In recent years, the logic of questions and dependencies has been investigated in the closely related frameworks of inquisitive logic and dependence logic. These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core. In this paper, we broaden the scope of these investigations by studying questions and dependency in the context of intuitionistic logic. We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models. The associated logic is a conservative extension of intuitionistic logic with questions and dependence formulas. We establish a number of results about this logic, including a normal form result, a completeness result, and translations to classical inquisitive logic and modal dependence logic.

In this paper, we axiomatize the negatable consequences in dependence and independence logic by extending the systems of natural deduction of the logics given in [22] and [11]. We prove a characterization theorem for negatable formulas in independence logic and negatable sentences in dependence logic, and identify an interesting class of formulas that are negatable in independence logic. Dependence and independence atoms, first-order formulas belong to this class. We also demonstrate our extended system of independence logic by giving explicit derivations for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms.

Both propositional dependence logic and inquisitive logic are expressively complete. As a consequence, every formula in the language of inquisitive logic with intuitionistic disjunction or intuitionistic implication can be translated equivalently into a formula in the language of propositional dependence logic without these two connectives. We show that although such a (noncompositional) translation exists, neither intuitionistic disjunction nor intuitionistic implication is uniformly definable in propositional dependence logic.

We consider team semantics for propositional logic, continuing In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We associate with every formula ϕ(symbol) based on finitely many propositional variables the set (left open bracket)ϕ(symbol)(right open bracket) of teams that satisfy ϕ(symbol). We define a maximal propositional team logic in which every set of teams is definable as (left open bracket)ϕ(symbol)(right open bracket) for suitable ϕ(symbol). This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the maximal propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.

In this paper, we define a multi-type calculus for inquisitive logic, which is sound, complete and enjoys Belnap-style cut-elimination and subformula property. Inquisitive logic is the logic of inquisitive semantics, a semantic framework developed by Groenendijk, Roelofsen and Ciardelli which captures both assertions and questions in natural language. Inquisitive logic adopts the so-called support semantics (also known as team semantics). The Hilbert-style presentation of inquisitive logic is not closed under uniform substitution, and some axioms are sound only for a certain subclass of formulas, called flat formulas. This and other features make the quest for analytic calculi for this logic not straightforward. We develop a certain algebraic and order-theoretic analysis of the team semantics, which provides the guidelines for the design of a multi-type environment accounting for two domains of interpretation, for flat and for general formulas, as well as for their interaction. This multi-type environment in its turn provides the semantic environment for the multi-type calculus for inquisitive logic we introduce in this paper.