When we refer to a "rigid designator," we refer to something that refers to a specific object in all the possible worlds in which the object exists. The term was coined by Saul Kripke, who famously argued that the identity statement "Hesperus = Phosphorus" is necessarily true (true in all possible worlds) even if it is a posteriori rather than a priori. This is quite unusual in the history of philosophy, since empirical or a posteriori truths are typically understood as only 'contingently' rather than necessarily true. For Kripke, the only meaningful sense in which one might say that it is 'contingently' true is epistemically, but not metaphysically. Thus, the contingency refers to our epistemic uncertainty, rather than being a metaphysical predicate of the object.
But does such necessity obtain for this a posteriori proposition? Because the identity statement is such that both names refer to the selfsame object. That is, "Hesperus" and "Phosphorus" are distinct names, they nonetheless refer to the same object. It is in this sense that they are "rigid":
"each designates just the object it actually designates in all possible worlds in which that object exists, and it designates nothing else in any possible world. The object that 'Hesperus' and 'Phosphorus' name in all possible weorlds is Venus. Since 'Hesperus' and "Phosphorus' both name Venus in all possible worlds, and since Venus = Venus in all possible worlds, 'Hesperus = Phosphorus' is true in all possible worlds"(LaPorte, 2011)
As LaPorte points out, however, a proposition affirming that Venus is the brightest non-lunar object in the evening sky is not "rigid." It may or may not be true (that is, it is contingently true), but it is certainly not necessarily true (LaPorte, 2011).
"While Hesperus is in fact the brightest object in the evening sky apart from the moon, Hesperus might have been dimmer: had, say, Hesperus been obscured by cosmic dust, Mars might have been the object designated by 'the brightest non-lunar object in the evening sky' rather than Hesperus. In that case, the above identity statement (H) would have been false. So the reason that (H) could have been false is that 'the brightest non-lunar object in the evening sky' does not designate Hesperus rigidly. It designates Hesperus in this world, which explains why (H) is true, but this description designates Mars in some other worlds, which explains why (H) could have been false: (H) would have been false had some other such world been actual"(LaPorte, 2011).
The predicate "is the brightest non-lunar object in the evening sky" is non-rigid because it may refer to any one of a number of objects, depending upon what conditions obtain in a given world. However, "Venus" and "Hesperus" are rigid designators because they always refer to a particular object in the worlds in which it exists. If the object which we use "Venus" and "Hesperus" to designate does not exist in a given world, then the designator designates nothing in those worlds (according to one formulation of the concept of the rigid designator).
The modal speculation surrounding the concept of the rigid designator can be quite difficult to fathom. Some philosophers clarify the concept by arguing that "a rigid designator designates the same object in all possible worlds as it is used in the actual world, not as it is used in other possible worlds in which the object gets picked out: for although we identify objects in other worlds by our own names, natives of some of these worlds use other names"(LaPorte, 2011). However, "A few philosophers resist this clarification. They find the idea of differentiating between the reference of terms in our world with respect to other worlds, on the one hand, and the reference of terms as used in other worlds, on the other, fatally confused"(LaPorte, 2011).
So a rigid designator designates the same object in our world that it does in other worlds, but we must keep in mind that natives of other worlds use different names for the same object. "Venus" designates the object we think of when we use the word, but the word "Venus" may refer to another object in another world. Likewise, natives of another world may use "Mars" to refer to the object we know of as Venus, and "Venus" to refer to the object we know in this actual world as 'Mars." The point, however, is that the designator designates the same object in every possible world in which the object exists. What gives a rigid designator its rigidity is its referential function, rather than a purely lexical consideration.
There does seem to be a tension in Kripke's work concerning the exact referential scope of the rigid designator. In some places, Kripke affirms that the rigid designator refers to its object in every possible world in which the referent actually exists. In other places, however, he seems to argue that the designator refers to its referent even in worlds in which the object does not exist at all (LaPorte, 2011). Suppose there is a possible world in which the relevant conditions do not obtain for Venus to exist. The rigid designators, in the latter formulation would still refer to the object we know as "Venus" even in possible worlds in which the object we know as Venus does not exist (LaPorte, 2011).
LaPorte notes that considerable controversy surrounds which notion of rigidity to adopt. "...some philosophers have held that true statements using a proper name to express that so and so might not have existed are unintelliglble unless the relevant name refers to the object in all worlds, period"(LaPorte, 2011). This, as LaPorte notes, has come to be known as "obstinate rigidity." As far as an 'obstinate' understanding of rigid designation is concerned, the designator refers to its designatum even in worlds in which the object does not exist. Likewise, there are differing degrees of necessity one can accord different conceptions of rigid designation. Kripke argues that the identity statement "Hesperus = Phosphorus" is, at the very least, "weakly" necessary, which means that it is true in all possible worlds in which its designatum exists. This is opposed to strong necessity, in which the identity statement is true in all possible worlds (LaPorte, 2011).
Another interesting controversy surrounding the notion of rigidity in designation has to do with indexicals such as "you." "You," in the sentence "you are eating," may refer to be specifically, or it may refer rather generally to the concept of an indefinite person eating. Likewise,
"if I say, pleasantly surprised, "You made good time," we could discuss reasons for this and conditions under which this might not have been the case, but in all of our considerations the same individual, and not anyone else who might have been at my door in good time or not, is the one in question. The individual in question is the one who is in fact identical to you (as any account could put it)"(LaPorte, 2011).
