Inquisitive semantics offers a unified analysis of declarative and interrogative sentences by construing information exchange as a process of raising and resolving issues. In this dissertation, I apply and extend inquisitive semantics in various new ways. On the one hand, I build upon the theoretical insight of inquisitive semantics and explore the prospect of incorporating other types of content into our conception of information exchange. On the other hand, the logical framework underlying inquisitive semantics is also of great interest in itself as it enjoys certain unique properties and is thus worth further investigation. In the first paper, I provide an account of live possibilities and model the dynamics of bringing a possibility to salience using inquisitive semantics. This account gives rise to a new dynamic analysis of conditionals, which is capable of capturing what I call the Extended Sobel Inference. In the second paper, drawing on the fact that disjunction in inquisitive semantics is understood as introducing a set of alternative answers to a question, I propose a Questions-Under-Discussion-based account of informational redundancy to tackle various Hurford sentences. In the third paper, I explore the prospect of cashing out the theoretical intuition behind inquisitive semantics using a non-bivalent framework. I develop a new logic which invalidates the Law of Excluded Middle just like inquisitive logic, but unlike inquisitive logic, it employs a negation that vindicates Double Negation Elimination.