A proposition is necessary if it could not have been false, contingent if it is true (or false) but might have been otherwise, possible if it is true in at least some way things could have gone, and impossible if there is no way things could have gone on which it obtains. Actuality is a separate axis: the actual is what is in fact the case as opposed to merely possible. Possible-worlds semantics (as developed by Saul Kripke, David Lewis, and others) provides an elegant formulation of these relations. Imagine the totality of ways things could coherently have been: each such way is a possible world, and the actual world is just one of them. A proposition is necessary if true in every possible world, possible if true in at least one possible world, impossible if true in no possible world, and contingent if true in at least one possible world and false in others. Necessity is universal quantification over worlds, possibility is existential quantification, and actuality is indexical: it picks out this world, the one we happen to inhabit.

Johannes Climacus, in the “Interlude” of Søren Kierkegaard’s Philosophical Fragments, operates with a version of these distinctions adapted from the work of Friedrich Adolf Trendelenburg. His central claim is that necessity and coming-into-existence are incompatible. Since what is necessary has no unrealized alternatives, there is nothing for a process of becoming to bring about. Whatever comes into being, by contrast, might not have done so or might have come about differently, and its actualization does eliminate alternatives. Thus everything historical is contingent in the strict modal sense even after it has happened: the fact that what’s done can’t be undone is not the same as necessity. The past is unrevisable in the actual world but remains false in countless other possible worlds, and that is enough to keep it contingent. From this Climacus infers that historical claims cannot be objects of demonstrative knowledge in the way logical or mathematical truths are. They must be taken up by belief, an organ specifically suited to apprehending the contingent with all of the uncertainty and ambiguity that entails.
Modal distinctions underwrite what Climacus calls the Absolute Paradox. The eternal (God, on the traditional picture) is necessary in the strongest sense: timeless, immutable, true in every possible world, incapable of not being and therefore incapable of coming-into-being. The Incarnation asserts that this eternal being enters time as a unique individual born at a particular time and place. That individual is contingent: he might not have existed and the events of his life could have unfolded otherwise. So the same item is being characterized in two modally incommensurable ways, necessary in its eternal aspect and contingent in its historical aspect. There is no inferential bridge from one term to the other: no amount of historical investigation could ever yield a necessary truth and no a priori reflection on the eternal could ever deliver a particular first-century Galilean. This is why the relation to the God-man can be neither a matter of historical scholarship nor rational demonstration but only of faith in the heightened religious sense, a faith whose object is a unification of modal categories that reason cannot reconcile.
Climacus’s argument depends on the tacit assumption that the necessity/contingency distinction lines up with the a priori/a posteriori distinction: necessary truths are knowable independently of experience, contingent truths only through experience. Kripke famously challenged this alignment. There are, he argued, necessary a posteriori truths: propositions that are true in every possible world but knowable only through empirical investigation. The standard examples involve identity statements with rigid designators: “Hesperus is Phosphorus,” “water is H₂O.” A rigid designator picks out the same individual in every world where that individual exists; so if “Hesperus” and “Phosphorus” both rigidly designate Venus then “Hesperus is Phosphorus” is true in every world and hence necessary, even though it took telescopes and astronomy not logical analysis to determine.

Suppose “Jesus” and “God” both function as rigid designators. Then “Jesus is God,” if true, is true in every possible world in which the referent exists, i.e., it is necessarily true. But no one could know this a priori; the only access to the identity claim we have consists of the Gospels. So the proposition would be necessary a posteriori: necessary in its modal status, historical in its mode of access. This doesn’t dissolve the Paradox, of course, because it’s still impossible ti understand how something could be both necessary and contingent in the metaphysical sense. But it does clarify another aspect of what Kierkegaard elsewhere calls Christianity’s “offense” against reason. The logic of rigid designation guarantees the impossibility of reasoning your way from the contingent vehicle to the necessary content; the unprovability is built into the semantics of reference itself. The leap goes from the a posteriori access to a rigid designator’s referent to the commitment to a necessary identity, and no rule of inference can deliver that decision.

This helps explain why Climacus denies that the contemporary witness has any advantage over the disciple at second hand. One might think the contemporary at least has an advantage in fixing the reference of “Jesus”: they get to point and use the name in ostensive contact with its bearer whereas the later receiver inherits the name through a long causally-connected chain of testimony. But in Kripke’s picture of rigid designation, later users of the name designate just as rigidly as the original users. Both the disciple at second hand and the contemporary are using “Jesus” to pick out the same individual in every world where he exists, and both face exactly the same gap between the available empirical information about him and the identity claim.

Climacus insists that what’s important in the transmission from the contemporary generation to later generations is not the eyewitness testimony but rather the “proclamation,” i.e. the sheer assertion of the identity of Jesus and God. On the Kripkean reading, he’s right to do so: what does the theological work in the chain is the transmission of the rigidly designating use of the names together with the identity claim, and that doesn’t depend on the volume of testimony that’s preserved. The proclamation is what carries the modal content; the testimony, however vivid for the contemporary, was never decisive for anyone.