Axiomatizing Groenendijk’s Logic of Interrogation

Jeroen Groenendijk (1999, reprinted in this book) introduced a logic, which he called the Logic of Interrogation (henceforth LoI), that can be used to analyze which linguistic answers are appropriate in response to a given question. Groenendijk gave only a semantic definition of his logic. For practical applications like building question-answering systems, however, we also need to understand the proof theory of this logic (Monz, 2003b, Section 2.4). A better understanding of the proof theory of LoI also enables us to better grasp the empirical predictions made by the model. In this chapter, we bridge this gap, by providing a sound and complete axiomatization for LoI. Furthermore, we will show that the entailment relation of LoI is closely related to the model-theoretic notion of definability. Roughly speaking, the question Who came to the party? entails the question Did anybody come to the party? in LoI because the proposition that someone came to the party is definable in terms of the property of having come to the party (in the same way that the first-order sentence ∃x.Px is definable in terms of the property P ). This connection between question entailment and definability also shows up in the fact that an answer to a natural-language question is typically built up from instances of the question.