In this paper we take up the study of Henkin quantifiers with boolean variables  also known as partially ordered connectives . We consider first-order formulae prefixed by partially ordered connectives, denoted D, on finite structures. We characterize D as a fragment of second-order existential logic Σ 1♥ whose formulae do not allow for existential variables being argument of predicate variables. We show that Σ 1♥ harbors a strict hierarchy induced by the arity of predicate variables and that it is not closed under complementation, by means of a game-theoretical argument. Admitting for at most one existential variable to appear as the argument of a predicate variable already yields a logic coinciding with full Σ 1 , thus we show.