Tuesday, October 29, 2019

Sameness without identity

Mike Rea’s numerical-sameness-without-identity solution to the problem of material constitution holds that the statue and the lump have numerical sameness but do not have identity. Rea explicitly says that numerical sameness implies sharing of all parts but not identity.

Does Rea here mean: sharing of all parts, proper or improper? It had better not be so. For improper parthood is transitive.

Proposition. If improper parthood is transitive and x and y share all their parts (proper and improper), then x = y.

Proof: But suppose that x and y share all parts. Then since x is a part of x, x is a part of y, and since y is a part of y, y is a part of x. Moreover, if x ≠ y, then x is a proper part of y and y is a proper part of x. Hence by transitivity, x would be a proper part of x, which is absurd, so we cannot have x ≠ y. □

So let’s assume charitably that Rea means the sharing of all proper parts. This is perhaps coherent, but it doesn’t allow Rea to preserve common sense in Tibbles/Tib cases. Suppose Tibbles the cat loses everything below the neck and becomes reduced to a head in a life support unit. Call the head “Head”. Then Head is a proper part of Tibbles. The two are not identical: the modal properties of heads and cats are different. (Cats can have normal tails; heads can’t.) This is precisely the kind of case where Rea’s sameness without identity mechanism should apply, so that Head and Tibbles are numerically the same without identity. But Tibbles has Head as a proper part and Head does not have Head as a proper part. But that means Tibbles and Head do not share all their proper parts.

Here may be what Rea should say: if x and y are numerically the same, then any part of the one is numerically the same as a part of the other. This does, however, have the cost that the sharing-of-parts condition now cannot be understood by someone who doesn’t already understand sameness without identity.


Brandon said...

While it's true in standard mereologies that proper parthood is defined in terms of parthood and identity, this is not the only way one can define proper parthood; one can also define it as "x is part of y and y is not part of x". While the two can be made equivalent with certain additional assumptions, it's noticeable that the latter blocks the argument here: since x is part of y and y is part of x, and proper parthood on this approach is not defined using nonidentity, the step using nonidentity to get a conclusion about proper parthood doesn't work. And that makes sense, I think; sameness without identity requires not relying on antisymmetry.

Philip Rand said...

You are correct no Aristotelian method can solve the problem. This points to the fact that any analysis using an Aristotelian system will always be incomplete.

The modal properties of Tibbles the "Head" and Tibbles the "Cat" are numerically the same.

Numerically, the modal propoerties of everything below Tibbles neck still exists. It is simply no longer accessible.

This is not an Aristotelian solution, but it is the solution and can be formalised.

Alexander R Pruss said...


That's a nice move, but to make it go requires Head to be a part of Tibbles and Tibbles to be a part of Head. The former seems right, but the latter seems wrong. Here is one argument against the latter:

* Nothing acquires a part by the annihilation of an object.

But the annihilation of Tibbles below the neck makes Head acquire a part, namely Tibbles. I suppose, though, one could say: granted, Head acquires a part, but the number of parts of Head doesn't rise (since the part is numerically the same as Head itself).

Another argument against this mutual parthood approach is this plausible thesis:

* If x is a part of y, then x depends on y or y depends on x.

But now imagine that two artifacts, with different modal properties, are formed out of the same chunk of metal. Maybe the chunk of metal is both a knife and a fork. Then on the above view, we will say that the knife is a part of the fork and the fork a part of the knife. But neither depends on the other.

Philip Rand said...

The knife and fork are dependent on the numerical relation between them.

One can formalise this relation without recourse to intuition.

Brandon said...


It could be that I am cross-eyed from grading ethics papers, but I don't follow the first argument; surely Head would already have Tibbles as a part? Perhaps I'm not understanding the assumed relation between Head and Tibbles.

On the second, since everything is part of itself, if 'part' is taken broadly, it seems it would make more sense for 'part' in the principle used here to indicate a proper part -- unless we are using 'depends' so broadly that we would say that everything depends on itself. If we were using 'depends' that broadly, though, I'm not sure why we couldn't also say that Fork and Knife depend on each other.

I'm not sure (this is just a thought that occurred to me in reading your first two sentences, not something I've thought out at all), but I think it might make sense for a sameness-without-identity theorist who wants to understand this in mereological terms to do it slightly indirectly, and take 'Tibbles is part of Head' as a shorthand for something like 'Being Tibbles is part of being Head', and I'm not sure that this is so absurd-sounding as the former (which perhaps conjures up imaginations of weird spatial deformations). The claim of sameness is not about sameness of components but about something more about 'partial being', if that makes any sense: what has being as Tibbles is the same as what has being as Head, although the properties of each are different.

Alexander R Pruss said...

Back when Tibbles is an ordinary cat, Tibbles is not a part of Head. For Tail is a part of Tibbles then, and Tail is not a part of Head.

Brandon said...

Tail is certainly not a proper part of Head; I'm not so sure that the mereological SWI should agree without qualification that Tail is not any kind of part of Head when they are both in the whole cat. Admittedly, 'Tibbles is part of every proper part of Tibbles' is surprising given usual space-based assumptions about mereology, and perhaps there's an oddity in holding that a proper part of X is not necessarily a part of what X is part of, but it's not really very far from how we would in real life talk about parts of Tibbles, and is also not too far from the scholastic totus in qualibet parte. In the annihilation of the body, Head loses a part (but not a proper part), namely, Tail; but Head and Tibbles were mutual parts all along.

Philip Rand said...

Take a sample of saliva from Tibbles "Cat"... take a sample of saliva from Tibbles "Head"...

Send the samples to a lab for DNA testing...

Ask the lab to determine if the samples are the same...

Dagmara Lizlovs said...

Poor Tibbles. :-)

Dagmara Lizlovs said...


It seems that Tibbles the cat is in good company. What can you say about this saint in reference to this post?


Dagmara Lizlovs said...

Bishop Dennis is gaining many converts from paganism. The Roman Governor in the region has Bishop Dennis beheaded. Bishop Dennis picks up his head and carries it while preaching a sermon on repentance. The head is Bishop Dennis. But what about the body? Is the body a proper part of the head now? On the last day will not both be raised?