Inquisitive semantics

Inquisitive semantics is a framework in logic and natural language semantics. In inquisitive semantics, the semantic content of a sentence captures both the information that the sentence conveys and the issue that it raises. The framework provides a foundation for the linguistic analysis of statements and questions. It was originally developed by Ivano Ciardelli, Jeroen Groenendijk, Salvador Mascarenhas, and Floris Roelofsen.
Basic notions.
The essential notion in inquisitive semantics is that of an "inquisitive proposition".
Inquisitive propositions encode informational content via the region of logical space that their information states cover. For instance, the inquisitive proposition formula_1 encodes the information that } is the actual world. The inquisitive proposition formula_2 encodes that the actual world is either formula_3 or formula_4. 
An inquisitive proposition encodes inquisitive content via its maximal elements, known as "alternatives". For instance, the inquisitive proposition formula_2 has two alternatives, namely formula_6 and formula_7. Thus, it raises the issue of whether the actual world is formula_3 or formula_4 while conveying the information that it must be one or the other. The inquisitive proposition formula_10 encodes the same information but does not raise an issue since it contains only one alternative.
The informational content of an inquisitive proposition can be isolated by pooling its constituent information states as shown below.
Inquisitive propositions can be used to provide a semantics for the connectives of propositional logic since they form a Heyting algebra when ordered by the subset relation. For instance, for every proposition "P" there exists a relative pseudocomplement formula_12, which amounts to formula_13. Similarly, any two propositions "P" and "Q" have a meet and a join, which amount to formula_14 and formula_15 respectively. Thus inquisitive propositions can be assigned to formulas of formula_16 as shown below.
Given a model formula_17 where "W" is a set of possible worlds and "V" is a valuation function:
The operators ! and ? are used as abbreviations in the manner shown below.
Conceptually, the !-operator can be thought of as cancelling the issues raised by whatever it applies to while leaving its informational content untouched. For any formula formula_24, the inquisitive proposition formula_25 expresses the same information as formula_26, but it may differ in that it raises no nontrivial issues. For example, if formula_26 is the inquisitive proposition "P" from a few paragraphs ago, then formula_25 is the inquisitive proposition "Q".
The ?-operator trivializes the information expressed by whatever it applies to, while converting information states that would establish that its issues are unresolvable into states that resolve it. This is very abstract, so consider another example. Imagine that logical space consists of four possible worlds, "w"1, "w"2, "w"3, and "w"4, and consider a formula formula_24 such that formula_26 contains }, }, and of course formula_31. This proposition conveys that the actual world is either "w"1 or "w"2 and raises the issue of which of those worlds it actually is. Therefore, the issue it raises would not be resolved if we learned that the actual world is in the information state }. Rather, learning this would show that the issue raised by our toy proposition is unresolvable. As a result, the proposition formula_32 contains all the states of formula_26, along with } and all of its subsets.