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Composition as abstraction

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“The whole is nothing over and above the parts.” This is a nice thought, but it turns out to be difficult to make precise. One attempt is the “composition as identity” thesis: if the Xs compose y, the Xs are y.

This won’t work, at least not without great cost. The United States is composed of fifty states, and it is also composed of 435 congressional districts. If composition is identity, then the U.S. is the states, and the U.S. is the districts; thus the states are the districts. This is bad: composition-as-identity collapses mereologically coextensive pluralities, which means now your plural logic can be no more powerful than your mereology. So you lose the value of even having plural quantifiers. That’s a big sacrifice. (This argument is basically Ted Sider’s, in “Parthood”.)

But the problem here isn’t that the fusion of the Xs is something more than the mere Xs: rather, the fusion is something less. Mereological sums are less fine-grained than pluralities, so if we require each plurality to be identical to a particular sum, we lose the (important!) distinctions that plural logic makes.

This suggests a better way: mereological sums are abstractions from pluralities. Roughly speaking, sums are pluralities with some distinctions ignored. In particular, sums are what you get by abstracting from pluralities on the relation of being coextensive. (Analogously: colors are what you get when you abstract from objects on the same-color relation. Numbers are what you get when you abstract from pluralities on equinumerosity.)

Let’s polish this up a bit. Take overlap as primitive, and define parthood in the standard way:

  • x is part of y iff everything that overlaps x overlaps y.

This has a natural plural generalization:

  • The Xs are covered by the Ys iff everything that overlaps some X overlaps some Y.

Parthood is the limiting case of being covered when there’s just one X and one Y. (I’ll identify each object with its singleton plurality.) We can also define an equivalence relation:

  • The Xs are coextensive with the Ys iff the Xs cover the Ys and the Ys cover the Xs.

Now we can state an abstraction principle. Let Fus be a new primitive function symbol taking one plural argument.

  1. Fus X = Fus Y iff the Xs are coextensive with the Ys.

(Compare Hume’s Principle: #X = #Y iff the Xs are equinumerous with the Ys.) This is the main principle governing composition. It isn’t the only principle we’ll need. For all I’ve said so far, fusions could live in Platonic heaven; but we need them to participate in mereological relations:

  1. The following are equivalent:
    1. The Ys cover the Xs.
    2. The Ys cover Fus X.
    3. Fus Y covers the Xs.

This guarantees that Fus X really is the fusion of the Xs by the standard definition of “fusion”. There is one final assumption needed to ensure that our mereology is standard:

  1. Parthood is antisymmetric. (If x is part of y and y is part of x, then x = y.)

Equivalently: Fus x = x. In the singular case, composition really is identity.

These three principles imply all of standard mereology. So just how innocent are they?

I think they’re fairly innocent, given the right conception of how abstraction works. I like a “tame” account of abstraction which doesn’t introduce any new ontological commitments. (This means tame abstraction is too weak for Frege arithmetic or for Frege’s Basic Law V—this is a good thing.) The basic idea is that abstract terms refer indefinitely to each of their instances. For example, the singular term “red” refers indefinitely to each red thing: we consider all red instances as if they were a single thing, without being specific as to which. (Semantically, you can understand indefinite reference in terms of supervaluations.) Red has the properties that all red things must share. E.g., if any red thing must be rosier than any taupe thing, then we can also say that red is rosier than taupe. Speaking of red doesn’t commit to any new entity—it’s just speaking of the old entities a new way.

As for colors, so for fusions. “The fusion of the Xs” doesn’t refer to some new exotic thing: it refers indefinitely to each plurality coextensive with the Xs. You could say it refers to the Xs, as long as you don’t mind the difference between coextensive pluralities. Furthermore, since whenever the Xs are coextensive with the Ys they stand in exactly the same covering relations, Principle 2 is justified.

Principle 3, on the other hand, is not entirely innocent. Given the definition of parthood, it amounts to extensionality: no distinct (singular) objects are coextensive. I think it’s right to consider this a separate, serious commitment, one that (unlike the rest of mereology) doesn’t flow from the mere conception of a mereological sum. It might, however, flow from the conception of an object. If you aren’t too worried about speaking completely fundamentally, antisymmetry can be had cheaply, by considering “objects” to be coextension-abstractions from the basic objects, in just the same way that sums are coextension-abstractions from the basic pluralities.

So, indeed, the whole is nothing more than its parts. It can’t be identified with any particular plurality of its parts, but it can be identified indefinitely with every plurality of its parts.

[There’s a technical issue for the semantics I’ve alluded to here. I’m treating Fus X as semantically plural (it refers indefinitely to pluralities), but it is syntactically singular. In particular, as a singular term it can be ascribed membership in pluralities. But this means that I need the semantics to allow pluralities to be members of pluralities—and so on—and this isn’t ordinarily allowed. So it looks like I’ll need to give the semantics in terms of “superplurals”. (See section 2.4 of the SEP article on plural quantifiers.) Whether this semantic richness should be reflected in the language is a separate issue—I’m inclined to think not, but I haven’t really thought it through.]

Written by Jeff

April 25, 2009 at 12:21 pm

Posted in Logic, Metaphysics

Tagged with , ,