## Composition as identity, part I

December 11, 2008

I’ve been thinking a bit about the (somewhat radical) thesis that an object is literally identical with its parts. So, for example, these things, my parts, are identical to me. One nice thing about this is that you seem to get unrestricted composition for free: you get it from the plural comprehension schema.

However, its main drawback is it requires you to be able to make sense of many one identity. Lewis notes one problem with this, namely: my parts are many, whereas I am not. There are a couple of responses out there: Baxter takes this to be a failure of Leibniz’s law, and Sider has a language where plurals and and singular terms are intersubstituteable. Predicates are polymorphic and you can say truly that I’m both one, and many.

Both these views have crazy consequences (see Sider’s paper “Parthood” to see why.) So I’ve been trying to come up with a more natural way for the composition as identity theorist to go.

Note firstly that Alice, Bob and Fred are human iff Alice is human, Bob is human, and Fred is human. ‘Human’ is a distributive property. Consequently, the atoms that compose me are human iff each atom individually is human. They’re not, so the atoms that compose me aren’t human. However, there is a non-distributive property my atoms have, being human*, which some things have, roughly, if they compose a human. Thus I am human iff my atoms are human*.

So that’s the first step: every monadic predicate of the language, F, has a pluralised homonym, F*. For example, ‘one*’ is short for ‘many’: I am one, the atoms that compose me are one* (they’re many.) The second step: for every singular variable (or name), x,  there is a pluralised version, x*. I shall follow the tradition in plural logic, and use xx for x*. So, for example, ‘Andrew*’ is short for ‘Andrew’s parts’. Finally identity. We have one-one identity, =, many-one identity, *=, one-many identity, =*, and many-many identity, *=*. For n-place relations, well, you can work out your own notation, but it’s the same idea as identity.

We are now in a position to state Leibniz’s law. There are actually lots of versions, I’ll just state a couple

• $\phi^*(xx), xx ^*\!\!=y \vdash \phi(y)$
• $\phi(x), x=^*yy \vdash \phi^*(yy)$

(you must also add suitable identity axioms such as $x =^* xx$, $xx ^*\!\!=x$, etc…). So, for example, Fred is one, Fred is Fred’s parts (that is, Fred =* Fred*), therefore Fred’s parts are one* (Fred* are one*.) So, Fred’s parts are many. I’m human, I’m my parts, so my parts are human*. That’s the idea.

So much for identity. How do we get mereology out of this? Define x is a part of y, iff the xx’s are among the yy’s. Supposing $\sqsubseteq$ is parthood, we have the following definition

• $xx ^*\!\!\sqsubseteq^* yy \leftrightarrow \forall z(z \prec xx \rightarrow z \prec yy)$

where $\prec$ is the ‘is one of’ relation from plural logic. Thus $^*\!\!\sqsubseteq^*$ is defineable in purely logical vocabulary, so if $\sqsubseteq$ is truly a homonym parthood is logical. What’s more, unrestricted composition falls out from plural comprehension as desired.

But the other good thing about this formulation is that it avoids some of the crazy consequences Sider claims they get. For example, allegedly the principle: x is one of $y_1, \ldots, y_n$ iff $x=y_1$ or … or $x=y_n$, fails. But his argument required moving between (in my language) ‘x is part of y’ and ‘x is part* of y’, rather than ‘x is a part* of yy’. Similarly he had to move between ‘x is-one-of xx’ and ‘x *is-one-of yy’ rather than ‘xx *is-one-of yy’ (his argument is just ungrammatical in this framework.)

Similarly, because he doesn’t pay attention to the difference between parthood, parthood*, *parthood and *parthood*, he gets all kinds of weird things coming out, e.g. ‘Tom, Dick and Harry carried the basket’ iff ‘Dom, Hick and Tarry carried the basket’, where Dom is the fusion of Dicks head and Toms body, Hick the fusion of Harry’s head and Dick’s bady, and Tarry the fusion of Toms head and Harry’s body. Following in the spirit of my rules, you can get from the LHS to ‘Tom*, Dick* and Harry* (carried the bucket)*’, where ‘(carried the bucket)*’ is a superplural predicate. But you can’t then swap bits from the plural terms ‘Tom*’, ‘Dick*’ and ‘Harry*’ and expect it to still satisfy (carried the basket)*.

Lastly, a predicate, P, is distributive iff $P(x_1, \ldots x_n)$ $\Leftrightarrow P(x_1) \wedge \ldots \wedge P(x_n)$. Sider claims there are no distributive predicates if you’re a composition as identity theorist. But again, the argument seems to rely on being able to freely move between plural and singular terms, without moving between the corresponding plural and singular predicates.

Ok, so it seems to be a natural way to formulate the position. That said, I think the position is ultimately incoherent, so I’ll talk a bit about that in the next post…

1. I’m actually working on a paper on composition as identity as we speak. My idea is not to distinguish predicate mates, although that idea is interesting. The one question I have with this approach is that it gives us no way real way to understand many-one identity or one-many identity. Leibniz’s law gives us no guide, because we’ve now split Leibniz’s law into many laws depending on which predicates we’re using (F or F*). One of Lewis’s and van Inwagen’s major challenges to the composition as identity theorist is this: make some semantic sense of many-one identity. I’m assuming your next post will have something to say here about why the view can’t be made sense of…. but maybe not.

My idea is to utilize the notion of a cover. In plurals, sometimes people endorse interpreting predicates wrt to cover. A cover is a set of sets (or it could be a hyperplural, or whatever) s.t. for every y that is one of the xx, y is a member of (one of) some member of the cover.

