It is a "common sense" principle that no two things can occupy the same place at the same time. But this assumption, in conjunction with some other plausible, "common sense" assumptions, leads to a contradiction. Something has to go.
A few words about this common sense principle, for there seem to be obvious counter-examples to it as I have first formulated it. The first problem is this: right now, I occupy the same place as my forearm. But this should not count as a counter-example to the common-sense principle, so the principle needs reformulating. What we want to say is that no two things can exactly occupy the same spatial region at the same time. Whereas there is a region that is occupied by both me and my forearm, this region is not exactly occupied by me: this region is a mere subregion of the region that I exactly occupy.
Furthermore, it would seem that events can exactly occupy the same region as other things at a given time. Consider a globe that undergoes a single rotation during interval I: the event of its rotation presumably exactly occupies the same region that the globe does during interval I, even though the event of the globe's rotation is not identical to the globe. Furthermore, if the globe heats up by 10 degrees during I, the event of its heating up 10 degrees presumably also exactly occupies the region occupied by the globe during I. It seems that we may have any number of events exactly occupying the same spatial region as a material object at any given time. But this should not count as a counterexample to the common sense thesis that no two things can exactly occupy the same place at the same time. And perhaps a ghostly lump of ectoplasm could exactly occupy the region occupied by the globe during I. But this shouldn't count as a counterexample either. Hence we should reformulate the principle as follows: no two material objects can exactly occupy the same place at the same time. (Things composed of ectoplasm, if there are such things, and events, do not count as material objects.)
But even this revised principle is in trouble, as is shown in Geach's "Tibbles the Cat" thought experiment, for it leads to contradiction in conjunction with some other common sense principles. (Essentially the same thought experiment was posed by the Stoic philosopher Chrysippus in ancient Greece.) Suppose that we have a cat named Tibbles. Let us name the part of Tibbles that consists of all of Tibbles but her tail (such a part is a "puss") "Tib." Now suppose that at time t, Tibbles meets with an unfortunate accident, and loses her tail. (To avoid the possibility that Tibbles continues to exist in "scattered" form, lets sat that the tail is completely annihilated at t.) Tibbles presumably did not cease to exist in this accident, nor did Tib. But after the accident, Tibbles and Tib, distinct material objects, seem to occupy exactly the same region.
We may formulate the problem as the following inconsistent set of sentences.
(1) Tibbles is a material object at any time at which she exists, and Tib is a material object at any time it exists.
(2) Tibbles is never identical to Tib.
(3) Tibbles exists after time t.
(4) Tib exists after time t.
(5) If Tibbles exists after t, she exactly occupies region R.
(6) If Tib exists after t, it exactly occupies region R.
(7) No two material objects can exactly occupy the same region at the same time.
This set of sentences is inconsistent, so at least one will have to be rejected.
(1) might be rejected by some (if one wished to claim that cats have an immaterial soul, perhaps), but an analogous difficulty will come up regarding analogous thought experiments involving things that surely are material objects: run the same thought experiment with the Venus de Milo, prior to the breaking off of its arms, and the torso of the Venus de Milo, which consists of every part of the statue except its arms. And (3) might be denied by those, such as Geach, who think that identity is relative to a sortal term, but relative identity is, in the opinion of many, an incoherent doctrine. (5) and (6) are merely details of the story of Tibbles as we have told it, so these premisses should not be questioned.
Some philosophers, such as Chisholm, reject premise (3), on the grounds that the parts of a thing are essential to it. This doctrine we may call mereological essentialism. But the thesis that material objects can gain and lose parts is even more a part of our common sense conception of material objects than is sentence (7), so this seems to be a desperate way to salvage (7).
There are two different ways in which one might wish to reject premise (4). One way, proposed by Michael Burke (echoing Chrysippus's solution to his original problem) is to say that at time t, Tib ceases to exist. Burke argues as follows, using the Kripkean doctrine of sortal essentialism.
(a) All cats are essentially cats.
(b) Therefore, non-cats are essentially non-cats.
(c) Tib is a non-cat.
(d) Therefore, Tib is essentially a non-cat.
(e) If Tib survived the annihilation of the tail, Tib would be a cat.
(f) Therefore, Tib ceased to exist when the tail was annihilated.
