This paradox is from Peter Geach, Reference and Generality, Cornell, 1980, 215. The following formulation is mine.
1. There is exactly one furry cat, Tibbles, on a mat.
2. Tibbles minus one hair is a proper part of Tibbles.
3. If Tibbles has n hairs, then there are at least n proper parts of Tibbles. For each of Tibbles' hairs, there is a proper part of Tibbles which is Tibbles minus that hair.
4. Each such proper part of Tibbles is a cat.
Therefore
5. There are n + 1 cats on the mat.
Therefore
6. (1) is false.
Something is wrong with this reasoning since it implies that if Tibbles has 1000 hairs, then there are 1001 cats on the mat, which contradicts (1). But where is the mistake?
The mistake may reside in line (4). It is true that if a cat loses a hair, then it is still a cat. And it is true that in a counterfactual situation in which a cat has one less hair than it actually has, it is still a cat. But it doesn't follow that a proper part of an actual cat is an actual cat. In actuality there is exactly one cat on the mat. So (4) is false.
Related Posts (on one page):
- Weak Supplementation and Unique Fusion
- Van Inwagen on Arbitrary Undetached Parts
- Van Inwagen and Lewis on Composition and Identity
- More on Qualitative Identity
- Puzzling Over Van Inwagen's Denial of Artifacts
- Van Inwagen on the Ship of Theseus
- Trinity and Relative Identity
- Dion and Theon
- Is Tibbles-Minus-One-Hair a Proper Part of Tibbles?
- Qualitative and Numerical Identity
- The Paradox of Tibbles the Cat
At first glance I would like to take issue with (2) and (3) and then make some distinctions regarding essential and accidental parts and wholes. But perhaps my discomfiture is simply with how you're defining "proper part". Because if each hair is a proper part (part [a]), and the whole composite ([a]+[b]) minus that hair is another proper part (part [b]), then each part [a] becomes a sub-part of a part [b] (ceasing to be a part [a]?) whenever a new part [a] is counted. Doesn't the aberrant multiplication of the whole come as a result of re-counting each part multiple times?
I'm unconvinced by your reasoning in the last paragraph. You make these claims:
7. If a cat loses a hair, then it is still a cat.
8. If a cat were to have one less hair than it has, it would still be a cat.
9. The claim that 'a proper part of an actual cat is an actual cat' doesn't follow from (7)&(8).
And the implication appears to be this:
10. (4) can be true only if (9) is false.
(Otherwise, how does (9) contribute to your case against (4)?) I agree with (7)-(9), though (10) is false: (4) can be true even if (9) is too. So you haven't really given us any reason to think (4) is the culprit. In fact, I believe that in affirming (7) and (8) you have committed yourself to the truth of (4). (Or at least you have if you grant (2) as well.) (4) says that "Each such proper part of Tibbles is a cat." The expression 'Each such proper part of Tibbles' refers to that thing which is Tibbles minus some hair. Hence, (4) is just the claim that 'Tibbles minus some hair is a cat'. (7) and (8) each entail that Tibbles would still be a cat if he were to lose a hair. Thus, (7) and (8) each entail that (4) is true.
The right thing to say is that (2) is false. This seems obvious:
11. If Tibbles starts with n hairs and then loses one, the result will still be (fully) Tibbles.
According to (2), however, the result of this change will be a proper part of Tibbles. Therefore, [(2)&(11)] entails that something which is (fully) Tibbles is identical to a proper part of Tibbles, which is absurd. Since (11) is obviously true, then, it follows that (2) is false.
I'm sorry if I was unclear. I was making the same point as you, i.e. that the problem is with (2) and (3) in that it makes no sense to call what is clearly a whole cat a part of a cat. In addition to this I was trying to point out some further absurdities that follow from the argument besides those Dr Vallicella mentioned.
This then spilled over into a huge debate I had with Brandon - unfortunately he is using a blog that loses comment threads so all that wisdom has been lost. I think he came round to my view eventually. My basic thrust was to deny Bill's
(4) Each proper part of Tibbles is a cat.
which is absurd, because, given that a proper part is not numerically identical to the whole, it would lead to the conclusion that there is more than one cat (which there isn't). Brandon naturally retorted that, if we pluck one hair from Tibbles, what is left is surely I cat. I replied with a 'tense' argument. Yes, after we have plucked one hair, there is still a cat (just one) on the mat. This cat is identical with a cat who used to have an extra hair. But he is not identical with any cat who, right now, lacks anything that Tibbles possesses. Any cat who is, right now, different in any respect from Tibbles, is not numerically identical with Tibbles.
Surprisingly, Bill, (given that we are in friendly disagreement about almost everything) you seem to agree with me in rejecting (4).
Spur:
>> The right thing to say is that (2) [2. Tibbles minus one hair is a proper part of Tibbles] is false.
Surely not. We have the concept of parts of Tibbles. E.g. liver, eyes, the hair. So why can't we refer to all the bits and pieces that make up Tibbles, excepting just one hair? Indeed, I just have so referred! Perhaps it's the idea of 'proper part' you don't like. Then remove it. Let X be the bits of Tibbles, constituted in the right way, excepting one hair. Then, the question is whether X is a cat or not. If X is a cat, the absurd conclusion follows.