Linguists like this approach for treating collective, distributive, and intermediary predicates as one and the same predicate. Anyway, an example is: Rogers, Hammerstein, and Hart wrote musicals. This is true with respect to the cover {{Rogers, Hammerstein}, {Rogers, Hart}} but none others.

The idea then, is that $xx = \sigma(xx)$ with respect to a certain cover. And that ties $\sigma(xx)$ to a unique decomposition. So we get around Sider’s weirdness. (And Yi’s).

But also we have Leibniz’s law so long as we do not shift covers. But we already needed that anyway, given $xx=yy \rightarrow (\varphi(xx) \leftrightarrow \varphi(yy))$ may not hold if we interpret the first and second predications wrt different covers.

Anyway, I agree that Sider, Yi, et al are shifting quickly between some things the comp. as identity theorist will want to be a bit more careful about.

I’ll be interested to see the next part!

2. In effect, your answer to the nature of many-one identity is this: it is the identity which holds between many parts and a whole which they compose. How does this respect Leibniz’ Law? Your answer is that we have homonymous predicates F* exemplified by some xx iff those xx compose a y which is F. But is the result something we should want to call Leibniz’ Law? You tell me that my atoms are identical to me, and I ask how this is possible since I am human and they are not. You answer that this is okay because my atoms compose me. So the explanation for when we have many-one identity is when we have composition. But that seems to be flipping the purpose of Composition as Identity on its head… it was a thesis meant to explain when we have *composition*.

3. Hi Colin,

I think this might just be a general problem with asking for explanations of identities.

“So the explanation for when we have many-one identity is when we have composition.”

Remember that this theorist thinks composition just *is* identity. Saying that the F’s compose x is saying that the F’s just *are* x. So yes, you have many-one identity iff you have composition, but its not an explanation since you have the same relation on each side of the equivalence.

Notice that an ordinary mereologist can also make sense of their notation. They will read “xx *= y” as “the fusion of the xx’s = y”, and “F*xx” as “the fusion of the xx’s is F”.

But for the ordinary mereologist F and F* are not synonymous: “being F” and “fusing to an F” are different properties. Similarly “fuses to x” and “identical to x” aren’t the same relation either. The composition as identity theorist on the other hand maintain that they really are synonymous, just grammatically different.

Similarly, for the ordinary mereologist, the statement of Leibniz’s law is not adequate. It doesn’t represent the principle that if x and y are identical, then they have the same properties, since F and F* aren’t the same properties. For the CI theorist F and F* really are the same property, so the statement of leibniz’s law represents correctly.

4. So yeah. I basically think to ask for an explanation of when two things are identical is wrong headed. In the many-one case just as much as in the one-one case.

Aaron:

That sounds very interesting. I had thought briefly about covers but didn’t think much about how the details would go, so I’d be interested in reading what you have at some point!

However, your approach seems to me to be more like a superplural account of many-one identity, where one side of the identity is flanked by a superplural term rather than a plural term. The only way I can make sense of ordinary plural many-one identity is if xx denotes the mereological atoms of x. (Rather than x’s parts, composite and atomic alike.)

5. Okay I think I get the idea now. Being a human and fusing to a human are the same property on your construal of Composition as Identity. I’d have to think on it some more, but it strikes me that there are still problems lurking. For instance, is a lot of work being done here by an unanalyzed notion of property identity, i.e. that some monadic property F is identical to the multigrade property F*?

6. Yes that sounds about right.

One note: I don’t think its just on my construal that ‘fusing to an F’ and ‘being an F’ are the same property. Since ‘fusing to’ is just ‘identical to’, ‘fusing to an F’ is just ‘being identical to an F’, which seems to be just logically equivalent to ‘being an F’.

As for the accusation that I’ve left property identity unexplained, I’d say a similar thing. F and F* are the same property by the argument above. So property identity can be explained in terms of many-one identity, which is already taken for granted.

7. Interesting post, but I’m a bit confused about several issues. You say you’re writing $xx$ for $x^*$, but then you write things like $xx^*$. Does that mean $(x^*)^*$? So we have starred versions not just of singular variables, but also of plural variables? How should I understand a starred plural variable? My guess is that plural-* is the inverse of singular-* (so $(x^*)^*$ is $x$), but I’m not really sure I’m following.

A related question: do we have two sorts of variables here, or are there more sorts, or are the variables not really sorted at all?

Finally, is extended identity transitive? The table’s atoms are *identical to the table, and the table is identical* to the legs and surface; do you also want to say that the atoms are *identical* to the legs and surface? If so, how do you express that they are different pluralities?

8. Thanks Jeff. Yeah that’s latex screwing me over. I wanted the * to prepend to =, rather than append to xx. (I tried using some \!’s to make it look right, but its not good.)

Good question about the sorts. I obviously haven’t worked any of this out in detail, but I intend you to be able to say stuff like $\forall xF^*xx$, and $\forall xx(x=^*xx)$. So really x and xx are the same variable, and can be bound by either. But the formation rules won’t allow Fxx, or F*x etc…

Lastly, yes it’s supposed to be transitive. Like I said to Aaron, the only way I can make sense of pluralising, is if xx means “x’s atoms” (so your second identity* isn’t quite right.) But this is touching on why I think the view is incoherent. Thinking about it, Aarons proposal above seems to do better on your example.

9. Ah, this explains another unvoiced confusion I had. So this means you don’t have any (first-order) pluralities of composites; right? (Incidentally, I would have thought if composition-as-identity made sense at all, it would be compatible with gunk. But this version wouldn’t be. Right?)

10. Right. Actually I was going to talk about both those things in the next post.

Ok, so basically, if you do go for the allowing pluralities of composites, I think you get a version of basic law V. But the other horn of the dilemma is the impossibility of gunk… (So now you can see why denying gunk seems more attractive…)

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