One concept, which I will call the concept of metaphysical modality in the narrow sense, crucially involves the subjunctive/indicative contrast, or the contrast between considering a scenario as counterfactual versus considering it as actual, and focuses on the subjunctive/counterfactual side. (This is why Chalmers is able, in 'The Foundations of Two-Dimensional Semantics' and other papers, to choose 'subjunctive necessity' as his preferred term for metaphysical necessity in the narrow sense.) It concerns how things could have been in a very broad sense. And so we can help fix the concept with familiar Kripkean talk like 'To be sure, we don't know a priori that Hesperus is Phosphorus. Given far-out enough empirical revelations, that could turn out to be wrong. But given that we're not mistaken about this - given that Hesperus is Phosphorus - then it could not have been otherwise. It is a necessary truth that Hesperus is Phosphorus'.
The other concept, which I will call the concept of metaphysical modality in the broad sense, doesn't involve this contrast. It may be roughly characterized as modality which is neither epistemic nor somehow conventional. Modal facts which are the way they are irrespective of anything to do with our knowledge, and irrespective of any conventions we might have, are metaphysical modal facts. And we might want to throw in something about the modality not being restricted as well.
To illustrate the difference, consider a proposition like 'This typewriter cannot have two of its keys depressed simultaneously' - or, to avoid the idea that this may be a case of some tacitly restricted modality, 'This typewriter cannot in the course of its proper functioning have two of its keys depressed simultaneously'. This proposition clearly has a modal element. Also, this modal element appears to have little to do with knowledge or some convention we have set up. If the proposition is true, then the typewriter in question has this modal property - that of not being able to have two of its keys depressed simulteneously in the course of its proper functioning - in virtue of the way it is, not in virtue of our state of knowledge or any convention we have set up. And so we might want to say that the modality in question is metaphysical in the broad sense. But it seems not to be an instance of metaphysical modality in the narrow sense. The subjunctive, or the consideration of scenarios as counterfactual, doesn't come into the matter; it is, we might say, about what the typewriter can actually do, not what it might have done had things gone differently (even if, in this case, there is a one-to-one correspondence between actual and counterfactual possibilities).
Another example of a proposition involving a modality which we should say is metaphysical in the broad sense but not in the narrow, is 'It is possible to win a game of chess in five moves'. Here the object of interest is something abstract (the game of chess), whereas in the first example the object of interest was a concrete mechanical thing.
Why is it important to realize that there are these two different concepts of metaphysical modality? One reason is that it seems very likely to be relevant to solving problems about the varieties of modality, a topic whose difficulty has become steadily more apparent in the decades following Naming and Necessity.
Another reason, which has been even closer to my concerns, is its relevance for the project of trying to analyze or give an account of metaphysical modality in the narrower sense. For instance, the account of this which I have been developing involves a notion which clearly has a modal component.
The account, which I will post on soon, says that a proposition is necessary iff it is, or is implied by, a proposition which is both inherently counterfactually invariant and true. And the notion of inherent counterfactual invariance is cashed out in terms of the counterfactual scenario descriptions producible by the language system to which the proposition in question belongs. Not those which it actually does produce in its career, but those which it can. (A proposition is inherently counterfactually invariant iff its negation does not appear in any of these producible counterfactual scenario descriptions.)
The question now arises: does the presence of this modal element - which should be a good sign to anyone who, like me, is suspicious that there could be any such thing as a reduction of a modal notion to non-modal notions - make the account circular? 'Circular' in this context seems like a dirty word, but note that if the answer is Yes, that wouldn't mean that the account is no good at all; it would still be far from obvious or trivial. It could then perhaps be seen as a recursive definition, presupposing some cases as a base, and explaining the rest in terms of it. But still, Yes might seem like the wrong answer. I suspect it is. Separating metaphysical modality in the broad sense from metaphysical modality in the narrow sense opens up a promising way of supporting a No; the account deals with metaphysical necessity in the narrow sense - subjunctive necessity, necessity when considering-as-counterfactual - and appeals, on the right hand side of the 'iff', to a distinct species of metaphysical modality in the broad sense. On this understanding, there is no circularity - or to put it more politely, recursiveness - in the account at all. Of course, it doesn't supply us with a key for analyzing modality away altogether, as some attempts at analyzing metaphysical necessity in the narrow sense (without perhaps isolating that sense sufficiently clearly) have tried to do, but that should probably be seen as one of its more important virtues.
Kripke, Saul A. (1980). Naming and Necessity. Harvard University Press.