Some philosophers argue that "you" means something indefinite and non-rigid. "You are eating," therefore, would refer to the fact that some indefinite person is eating. Others insist that indexicals are rigid and refer necessarily to the specific person being referred to at that time. Others hold that terms for "natural kinds" such as "cat," "dog," "silver," sometimes refer to rigid, singular terms, and other times, refer to indefinite classes.
I believe that the question of worldview is a crucial respect in which the concept of the rigid designator must be examined. This is clear when LaPorte notes that "‘Hesperus’ is a name that was given to a heavenly body seen in the evening, and ‘Phosphorus’ is a name that was, unknown to the first users of the name, given to that same heavenly body seen in the morning. The heavenly body is Venus"(LaPorte, 2011). So do the two terms refer to the same thing? Yes and no. It depends on what we mean when we refer to reference. If by reference we refer to correspondence with a metaphysical entity, then yes, Hesperus and Phosphorus refer to the same thing.
If by reference we are merely referring to a name that constitutes an output of a a worldview, then clearly Hesperus and Phosphorus do not refer to the same thing, because considering the use of the name as a function or output of a worldview involves considering it in purely epistemic or doxastic terms. The question then turns from "Do the words "Phosphorus," "Hesperus," and "Venus" all refer to the same metaphysical entity?" to the epistemic or doxastic question of "What did those who referred to the object we now know of as "Venus" as "Phosphorus" and "Hesperus" mean by those terms?" These are quite distinct questions, and we must always be very clear as to which question we are asking and which question we are answering, if we are to avoid confusion. So Cristina Lafont on the fundamental differences in atttiude and orientation between the Anglo-American and German approaches to the linguistic turn in philosophy. :
"This German tradition exhibits specific features that distinguish it clearly from the Anglo-American philosophy of language. Perhaps its most important feature is the explicit attempt, found in all the authors of this tradition, to break with the assimilation of all functions of language to the cognitive function (language as a vehicle of knowledge) at the expense of its communicative function (langauge as a means of understanding). In other words, it is a central aim of this tradition to end what Humboldt terms "the primacy of logic over grammar," a primacy that the authors in question trace to the very beginnings of Greek philosophy. The basic orientation of this tradition toward social and cultural phenomena rather than natural ones (toward the social rather than the natural sciences) explains this common motif among its authors. In keeping with this focus, the German tradition has always concentrated on the analysis of natural languages and it has regarded these as constitutive of the relationship of human beings with the world at large. That is to say, this tradition's philosophical interest in the analysis of language does not stem only from the crucial role played by language in our relationship with the objective world (by allowing us to have propositional knoweldge of it). Rather, language is also held to be pivotal to our relation with the social world (which is essentially dependent on intersubjective communication), and even to our experience of our own subjective worlds (which are expressible only through linguistic articulation). In this way, language is considered in its multidimensional world-disclosing function"(Lafont, preface).
Therefore, consideration of an utterance as a function of a worldview, Hesperus does not equal Phosphorus because the original users of the language did not intend to refer to the same thing by "Hesperus" as they did by "Phosphorus." The two words meant different things for them. Their utterances reflected an incorrect understanding of the natural world, of course, but it is perfectly legitimate to examine such utterances as functions of a holistic worldview rather than solely in terms of rather or not a proposition corresponds with an external metaphysical reality. This latter course is also a perfectly legitimate avenue of investigation, of course. We just need to be clear about when we are conducting one sort of investigation rather than the other.
It is furthermore difficult to imagine what it would mean for a rigid designator to refer to an object even in worlds in which the object did not exist. As Lafont pointed out, language is an inherently intersubjective phenomenon. If there were a world in which Venus did not exist, would the natives of that world even have a word for it? Perhaps they would, but perhaps they would not. In any case, it seems like an exceedingly strange question to ask. Could we conceive of natives of a possible world in which Venus did not exist as having a concept of Venus that was exactly identical to ours? Perhaps, but would we really expect them to have such a concept? Can we speak intelligibly of a word that does not actually have any referent?
The question of temporality and historicity is also worth considering. If Venus explodes or is engulfed by a star, such that it no longer exists, presumably the rigid designator for the object will still refer to it to future generations who are aware that there once existed a planet called Venus. But if by "possible world" we are referring to a kind of parallel universe whose inhabitants have never known of the planet "Venus" because the planet does not exist in their parallel universe, it is difficult to see what it would mean for the rigid designator to be meaningful in such a word. Language is intersubjective, and because it is intersubjective, it is relative. Its meaning and meaningfulness is relative to actual language-users, which includes individuals and groups of people. When we ask whether or not the rigid designator refers to the same thing in possible worlds in which an object may or may not exist, the answer seems to be "it depends on whom you ask. It may mean something to some of the inhabitants of such a world, and it might not mean anything whatsoever to its other inhabitants." The concept of the rigid designator thus seems to run the risk of allying with a very strange sort of Platonic essentialism that fails to take adequate account of the inherently intersubjective nature of language.
LaPorte, Joseph, "Rigid Designators", The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL = <plato.stanford.edu/archives/sum2011/entries/rigid-designators/>.
Lafont, Cristina. "The Linguistic Turn in Hermeneutic Philosophy." The MIT Press (August 7, 2002).