SUBTRACTABILITY AND CONCRETENESS
Descripción
The Philosophical Quarterly, Vol. , No. ISSN –
April doi: ./j.-...x
DISCUSSION
SUBTRACTABILITY AND CONCRETENESS B R P. C I consider David Efird and Tom Stoneham’s recent version of the subtraction argument for metaphysical nihilism, the view that there could have been no concrete objects at all. I argue that the two premises of their argument are only jointly acceptable if the quantifiers in one range over a different set of objects from those which the quantifiers in the other range over, in which case the argument is invalid. So either the argument is invalid or we should not accept both its premises.
Could there have been nothing at all? Probably not, think many philosophers, because there are some necessarily existing abstract objects. But could there have been no concrete objects? The view that this is possible, known as ‘metaphysical nihilism’, may seem very plausible;1 but is there a convincing argument for it? Tom Baldwin has argued for metaphysical nihilism as follows: since each concrete object might not have existed, we can remove them one by one, and end with a world in which no concrete object exists.2 The idea is that if one of the concrete objects did not exist, nothing would have to replace it. The world could be just as it is, but smaller, lacking this concrete object (and any other objects which depend for their existence on it). So we can infer the existence of a series of smaller and smaller worlds, ending in a world entirely lacking in concreta. This subtraction argument has been much discussed in the literature, with problems raised and defences advanced.3 In a recent article, David Efird and Tom Stoneham defend what I think is the best version to date of the subtraction argument for 1 It has its detractors: see, e.g., E.J. Lowe, ‘Metaphysical Nihilism and the Subtraction Argument’, Analysis, (), pp. –, and The Possibility of Metaphysics (Oxford: Clarendon Press, ), pp. –. 2 See T. Baldwin, ‘There Might Be Nothing’, Analysis, (), pp. –. 3 See G. Rodriguez-Pereyra, ‘There Might Be Nothing: the Subtraction Argument Improved’, Analysis, (), pp. –, and ‘Metaphysical Nihilism Defended: Reply to Lowe and Paseau’, Analysis, (), pp. –; A. Paseau, ‘Why the Subtraction Argument Does Not Add Up’, Analysis, (), pp. –; R.P. Cameron, ‘Much Ado about Nothing: a Study of Metaphysical Nihilism’, Erkenntnis, (), pp. –. 4 D. Efird and T. Stoneham, ‘The Subtraction Argument for Metaphysical Nihilism’, Journal of Philosophy, (), pp. –.
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metaphysical nihilism.4 Their argument has the virtue of giving a formally rigorous argument, and none of the premises is tantamount to the very claim of nihilism which is to be proved. They use the following premise: B.
∀w1∀x{E!xw1 → ∃w2[¬E!xw2 ∧ ∀y(E!yw2 → E!yw1)]}
where ‘E!aw’ is to be read as ‘a exists at world w’, w1 and w2 range over possible worlds, and x and y range over possible concrete objects (i.e., those objects among the possibilia which are concrete). Premise (B), together with the premise (A) that there could have been finitely many concrete objects,5 seems to entail that ∃w∀x¬E!xw, i.e., that there is a world in which none of the possible concrete objects exists. This, of course, is a world in which no concrete objects exist. (Proof: if a concrete object existed at this world, then it would not be a possible object, in which case the world would not be a possible world; but this is ruled out because w only ranges over possible worlds.) In this paper I shall argue that this argument is either invalid or unsound. My claim is that when the premises are understood in such a way that the argument is valid, there is a tension between the premises, and hence we should not hold both together, in which case the argument is unsound. On the other hand, if we understand the premises in such a way that they are both true, then the argument is invalid. How can there be different ways to understand the premises, given that they are in a formal language? The different ways arise because there are different ways of understanding the domain of the quantifiers. I shall argue that in order for (A) to be acceptable we need a particular understanding of what it is for an object to be concrete (with this much Efird and Stoneham agree), but that when concreteness is understood in this way, we lose the supposed reason for thinking (B) true. If premises (A) and (B) are both to be accepted, then, I claim, the objects over which the nonworld variables in (A) range are not the same objects as those over which the non-world variables in (B) range; and if this is true then the argument is invalid. I shall deal first with (A). Is it true that there could have been finitely many concrete objects? What about the infinitely many concrete proper parts of the concrete objects you have? The worry here is that if you have one concrete object then you have infinitely many, because the object can be infinitely divided into infinitely many parts, each of which will also be concrete.6 There are two bad ways in which the defender of (A) could respond to this worry. The most obvious response is that it is only extended concrete objects that are made up of infinitely many concrete parts. A world in which there are no extended objects, but merely finitely many concrete simples (and their mereological sums), is a world in which there is a finite number of concrete objects. This is true; but it makes a poor justification of (A), because the possibility of finitely many concrete simples floating in the void is on an epistemic footing no stronger than the possibility of there being no concrete objects at all, so an argument from this suggestion to metaphysical nihilism will not be suasive. 5 The particular premise Efird and Stoneham use is that there could have been two concrete objects. But their argument is valid for any natural number, if it is valid in this case. 6 See Rodriguez-Pereyra, ‘There Might Be Nothing’, p. .
