Wo is on a roll at the moment. This post about rigidity and
counterpart theory is actually I think much more interesting than it may
first look. (Or, I should say, may first look if you’re a lazy reader who reflexively assumes that counterpart theory can solve all the traditional problems concerning de re modality without breaking a sweat.) So at first, I thought the issue of how do you define rigidity without assuming trans-world identity wasn’t that deep. Here’s how you
define rigidity if you’re a counterpart theorist

(CR)    t rigidly denotes a iff for
all a´, w´, if t denotes a´ in w´, then a´
is a counterpart of a in w´.

This doesn’t cover the issue about how to
define rigidity for natural kind terms if you’re a trope theorist, but I
suppose something similar could be done. So on first glance I thought, nothing
to see here, move along. But on second thought I thought maybe I’d gone a
little too far. After all, any fool can put the definition of rigidity through
the ‘counterpart-theoretic-equivalent’ generating machine and come up with
something like (CR). The philosophically interesting question is whether what we
get out really deserves to be called rigidity. And on third thought, I
came up with a little argument that it doesn’t. To get to it, consider the
following argument:

The Rigidity Argument

R1.     Possibly, t₁ is t₂

R2.     ‘t₁’ is a rigid designator

R3.     ‘t₂’ is a rigid designator

RC.     So,
t₁ is t₂

(The round quotes here should be corner
quotes, but I’ve got no idea about how to do them in HTML.)

I think that it’s a platitude about rigidity
that The Rigidity Argument is valid. If you don’t immediately agree, here’s a little argument to convince you. Assume that possibly t₁
is t₂. So in some world, t₁ and t₂
denote the same object. Since t₁ is rigid, it denotes the
same object here as it does in that world. And since t₂ is
rigid it denotes the same object here as it does in that world. But that means
that t₁ and t₂ denote the same object here,
so the sentence ‘t₁ is t₂’ must be true, and
by semantic descent t₁ is t₂.

Now let’s recall some of the special
features of counterpart theory. Two objects, a and b, in this
world, can have the same counterpart, c in another world w. Assume
that this is the case, and assume that F is some (quite precise)
property instantiated only by a in this world, and G another
(also quite precise) property instantiated only by b. Then for the
counterpart theorist, it’s true in w that the actual F is the
actual G. So, it’s possible that the actual F is the actual G.
But it’s not true that the actual F is the actual G. But I
thought ‘the actual F’ and ‘the actual G’ were rigid designators
if anything was. It looks like one of the following three things has to go.

1        A
counterpart-theoretic treatment of de re modality

2        The
view that ‘the actual F’ is a rigid designator

3        The
view that The Rigidity Argument is valid

It’s a pity that we have to give up one of
these, because until a few hours ago I’d have probably endorsed all three. And
I’d still like to endorse all three, if I didn’t think they were mutually contradictory.

It would be no good to ‘solve’ the problem
by finding a version of ‘counterpart theory’ that eliminated the possibility
that a and b could have a counterpart in common. I know this
would be technically possible, and I could even whip up a theory that did it quicker
than I could fry an egg (though maybe not quicker than you could fry an
egg) but it’s hard to see the philosophical point. Counterpart theory is a
substantive philosophical view, not a technical apparatus, and if you start
putting those kind of restrictions into the formalism, saying that a and
b can’t have a common counterpart, you’ve effectively abandoned the
philosophical view. As I said, this could be done technically, and could even
be done technically with the veneer of counterpart theory, but I’d regard it as
giving up 1.

So what should we do? I think we should
recognise two notions of rigidity. There’s a strong notion of rigidity,
according to which a rigid designator really names the same damn object in
every possible world. On that notion of rigidity, The Rigidity Argument is
valid, and ‘the actual F’ simply isn’t a rigid designator according to
counterpart theory. And there’s a weak notion of rigidity, given by (CR) on
which ‘the actual F’ is rigid, and The Rigidity Argument is invalid. And
it’s a substantive thesis, one of the many substantive yet somehow plausible theses
advocated by counterpart theorists, that apparently rigid designators are only
weakly rigid, not strongly rigid. (Some terms, such as numerals, may be
strongly rigid de facto, but no term is strongly rigid de jure.)
This ‘two kinds of rigidity’ solution may look pretty ugly, but I can’t try and
be Solomon all the time without sometimes splitting the baby. (I got the last
bit of pop philosophy from Law & Order, which is I think a step up from stealing
philosophical examples from Budweiser commercials.)

Anyway, this is all a mess, but if anything
things get worse. I’d have sworn, before thinking too hard about this, that (4)
and (5) were equivalent, but of course this isn’t true. (4) is a contingent falsehood
and (5) a necessary falsehood.

(4)      The
actual F is the actual G.

(5)      Actually,
the F is the G.

If it makes you feel any better, the
biconditional (4) iff (5) is still a priori, last I checked, just not
necessarily true.

I planned to write something more on the
paradoxes, but it’s late and I’m tired, and in a little philosophical shock.