Description logics have been extended in a number of ways to support defeasible reasoning in the KLM tradition. Such features include preferential or rational defeasible concept inclusion, and defeasible roles in complex concept descriptions. Semantically, defeasible subsumption is obtained by means of a preference order on objects, while defeasible roles are obtained by adding a preference order to role interpretations. In this paper, we address an important limitation in defeasible extensions of description logics, namely the restriction in the semantics of defeasible concept inclusion to a single preference order on objects. We do this by inducing a modular preference order on objects from each modular preference order on roles, and using these to relativise defeasible subsumption. This yields a notion of contextual rational defeasible subsumption, with contexts described by roles. We also provide a semantic construction for rational closure and a method for its computation, and present a correspondence result between the two.
Given the special status of concept inclusion in description logics (DLs), and
the historical importance of entailment in logic in general, past research efforts
to extend DLs with non-monotonic reasoning capabilities have focused
primarily on accounts of defeasible subsumption and the characterisation of defeasible
entailment. Semantically, the latter usually takes as point of departure an
ordering on a class of first-order interpretations [
Recently, we proposed a new direction to introduce defeasibility to DLs [
We achieved this by introducing defeasibility in the concept language via
an intuitive notion of normality for roles [
Britz and Varzinczak
of the form “ C is usually subsumed by D”, for example, “Chenin blanc wines are
usually unwooded”. In the extended language, one can also refer directly to, for
example, “Chenin blanc wines that usually have a wood aroma”. These notions
can also be combined seamlessly, as in: “Chenin blanc wines that usually have a
wood aroma are usually wooded”. This cannot be expressed in terms of
defeasible subsumption alone, nor can it be expressed w.l.o.g. using typicality-based
operators [
Nevertheless, even this generalisation left open the question of different, possibly incompatible, semantic characterisations of defeasibility in subsumption. A single ordering on individuals, as is usually assumed, does not suffice when preferences depend on context. In this paper, we therefore propose to induce multiple preference orders on objects derived from preference orders on role interpretations, and use these to relativise defeasible subsumption. This yields a notion of contextual defeasible subsumption, with contexts indicated by role names.
Building on previous work in the KLM tradition, we show that restricting the
preferential semantics to a modular semantics allows us to define and compute a
notion of rational entailment from a defeasible knowledge base. Rational closure
is a form of inferential closure studied by Lehmann and Magidor [
Defeasibility introduces a new facet of contextual reasoning not present in
deductive reasoning in that preference orders can be used to indicate context. While
an account of deductive reasoning with contexts in knowledge representation is
not intrinsically linked to defeasible reasoning, its integration with contextual
defeasible description logics warrants further investigation [
Contextual rational closure for defeasible ALC