entailments of imperatives and modals

I take it almost as a datum that, for each pair of questions below, your intra-pair answers should be the same. (‘!p’ should be read as ‘make it that p’, and ‘must’ is some sort of generic deontic modal, ‘may’ is its dual.)

Pair 1.

Q1. Does !(p and q) entail !p?

Q2. Does must(p and q) entail must(p)?

Pair 2.

Q3. Does !(p or q) entail may(p)?

Q4. Does must(p or q) entail may(p)?

One nice feature of views (like Magdalena Schwager’s) that treat imperatives as a species of necessity modal is that intra-pair consistency is automatic.

Somewhat oddly, however, many other views I’ve looked at require giving different answers for at least one of the pairs. Krister Segerberg’s logic of imperatives, for example, says ‘no’ to Q1 and ‘yes’ to Q2; his ‘no’ to Q1 follows directly from his favored resolution of Ross’s Paradox. Maria Aloni’s solution to the Free Choice Permission problem involves saying ‘yes’ to Q3 and ‘no’ to Q4. (Aloni treats ‘!’ as a modal, but as something much stronger than an ordinary necessity modal — very roughly, ∇.)

This seems to me like a rather large problem for such views. Am I wrong about that?

Addendum: there’s an immediate problem with letting your answers to Q3 and Q4 come apart, if, like Aloni, you go in for a semantic solution to the Free Choice Permission problem, i.e., you hold that may(p or q) entails may(p). Suppose, safely enough, that must(p or q) entails may(p or q) (actually, Aloni can’t allow this, but this seems like a strike against her view). Then must(p or q) will entail may(p). Contradiction.