FOM: Axiom of Extensionality

[Charles Parsons]

>What is this axiom?  Does it say that a set A is the same as set B iff they
>have the same members?
>
>But then, if a set is something apart from its members, and perhaps defined
>by properties they have in common, how is that possible?  There is a set of
>unicorns, which is empty.  If there were unicorns, would this be a different
>set?  But then how could both be the set "of unicorns"?
>
>Or take the set {Alice, Bob, Carol}.  Does this continue to exist even if
>they cease to?  Is it that it currently has the number 3, but the number 0
>if they do not exist?
>
>Either (you would think) the set is one and the same with its members, in
>which case the axiom reduces to {Alice, Bob, Carol} = {Alice, Bob, Carol},
>and hardly seems necessary.
>
>Or it is something different, which raises the possibilty of its changing
>its membership, while remaining the same.  E.g. the set of people who live
>in Spencer Walk, which changes slowly through the years, bit by bit, but
>which (presumably) remains the same set.  In which case the axiom is clearly
>false.
>
>Or is the formulation I have used incorrent?  I have seen a number of
>different ones, and they are all subtly different.  One says that "two" sets
>are one and the same iff...  what?  Iff they are not two sets after all?
>

Dean Buckner's remarks raise questions about talk of sets in modal 
and other intensional contexts, a matter about which there is some 
literature which neither he nor others who have commented seems to 
know.

I would recommend particularly Kit Fine, "First-order modal theories 
I: Sets," Noûs 15 (1981), 177-205. The theories Fine develops employ 
three principles that came to be widely accepted:

(1) x e y -> N(x e y)
(2) ¬(x e y) -> N ¬(x e y)
(3) (Ey & x e y) -> Ex.

(I use 'e' for membership, 'N' for necessity, and 'E' for existence. 
(3) only naturally makes sense in the context of a free logic.)

Exposition of the ideas involved, and citations of some other 
literature up to the time, are to be found in my book _Mathematics in 
Philosophy_ (Cornell, 1983), pp. 298-308.

Now assuming it to be possible that there might have been unicorns, 
then "the set of unicorns" is not a rigid designator. For (2) implies 
that the empty set is necessarily empty. But "the set of unicorns" 
designates the empty set, but in a possible world containing 
unicorns, it designates some non-empty set.

(The assumption is not obviously true; see the well-known remarks of 
Saul Kripke, _Naming and Necessity_ (Harvard, 1980), pp. 156-58.)

About {Alice, Bob, Carol}, (3) implies that if one of them does not 
exist then the set does not exist either.

"the set of people who live in Spencer Walk" is not a (temporally) 
rigid designator, for essentially the same reasons for which "the set 
of unicorns" is not (on the above assumption). The tense-logical 
equivalent of (1) and (2) implies that the set of people who live on 
Spencer Walk is no longer the same set when someone has moved in or 
out.

That the axiom of extensionality is not redundant is evident once we 
distinguish sets from properties or attributes, which do not obey 
extensionality.

Charles Parsons