Formal similarities and differences among qualitative conditional semantics

Various semantics have been used for conditionals in the area of knowledge representation and reasoning. In this paper, we study similarities and differences between a purely qualitative semantics based on the popular system-of-spheres semantics of Lewis, an ordinal semantics making use of rankings, a possibilistic semantics, and a semantics representing conditionals by probabilities in a qualitative way. As a common framework for the corresponding logics, we use Goguen and Burstall’s notion of institutions whose central motto is that truth is invariant under the change of notation. The institution framework provides the formal rigidity needed for our investigation, but leaves enough abstract freedom to formalize and compare quite different logics. We show precisely in which sense the conditional semantics mentioned above are logically similar, and point out the semantical subtleties each semantics allows.