Armstrong, Quine, Universals, Abstract Objects, and Naturalism

A Serbian reader inquires,

I have read your latest post on truthmakers. Among other things, you mention [David] Armstrong's view on abstract objects. As I read elsewhere (not in Armstrong own works, I have not read anything by him yet) he was realist about universals and gives a very voluminous defense of his view. Does this view entail realism about abstract objects?

I think that Quine was realist about abstract objects and at the same time naturalist and also holds that his Platonism was consequence of his naturalized ontology.  Moreover, I have the impression that several preeminent analytic philosophers hold realist views on abstract objects, mostly under influences from Quine and in a smaller degree from Putnam.

Do Armstrong's views about universals entail realism about abstract objects?

No, they do not.  Rejecting extreme nominalism, Armstrong maintains that there are properties.  (I find it  obvious that there properties, a Moorean fact, though I grant that it is not entirely obvious what is obvious.)  Armstrong further maintains that properties are universals (repeatables), not particulars (unrepeatables) as they would be if properties were tropes.  But his is a theory of immanent universals.  This means two things.  First, it means that there are no unexemplified universals. Second, it means that universals are constituents of the individuals (thick particulars) that 'have' them.  In Wolterstorff's terminology, Armstrong is a constituent ontologist as opposed to a relation ontologist.  His universals are ontological parts of the things that 'have' them; they are not denizens of a realm apart only related by an asymmetrical exemplification tie to the things that have them. 

So for Armstrong universals are immanent in two senses: (a) they cannot exist unexemplified, and (b) they enter into the structure of ordinary (thick) particulars.  It follows that his universals are not abstract objects on the Quinean understanding of abstract objects as neither spatial nor temporal nor causally active/passive.  For given (b), universals are where and when the things that have them are, and induce causal powers in these things.  And yet they are universals, immanent universals: ones-in-many, not ones-over-many.  Some philosophers, including Armstrong, who are not much concerned with historical accuracy, call them 'Aristotelian' universals.

Does Armstrong reject all abstract objects?

Yes he does.  Armstrong is a thorough-going naturalist.  Reality is exhausted by space-time and the matter that fills it.  Hence there is nothing outside of space-time, whether abstract (causally inert) or concrete (causally active/passive).  No God, no soul capable of disembodied existence, or embodied existence for that matter, no unexemplified universals, not even exemplified nonconstituent universals, no Fregean propositions, no numbers, no mathematical sets, and of course no Meinongian nonenties. 

How do Armstrong and Quine differ on sets or classes?

For Quine, sets are abstract entities outside space and time.  They are an addition to being, even in those cases in which the members of a set are concreta.  Thus for Quine, Socrates' singleton is an abstract object in addition to the concrete Socrates.  For Armstrong, sets supervene upon their members.  They are not additions to being.  Given the members, the class or set adds nothing ontologically.  Sets are no threat to a space-time ontology.  (See D. M. Armstrong, Sketch for a Systematic Metaphysics, Oxford UP, 2010, p. 8.)

What about the null set or empty class?

For Armstrong, there is no such entity.  "It would be a strange addition to space-time!" he blusters. (Sketch, p. 8, n. 1).  Armstrong makes a bad mistake in that footnote.  He writes, "Wade Martin has reminded me about the empty class which logicians make a member of every class."  Explain the mistake in the ComBox.  Explain it correctly and I'll buy you dinner at Tres Banderas. 

Are both Quine and Armstrong naturalists?

Yes.  The Australian is a thorough-going naturalist: there is nothing that is not a denizen of space-time.  The American, for reasons I can't go into, countenances some abstract objects, sets.   It is a nice question, which is more the lover of desert landscapes.

Related articles

Constituent Ontology Versus Relational Ontology and an Argument Against the Latter

Properties as Parts: More on Constituent Ontology

Relational Ontology, Constituent Ontology, and Divine Simplicity

What Exactly is an Ontological Constituent?

Comments

[Peter Lupu]

Two Blunders:

I suppose that Armstrong means here that the "empty set" belongs to every "set" and not class. The mistake here is to conflate classes with sets. While all sets are classes, the converse is not the case. Since a class is any collection of objects, whereas a set is not, conflating sets with classes will lead directly to Russell's paradox. Thus, if all classes were sets, then the set (collection) of all sets that are not members of themselves would be a set. But, of course, this will lead to the conclusion that this collection: i.e., set, is both a member of itself and it is not a member of itself. Therefore, not all classes are sets and Russell's class is not a set.

Second, it makes no sense to say, as Armstrong says in the above quotation, that logicians "make" the empty set a member of every set. I am not even sure what could such a phrase mean. It is like saying that mathematicians "make" the *number* two even; as if they could have alternatively "make" the *number* two odd.

[John]

The empty class is not a member of every class; it is a subset of every class.

[John]

I should have elaborated on my previous comment. The empty class is not a member of every class; it is a subset of every class. This is easily demonstrated. The set of all apples does not have the empty class as a member. Its only members are apples. But just as the set of Granny Smith apples is a subset of the set of all apples, so too is the empty set a subset of the set of all apples. Set membership is not the same relation as the subset relation. The latter is, for example, transitive; the former is not.

[G. Rodrigues]

"Armstrong makes a bad mistake in that footnote. He writes, "Wade Martin has reminded me about the empty class which logicians make a member of every class." Explain the mistake in the ComBox."

I am even afraid of writing this, as the mistake is elementary enough that I suspect that the *real* one is located elsewhere (I do not have Armstrong's book by me, so cannot check).

If the empty set, denoted by 0, were a member of every set, it would also be a member of itself, and therefore 0 would have at least one member and be non-empty. Contradiction. Ergo 0 cannot be a member of every class.

The empty set is not a member of every set (for every x, 0 e x) but rather is contained in every set (for every x, 0 <= x). The membership and containment relations are very different. For one, the latter is a partial order, therefore it is transitive, but the membership relation is not transitive.

[BV]

John,

Exactly right.

G. R.,

You are quite right as well. And I like your reductio.

Peter,

The context is very informal, so no distinction is being made between classes and sets. Both of your points are good, even if the second is a bit pedantic, but I think you missed the really glaring mistake pointed out by John and G. R.

Of course, Armstrong is well aware or perhaps I should say was well aware -- he died recently -- of the distinction between the membership relation and the subset relation. He was just having a 'senior moment.'

[Nigel PJ]

Bill, I enjoy reading your blog. I've felt like making a comment or two to some of your posts however very few of them have a comment box. Is that intentional? As you asked for comments on one post, and there was no available place for remarks, that seems unlikely. I'm putting this message in the most recent post I can find that accepts comments. Im using Chrome but have tried IE.
Nigel

[BV]

Nigel,

Yes, it is intentional. You can shoot me an e-mail if you like. Thanks for your interest.

[Nigel PJ]

Bill,
Hmmm...obtuse man looks for an email link but finds none.
Nigel

[BV]

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