S4

Here is an argument for S4. We want metaphysical necessity to be the strongest kind of necessity without arbitrary restrictions. If one responds that conceptual or strictly logical necessity are stronger, the answer is that they are, nonetheless, arbitrarily restricted, being dependent on a particular set of rules of inference and axioms. (The only non-arbitrary way to specify which which axioms are permitted is to say that it is all the fundamental metaphysically necessary propositions that are axioms, and then we presumably get metaphysical necessity.) Now, if L is a necessity operator, then LL is also a necessity operator. If LL is not equivalent to L, then LL is a stronger necessity operator. If LL counts as arbitrarily restricted, then we have reason to think that so does L, since L is even more restricted than LL, and it seems arbitrary to work with L instead of LL or LLL. And if LL doesn't count as arbitrarily restricted, then L is not the strongest non-arbitrarily restricted necessity operator. So if L is metaphysical necessity, L and LL are equivalent.

The dual of this argument is that metaphysical possibility is the most fundamental sort of possibility. But if M is metaphysical possibility, and MM is not equivalent to M, then MM will be a more fundamental possibility. So, if M is metaphysical possibility, M and MM are equivalent.