In this paper we consider conditional prevision assessments on random quantities with nite set of possible values. After some preliminaries, we give the notions of generalized coherence and total coherence for imprecise conditional prevision assessments on finite families of conditional random quantities. Then, we examine some results on total coherence of such conditional previsions under different assumptions for the conditioning events. We first consider the case of logically incompatible conditioning events; then, we examine the case of logical independence. Finally, we examine the general case in which there may be some logical dependencies among the conditioning events. We show that in this case the property of total coherence is generally lost, while it is always valid a connection property. By exploiting such property, we obtain suitable totally coherent sets of conditional prevision assessments. We also give a necessary and sufficient condition of total coherence for interval-valued conditional prevision assessments.