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SUBTRACTABILITY AND CONCRETENESS
Another natural response, which Rodriguez-Pereyra himself makes, is that when we perform the subtraction and get rid of a thing, we thereby get rid of all its parts along with it. So when we subtract some extended concrete object a we also subtract all of a’s (infinitely many) parts. But this line of thought will not work, at least not without changing the premises of the subtraction argument. It does not follow from the contingency of an object that all its parts might not exist, since an object can fail to exist even if some of its parts exist. Indeed, most objects can fail to exist without any of their parts failing to exist, since scattering its parts is often enough to destroy an object. So for the subtraction argument to work we would need to abandon (B) in favour of a modified premise to the effect that for every concrete object, that object can be wholly subtracted, i.e., removed with all its parts: B*. ∀w1∀x{E!xw1 → ∃w2[∀y(Pyxw1 → ¬E!yw2) ∧ ∀y (E!yw2 → E!yw1)]} where ‘Pabw’ is to be read as ‘a is a (proper or non-proper) part of b at w’. But the resulting argument is no good. The problem is that if we rely on (B*), then the argument is no longer a subtraction argument: the notion of subtraction does no work in it. For if we can get rid of any concrete object and thereby get rid of all its parts, then why not simply, in one step, get rid of that object which is the mereological sum of all the concrete objects? It is itself concrete, presumably; for how could a collection of concrete objects compose an abstract object? Well, if it is concrete, and if concrete objects can be subtracted along with their parts, then every concrete object can be removed in one go. Given (B*), that will get us to the empty world immediately; there is no need to use the notion of subtraction. And this new argument would not be a good one. The premise that one can remove a concrete object and all its parts is unacceptable if it gets us to the world devoid of concrete beings so easily. The complaint is not that the premise (B*) is false, but that it is too close to the conclusion to be dialectically effective. We would, in effect, be arguing that there could have been no concrete objects, on the ground that there is a world in which none of the concrete objects exists and neither does anything else. This just begs the question.7 So neither of the two obvious responses to the objection to (A) results in a suasive argument: both responses rely on presuppositions which anyone who is yet to be convinced of metaphysical nihilism is likely to question. Efird and Stoneham offer another solution, which they intend to be in line with their project of showing that our pre-theoretic modal intuitions commit us to the premises from which nihilism can be derived. Their solution lies in their definition of concreteness. They take an object to be concrete iff . . .
It is spatiotemporally located It has some intrinsic quality It has a natural boundary.
As a preliminary, this definition of concreteness needs to be improved, for it gives what is uncontroversially the wrong judgement about whether some worlds contain 7 I make much the same point against Rodriguez-Pereyra in my ‘Much Ado about Nothing’.
