Modal logic for subjective default reasoning

In this paper we introduce DML: Default Modal Logic. DML is a logic endowed with a two-place modal connective that has the intended meaning of 'If α, then normally β'. On top of providing a well-defined tool for analyzing common default reasoning, DML allows nesting of the default operator. We present a semantic framework in which many of the known default proof systems can be naturally characterized, and prove soundness and completeness theorems for several such proof systems. Our semantics is a 'neighborhood modal semantics', and it allows for subjective defaults, that is, defaults may vary among different worlds within the same model. The semantics has an appealing intuitive interpretation and may be viewed as a set-theoretic generalization of the probabilistic interpretations of default reasoning. We show that our semantics is most general in the sense that any modal semantics that is sound for some basic axioms for default reasoning is a special case of our semantics. Such a generality result may serve to provide a semantical analysis of the relative strength of different proof systems and to show the nonexistence of semantics with certain properties.