A rigid designator is one that denotes the same thing in all possible worlds, or in all possible worlds in which that thing exists. The term was introduced by Kripke, to label the fact that if we specify a possibility using a singular term we intend the singular terms to have the same reference in the imagined, or possible, situation as they have in the actual world. For definite descriptions this is not necessarily so: in ‘had Hume been English, the town he died in would not have been Edinburgh’, both ‘Hume’ and ‘Edinburgh’ refer to what they always do: the philosopher Hume and the town Edinburgh. But we cannot suppose that the phrase ‘the town he died in’ is referring to the town he actually died in, namely Edinburgh. The sentence is not trying to say that had Hume been English, Edinburgh would not have been Edinburgh. Kripke used the point in his influential criticism of the theory that names function as disguised definite descriptions. Complexity arises in handling cases where we wish to evaluate a sentence in a world in which the normal reference would not have existed, such as ‘had his mother died young, Hume would not have existed’. Here we cannot intend the term to have the same reference as usual in the imagined possible situation, since in that situation there would be no Hume to refer to, yet intuitively we are still talking about Hume. The other classic modern discussion is David Kaplan, ‘Demonstratives’, Pt. III, in Themes from Kaplan (1989).