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concrete objects. A world in which there is just one big hunk of homogeneous matter unbounded in every direction, spatial and temporal, is not concrete, according to Efird and Stoneham, because it has no natural boundary (because it has no boundary). No one should count as a metaphysical nihilist in virtue of believing in that possibility. But it should not be too hard to alter the definition to avoid this problem. Perhaps the condition on concreteness should not be that concrete objects have a natural boundary, but that they have no unnatural boundary. Then objects which are spatiotemporally located and have intrinsic character, but which have no boundary at all, will count as concrete. The notion of a natural boundary is taken from Ted Sider.8 If a white sheet of paper has a red circle on it, both the paper and the circle have a natural boundary. In contrast, the red circle whose centre is the same as that of the main circle, but whose radius is, say, half as great, does not have a natural boundary: this inner circle does not contrast with its surroundings in the appropriate way. In a world in which the only thing that exists is a white sheet of paper with a red circle on it, this thing is extended in space, and so it has infinitely many proper parts. But there are not infinitely many concrete objects in this world, given Efird and Stoneham’s definition of concrete: there are only two concrete objects, for there are only two objects which have natural boundaries – the paper, and the part of the paper on which the circle is drawn. The other parts of the paper do not count as concrete objects, and hence do not fall under the quantifiers in (A). My problem is that if this is what it is to be concrete, then I think we should deny (B). (B) is false when the quantifiers range only over those things which have natural boundaries. I can perhaps bring out this point best by describing the rather strange tribe of people known as the Qube. The Qube worship their god Qubec, which is a large cube made of homogeneous matter. The Qube believe that Qubec has its size essentially: in particular, they believe that anything smaller than Qubec would not be Qubec. Part of the reason for this is that Qubec contains within it an infinite number of other gods. Suppose you chip off some of the matter from the surface of Qubec in such a way that you are left with a smaller cube. This cube is not Qubec, but rather one of the other gods contained within Qubec, which I shall call Qubec*. Qubec* has its size essentially as well; if we again chipped off some matter from it, we would destroy it and be left with yet another god, Qubec**. The Qube believe this process can go on indefinitely: you can get rid of gods one by one without ever getting rid of all the gods. Since the biggest god is a concrete object, this means that the Qube believe you can keep on removing concrete objects from the world without getting to a world devoid of concrete objects. So far, all this is consistent with metaphysical nihilism, because Efird and Stoneham can accept that there are some processes of subtraction which do not lead to an empty world; all they require is that there is some process which does. But unfortunately, while the Qube believe that any one of the gods might not have existed (and hence they agree with Efird and Stoneham that, necessarily, every concrete 8 T. Sider, ‘Maximality and Intrinsic Properties’, Philosophy and Phenomenological Research, (), pp. –.
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SUBTRACTABILITY AND CONCRETENESS
object exists as a matter of mere contingency), they think it is necessary that there are some gods. After all, if there were no gods, what would explain the remarkable design evident in the fact that a cube has six square sides all of equal size? The Qube believe that any particular god might not have existed, because they can destroy any god by chipping away deep enough at Qubec. Can they consistently combine this belief with their belief that necessarily there is some god? If they can destroy any part of Qubec, why not every part? To show why not, I need to deliver a quick course in Qube theology. No god wants to be destroyed. So any god who anticipates an event which will destroy it will attempt to prevent that event, and if no other god intervenes, it will succeed. Unfortunately, every god desires the annihilation of every larger god, and will attempt to prevent the prevention of any event that will destroy a larger god but not itself. All these gods are of equal power: if some gods attempt to bring about one state of affairs, and others attempt to bring about a conflicting state, then the majority rules. The result is the state which has more gods trying to bring it about. (If the numbers are equal, the result is that which would have occurred without godly intervention.) Any event which makes the cube smaller destroys only a finite number of gods, leaving an infinite number unharmed. So the larger gods’ attempts to prevent their own destruction will be outweighed by the smaller gods’ attempts to prevent this prevention. But no event can destroy the entire cube, since infinitely many gods would attempt to prevent this event, and none would attempt to prevent its prevention. The Qube, then, are committed to the denial of metaphysical nihilism, for they think that in every possible world there is a concrete object. In every world there is a largest god, which has a natural boundary, and which, like every other god, is spatiotemporally located and has intrinsic properties. So in every world some god or other is concrete. The Qube must deny either (A) or (B). Clearly it is (B) the Qube deny. They think that removing a concrete object can necessitate the introduction of some new concrete object. But are the Qube, in thinking this, denying some pre-theoretic modal intuitions? I do not think so. Efird and Stoneham locate the justification for (B) in the following principle, which they call (): .
Necessarily, if there are some concrete objects, there could have been fewer of those concrete objects (and no other concrete objects).
