Inquisitive first-order logic is neither compact nor recursively axiomatizable

Inquisitive logic is a research program that extends the scope of logic to cover not only statements, but also questions. In the context of this program, a logic that plays a prominent role is inquisitive first-order logic, InqBQ, which extends classical first-order logic with a question-forming disjunction and a question-forming existential quantifier. This logic makes it possible to formalize a broad range of questions, and to capture their logical relations to each other and to statements. Since its introduction in 2009, two central questions about the meta-theoretic properties of InqBQ have been open: the first is whether entailment is compact, in the sense that any conclusion that follows from a set of premises already follows from a finite subset of these premises; the second is whether the set of validities is recursively enumerable and, thus, whether the logic admits a recursive axiomatization. We settle these questions in the negative: entailment in InqBQ is not compact, and the set of validities of InqBQ is not recursively enumerable.