We were considering the logical conception of set, according to which a set is the extension of a property. But how are we to understand ‘property’ here? In the last post, I mentioned David Lewis’s well-known theory of properties. If we adopted that theory, which sorts of property would sets the extensions of? The ‘natural’ ones? — no, too few. The ‘abundant’ ones? — too many, it seems, unless we are just to fall back into the combinatorial conception. OK, perhaps Lewis’s isn’t the right choice of a theory of properties! But then what other account of properties gives us a suitable setting for developing a distinctive logical conception of set? Now read on …
Luca does mention the problem just noted about Lewisian abundant properties in his §1.8; but having remarked that this notion of property won’t serve the cause of a logical conception of set, he doesn’t I think offer much guidance about what notion of property will be appropriate. This seems a rather significant gap. (Given a prior conception of sets, we might aim to reverse-engineer a conception of properties such that sets can be treated as extensions of properties so conceived, as in effect Lewis does for his abundant properties: but we are here trying to go in the opposite direction, elucidating a conception of sets in terms of a prior notion of property that will surely itself need some clarification.)
Be that as it may. Let’s suppose we have settled on a suitable story about properties (which will presumably be type-disciplined, distinguishing the type of properties of objects from the type of properties of properties from the type of properties of properties of properties, etc.).
Now on the type-theoretic conception of the universe, the types are incommensurable. As Quine pointed out, this is an ontological division. But, at least on an immediate reading, when the types are collapsed [as in NF] this ontological division is removed: properties (of whatever order) are now objects, entities in the first-order domain. Thus, on this reading, NF becomes a theory of [objectified] properties and ∈ becomes a predication relation, by which a property can be predicated of other objects: x ∈ y is to be read as x has property y.
So the idea is that we in particular are to move from (i) a claim attributing a property P to the object a to the derivative type-shifted claim (ii) that a stands in the membership relation to an object (an extension, or as Luca says an objectified property) associated with the property P.
But how tight is the association between a property and this associated object, the objectified property? The rhetoric of “objectification” might well suggest a non-arbitrary correlation between items of different types (as non-arbitrary as another type-shifting correlation, that between an equivalence relation and the objects introduced by an abstraction principle — prescinding from Caesar problems, it is surely not an accident that the equivalence relation is parallel to gives rise by abstraction to directions rather than e.g. numbers). Luca suggests a different sort of comparison: we can think of the introduction of objectified properties as an ontological counterpart of the linguistic process of nominalization, where we go from e.g. the property-ascribing predicate runs to the nominal expression running. This model too suggests some kind of internal connection between a property and its objectification — after all, it isn’t arbitrary that runs goes with running as opposed to e.g. sitting! If we are going to run with this model(!) then there should similarly be a non-arbitrary connection between the property you have when you run and the object that is its objectivization.
A page later, however, we get what seems to be a crashing of the gears. Luca tells us that sets are objectified properties in the sense of proxies for properties — and
a particular association of properties with objects is arbitrary: there is no reason for thinking of an object as a proxy for a certain property rather than another one.
Really? Well, we don’t want to be quibbling about terminology, but it does still seem to me a bit of a stretch to call a mere proxy an objectification (for that surely does still sound like some kind of internal ontological relationship). If I arbitrarily associate the properties of being red, being blue, and being yellow with respectively the numbers 1, 2, and 3 as proxies, aren’t the numbers more like mere labels? And this now suggests a picture introduced by Randall Holmes in motivating NF: a singleton is like a label for its ‘member’ (different objects get different labels), and a set comprising some objects, having their singletons as parts, is thus like a catalogue of these objects. Now, this conception gives rise to the thought that the resulting set-theoretic truths ought to be invariant under permutations of labels (since labellings in forming catalogues are indeed arbitrary). And then we can argue that, with a few extra assumptions in play, the desired permutation-invariance is reflected by NF’s requirement of stratification in its comprehension principle. For some details, see Ch. 8 of Holmes’s book.
