Tuesday, February 7, 2017

Counterparts and singletons

Here is an interesting problem for Lewis. Lewis says that sets are necessary beings, and hence count as existing in all worlds. Very plausibly then:

  1. If A is a set and w1 and w2 are worlds, then A in w2 is a counterpart of A in w1.

After all, if identity isn’t good enough for being a counterpart, nothing is. Note that (1) does not say that A at w2 is the only counterpart of A in w1. To handle some identical twin scenarios, Lewis may need to allow a world to have more than one counterpart of an object.

Let α be Aristotle. Let A = {α} be the singleton of α. Lewis is now committed to the truth of:

  1. Possibly α is not a member of A.

For Lewis’s criterion for whether F(a, b) is possible is whether there is a world w with counterparts a′ and b′ of a and b respectively such that F(a′,b′) holds at w. Let w be a non-actual world where there is a counterpart β of Aristotle. Since individuals are world-bound, β ≠ α. Moreover set membership is necessary, so:

  1. β is not a member of A at w.

Since β is a counterpart of α and A is a counterpart of A by (1), it follows that (2) is true. But (2) seems clearly wrong: it is impossible for Aristotle not to be a member of A.

Here’s what seems to me to be the best way out for Lewis: Require pairwise counterparts rather than individual counterparts (in fact, I vaguely remember that Lewis may do that somewhere) for possibility claims involving two objects. Thus that β is not a member of A and β and A are individually counterparts of α and A isn’t enough to make it be that possibly α is not a member of A. One would need β and A to be pairwise counterparts of α and A. But perhaps they’re not. Perhaps, rather, it is β and B = {β} that are pairwise counterparts of α and A. However, this greatly complicates the counterpart relation as well as Lewis’s identification of properties with sets.


Unknown said...

Why wouldn't Aristotle in our world just be a member of A in every world?

It seems like cross world membership would be open to the modal realist.

wo said...

Yes, Lewis invokes pairwise counterparts in the Postscript to the 1968 paper to deal with somewhat similar problems involving relations between individuals.

Another way out is to deny 1 and say that on one salient resolution of the counterpart relation, the unique counterpart of {a} at w is {b}, if b is a's unique counterpart at w. I think Lewis makes a suggestion along these lines in "Counterparts of states of affairs". (To what extent this conflicts with treating counterparthood as a matter of qualitative similarity might depend on what one says about sets. On Lewis's 1994 account, {a} and {b} are primitive abstract objects that don't have any intrinsic qualitative properties at all. Things are trickier on the 1998 account where talk about sets is ultimately eliminated. In any case, it's not really important for Lewis that counterparthood is always a matter of similarity; at some point, he allows one individual to be a counterpart of another individual at the same world, even though the individuals are qualitatively distinct.)

A third way out is to hold that modal talk about sets does not get a counterpart theoretic interpretation in the first place. In the 1968 paper, counterparts must be "in" worlds, and to be "in" a world is to be a mereological part of a world, which sets are not.

Alexander R Pruss said...


I don't think saying that modal talk doesn't get a counterpart theoretic interpretation is a good way out. Sentences like "Possibly, a is not a member of A" become true, and sets now have their elements accidentally. That seems implausible. It also means that properties--since properties can be identified with sets--vary in what falls under them across worlds.

Now, taking {b} to be the counterpart of {a} (either simpliciter or at least when dealing with modal sentences that also use "a") requires quite a lot of adjustment of counterparts of other sets throughout the set-theoretic hierarchy (e.g., the counterpart of { {{a}, 3}, 4 } will sometimes be { {{b}, 3}, 4 }). And sometimes we will have sentences that not only have {a} in them but also {b} and other sets that bottom out in a AND b, for instance: "Necessarily, a is not a member of {a} and {b} is disjoint from {a} and both {a} and {b} are proper subsets of {a,b}."

Maybe the way to do this is that when one needs to take {b} as a counterpart of {a}, one swaps a with b everywhere in the set-theoretic hierarchy (and in proper classes, too). Except that that's not right, either. Suppose b is bearded and a is not. Let C be the set of all bearded things in all worlds. This includes b but not a. Then we want to say that possibly a is bearded. But, necessarily, a is bearded iff a is in C. But "possibly a is in C" comes out false if we swap a with b throughout the set-theoretic hierarchy when evaluating the counterparts, since then we take C's counterpart to be C*=(C-{b})u{a}, and b, which is the counterpart of a, is not a member of C*.

Intuitively, what Lewis needs to do is something like this when looking at counterparts of sets is something like this:
- keep sets defined by properties that are definable without _de re_ reference to a and b fixed
- swap a and b when embedded (i.e., member of .... of a member of) in other sets.

But that might not be quite right. After all, {a} might turn out to be definable without _de re_ reference to a or b (there may be worlds that are, together with their denizens, uniquely specified by a finite purely qualitative description -- this depends on the question whether there are indiscernible worlds).

In any case, this is all really complicated.

So maybe what I rejected in the first para of this comment is the way to go for Lewis, after all?