Information types in intuitionistic predicate logic with constant domains

In this paper, we study two operators, inquisitive disjunction and inquisitive existential quantifier, in the context of first-order intuitionistic logic with constant domains. We explain that these operators allow us to express types of intuitionistic propositions. We first provide this language with a relational semantics and formulate a sound axiomatic system. Completeness of the system for the full language is presented as an open problem but we prove completeness for a rich fragment of the language adapting the methods developed by Gianluca Grilleti in the context of classical inquisitive logic. We also develop a general algebraic framework in which we characterize the class of “Kripkean algebras” generated by the relational semantics. In this way, we obtain our first algebraic characterization of the logic of intuitionistic types. We further characterize the class of all homomorphic images of “Kripkean algebras” that we call “inquisitive algebras”, thus obtaining a second, more general algebraic semantics for this logic.