We formalise and study the concepts of Boolean determinacy and Boolean independence of a logical statement B with respect to a list of logical statements A1,…,An.  
Boolean determinacy means that the truth value of B is a Boolean function of the truth values of A1,…,An,  defined on a class of models construed as possible worlds. More precisely, this means that in every two models (possible worlds) where each of A1,…,An has the same truth values, B must have the same truth value, too. It can be viewed as a simplified and formalised version of the notion of supervenience.
Respectively, Boolean independence means that the truth value of B is functionally independent from the truth values of A1,…,An, defined on a class of models construed as possible worlds. 
We give explicit semantic definitions of these, following Väänänen’s logics of dependence and independence 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. 
On the technical side, we extend propositional logic with additional operators of determinacy and of independence, applied to lists of formulae. Thus, we construct logics of Boolean determinacy and of Boolean independence, which we provide with the semantics described above. We compare and analyse their expressiveness, obtain complete axiomatisations for them, and discuss some applications and extensions to modal logic. In particular, we discuss the epistemic interpretation of the concept of Boolean determinacy, relating to conditional knowledge.