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

(you must also add suitable identity axioms such as , , 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 is parthood, we have the following definition

where is the ‘is one of’ relation from plural logic. Thus is defineable in purely logical vocabulary, so if 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 iff or … or , 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 . 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…