Because Luca also makes the association of sets with properties arbitrary, he too wants a similar permutation invariance of the resulting truths about sets, and so he claims he can too use an argument that this invariance will be reflected by an NF-style theory: “The stratification requirement, far from being ad hoc, turns out to be naturally motivated by the idea that sets are objectified properties.” (Luca’s story seems to have less moving parts than Holmes’s, for on the latter story it seems to be important that sets not only have labels as parts but are themselves labelled. I haven’t worked out whether this matters for Luca’s argument from permutation invariance.)
So where does this leave us? Given the linkage just argued for, Luca can call his picture of arbitrary proxies for properties the ‘stratified conception’, and he writes:
If we accept that there is a sensible distinction between a logical and a combinatorial conception of a collection, this opens up the way for regarding the stratified conception as existing alongside the iterative conception. According to this proposal, the sets – the entities that we use in our foundations for mathematics – are provided by the iterative conception. This conception is often taken to be, and certainly can be spelled out as, a combinatorial conception of collection. By contrast, objectified properties – the entities that we use in the process of nominalization – are provided by the stratified conception. This conception is a logical conception of collection. … [If] the stratified conception is best regarded as a conception of objectified properties, i.e. extensions, it seems possible for the NF and NFU collections to exist alongside iterative sets.
I’m not sure the NF-istes would be too happy about this proposal: their usual view is that the NF universe includes the iterative hierarchy as a part — they just believe in more sets that the ZF-istes, more sets of the same ontological kind (i.e. they don’t see themselves as changing the subject, and talking about something different). But let that pass. What you make of all this will depend in part on what you think of this talk of objectified properties as mere arbitrary proxies. Holmes’s talk of sets-as-catalogues-based-on-arbitrary-labelling does seem a franker version of the same basic conception. Does that make it more or less attractive?
6 thoughts on “Luca Incurvati’s Conceptions of Set, 14”
Thanks, Luca — that seems very helpful!
A few clarifications:
1) I chose to remain relatively neutral on properties since all I need is that properties satisfy second-order Comprehension. However, there is a conception of property in the background, one that fits with the way I describe the property-theoretic hierarchy as including at type n the entities that an nth-order quantifier ranges over. This conception of property is the logical conception of property: a property is whatever is in the range of the higher-order quantifiers. Compare with the logical conception of an object (Dummett, Parsons, etc.), according to which an object is whatever is in the range of the first-order quantifier. It is the need to ask for more that leads to nominalism or the need to construct properties as natural properties. This, of course, goes back to Frege and then Dummett.
This conception of property could perhaps be characterized as an abundant conception (but this doesn’t imply, to stress, that it needs to be spelled out in terms of a combinatorial conception of set). In Section 1.8 I never say that what you describe as a problem for the notion of an abundant property is a problem. Nor do I say that this notion won’t serve the cause of the logical conception. All I say is that the claim that the combinatorial conception is more generous than the logical conception depends on a particular way of understanding the logical conception. Disjunctive properties should be fine with anybody who accepts SOL in its property-theoretic interpretation. In any case, even if one takes the issue I describe in Section 1.8 to be a problem, this does not affect what I do in this chapter since the example of Section 1.8 concerns an *infinite* disjunction, which is a different matter. Second-order Comprehension asserts the existence of properties corresponding to disjunctive conditions, but these conditions must be finitely expressed of course.
2) The paragraph beginning with ‘Now’ that you quote presents one of the two possible understandings of collapsing the property-theoretic hierarchy, but it is the one I reject. I am going Fregean on behalf of the stratified conception: one can’t make a property an object. But, again following Frege, one can *associate* properties with objects, which is the view I go on to describe and endorse on behalf of the NF theorist. (Note that that’s why I am saying ‘Thus, on this reading, NF becomes a theory of properties’ and not ‘Thus, on this reading, NF becomes a theory of *objectified* properties’ as the material in square brackets you’ve added would have me say.)
