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Luca Incurvati’s Conceptions of Set, 13

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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. . .

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News source: Logic Matters

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