The notion of rigid designation  figures prominently in Saul Kripke's Naming and Necessity. (See Naming and Necessity p. 48.) There has been much confusion about rigidity, even among professional philosophers. I hope this brief note helps to eliminate some of that confusion.

Kripke holds the following rigidity thesis:

KRT    Proper names are rigid designators.

It's a simple enough thesis, but the concept of rigidity has made it difficult for some to understand. What would it be for a name to be a rigid designator? Let's focus on a particular individual, Jill Beck, and her name, "Jill Beck." Presumably, Jill Beck is, like the rest of us, merely contingent; there are worlds in which she exists and worlds in which she does not exist. We, here, in this the actual world can talk about her and what she does in some of the other possible worlds in which she exists. Let w1 be a possible world in which she exists, is not the president of L.U., but is the president of Carleton College. Here in the actual world, if we wanted to say something about her in w1, we can use her name: we can say, for example,

(1) In w1 Jill Beck lives in Minnesota.

When we, here in the actual world, make such a claim, KRT entails that our utterance of "Jill Beck" in our utterance of (1) refers to, or designates, Jill Beck--the woman who is the actual president of L.U.. Contrast this with

(2) In w1 the president of L.U. lives in Minnesota.

Let's suppose that in w1 the president of L.U. is our current actual Provost, David Burrows, and that, like every other provost of L.U., David Burrows lives in Wisconsin. Hence, (1) truly attributes to Jill Beck the property of living in Minnesota in w1, and (2) falsely attributes to David Burrows the property of living in Minnesota in w1. Now, in the actual world terms, "Jill Beck" and "the president of L.U." codesignate, but, as we here in the actual world use them to talk about w1 they do not co-designate: "Jill Beck" continues to be used by us to designate what we use it to designate when we talk about the actual world, but "the president of L.U." does not continue to be used by us to designate what we use it to designate when we talk about the actual world. That's because the proper name "Jill Beck" is functioning as a rigid designator, and the definite description "the president of L.U." is not. Let's try the following definition

Def       Singular term, T, rigidly designates object, x (at world w) =df T designates x at w & as used by inhabitants of w to talk about another possible world, w', T designates a inhabitant of w' iff T designates x at w'.

Hence, according to KRT, since "Jill Beck" is a rigid designator, when we use "Jill Beck" to talk about other worlds, if our use of "Jill Beck" designates something, it designates the very person it actually designates. This is not the case with "the president of L.U." It actually designates Jill Beck, but it varies in what it designates as we use it to talk about certain possible worlds.

To be clear about this notion, let's make sure that we understand that KRT does not imply certain things that others have mistakenly thought that it implies.

First, some have mistakenly thought that KRT implies both the falsehood that Jill Beck is named "Jill Beck" in every world in which she exists and the falsehood that there is no possible world in which something other than Jill Beck is named "Jill Beck."  Here is an example,

Kripke argues against this view [i.e., contingent identity] using the concept of rigid designator--if a term (or name) refers to an object, then it necessarily refers to that object. (Jack S. Crumley II, in Problems in Mind: Readings in Contemporary Philosophy  of Mind, Mayfield, 2000, p. 95)

(3) In w2 Jill Beck is not named "Jill Beck,"

is true. Whereas both

(4)     In w2 the person named "Jill Beck" is not named "Jill Beck,"

and

(5) In w2 Jill Beck is not Jill Beck,