3) Yes, which object we chose as the proxy for a given property is arbitrary. But it is not arbitrary what we take as belonging to this object depending on the property we have taken it to be the objectification of. So the inference from ‘which object we choose as objectification of the property is arbitrary’ to ‘it is arbitrary which things belong to the objectified property’ fails. All that is needed is that we define membership in the objectified property in terms of the property it is the objectification of. This is done in the usual way (going back to Frege, which I describe in chapter 5 for instance, and which is the way in which Cocchiarella defines it for the results I state): y belongs to x iff there’s an F such that x is the extension of F and y has the property F. So there is a correlation between, say, *runs* and *running*, as required, but it is not given by the object we choose *running* to stand for, but by how membership in *running* is determined (in terms of what runs). Again, all of this goes back to Frege: the extension of F for Frege has no internal structure (although often it is wrongly assumed otherwise). But we can *derivatively* define membership in terms of falling under F.
A note on abstraction principles. I think we might disagree about how they work, but in any case, they do fit with my model. You say: ‘it is surely not an accident that the equivalence relation *is parallel to* gives rise by abstraction to *directions* rather than e.g. *numbers*’. In my view, it is indeed arbitrary which property (Fregean concept) a *particular* object is associated with by, say, HP. For the object could be associated by HP with the property of being a planet just as well as it could be associated with the property of being a day of the week. But this does not mean that it’s an ‘accident’ that the equivalence relation of equinumerosity gives rise to numbers rather than other abstracta since numbers are the things that fall under the concept of number, which was introduced by stipulating HP.
4) About the traditional NF-iste not being happy. We agree on that. But it was never my aim to make the traditional NF-iste happy. What I’m saying is that giving a theory of sets has been the wrong strategy, both in theory and in practice. In theory: one should see NF for what it is, namely a theory of logical collections which does something rather different than ZF (and Quine himself went very close to saying that; it’s only that his qualms about properties prevented him from doing so; and this also fits with his idea that the only intuitive conception is the naïve conception of set and that the best we can do is try to approximate that conception). Also, taking the theory to be a theory of logical collections helps overcome what I take to be rather fundamental problems for NF as a theory of *sets*, which are the entities we use in the foundations of mathematics (that’s what the rest of the chapter is arguing anyway). In practice: as a matter of sociological fact, when I’ve presented this material to audiences, even ZF traditionalists have been willing to consider NF now understood as a theory of logical collections. So maybe the imperialist aims of the traditional NF-iste might have got in the way of the theory getting the credit it deserves. (By the way, the same might be true of the iterative conception. After all, as I say in Chapter 3, one way of tackling the no semantics objection is by not doing the semantics of set theory set-theoretically, but rather using higher-order resources. As I say in the concluding chapter, different conceptions and systems might vary depending on one’s goal. There’s no reason to expect that a theory will be the best suited for tasks as different as providing a foundation of mathematics and providing the semantic values of property-talk.)
5) The comparison with Holmes on labels: what I miss from his account is some reason for even thinking of sets as labels. What I tried to do in this chapter is describe a natural story that leads from the property-theoretic hierarchy to the stratified conception (via the need to have proxies for properties in the first-order domain). Of course, the end of the story is the same, since we need to get to permutation invariance. But that means that the story leads where it should. Also, it is a natural story, not a conclusive argument for the stratified conception. (I take it it should be clear by now that I think that such an argument isn’t readily available for any conception of set anyway.)
6) With regards to the details of the permutation invariance argument. We associate a property with an object, but the properties P we can associate in this way are those that have the following feature: P is had (or not had) by the object it is associated with regardless of which object that is (that is, regardless of the details of the implementation process). Obviously, this lets in self-identity and non-self-identity, but leaves out self-membership and non-self-membership. The Petry-Henson-Forster Theorem tells us that this generalizes so as to give us NF: the properties that have the relevant feature are exactly those picked out by a stratified condition.
Something seems to be going wrong with my comment on this post. Here’s another attempt.
I think you are right that the proxy objects are mere labels, and arbitrary labels as well.
