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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 , ,

9 Responses

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  1. Hi Jeff,

    I think it’s a bit confusing to call that relation “coextensiveness” – which is usually used to mean “has exactly the same members” (abstracting this relation gets you BLV which is inconsistent.)

    By coextensive you mean some mereological notion. The RHS of (1) therefore contains mereological terminology so it seems you don’t get a reductive account of fusions in the same way as, say, you’d get with numbers with HP, or with the composition as identity thesis – which is what I thought you where going for.

    (BTW, I like the idea that abstraction can be thought of in terms of arbitrary reference. How would you cash out “a is red” supervaluationally though? As \neg \Delta \neg a = red?)

    Andrew Bacon

    April 27, 2009 at 8:20 am

  2. Hmm, actually, I thought the point of doing arbitrary reference supervaluationally would be so you could say stuff like (letting a be an arbitrary American): “a is American” is supertrue, “a is Obama” comes out indeterminate and “a is Gordon Brown” superfalse. This thought doesn’t seem to transfer easily to the “red” case.

    Andrew Bacon

    April 27, 2009 at 8:27 am

  3. Hi Andrew,

    Thanks for the comments.

    I think it’s a bit confusing to call that relation “coextensiveness”

    You’re right that “extension” is a pretty overloaded word, but I think it does standardly have this use in mereology—that’s why mereologies that affirm strong supplementation are “extensional”. But if you don’t like it, we could use “total overlap” instead.

    The RHS of (1) therefore contains mereological terminology so it seems you don’t get a reductive account of fusions in the same way as, say, you’d get with numbers with HP, or with the composition as identity thesis – which is what I thought you where going for.

    You’re right that, since I’m taking overlap as primitive here, I don’t get a complete conceptual reduction of mereology. But I still get an ontological reduction of fusions.

    I wonder, though, do you really get conceptual reduction from the composition-as-identity principle either? Say you state the claim in the most obvious way: “the fusion of the Xs is the Xs”. Either you mean “fusion” in its standard sense, in which case this takes mereological concepts for granted, or else you treat the principle as its definition, in which case I don’t see how you’re going to derive interesting consequences from it. Is that wrong?

    I suspect there’s a way of cleaning my proposal up so that it doesn’t take overlap as primitive. (This should also independently motivate the principle that Fus x = x.) One way to go would be to assume that the “base ontology” (before abstraction) is a bunch of atoms, and the “base ideology” is singular overlap, which for atoms is just identity. With a rich plural logic and the abstraction principle, we then derive the rest of mereology. But what I said in the main post should be (I think) compatible with gunk.

    How would you cash out “a is red” supervaluationally though?

    I didn’t have any aspirations to analyze predication in terms of abstracta; that gets it backward, since certain things being red is the base we abstract from. Am I misunderstanding the question?

    Hmm, actually, I thought the point of doing arbitrary reference supervaluationally would be so you could say stuff like (letting a be an arbitrary American): “a is American” is supertrue, “a is Obama” comes out indeterminate and “a is Gordon Brown” superfalse. This thought doesn’t seem to transfer easily to the “red” case.

    I wasn’t thinking that was the point of an account like this (though I also don’t see it as a demerit to say that “Redness is this apple” is indeterminate, or that “Red is brightly colored” is determinately true). I figured the account was mainly motivated by its metaphysical tidiness. I have a secondary motivation from ordinary language in the case of numbers, but that would take some time to explain.


    April 27, 2009 at 1:15 pm

  4. Ah, that was another question I wanted to ask. I like the thought that if we started off with a nihilist parthood relation, P, we could extend it to a univeralist parthood relation P* by adding abstracted objects. That *would* be reductive in the sense that HP is, because as you said, overlap could be replaced by identity. But it wouldn’t collapse into BLV provided the RHS quantifiers were restricted to fundamental objects (those not created by abstraction, atoms in this case.)

    Using the same notion of parthood on both sides of the abstraction principle seemed to me to already beg the question against nihilists, since your principle could only hold if the parthood relation in question was already universalist. If it’s just a reformulation of the standard mereological axioms, I see no reason to say fusions are abstractions than I would in the original case when the axioms werent rearranged. (I’m not sure that being compatible with gunk is a decisive reason to favour if we’re concerned about whether there are fusions. Given weak supplementation, gunk already entails the existence of fusions.)


    April 28, 2009 at 1:25 pm

  5. I just realised you don’t even need weak supplementation to see gunk entails the existence of fusions (if x is gunky, and the xx are x’s parts, then x is a (possibly not unique) fusion of the xx’s.)

    Regarding the predication thing. You said that “red” referred indefinitely to each red obect. So it seems you won’t have many options when trying to say what the truth conditions of “a is red” are, given you’ve specified how “red” is to be interpreted.


