Conditional inference under disjunctive rationality

The question of conditional inference, i.e., of which conditional sentences of the form "if ? then, normally, ?" should follow from a set KB of such sentences, has been one of the classic questions of AI, with several well-known solutions proposed. Perhaps the most notable is the rational closure construction of Lehmann and Magidor, under which the set of inferred conditionals forms a rational consequence relation, i.e., satisfies all the rules of preferential reasoning, plus Rational Monotonicity. However, this last named rule is not universally accepted, and other researchers have advocated working within the larger class of disjunctive consequence relations, which satisfy the weaker requirement of Disjunctive Rationality. While there are convincing arguments that the rational closure forms the "simplest" rational consequence relation extending a given set of conditionals, the question of what is the simplest disjunctive consequence relation has not been explored. In this paper, we propose a solution to this question and explore some of its properties.