We formalise and study the concepts of Boolean determinacy and Boolean
independence of propositions represented by logical formulae. Technically, the logics we propose extend the propositional logic with additional operators of determinacy and of independence, applied to tuples of formulae. Intuitively, Boolean determinacy means that the truth value of ψ is a Boolean function of the truth values of the formulae in Γ, defined in models built on sets of possible worlds. More precisely, this means that in every two possible worlds in the model where each of the formulae in Γ has the same truth values, ψ must have the same truth value, too. Thus, Boolean determinacy can be viewed as a formalised version of the notion of supervenience, as well as relativised knowledge.  Respectively, Boolean independence of tuples of formulae Γ and ∆ means that their respective tuples of truth functions are informationally independent (in a precise sense) from each other. These semantic definitions derive from Väänänen’s logics of dependence and independence developed in the context of team semantics, but we provide an alternative, possible worlds semantics for them. We argue that our approach is more natural and resolves some conceptual problems arising in the approach based on team semantics. We then compare and analyse the expressiveness of these logics, obtain complete axiomatisations for them, and discuss some applications and
extensions over modal logics.