Questions & quantification A study of first order inquisitive logic

This dissertation focuses on the study of inquisitive first order logic InqBQ, a logical formalism encompassing questions in the presence of quantification. In particular, we focus on developing tools and techniques to study the expressive power of InqBQ and its entailment. The dissertation can be divided in four parts, each considering a different approach to study the logic.
In the first part we adapt Ehrenfeucht-Fraïssé games to InqBQ. Using the game, we achieve a characterization of the cardinality quantifiers definable in InqBQ, generalizing the analogous result for classical logic.
The second part presents several ways to manipulate and combine models of InqBQ. The theory developed allows us to prove two hallmarks of constructive logics: the Disjunction and Existence properties. The semantic proof we give allows us to further generalize the result to several classes of theories.
In the third part we shift our attention to axiomatizing fragments and variations of InqBQ. We first focus on the classical antecedent fragment, which contains—modulo logical equivalence—all the formulas corresponding to natural language sentences, and for which we provide a strongly complete axiomatization. Afterwards, we focus on the finite-width inquisitive logics, introduced by Sano [2011]. We answer two open problems: whether the logics are axiomatizable (they are), and whether InqBQ is their limit (it is not). Finally we introduce and study the bounded-width fragment, building on the completeness result for the finite-width logics.
The fourth part is an exploratory work developed for the propositional case. Firstly we present an algebraic semantics for inquisitive logic based on the so-called inquisitive (Heyting) algebras. Secondly we characterize inquisitive algebras in terms of their dual topological UV-spaces and define a novel topological semantics for inquisitive logic.