I'm inclined to disagree. In fact I wonder if it doesn't obscure the real problem. It's true the Tibbles Paradox destroys the notion of identity across change, but I think it also muddles what it is to be Tibbles here and now. Any one hair is irrelevant to Tibbles being Tibbles, it is right now irrelevant to what and "who" Tibbles is that he has a given hair or that he doesn't. These particular hairs, these skin cells, these blood cells, this air in his lungs, etc., etc., are not what Tibbles is. Each of these individual parts are expendable, for Tibbles was himself before his body formed or absorbed them and will be himself when they have been expelled or sloughed off. They form then an accidental unity with Tibbles, whereas the cat himself is an essential unity.
Not all parts are created equal. Without some given hair he's still a cat; without his spine he isn't.
Actually (4) does not by itself (or even with the assumption that no proper part = the whole) lead to the absurd conclusion. For instance, if there were no such proper parts of Tibbles, then it wouldn't follow that there's more than one cat. In order to entail that, (4) would have to be combined with something like the claim that the result of removing a hair from Tibbles is a proper part of Tibbles, i.e., (2). So the absurdity of there being multiple cats only implicates (2) or (4). What is the argument that the culprit is (4) rather than (2)?
As I pointed out above, (2) together with (11) entails an absurdity. (11) is obviously true; hence, (2) is false. So (2) is at least one of the culprits.
We can, and I never suggested otherwise. My point is that some such sums of Tibble-pieces are not proper parts of Tibbles.
Okay, let's try that. Suppose we re-formulate the argument this way:
1. There is exactly one furry cat, Tibbles, on a mat.
3'. For each of Tibbles' n hairs, there is a distinct sum which is the sum of all its individual parts (constituted in the right way) save one of those hairs.
4'. Each such sum of Tibble-parts is a cat.
Therefore
5. There are n + 1 cats on the mat.
Therefore
6. (1) is false.
In this case I would agree that (4') is the problem and the trouble has something to do with tense. (4') is really this:
4'. Each such sum of Tibble-parts is [now] a cat.
This is clearly false, even though any such sum could become a cat if the right hair were to be removed.
So what I want to say is that the most obvious problem with the "paradox" as originally formulated is (2), but that if we re-formulate it to fix (2), we can see that the fundamental problem is (4).
This is basically right, though if we grant that the sum of all individual Tibble-parts save some hair is a proper part of the "here and now Tibbles," we commit ourselves to an absurdity. For suppose that Tibbles were to lose the very hair that is not included in that sum. Then the sum--a proper part of the here and now Tibbles--would be numerically identical to that future Tibbles. But that future Tibbles is also numerically identical to the here and now Tibbles. It follows that the here and now Tibbles is numerically identical to one of its proper parts, which is absurd. This is why we must object to the talk of such sums as proper parts.
Spur,
Excellent comment. You would solve the problem by rejecting
2. Tibbles minus one hair is a proper part of Tibbles.
But note that (2) says nothing about change. You are to think of Tibbles at a time and then remove (in thought)one of his hairs. Now it seems obvious to me that if you do this, the result, namely, Tibbles-minus-one-hair, is a proper part of Tibbles.
We will agree, however, that when a cat suffers hair loss it does not change into a proper part of itself. Thus if Tibbles at time t has 1000 hairs and at t* has 999 hairs, this is not to be understood as a whole cat changing into a proper part of itself. A whole cat exists at both times.
I could have been clearer in the post. What I was getting at was that (4) is a false and confused way of expressing one or both of the truths that numerical identity is consistent with change across times and change across possible worlds: T. can lose a hair without prejudice to his being numerically the same cat, and the possibility of his having a different number of hairs at time t than he has actually at t is consistent with his being the same cat.
I suppose I am assuming some such maximality or completeness principle as the following: An F at time t considered in abstraction from one of its proper parts is not an F at t. So a cat at t minus one of the hairs it actually has at t is not a cat. Does this seem intuitively right?
It is nice to find a point of agreement. Here we join our decrepit forces against the mighty and youthful Spur. It seems obvious to me that (in thought) one can abstract from or prescind from some proper part of a thing such as a hair of a cat and the result of this 'precision' will be a proper part of a cat. A cat minus one hair is a proper part of a cat. But no proper part of a cat is a cat, not even a proper part that differs by a hair from a whole cat.
Your second-to-last comment seems 'spot on.' 'Tibbles minus one hair' is to be understood statically, (synchronically not diachronically) not as referring to the result of the plucking of a hair from the poor feline.
Wouldn't you say that you, the one reading this right now, are numerically identical to the one who wrote the comment at 10:37am, even though you do not share all the same properties? Most philosophers would say yes.