Many have found it strange, however, to think that a mere Cambridge change (non-intrinsic change) of an object could make it cease to exist (These philosophers would deny either (a) or (e) in Burke's argument.) Tib is not changed in any intrinsic way by the annihilation of the tail, so how could the annihilation of the tail cause Tib to cease to exist.
Peter van Inwagen is one such philosopher. Van Inwagen rejects premise (4), not on the grounds that Tib ceased to exist at t, but on the ground that Tib never existed in the first place. Denying what he calls the "Doctrine of Arbitrary Undetached Parts," van Inwagen claims that there is no such thing as "all of a cat except its tail," given that the cat does indeed have a tail.
Other philosophers, such as Wiggins and David Lewis, would deny (7), although they differ in what they would replace (7) with. Wiggins claims that (7) must be amended as follows: No two material objects that both satisfy a sortal F can exactly occupy the same place at the same time. Tibbles satisfies the sortal "cat", but Tib does not. Tib satisfies the sortal "puss", but Tibbles does not. So there is no problem in there occupying exactly the same region at a given time.
Lewis would make use of the doctrine of temporal parts in explaining how (7) is false, and how Tibbles and Tib can exactly occupy R after time t. (Lewis has not, to my knowledge, explicitly commented on this case, but this is what he would say. It's what I say about this case, anyway.) The temporal part of Tibbles after time T is identical to the temporal part of Tib after time t, even though Tibbles and Tib are not identical. There is no more mystery about how Tibbles and Tib can both exactly occupy region R after time t than there is about how I and my forearm can occupy the same region at a given time: they occupy the same region by having a part in common. Thus the principle of (7) should be amended to read "No two material objects can exactly occupy the same region at every time at which they exist," or "No two material objects can exactly occupy the same region of space-time."
Some have claimed that the doctrine of temporal parts does not resolve the difficulty, and that even the revised principle is false. For, these philosophers claim, two distinct objects might exactly occupy the same region at every time at which they exist, or (what amounts to the same thing) exactly occupy the same region of space-time. (These are generally the philosophers who prefer Wiggins relativising the principle to sortals.) Here's an example from Gibbard's "Contingent Identity" that purports to show this. (This is not exactly the use Gibbard puts the case to, but similar cases have been so used.) Consider a statue of Goliath, call it "Goliath", and the lump of clay from which it is made, call it "Lump1." Suppose, as seems possible, that Goliath and Lump1 come into existence at the same time, coincide (exactly occupy the same region) at all moments of their existence, and are both annihilated at the same time. Goliath and Lump1 exactly occupy the same space-time region. Nevertheless, these philosophers say, Goliath is not identical to Lump1, since they differ in their modal properties. If at some time t, the statue had been remolded into a sphere, Goliath would have ceased to exist, but Lump1 would not have. Lump1 has the property "possibly surviving a deformation", but Goliath does not. Therefore, by Leibniz' Law, they are not identical. So two distinct material objects can exactly occupy the same space-time region.
(Personally, I think it questionable that Goliath would cease to exist when the statue was remolded, for we could truthfully say of the sphere: "That used to be a statue". And the only statue it could have been was Goliath. But I will let this objection rest.)
Lewis would avoid this difficulty by appeal to his counterpart theory of modal properties. Whenever we attribute a modal property to an object, we make implicit use of a counterpart relation, and when we appear to attribute contradictory modal properties to the very same object (Goliath = Lump1), we are making use of different counterpart relations, fixed by the context. When we say "Lump1 could have survived deformation into a sphere," this should be analyzed as "Lump1 (=Goliath) has an other-worldly lump-counterpart that survives the deformation into a sphere." But when we say "Goliath could not have survived deformation into a sphere," this should be analyzed as "Goliath (=Lump1) has an other-worldly statue-counterpart that survives deformation into a sphere." Since we make implicit appeal to two different counterpart relations, we do not contradict ourselves in saying that Goliath could not have survived deformation, but Lump1 could have.
Chisholm, Roderick. Person and Object. Open Court, 1976.
Gibbard, Alan. "Contingent Identity," Journal of Philosophical Logic 4 (1975), 187-221.
Van Inwagen, Peter. "The Doctrine of Arbitrary Undetached Parts," Pacific Philosophical Quarterly 62 (1981), 123-37.
Wiggins, David. "On Being in the Same Place at the Same Time," Philosophical Review 77 (1968), 90-5.
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