Linking Makinson and Kraus-Lehmann-Magidor preferential entailments

Twelve years ago, various notions of preferential entailment have been introduced. The main reference is a paper by Kraus, Lehmann and Magidor (KLM), one of the main competitor being a more general version defined by Makinson (MAK). These two versions have already been compared, but it is time to revisit these comparisons. Here are our main results: (1) These two notions are equivalent, provided that we restrict our attention, as done in KLM, to the cases where the entailment respects logical equivalence (on the left and on the right). (2) A serious simplification of the description of the fundamental cases in which MAK is equivalent to KLM, including a natural passage in both ways. (3) The two previous results are given for preferential entailments more general than considered in some texts, but they apply also to the original definitions and, for this particular case also, the models can be simplified.