Conditionals, information, and inference

Conditionals are fascinating and versatile objects of knowledge representation. On the one hand, they may express rules in a very general sense, representing, for example, plausible relationships, physical laws, and social norms. On the other hand, as default rules or general implications, they constitute a basic tool for reasoning, even in the presence of uncertainty. In this sense, conditionals are intimately connected both to information and inference. Due to their non-Boolean nature, however, conditionals are not easily dealt with. They are not simply true or false — rather, a conditional “if A then B” provides a context, A, for B to be plausible (or true) and must not be confused with “A entails B” or with the material implication “not A or B.” This ill- trates how conditionals represent information, understood in its strict sense as reduction of uncertainty. To learn that, in the context A, the proposition B is plausible, may reduce uncertainty about B and hence is information. The ab- ity to predict such conditioned propositions is knowledge and as such (earlier) acquired information. The ? rst work on conditional objects dates back to Boole in the 19th c- tury, and the interest in conditionals was revived in the second half of the 20th century, when the emerging Arti? cial Intelligence made claims for appropriate formaltoolstohandle“generalizedrules.”Sincethen, conditionalshavebeenthe topic of countless publications, each emphasizing their relevance for knowledge representation, plausible reasoning, nonmonotonic inference, and belief revision.