A Logical Study on Qualitative Default Reasoning with Probabilities

Only very special subclasses of probability distributions can be used for qualitative reasoning that meets basic logical demands. Snow’s atomic bound systems (big-stepped probabilities) provide one positive example for such a subclass. This paper presents a thorough investigation of the formal logical relationships between qualitative and probabilistic default reasoning. We start with formalizing qualitative conditional logic, as well as both standard and big-stepped probabilistic logic as abstract logical systems, using the notion of institutions. The institution of big-stepped probabilities turns out to be a proper combination of the other two. Moreover, the framework of institutions offers the possibility to elaborate exactly the properties that make probability distributions suitable for qualitative reasoning.