Are there possible worlds that differ only in the assignment of rigid designators?

Question

Is there an implicit idea in Kripe's definition of rigid designators that rules out the following possible world:

Let A, B rigidly designate two things in the real world. Imagine the possible world where A has all of the properties that B has in the real world, and vice versa.

i.e. make a world that is indistinguishable in a descriptivist sense, but differ in terms of the identity of objects therein.

As far as I can tell, there is nothing ruling this out, so there's nothing ruling out making more general permutations of rigid designators.

Answer 1

There is such an implicit assumption with Kripke.

This is a central assumption operative in deriving essentialism from the possible world's apparatus, and was isolated in such a context by Nathan Salmon in his book Reference and Essence. The assumption also plays a role in the Four Worlds' Paradox.

Answer 2

It depends on the comparison operator which would belong to the metalanguage the concept of possible world is defined in. If it is classical logic (i.e. with single equality and no way to distinguish equal objects using a custom operator) it would be true. On the other hand if it were natural language without any additional logical constrains it would be undefined (but pure natural language is not consistent so we do not want to use it this way). You could also define custom metalanguage with nonstandard equality.

But it does not lead to any use full results. Even if we assume there are such different possible worlds we still cannot reach one from the another and there always is an autoisomorphism of set of possible worlds with maps A on B without losing their features.

It is like saying that swapping p and q in axioms of logic creates a new, different logic. This is in formal sense true but does not create new concepts in everyday life sense.

Answer 3

The only limit to the 1st case you describe is given by Kripke's idea of essential properties. Basically, A and B would have to share all of their essential properties if the possible world with swapped names is really possible.

This limit can be really tight. If, for instance, an essential property of people is having a particular DNA, or being born from a particular set of gametes, name swapping can only work for identical twins.

Now, general permutations of rigid designators would require you to start by descriptively describing different possible worlds and then to assign rigid designators to the objects in those worlds. This way of thinking is, according to Kripke, mistaken.

Kripke says in NaN that "you don't see possible worlds through a telescope" and "possible worlds are given by the descriptive conditions we associate with them" (1st conference, not literal quotes, I only have a Spanish edition and am quoting from memory); this remarks mean that, since there's not anything more to possible worlds than the descriptions we use when we describe them, the rigid designators that apply in a possible world have to be already given once we describe it, and it makes no sense to ask which objects in a descriptively specified world are to be called with such and such rigid designators.