Inquisitive first-order logic, InqBQ, is a system which extends classical first-order logic with formulas expressing questions. From a mathematical point of view, formulas in this logic express properties of sets of relational structures. This paper makes two contributions to the study of this logic. First, we describe an Ehrenfeucht–Fraïssé game for InqBQ and show that it characterizes the distinguishing power of the logic. Second, we use the game to study cardinality quantifiers in the inquisitive setting. That is, we study what (...) statements and questions can be expressed in InqBQ about the number of individuals satisfying a given predicate. As special cases, we show that several variants of the question how many individuals satisfy α(x) are not expressible in InqBQ, both in the general case and in restriction to finite models. (shrink)

We present a natural standard translation of inquisitive modal logic $\mathrm{InqML}$ into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of $\mathrm{InqML}$. This translation is based on a graded notion of flatness that ties the inherent second-order, team-semantic features of $\mathrm{InqML}$ over information states to subsets or tuples of bounded size. A natural notion of pseudo-models, which relaxes the non-elementary constraints on the intended models, gives rise to an elementary, purely (...) model-theoretic proof of the compactness property for $\mathrm{InqML}$. Moreover, we prove a Hennessy-Milner theorem for $\mathrm{InqML}$, which crucially uses $\omega $ -saturated pseudo-models and the new standard translation. As corollaries we also obtain van Benthem style characterisation theorems. (shrink)

Inquisitive first order logic is an extension of first order classical logic, introducing questions and studying the logical relations between questions and quantifiers. It is not known whether is recursively axiomatizable, even though an axiomatization has been found for fragments of the logic. In this paper we define the \—classical antecedent—fragment, together with an axiomatization and a proof of its strong completeness. This result extends the ones presented in the literature and introduces a new approach to study the axiomatization problem (...) for fragments of the logic. (shrink)