For better, for worse: Comparative orderings on states and theories

In logic, including the designer logics of artificial intelligence, and in the philosophy of science, one is often concerned with qualitative, comparative orderings on the states of a system, or on theories expressing information about the system. States may be compared with respect to normality, or some preference criterium, or similarity to some given (set of) state(s). Theories may be compared with respect to logical power, or to truthlikeness, or to how well they capture certain information. We explain a number of these relations, study their properties, and unravel some of their interrelationships.
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