Explaining decisions of black-box classifiers is both important and computationally challenging. In this paper, we scrutinize explainers that generate feature-based explanations from samples or datasets. We start by presenting a set of desirable properties that explainers would ideally satisfy, delve into their relationships, and highlight incompatibilities of some of them. We identify the entire family of explainers that satisfy two key properties which are compatible with all the others. Its instances provide sufficient reasons, called weak abductive explanations. We then unravel its various subfamilies that satisfy subsets of compatible properties. Indeed, we fully characterize all the explainers that satisfy any subset of compatible properties. In particular, we introduce the first (broad family of) explainers that guarantee the existence of explanations and their global consistency. We discuss some of its instances including the irrefutable explainer and the surrogate explainer whose explanations can be found in polynomial time.