    April 28, 2009 at 1:36 pm

  6. I don’t think I understand your objection. (Is it an objection?) I didn’t think I did use “the same notion of parthood on both sides of the abstraction principle”: the symbol “Fus” wasn’t supposed to encode anything mereological; its meaning is supposed to just be fixed in terms of abstraction. Can you spell out what you mean?

    I don’t think gunk ensures the existence of fusions in the important sense. If by “the existence of fusions” you just mean “something is a fusion”, then it’s trivial, since everything is a fusion of itself by definition. But if you mean unrestricted composition, then gunk, even with supplementation principles, won’t ensure it: there could be two gunky objects that don’t have a fusion. Did you mean something else?

    On predication: the word “red” is ambiguous between a predicate and a name. (I would have been better off using “redness” for the latter.) I take the predicate use as primitive, and explain the name use by way of indefinite reference. But the indefinite reference stuff makes no difference when it comes to predication, as in “a is red”—that’s the primitive predicate’s job, not the name’s.


    April 29, 2009 at 7:12 pm

  7. I assume you accept the principle that x overlaps Fus(xx) iff it ovelaps one of the xx’s? If so you’re abstraction principle isn’t conservative over the language of pure parthood since it implies unrestricted fusion (the principle stated purely using parthood and not Fus.) I.e. it doesn’t just add abstracted objects to the domain, but it also forces the initial parthood relation to change to be universalist as well. (For example, a nihilist would presumably reject this, even if he were ok with abstracted objects.)

    But more generally, suppose I rearrange my mereological axioms (perhaps adding new primitives as well, like you suggest.) What is it about doing this that justifies me in saying fusions are abstractions? It can’t be that I can rearrange them into the form of an abstraction principle, as people have shown you can do this to almost anything. If we can distinguish between the fundamental objects and the fundamental parthood relation and the abstracted objects and the extended parthood relation, we’re in a position to make this distinction.

    The question of whether there are fusions is what is at stake, for example, in the nihilism debate – i.e. whether two or more things ever fuse. (I’m not sure either side of this debate would reject the existence of one object fusions, but it’s still a substantial debate.) The question of whether there are fusions is, of course, different from the question of whether there are always fusions.


    April 30, 2009 at 9:13 am

  8. Hi Andrew,

    Sorry for the hiatus.

    Yes, the principle that x overlaps Fus X iff it overlaps an X is equivalent to my principle 2. I was sort of thinking of Principle 2 as a rule for extending mereological relations to apply to Fus X. One way to bear down on this is to think of it as constructing a new relation P* from the old parthood relation P. But if you think about abstraction my way, that isn’t the only way to go. If “Fus X” refers indefinitely to things that bear parthood relations, then Fus X also bears parthood relations in just the same sense of “parthood”—in particular, it bears the relations given by Principle 2.

    So I have two ways of thinking about this. The clean way is the one I think we both are happy with, where nihilism is the fundamental story and everything else comes from iterative abstraction and extensions of the parthood relation. In the hazier picture, everything is “created all at once”, so to speak, but the abstraction principle still tells us about the nature of composites. I’m not totally sure the hazier picture is tenable, but I want to hang on to it as long as I can.

    (Do you know what neo-Fregeans generally say about fundamentality? Do they think of all the numbers as “created all at once”, or does Hume’s Principle induce a kind of hierarchy of basicness, where, say, 1 is grounded in 0, etc.? Certainly that’s got to be the picture for abstractionist set theory, I would think.)


    May 9, 2009 at 8:56 am

  9. Hi Jeff,

    Yeah – I think that’s pretty much how I see things right now.

    The thing to get clear on next, in my view, is what exactly it is about the second view that does better than ordinary universalism. You could imagine a conversation between a universalist and this kind of nihilist. They’ll both agree about every sentence in the language of pure parthood – they both believe in the fusion of alpha centuri and my left hand, trout-turkeys, and everything that is so counterintuitive about universalism. And they’ll both insist that it’s *genuinely* the same notion of parthood that holds in both cases. I find it hard to see how these views differ. The nihilist will insist that fusions are “abstractions” – but unless she can make this difference sound substantial (and more importantly, show how makes all these weird claims about trout-turkeys sound better) I can’t see a reason to take her seriously. As far as I can see “abstraction” is just philosophical jargon – and I can’t see how to views differing only in jargon language can be substantially different.

    Interesting question about neo-Fregeanism. The standard view, I think, is the ‘all at once’ picture. I guess this is why people often object that you’re getting something for nothing, and feel uncomfortable about the impredicativity. (A relevant difference might be that, unlike the mereological case, you can’t generate all the numbers in one iteration.)

    Andrew Bacon

    May 10, 2009 at 10:43 am

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