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A proposition is metaphysically necessary if it could not have been false — it holds in every possible world — and contingent if it is true but could have been otherwise. The distinction seems straightforward until Kripke showed that necessity and a priority come apart: 'water is H2O' is necessary (true in all possible worlds) yet knowable only a posteriori (through empirical investigation). This shattered the traditional assumption that necessary truths are always known from the armchair. Kripke also argued for the necessity of origin (this table, if it exists, must have come from this very block of wood) and the necessity of identity (if Hesperus is Phosphorus, it is necessarily so). These results transformed metaphysics by showing that the structure of reality constrains what is possible in ways that outstrip what we can know a priori.
Read Kripke's Naming and Necessity lectures I and III. Work through the Hesperus/Phosphorus case carefully: why is the identity necessary despite being an empirical discovery? Then compare with Fine's critique in 'Essence and Modality,' which argues that necessity should be grounded in essence, not the other way around.
From your study of possible worlds semantics, you have a precise formal definition to work with: a proposition is necessarily true if it is true in every possible world, and contingently true if it is true in the actual world but false in at least one other possible world. The proposition "2 + 2 = 4" is true in every world — it could not have been false. The proposition "Napoleon was exiled to Saint Helena" is true in the actual world but false in worlds where the battle of Waterloo goes differently. These seem like clear cases. The interesting philosophy begins when you ask whether the formal distinction maps neatly onto the epistemic distinction between what we can know from the armchair and what we must investigate.
Before Kripke, the dominant view was that the necessary/contingent distinction coincided with the a priori/a posteriori distinction. What is necessary, the thought went, is knowable by pure reason; what is contingent must be learned from experience. Kripke's lectures *Naming and Necessity* dismantled this alignment with a series of brilliant examples. The identity "water is H₂O" can only be known through empirical chemistry — it is firmly a posteriori. Yet once we know it, we see that it could not have been otherwise. Water just *is* H₂O; a substance with a different molecular structure would not be water, regardless of how much it looked and tasted like the stuff in our rivers. This is a case of necessary a posteriori truth — necessary in the modal dimension (true in all possible worlds), yet not knowable from the armchair.
Kripke developed two further examples of necessity that resist a priori access. The necessity of identity holds that if two names designate the same thing, the identity is necessary: "Hesperus is Phosphorus" is necessarily true once we discover it is true, because both names rigidly designate Venus — the same object in every world. The necessity of origin holds that an object could not have originated from a substantially different origin: this table, if it exists at all, could not have been made from a completely different piece of wood. These claims are metaphysical rather than merely linguistic; they tell us that the structure of reality — not just our concepts — constrains what is possible.
Kit Fine's later work introduces a challenge to this picture. Fine argues that necessity should be grounded in essence rather than the other way around. On Kripke's view, Socrates is necessarily not a number because a world where Socrates is a number is too different from the actual world to count as one where Socrates exists. But Fine points out that the number 2 is necessarily not identical to Socrates — yet this seems to have nothing to do with Socrates's essence. Socrates's nature, Fine argues, is what grounds what must be true *of Socrates*, and essence is not reducible to necessity. This refinement does not undermine Kripke's insight that necessity and a priority come apart; it pushes more deeply into what explains modal facts in the first place.
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