Propositional possibilistic logic handles pairs made of a proposition and a level expressing a degree of certainty; these levels belong to a totally ordered scale. This basic possibilistic logic only allows for the conjunction of possibilistic formulas, in agreement with the min decomposability of necessity measures for this connective. Generalized possibilistic logic extends this formalism to negation and disjunctions of weighted pairs of formulas. In this paper, we consider a class of possibilistic logics where propositions are labeled by elements of a Boolean algebra. We first consider the example where propositions are associated with groups of agents that believe in them. This multiagent logic is then extended by attaching degrees of necessity to pairs (proposition, set of agents), and a multiagent counterpart of generalized possibilistic logic is proposed as well. Other examples of Boolean-valued formulas are discussed, where the Boolean labels represent time intervals, or yet other propositional formulas representing reasons to believe propositions.