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Kripke introduced the concept of rigid designators—terms that refer to the same object in all possible worlds in which that object exists. Proper names are typically rigid; they pick out an individual essentially. Definite descriptions are usually non-rigid: 'the tallest person' designates different individuals in different possible worlds. This distinction has profound implications: statements like 'Hesperus could have not existed' become necessary truths, not contingent discoveries.
Study the Hesperus/Phosphorus example: both are names of Venus, so the identity 'Hesperus = Phosphorus' is necessary (true in all possible worlds), not contingent. Compare with non-rigid descriptions.
Rigid designators must pick out the same thing in all worlds—they only need to pick out the same thing in worlds where they pick out anything. Necessity must be knowable a priori—Kripke's distinction between metaphysical necessity and epistemic necessity allows necessary truths to be discovered empirically.
From your study of Kripke's causal theory of naming, you know that proper names get their reference not through associated descriptions (as Frege and Russell thought) but through a causal-historical chain connecting the name back to an initial baptism event. "Aristotle" refers to a particular person not because competent speakers associate the name with "the teacher of Alexander" or "the author of the Nicomachean Ethics," but because their use of the name connects back, through a chain of transmission, to the original introduction of that name to that person. This causal account tells you *how* names refer. Rigid designation tells you *what* that reference consists in across possible worlds.
A rigid designator is a term that picks out the same object in every possible world in which that object exists. Proper names, on Kripke's view, are the paradigm case. "Aristotle" refers to Aristotle in the actual world; it also refers to Aristotle in the possible world where he became a farmer instead of a philosopher, and in the world where he died in infancy. In every world where Aristotle exists at all, the name "Aristotle" tracks *him*. Non-rigid designators contrast sharply. The definite description "the teacher of Alexander" picks out whoever happens to fill that role in a given world — in the actual world, Aristotle; in a world where Aristotle was never hired, perhaps someone else entirely. Same words, different referents in different possible worlds.
This distinction has a striking consequence for modal statements. If both "Hesperus" and "Phosphorus" are rigid names — both referring to Venus in every world — then "Hesperus is Phosphorus" expresses a necessary identity. The names co-refer in every possible world. There is no world where Hesperus exists and Phosphorus exists but they are different things. The identity is necessary even though it was an empirical astronomical discovery. This is Kripke's key result: necessity and a priority come apart. The statement "Hesperus is Phosphorus" is metaphysically necessary but not knowable a priori — it required telescope and observation to establish. The pre-Kripkean assumption that all necessary truths are knowable from the armchair was false.
The same logic extends to natural kind terms — words like "water," "gold," and "tiger." These terms, Kripke argued (along with Hilary Putnam), refer rigidly to their underlying essential nature: "water" refers to H₂O in every possible world. A substance that appeared exactly like water but had a different molecular structure would not be water — it would be something else, regardless of how it looked and tasted. This is why "water is H₂O" is necessarily true once discovered, not merely contingently true as a definition. Together, rigid designation and natural kind reference transform how we understand the relationship between language, modal reality, and scientific discovery: names and kind terms hook directly onto the world, and the necessities they reveal are in the world, not just in our concepts.
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