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Luca Incurvati’s Conceptions of Set, 14

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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,. . .

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News source: Logic Matters

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