A rigid designator is one that designates the same object in every possible world. Thus “Plato” designates Plato in every world; in no world does it designate anyone else. We must hasten to add that names are only rigid with respect to a language, i.e. under a particular assignment of meaning; no name is rigid in virtue of being the sound or mark that it is. Words are conventionally attached to meanings, so that they only contingently denote whatever it is they actually denote. Clarity might be served by saying that the meaning of a name is what is properly rigid (similarly the meaning of a description is what is properly non-rigid). The meaning or sense of a name rigidly designates its reference. It doesn’t follow that the mode of presentation associated with a name is rigid, if that concept it taken qualitatively, i.e. how the reference seems to the speaker. And that would not be a plausible view given that numerically distinct objects can appear the same way. Nor do the ideas in the speaker’s mind rigidly designate (this is one reason description theories of names run into trouble). Names have a special kind of meaning that ties them to their actual bearer across possible worlds. The standard view of this is that the meaning of the name is its bearer, so that constancy of meaning guarantees constancy of bearer, by virtue of strict identity. If the meaning of “Plato” is Plato, then of course it designates the same person in every world, since the meaning just is the reference: this is like saying that Plato is Plato in every world. The statement “The meaning of ‘Plato’ is identical to the reference of ‘Plato’” is true, and identities hold necessarily. Nothing like this can be said of definite descriptions, so they fail to be rigid. We could say that the general terms forming the description rigidly designate the properties they actually designate, but not that the description rigidly designates the object that contingently satisfies it.
So far, so orthodox: but now I want to raise the question of what kind of necessity is in play here. We could rephrase the concept of rigidity as follows: names have the property of necessarily denoting what they actually denote. It is part of the essence of “Plato” (its meaning) that it denotes Plato; in no world does it denote anyone else. So names have essences just as objects have essences: Plato is necessarily a man and “Plato” necessarily denotes this man. The name “Plato” has other essential properties (remember we mean its meaning) such as that it is a name or is part of language or is not identical to the name “Aristotle”: meanings have essences too. It is part of the essence of the meaning of “Plato” that it designates Plato—but it is not part of the essence of the meaning of “the teacher of Aristotle” that it designates Plato, even though he did teach Aristotle. So we can say that the same notion of necessity is used to characterize rigidity as is used to characterize the essence of objects—good old metaphysical necessity. We say that a person necessarily has the parents she actually has, and we can equally say that a name necessarily refers to what it actually refers to; while a person does not necessarily attend the school she actually attends, and does not necessarily satisfy the descriptions she actually satisfies. Semantic properties can be essential (or contingent) properties too. Languages are bearers of modality just as non-linguistic reality is. Rigidity is just another species of necessity.
Now I can raise the following heterodox question: is the necessity involved in rigidity reducible to other categories of necessity? Kripke gave us four categories of necessity: identity, kind, constitution, and origin. Is the rigidity of names a special case of one of these categories? The alternative is that it is not but is a sui generiscategory of necessity that we need to add to our inventory of categories of metaphysical necessity (“necessities of reference”). I am going to suggest that rigidity is reducible to the necessity of constitution plus the necessity of origin: a name’s having a certain reference essentially is the upshot of a particular type of necessity of constitution plus necessity of origin. Thus we can explain these necessities of language in terms of more general types of necessity applicable to the non-linguistic world. This may sound strange, but on reflection it is quite intuitive, once we understand how general the notions of constitution and origin are. Suppose we say that the meaning of a name is constituted by its bearer; and we compare this to saying that this table is constituted by a particular piece of wood. In the latter case it is right to say that the table is essentially so constituted—in every world in which the table exists it is made of the same piece of wood as in the actual world. Similarly, in the former case, if the meaning of the name is constituted by its actual bearer, then it is so constituted in any world, since constitution generates necessities. If x is made of y, then you can’t have x without y. Of course, you can have something that is like x that is not made of y, but not that very thing—a table that looks like x, say. Likewise, you can have a meaning that resembles the meaning of “Plato” and it not be constituted by Plato, but you can’t have that meaning without Plato. Thus two speakers may be exactly alike physically and mentally and use a name “Plato” but refer to different people by that name, because the meaning is constituted by different references in the two cases. Two meanings can seem the same but not be the same because of a difference in actual constitution—just like two tables. According to the direct reference conception of names (the “Millian” view), the meaning of a name is constituted by its bearer; but then it is necessarily so constituted, by the necessity of constitution, in which case it will be rigid. We could say metaphorically that the table “rigidly designates” the piece of wood it is actually made from, just as a name literally rigidly designates its actual bearer: the necessity of constitution is at work in both cases. And don’t object that this latter must be metaphorical because only physical objects have constitutions: clearly the concept of constitution can be applied outside of the physical realm, for example to states of mind and to mathematical entities (emotions and geometric figures, for example). Identity can be applied with this kind of generality (and is often invoked to express the Millian view), and there is no metaphysical reason to restrict the idea of constitution to material objects (the Constitution is not wrongly named). Thus the referential rigidity of names falls out as a consequence of the necessity of constitution: the former follows from the latter.
