Multiple philosophers have suggested that this stance seems similar to "rigid designation", i.e., when I say 'fair' it intrinsically, rigidly refers to something-to-do-with-equal-division. I confess I don't see it that way myself - if somebody thinks of Euclidean geometry when you utter the sound "num-berz" they're not doing anything false, they're associating the sound to a different logical thingy. It's not about words with intrinsically rigid referential power, it's that the words are window dressing on the underlying entities. I want to talk about a particular logical entity, as it might be defined by either axioms or inchoate images, regardless of which word-sounds may be associated to it. If you want to call that "rigid designation", that seems to me like adding a level of indirection; I don't care about the word 'fair' in the first place, I care about the logical entity of fairness. (Or to put it even more sharply: since my ontology does not have room for physics, logic, plus designation, I'm not very interested in discussing this 'rigid designation' business unless it's being reduced to something else.)

He seems to have thought Rigid Designation was about a magic connection between sound wave patterns and objects, such that the sound waves would always refer to the same object, rather than that those sound waves, when spoken by such a speaker in such a context, would always refer to the same object, regardless of which possible world that object was in.

I'm sorry if that explanation was a little unclear; it was aimed at non-philosophers, but I suspect you could explain it better.

EDIT: see also prior discussion