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David Lewis' conception of abundant properties which he identifies with sets/classes ("no matter how oddly defined") is not quite clear to me. Since according to this position there cannot be - in no possible world - two different properties with the same extensions (according to the extensionality principle). Especially there cannot be two different properties that adhere to no object (resulting in the empty set).

I am sure that Lewis did not oversee this problem. So what is its solution?

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    Is your question: "Is the empty set uniquely defined in Lewis' theory?" – Xodarap May 5 '12 at 20:00
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    Among others. My question is "Is a property uniquely defined by the objects that have that property?" If a property is identified with the set/class of objects that have this property, this cannot go together with the extensionality principle by which every set/class is determined by the objects that belong to it. – Hans-Peter Stricker May 5 '12 at 21:41
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[A]ccording to this position there cannot be - in no possible world - two different properties with the same extensions (according to the extensionality principle).

The first thing to note is that the extensionality principle does not depend on the abundant understanding of properties: any other, however sparse, conception should, according to Lewis, likewise abide by the extensionality principle.

Notice that the objects that can be members of properties need not be actually existing. They can also be mere possibilia. This goes some way towards alleviating the problem you raise: what distinguishes two uninstantiated properties, say, Being a golden mountain and Being a flying man? Well, the first has (merely possible) golden mountains as members; the second (merely possible) flying men.

Another, more pressing problem is that of hyperintensionality: how do we distinguish the properties Being the largest prime and Being such that 2+2=5? Not in terms of possibilia, or actual entities: both these properties are empty in every possible world. The possible-world treatment of hyperintensionality is, as you say, unsatisfactory as it stands. There are a number of solutions, non of them particularly Lewisian:

  • Postulating the existence of impossible worlds. Some of those will contain a largest prime. Some, different ones, will make it true that 2+2=5.

  • Modelling hyperintensional properties as not just a set of entities, but as a pair of a set of entities and something like a syntactic tree. Thus, the two necessarily uninstantiated properties above would be associated to the same set, but with different syntactic trees (one would have "largest" and "prime" at some of its nodes, the other would have "2" and "equals" at some of its nodes.

In any event, probably the right way for a Lewisian to think of these things is as follows: there is only one necessarily uninstantiable property, and the hyperintentional differences should be thought not as metaphysical, but as merely semantic: we should postulate intentional (with a "t") objects of thought that allow us to distinguishbetween thinking, or talking, about largest primes and about entities such that 2+2=5. The objects of thought, according to this suggestion, cannot be properties.

Finally, in your ensuing comment you seem to identify a circularity problem in Lewis's extensionality principle:

If a property is identified with the set/class of objects that have this property, this cannot go together with the extensionality principle by which every set/class is determined by the objects that belong to it.

There is no circularity: a property is individuated by a set. A set is individuated by its members. There would only be circularity if a set's members were individuated by their properties, but they are not.

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