Default reasoning about probabilities

Acknowledgments First and foremost I would like to thank Harald Ganzinger and Hans J urgen Ohlbach who provided me with the opportunity to conduct this research at the Max-Planck-Institute for Computer Science. Here I found a perfect environment that gave me all the freedom I wanted and every support I needed. Hans J urgen put me on the track of probabilistic reasoning in terminological logics. The longer I followed this track, the more fascinating I found the questions I encountered, and the more convinced I became of doing just the research that suited me best. Alas, in this process I also left behind terminological logics and what Hans J urgen originally had in mind that my thesis should be about. More than anybody else, Emil Weydert has kept a close look on the progress of my work. While not sparing his critical comments, he showed an interest in this work that gave me much encouragement, and asked the questions that helped me nd the right directions for my investigations. I have much beneetted from the expertise of Martin Beibel and Daniel Hug in the elds of probability theory and convex geometry, as well as their direct access to the well-stocked mathematical library at the University of Freiburg. At one stage of my work I shamelessly exploited Peter Barth's experience in C ++-programming. I don't know how I should ever have minimized a single cross-entropy without his help. My roommate Luca Vigann o has provided me with additional intellectual challenges by trying to teach me Italian and introducing me to a silly computer game called xjewel. In neither discipline have I come close to his mastery.