Thanks for the kind words. You write:
What you are calling Tibbles-minus-one-hair is what I described as "the sum of all individual Tibble-parts save some hair." The problem with thinking of this thing as a proper part of Tibbles, as I pointed out in my previous comment to David, is this. If Tibbles were to lose the very hair that is not included in Tibbles-minus-one-hair, then that future Tibbles would be numerically identical to Tibbles-minus-one-hair. But that future Tibbles is also numerically identical to the present Tibbles, of which on your view Tibbles-minus-one-hair is a proper part. It follows that in this situation the present Tibbles would be numerically identical to one of its proper parts, which is absurd. The way to avoid this absurdity, on my view, is to deny that Tibbles-minus-one-hair is a proper part of the present Tibbles.
Suppose you walk into an empty room and then exit ten minutes later. How many people have been in that room during the ten minutes? Most people would say that the one who exits is numerically identical to the one who entered, so there was only one person in the room. This means that numerical identity does not require qualitative identity. If you say that numerical identity does require qualitative identity, then you'd have to say that during the ten minutes, there were many people in the room, perhaps millions or billions, though never more than one at a time. That's an odd thing to say, isn't it?
That's why very few people think numerical identity requires qualitative identity.
Astonishingly, we agree again. Why 'youthful'. I always had the impression that practically everyone who contributes here was born around 1955. Is that wrong?
I'm happy, of course, to say that the same person leaves the room as entered it ten minutes earlier, but it seems to me that that person has indeed undergone billions of changes (this raises the question of identifying discrete changes) during the ten minutes and does not emerge with all qualities unchanged, and so is no longer numerically identical to his former self. Hence my view that the criterion for sameness of person, cat, object, etc, must be weaker than numerical identity. Perhaps another way of putting it is to say that I'm looking for objective criteria to distinguish accidental change from essential change.
At least there's only one person in the room at any moment!
I was born in 1980.
I was born in 1980.
David,
Noonan's SEP article puzzles me. He suggests that numerical identity "is the only identity relation in accordance with which we can properly count (or number) things: x and y are to be properly counted as one just in case they are numerically identical (Geach 1973)." I agree with that. Numerical identity is the sort of identity that makes a difference to how many things there are. If x and y are numerically identical, then x and y count as just one thing. If they are numerically distinct, they count as two. And so forth. But why would he then say that numerical identity "requires absolute, or total, qualitative identity"? Clearly a thing can undergo qualitative change over time even while remaining numerically the same in the sense just specified. I agree with you that the person in the room will undergo many changes in the ten minutes, but even so the one who enters the room will be numerically identical to the one in the room at any point during those ten minutes and to the one who exits. That is because the correct answer to the questions How many people entered or left the room? and How many people were in that room during the ten minutes? is not 'two' or 'billions' but 'just one'.
I also take issue with his claim that "things can be more or less qualitatively identical." On my view, x and y are qualitatively identical just in case this condition is met: for any quality q, x has q iff y has q. If x and y have some but not all qualities in common, then I say that they are qualitatively similar but not qualitatively identical. It seems quite wrong to me to say, as Noonan does, that "Poodles and Great Danes are qualitatively identical because they share the property of being a dog, and such properties as go along with that." At best, poodles and great danes are qualitatively similar. (Incidentally, qualitative similarity is the one that admits of degrees. Two things can be more or less qualitatively similar. But they can't be more or less qualitatively identical. They're either identical in that sense or they aren't.)
Then we have
1a. There is exactly one sum of parts P of Tibbles.
1b. There is exactly one furry cat, Tibbles, viewed as an enduring whole.
2a. P minus one hair is a proper part of P.
2b. Tibbles minus one hair is still Tibbles.
3a. If Tibbles has n hairs, then there are at least n proper parts of P. For each of Tibbles' hairs, there is a proper part of P which is P minus that hair.
3b. If Tibbles has n hairs, then there is just one Tibbles.
4a. Each such proper part of P is a cat.
4b. Each such proper part of Tibbles is a cat.
Therefore
5. There are n + 1 cats on the mat.
Here, 4a is false and 4b is incoherent since we haven't come up with any proper parts of Tibbles. There is no remaining basis for concluding 5.
"But if Tibbles is something else, an enduring whole, then it is less obvious what would constitute a proper part of Tibbles..."
The question of what is an enduring whole does seem to be different from the subject of wholes and parts atemporally considered. Consider a flame. Molecules are entering it, undergo chemical reactions, and leave. The flame is something we would like to call an enduring whole, but its boundaries are indeterminate. At any given time, there will be molecules whose participation in the flame is indeterminate. Then if we were to suppose that there is a fact of the matter as to what set of molecules constitutes the flame, S, at a particular time, we can also consider another set S', that is the same as S, but minus one of the molecules from the indeterminate boundary. Then it is indeterminate which set, S or S', lists the parts of the flame. If W is the whole composed of the parts in S, then the entity composed of the parts in S' is a proper part of W. But there seems to be no fact of the matter as to which of these is THE flame, or whether both are flames.
But if numerical identity is about counting then we had better decide what kinds of things we are counting. To go back to Dave Gudeman's earlier example, if we are counting cars then my car is numerically identical to my car minus a hubcap. If we are counting car parts then clearly it is not. So 'A is numerically identical to B' is potentially ambiguous. We must also say what kind of thing we are counting.
What kinds of thing are you counting when you say that the present Tibbles-minus-a-hair is numerically identical to the future Tibbles?