Incurvati likens the proxies to implementations, something I assume is meant to helpful but which I’ve found confusing instead. Consider this from p 176:
“The implementation … should be as independent as possible of the details of the implementation”? If that were standing on its own, rather than in the context of the rest of the chapter, I could guess at what that might be meant to say, in order to make sense, but what it actually seems to be saying is that the implementation should be independent of itself. The usual idea is instead that the implementation can make use of its own details internally but should not let them be visible from outside.
Incurvati doesn’t make an internal / external distinction, and so instead of treating “x ∈ y, read as x has property y” abstractly, he has ∈ still mean set-membership too. Stratification seems to be a way to rescue this by ruling out properties that would ‘misuse’ ∈ by using it for the set-membership of a proxy:
BTW, another thing I find confusing, and which also makes it difficult to talk about what Incurvati is doing, is this chapter’s use of the word “extension”.
Before reading this chapter, I have always been able to understand the “extension” of a property, when sets are involved, as the set of things that have the property. So for the property of being a cat, the “extension” would be a set, each element of the set would be a cat, and every cat would be an element.
But now it seems “extension” is also being used in a different way, in which an “extension” is an arbitrary object that we’re meant to treat opaquely. So the “extension” of ‘cat’ would not be the set of all cats, except by accident. It could just as well be an apple. Even if an “extension” has to be a set, it could just as well be a set of apples, or a set of two figs. In effect, “extension” is this sense can just be a proxy or label.
A propos of the last point. I’d put the worry like this. Supposedly we fix the extension of “cat” as being the object X on the basis of an arbitrary choice. Relative to that choice, X has cats as “members”, and X counts as the set of cats, and similarly another object Y has cherries as “members”, and Y counts as the set of cherries. But we could equally have chosen Y to be the extension of “cats” and then Y would count as the set of cats and would have cats as its members. Which object counts as the set of cats, which as the set of cherries, is up for grabs. And that strikes us a distinctly odd. Given a moggie, and a set, we certainly don’t ordinarily think that it us a matter of arbitrary choice whether the moggie is in the set or not! [Footnote, then: rather than your “So the “extension” of ‘cat’ would not be the set of all cats, except by accident” I’d rather have “So the “extension” of ‘cat’, i.e. the set of cats, would not be the object it is, except by the accident of choice”.]
This is extremely frustrating.
When I wrote my earlier comment, I thought I understood what Incurvati was doing in the last paragraph on p 176, but looking at it again now, I can see that I didn’t.
There must be some clearer way to explain it.
Here’s where I run into trouble. The problem for the condition x ∈ x goes like this:
* Consider the condition x ∈ x and the property it picks out, P.
* Suppose the object a is the proxy for P.
* Then a ∈ a would be true because … ?
* But if a were the proxy for a different property, a ∈ a might not be true.
What goes in the “…”?
I could see x ∈ a meaning x has the property, and so x ∈ x. But why would x = a mean x has the property?
I can imagine that a just happens to contain itself as a member, but then it plainly does have the property and ought to have it regardless of what object was the proxy.
So what about a being the proxy means that a ∈ a?
Distinguishing between “set” and “member” in quotes and those words without quotes (or marked in some other way) results in something easier to understand. It sets up a distinction similar to the internal / external distinction that applies when implementing data types abstractly.
If Z is the proxy / implementation for ‘cat’, then Z can be regarded as the “set” of cats and has cats as “members”; however, Z might not have cats as members. Z might not even be a set.
The “extension” of ‘cat’ would be the “set” of all cats; but it would not be the set of all cats, except by accident. And when asking whether Mog was a “member” of the “set”, we wouldn’t answer that by looking at the actual members of Z (if Z has any). Instead, we’d just ask whether Mog is a cat.
(Indeed, we don’t look at anything at all about Z itself, which is why it makes more sense to see Z as a label or proxy than as an implementation.)
However, Incurvati can’t use that sort of distinction, because he needs to have one notion of set and member, not two.
“This is extremely frustrating.” I share your frustration …!