Topological Semantics for Default Conditional Logic

Conditional logics have played an important role in the attempts to investigate the foundations of default reasoning. The conditional connective alpha double right arrow beta is interpreted as a `normality' statement (`if.., then normally.. ') or a `typicality' statement (assertion about prototypical properties of objects). Several conditional logics for defeasible reasoning have been proposed in the literature and the relation of well-known conditional systems to the KLM machinery of logics has been investigated. In this preliminary report, we contribute to the study of topological semantics for Default Conditional Logics. The perspective seems meaningful and promising, once somebody observes two facts: (i) the privileged relation of modal logic S4 to the normality conditionals - the relation has been discovered already from the '90s, and (ii) the prominent relation of S4 to topology, a result that dates back to the work of McKinsey & Tarski from the '40s. We introduce topological semantics for conditional logic, in the style of the Scott-Montague models introduced by B. Chellas in the '70's, further exploited in the filter-based models for conditional of S. Ben-David and R. Ben-Eliyahu-Zohary. We provide some first results on the expressive power of the emerging logics and point directions for future research.