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Rigid designators—names and natural-kind terms referring to the same object across all possible worlds—provide a framework for understanding essentialism. If a name rigidly designates an object, then statements like 'Aristotle is necessarily human' can be true: Aristotle is human in all possible worlds where he exists. This connects semantics of reference to metaphysics of essence.
From your study of rigid designators and modal reference, you know that a term is rigid if it refers to the same object in every possible world in which that object exists. From your study of essentialism, you know that an object's essential properties are those it could not lack while still existing — as opposed to accidental properties, which it happens to have but could have lacked. What you're learning now is how the semantics of rigidity and the metaphysics of essence are not merely analogous — they are deeply connected through the machinery of possible-worlds reasoning.
Here is the bridge: once you fix that a name like "Aristotle" rigidly designates a particular individual, you can ask what is true of that individual in *every* possible world where he exists. Whatever is true of him in all such worlds is necessarily true of him — which is just what it means for a property to be essential. Kripke's contribution, building on rigid designation, was to argue that identity statements involving two rigid designators are necessary if true. "Hesperus is Phosphorus" (both rigid names for Venus) is, if true, necessarily true — there is no possible world where they are distinct, because both names pick out the very same object everywhere. This was striking: it meant that empirical discoveries could turn out to be necessary truths.
The same logic applies to natural-kind terms like "water," "gold," and "tiger." Kripke and Putnam argued these terms are rigid: "water" refers to H₂O in every possible world, not just in ours. Before we discovered the chemical composition, we used the term to refer to the stuff — whatever its inner nature turned out to be. Once discovered, the identity "water = H₂O" is necessary: there is no possible world where water is not H₂O (though there could be a world with a watery-looking, watery-tasting substance that is XYZ — but *that* wouldn't be water). This generates a posteriori necessary truths: claims that are necessary but only discoverable through empirical investigation.
Essentialism for kinds follows similarly. If "tiger" rigidly designates the natural kind, then whatever is essential to the kind — having a certain biological nature, DNA structure, evolutionary lineage — will be necessary in all possible worlds containing tigers. A creature that looks and behaves like a tiger but lacks the biological nature isn't really a tiger; it's a tiger-duplicate. This distinguishes Kripkean essentialism from superficial or nominal essentialism: the essence is given by the internal structure discovered by science, not by the description that first fixed the reference.
This framework puts semantic and metaphysical questions in direct contact. Questions about what properties a name's referent necessarily has — what Aristotle could not have lacked, whether the substance called "water" could have been something other than H₂O — are simultaneously questions about the semantics of rigidity and the metaphysics of essence. The transworld identity work you do next (asking *which* object in another possible world is the same as this one) depends on having a grip on which properties are identity-preserving, and rigid designation provides the semantic foundation for that inquiry.
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