It is sometimes suggested that the modifiers ‘metaphysically’, ‘physically’, ‘epistemically’, etc., in phrases like ‘metaphysically possible’ act as restrictors on the more general, univocal, property of possibility. (Compare: someone can be surprisingly wealthy, unjustly wealthy, or extremely wealthy — these are all just more specific ways of being wealthy.) On this model, there is a property — possibility — and modal language attributes it, often specifying the way in which the relevant situation is said to be possible. When we say that it is physically possible that a man should run the 100m-dash in 9.50 seconds, we say that this achievement is among a particular subset of the possible: the physically possible.
This isn't exactly how I think about possibility language, but it's not too far off.
One question raised by this approach concerns just what the unrestricted modality includes. It’s clear enough that the physically possible is a restriction on the metaphysically possible. One might continue to suggest that the metaphysically possible is a subset of the conceptual or logically possible—and perhaps further into various logically impossible ‘possibilities’. Or one might somewhere draw the line. George Bealer, in his contribute to the Gendler & Hawthorne Conceivability and Possibility anthology, draws the line at the metaphysically possible. He writes:
[S]ome people insist on distinguishing logical possibility and metaphysical possibility and so are led to the following: p is logically possible iff p is merely consistent with the laws of logic (i.e., not ruled out by logic alone). This usage, however, invites confusion. There are many logically consistent sentences that express obvious impossibilities (e.g., ‘Bachelors are necessarily women’, ‘Triangles are necessarily circles’, ‘Water contains no hydrogen’). If you buy into calling mere logical consistency a kind of possibility, why not keep going? For example: p is ‘sententially possible’ iff p is consistent with the laws of sentential logic. Then, since ‘Everything is both F and not F’ is not ruled out by sentential logic (quantifier logic is what rules it out), would it be possible in some sense (i.e., sententially possible) that everything is both F and not F?! Certainly not my ear! (78-79)
I don't think this comprises a very good argument, for at least two reasons.
First, the logical structure of the argument for drawing the line at metaphysical possibility is suspect. It follows a particular erroneous form of a slippery slope argument: if you permit X (which doesn’t seem too bad), then what’s to stop you from going on to permit Y (which seems terrible)? The difference in felt terribleness, if there is one, would provide just the needed traction between X and Y in order to avoid the slip. Remember that drawing the line at metaphysical possibility represents a substantive choice; one might try draw it even more narrowly — at physical possibility, say. Imagine a philosopher who refuses to countenance those ‘metaphysical possibilities’ which violate the laws of physics; he will insist that they’re in no sense possible. Against someone like Bealer, who believes in such physically impossible possibilities, he might offer just the same retort: “if you buy into calling mere metaphysical consistency a kind of possibility, then why not keep going?” If there is reason to countenance metaphysical possibilities—and I agree with Bealer that there is—then presumably, we will justify it by reference to the useful work to which thinking about metaphysical modality can be put. But for all Bealer has said, it may well be that further conceptual or logical possibilities can be put to similar work. (This seems to me very plausible.)
The second reason to be concerned with Bealer’s argument is that he makes an insufficient case for the undesirability of the bottom of his slippery slope. Bealer apparently finds the suggestion that there is a sense in which this contradiction is possible implausible. He doesn’t say why, but the invocation of how it strikes his ear suggests it may be based in a linguistic intuition; it just sounds terrible, perhaps, to say that there’s a sense in which it is possible that everything is both F and not F. But that there is some sense in which it possible does not, of course, imply that it’s a very interesting sense, or one that ordinary speakers are used to thinking about. Bealer ostends a notion of ‘sentential possibility’, abstracting away from any use to which thinking about it might be put. It shouldn’t come as any surprise, then, that, so presented, we shouldn’t have any interest in thinking about such ‘possibilities’. That doesn’t mean they’re not there, ready for us to take them up if and when the course of inquiry demands it. (Again, a philosopher who thought of physical possibility the way Bealer feels about metaphysical possibility might respond just the same way Bealer does, in response to what is conventionally recognized as the physically impossible metaphysically possible. "It's just not possible in any sense for humans to travel faster than light." Actually, I suspect that most people who haven't studied metaphysics are likely to respond this way.)