Inquisitive semantics takes sentences to express a proposal to the  participants in the conversation to update the common ground of the conversation (CG) in one or more ways. The question in (1a) proposes two alternative ways to update the CG, which correspond to the two responses (1b-c).

(1)	a. If Alf goes to the party, will Bea go too?	p -> ?q
	b. If Alf goes, then Bea will go as well.	p -> q
	c. If Alf goes, then Bea will not go.	p -> ¬q
	d. Alf will not go to the party.	¬p

Basic inquisitive semantics (InqB) accounts for the intuition that (1b-c) are responses that, if accepted by the other conversational participants, yield a CG that supports the question in (1a), settling the proposal that it expresses.

InqB does not account for the intuition that (1c)  rejects the proposal expressed by (1b), and vice versa. Radical inquisitive semantics (InqR) does account for this. It achieves this by not only specifying  support-conditions, as InqB does, but simultaneously also rejection-conditions.

InqB and InqR do not account for the intuition that (1d) dismisses a supposition that is shared by (1a)-(1c). This is just as much a way of settling the proposals that these sentences express, on a par with support and rejection. Suppositional inquisitive semantics  aims to characterize when a response suppositionally dismisses a given proposal. To achieve this, it does not only specify conditions for support and rejection, but  also for supposition dismissal.

We will present the semantics for a propositional language, extended with epistemic might and must.