Inquisitive first-order logic is neither compact nor recursively axiomatizable

TL;DR

This paper proves that inquisitive first-order logic (InqBQ) is neither compact nor recursively axiomatizable, showing fundamental limitations in its meta-theoretic properties.

Contribution

It establishes that entailment in InqBQ is not compact and its validities are not recursively enumerable, answering longstanding open questions.

Findings

Entailment in InqBQ is not compact.

The set of validities of InqBQ is not recursively enumerable.

InqBQ cannot be recursively axiomatized.

Abstract

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.