Building on ideas from inquisitive semantics, the recently proposed framework of in-quisitive epistemic logic (IEL) provides the tools to model and reason about scenarios in which agents do not only have information, but also entertain issues. This frame-work has been shown to allow for a generalization to issues of important notions, such as common knowledge and public announcements, and it has been argued to form a suitable basis for the analysis of information exchange as an interactive process of raising and resolving issues. From an abstract point of view, the system is interesting, in that it implies extending the logical operations, including the modalities, beyond the truth-conditional realm, in such a way that they can embed not only standard declarative formulas, but also interrogatives. The present paper investigates the logic of IEL, building up to a completeness result. It is shown that the standard logical features of the logical constants extend smoothly beyond the truth-conditional realm, except for double negation, which is the hallmark of truth-conditionality. In partic-ular, while the modalities of IEL operate in a crucially richer semantic space than Kripke modalities do, they retain entirely standard logical features.