But it seems to me that the intuition behind () relies on thinking of concrete objects differently from the way Efird and Stoneham do. The idea behind () is, I take it, that the non-existence of one of the concreta should not magically necessitate the coming into being of some new thing. But the Qube do not believe that anything comes into being to replace the missing concrete object. They believe that one of the things that existed in the previous world, but which was not concrete in that world, becomes concrete in the new world. Nothing comes into being as a replacement for the non-existing concrete entity: rather, something which was not concrete becomes concrete. But there is nothing mysterious going on here: there is no mystery as to how the non-existence of some thing can affect whether some distinct thing is concrete or not, precisely because concreteness is, on Efird and Stoneham’s account, an © The Author
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extrinsic property of things. Whether or not a thing is concrete depends on its surroundings, because whether or not a thing has a natural boundary depends on its surroundings. So we should not be surprised in the slightest that the non-existence of Qubec necessitates that there is some ‘new’ concrete object; this is only because one thing which does not actually fall under the quantifiers in (B) (since it does not actually have a natural boundary) falls under the quantifiers in the world in which Qubec does not exist, and this is only because which of the cube-like objects has a natural boundary varies from world to world. Premise () is very plausible when we do not demand that concreta have natural boundaries.9 In that case it looks as though whether or not a thing is concrete is essential to it, in which case the only way in which () can fail is if the non-existence of some concrete thing necessitates the existence of some concrete thing which does not in fact exist (not some particular concrete thing: just some concrete thing or other). So it is not plausible that the non-existence of some concrete thing could necessitate the existence of some new concrete thing by necessitating the existence of some concrete new existent, something that did not exist in the world we started from. But it is not at all unplausible that the non-existence of some concrete thing should necessitate the existence of some new concrete thing by necessitating that something is newly concrete, i.e., concrete in the new world but not concrete in the world we started from. At least, this is plausible if concreteness is the extrinsic property Efird and Stoneham think it is. So the Qube do not seem to me to be denying () in the sense in which () is a pre-theoretical modal intuition. They deny () only in the sense in which it should be denied: they deny that subtracting something from a world cannot change how the rest of the things in that world are with respect to a certain extrinsic property, i.e., concreteness as understood by Efird and Stoneham. They do not deny the very strong intuitive thought behind (), for they accept that the non-existence of something cannot necessitate the existence of something which does not in fact exist. There remains a possible concern I need to address. I have been attempting to cast doubt on (B) by means of a fanciful thought-experiment, that of the Qube; but one might doubt whether the situation I have been describing is a possibility. In particular, one might doubt whether there could be concrete objects that have their size essentially, as the Qube’s gods are said to. But really this concern misses the point. The question is not whether the Qube’s beliefs could be true, but whether they are in conflict with some basic modal intuition. Even if the Qube believe a proposition which could not possibly be true, do their beliefs commit them to denying some basic intuition concerning the space of possible worlds? Efird and Stoneham must think that this is so, since they claim that the premises of their subtraction argument are justified by basic modal intuitions. But, I have argued, the basic modal intuition they take to justify (B), namely, that the removal of a concrete object need not result in the introduction of a new thing into the world, is perfectly consistent 9 Premise () is by no means undeniable. Lowe denies it, along with metaphysical nihilism (see references in fn. ). For criticism of Lowe’s arguments, see the second half of my ‘Much Ado about Nothing’.
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with the Qube’s beliefs. This is the crucial point – not that there is a possible situation in which (B) is false, but that (B) does not follow from the basic modal intuitions which Efird and Stoneham think justify it. So even if you think the Qube scenario is impossible, you should still deny Efird and Stoneham’s claim that the premises of their subtraction argument are justified by our basic modal intuitions. (B) may be a necessary truth, for all I have said; I am simply arguing that Efird and Stoneham have done nothing to justify it. I conclude that Efird and Stoneham’s subtraction argument is unsuccessful. They solve the problem for (A) by appealing to a notion of concreteness which makes being concrete an extrinsic and accidental property of things; but with concreteness thus understood, (B) appears to be false, and not supported by (). (B) appears to be true if we take concreteness to be an intrinsic property which all the infinitely many parts of concrete things have, and which as a matter of necessity is had essentially or lacked essentially. But when we understand concreteness in this way, the possibility of there being finitely many concrete objects is not immediately obvious, and is certainly not more obvious than the possibility of there being no concrete objects; and so the argument with (A) construed thus is not going to be suasive. Premises (B) and (A) can both be secured, then, only if the quantifiers in (B) range over all and only the things which are concrete in the intrinsic and essential sense, and the quantifiers in (A) range over all and only the things that are concrete in the extrinsic and accidental sense. In that case Efird and Stoneham’s argument will be invalid, because they make the illegitimate move of instantiating a quantifier in (A) and then using that constant to instantiate a quantifier in (B). Since the quantifiers do not have the same domain, this step is invalid. University of Leeds
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