Luca Incurvati’s Conceptions of Set, 13

Among other things, I need to get more answers to the exercises in IFL2 online before publication, and that’s a ridiculously time-consuming task, which is no doubt why I’ve been rather putting it off! Doing some of the needed work partly explains the hiatus in getting back to Luca’s book. But there’s another reason for the delay too. I’ve found it quite difficult to arrive at a clear view of the second half of his Chapter 6 on NF. However, I must move on, so these remarks will remain tentative:

Early on, in §1.8, Luca distinguished what he called logical and combinatorial conceptions of set. And now in §6.7, he tells us that NF can be treated as a theory of logical collections.

It is a familiar claim that some such distinction between logical and combinatorial collections is to be made. And it seems tolerably clear at least how to make a start on elaborating a combinatorial conception: the initial idea is that, take any objects, however assorted and however arbitrarily selected they might be, they can be combined to form a set with just those objects as members. And then it is reasonable to argue that the iterative conception of set is a natural development of this idea. It is much less clear, however, even how to make a start on elaborating the so-called logical conception. Let’s pause over this again before turning to the details of §6.7.

In §1.8, Luca suggests “Membership in a logical collection is determined by the satisfaction of the relevant condition, falling under the relevant concept or having the relevant property. Membership is, in a sense, derivative: we can say that an object a is a member of a [logical] collection b just in case b is the extension of some predicate, concept, or property that applies to a.” But are there going to be enough actual predicates (linguistic items) to go around to give us the sets we want? Which language supplies the predicates? If we say ‘a logical set is the extension of a possible predicate’ then we are owed an account of possible predicates — and in any case, this doesn’t really seem to tally with the idea of membership as a derivate notion: the picture now would rather seem to be that here already are the sets with their members, and they are (as it were) waiting to be available to serve as extensions for any predicates that we might care to cook up in this or that possible language.

So maybe we need to concentrate on the concepts or properties? The notion of a concept here is slippery to say the least, so let’s think about properties for a bit. How plentiful are properties? We don’t want to get too bogged down in metaphysical discussions here, but for orientation let’s recall that David Lewis famously makes a distinction between the sparse natural ‘elite’ properties (which can appear in laws, where sharing such a property makes for real resemblance, etc., etc.) and the abundant non-natural properties (where Lewis explicitly explains that for any combinatorial set of actual and possible objects, however gerrymandered, there is an abundant property, namely the property an object has just in case it is a member of the given set). Now, taking logical sets to be the extensions of properties in an abundant sense which is anything like Lewis’s will just collapse the supposed logical/combinatorial distinction.

But on the other hand, taking logical sets to be always the extensions of properties in some narrower sense would again seem to be in danger of giving us too few sets. It is a common argument, for example, that x’s being, as we might casually say, F or G is really just a matter of x’s being F or x’s being G — the world doesn’t contain, as part of its furniture, as well as the property F and the property G the disjunctive property F-or-G. In other words, even if F and G are real properties which have logical sets as extensions, there is no real property F-or-G to have the union of those sets as its extension. Drat! So perhaps we do want to be thinking in terms of predicates after all, since we can apply Boolean operations of predicates (and get unions and intersections of extensions) in a way we can’t apply them to natural properties (at least on some popular and well-defended views). But defining sets in terms of predicates wasn’t looking a great idea …

The nagging suspicion begins to emerge that the idea that we can (i) characterize a logical set as “the extension of some predicate … or property” while (ii) not collapsing the idea of a logical set into the notion of a combinatorial set depends on cheating a bit by blurring the predicate/property distinction. We need something predicate-like to give us e.g. the Boolean operations; we need something sufficiently natural-property-like to avoid the unwanted collapse. So what’s the honest story about logical collections as extensions going to be? Let’s see what more Luca has to say!

To be continued

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