How does the necessity of origin enter the picture? First we must note the generality of that notion; it isn’t just parents and children but any generative historical relation. Clearly it applies to any organism and its ancestors: each organism necessarily has the ancestors it actually has, going back to the origin of life on Earth. But also historical events fall under the necessity of origin as well as human artifacts: WWI (that war) could not exist in a world in which its actual antecedents do not exist (though there could be a war similar to it but differently caused), and no one other than Leonardo could have painted the Mona Lisa (that very painting). We can’t completely change history and leave the identity of the objects and events intact. In the case at hand, a name has a certain history originating in the initial baptism of a particular object (say, baby Plato): then a chain of linguistic events connects this origin to later uses of the name. Let us then say that the name “Plato” has origin O: accordingly, it (that name) could not exist without O. The name (its meaning) owes its identity to its actual origin: just as Plato has to come from his actual parents, so the name “Plato” (with its actual meaning) has to come from baby Plato in an act of baptism. If we substitute another baby in a possible world, we get a different name (a different meaning), despite any resemblance of baby—just as we get a different child if we substitute different parents, despite resemblance of progeny. If so, rigidity follows from origin: the name could not refer to anyone not at the origin of the causal chain that exists in the actual world, i.e. the one culminating in baby Plato. If the origin of “Plato” is baby Plato, then it could not have had any other origin, by the necessity of origin; but then the name must designate the same person in all worlds. That name requires that origin, so there is no world in which that name exists but is anchored to a different origin (as it might be, baby Aristotle). Nothing like this is true of descriptions, of course, since they are not individuated by origin at all: they don’t refer in virtue of a causal chain leading back to an object’s baptism. Accordingly, descriptions are free to be non-rigid, as flexible as the occasion demands. But names are strictly tied to their historical antecedents in babies, baptisms, and the like. If so, rigidity follows from the necessity of origin, and is a special case of it.
There is a question, then, about which of these two necessities is basic in the modal semantic of names. We need not take a firm stand, but I am inclined to think that origin is basic: it is because names are introduced in the way they are that their meaning is constituted in the way it is. Having that origin determines what constitutes the name’s meaning: there is nothing else for the meaning to be given that names originate as they do. Tracing back to a particular object is what fixes their meaning (not a cluster of associated descriptions), and hence we say that the meaning is constituted by the object. So origin is primary, though both are equally correct as modal claims. The important point is that semantic rigidity is not some new type of necessity but is a special case of necessities already recognized. We don’t have Kripke’s four categories and the necessity of reference as an additional primitive category; the latter is an instance of the former. It is what the necessity of constitution and origin look like when manifested in language. This is good because it is not clear what else referential necessity might be given that we seem to have covered the bases with the four categories. Rigidity is a type of essence found in language, but what other types of essence are there other than the big four? They seem to exhaust the field, in which case linguistic essence needs to emerge as a form of one of more of those. Clearly the concepts of constitution and origin apply to language, and are so employed quite spontaneously by theorists, so it is in the cards that we can explain referential necessity by appeal to these concepts. Referential necessity thus arises from a combination of the necessity of constitution and the necessity of origin.
 It can also be applied to phrases and sentences: a string of words is constituted by the individual words that compose it. As a consequence, we can say things like, “The sentence ‘snow is white’ is necessarily constituted by the words ‘snow’, ‘is’, and ‘white’”. The same applies to thoughts and their constitutive concepts.
 We can also define the notion of rigid portrayal: a painting rigidly portrays a certain individual if it portrays the same individual in every possible world. The claim that paintings are sometimes rigid portrayers is plausible: the painting must portray the same individual in any world in which it (that painting) exists—no Mona Lisa no Mona Lisa painting (her twin will not do). This is different from a painting just happening to fit a certain individual. In this kind of case the origin theory of the necessity seems very plausible; so the Mona Lisa painting needs both Leonardo and Mona herself in order to exist in any world.
 It might be said that there are strictly five categories: in addition to constitution by a particular object we need to recognize the type of the constituting object. Thus we can say that this table is necessarily made of wood not just this piece of wood (as is the piece itself). The analogue for names would be the fact that the meaning of the name is necessarily of the human being type: the name “Plato” must refer to a human being and not (say) to a goat, in addition to necessarily referring to Plato (who is himself necessarily a human being). But this isn’t to accept that there are irreducibly semantic types of de re necessity. The name itself might be necessarily composed of certain sounds, which are necessarily sounds. We just have iterations of the same metaphysical necessities we had before we got to the de re necessities of language.
 If we choose to say that predicates rigidly designate their corresponding properties, we can give the same type of explanation of this semantic fact, namely that the properties denoted form the constitution and origin of (the meaning of) their denoting predicates. The property of being red constitutes the meaning of “red” and is the origin of that predicate (in its actual meaning): hence “red” rigidly designates the property red. The meaning of “red” could not be constituted by any other property or originate in any other property.