t think we know that fissioners are *impossible objects*.

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