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?