There is, I think, a very good, but generally overlooked, case for the claim that necessity implies normativity; that is, that necessary claims are, ipso facto, normative claims as well.
(1)
Role in practical planning. Suppose that you are an engineer building a bridge. There are certain things to which your project must conform; among these are principles of mathematics directly relevant to the project of building a structure spanning a space. But this is nothing other than for relevant necessary claims (in this case mathematics) to function as relevant normative claims, since they impose a standard or requirement on your plans. But what is true of engineering in building a bridge is true of any sort of practical plans; there are always at least some necessary claims relevant to your plans, and these claims work just like any other kind of normative claim.
(2)
Role in rational discourse. The principle of noncontradiction is a necessary principle; what does that mean
in practice. In practice it means that its being necessary makes it function in rational discourse (say, in the evaluation of an argument) just like anything we would consider a normative principle. All discussion of rational argument, in fact, treats necessity as implying normativity for how one ought to act and reason.
(3)
Convergence of vocabularies. We in fact talk about necessary truths in normative terms. For instance, we can phrase the principle of noncontradiction as the norm that you shouldn't hold something to be and not to be in the same way at the same time. Likewise, we talk about at least some norms in necessity terms: they are things to which you must conform. This indicates that there is at least a strong analogy between the two. But that it is more than an analogy is suggested by the fact that we get absurdities if we try to treat the necessary as not normative, e.g., "This principle is certainly necessary but nobody has to act accordingly."
(4)
Deontic logic. In deontic logic, we have the well-known problem of deontic necessitation. But the whole of the problem just consists in assuming that it is wrong that the necessary should be 'obligatory';
as I've noted, this is probably not a fair assessment of the deontic logic itself, but if we look at how we use strong normative terms, we find that, in fact, there is in real life
some kind of connection between necessity and normativity, and so we should be putting that assumption in doubt, anyway. In any case, assuming that necessity implies normativity allows the most elegant and easy-to-work with kinds of deontic system. This would not be definitive on its own, but the only reason for not taking it to be a reason is if it were actually a bad model for our discourse. But, as noted above, this does not seem to be true.
As there seem to be contingently normative claims, it is not true that normativity implies necessity; necessity is the stronger modality. But it does appear that it is strong enough that it is reasonable to say that necessity implies normativity. And it is noteworthy that contingent normativities work a lot like conditional necessities.
There may, of course, be some kind of qualification. For instance, we don't usually pay attention to details of what is relevant to what when dealing with necessary truths (we just assume that they are relevant, somehow), but one might argue that normative claims require definite relevance to practical contexts; in which case it might be better to say that necessity, in a practical context, implies normativity. But this does not affect the fundamental point, which is that learning that something is a necessary truth would ipso facto be learning that it is normative for certain kinds of practical contexts.
