This article compares Defeasible Description Logics (DDL) and Topological Approach to reason on Ontologies with exceptions. DDL is integration between Description Logics and Defeasible Logics to deal with monotonic and non-monotonic parts of the knowledge bases respectively. Topological approach tries to reason on inconsistent knowledge bases using the conventional topological operators e.g., interior, exterior, border and closure. We develop neo-Topology based on topological operators and we make major development and improvements of current Topological approach by properly introducing the ``Thickness Border'' with strong inference rules. We proof the validity of the inference rules using set operations. We demonstrate both approaches with appropriate example. We show the advantages and disadvantages of both approaches.

The difference between typical instances and atypical instances in a natural categorization process has been introduced by E. Rosh and studied by cognitive psychology and AI. A lot of the knowledge representation systems are expressed in using fuzzy concepts but a degree of membership raises some problem for natural categorizations (especially to classification problems in anthropology, ethnology, archeology, linguistics but also in ontologies), but atypical instances of a concept cannot be apprehended adequately by different degrees from a prototype. Other formal approaches, as paraconsistent logics or non monotonic logics, conceptualize often atypical objects as exceptions. It had yet been developed an alternative way with the logics of determination of the objects (LDO). In this paper, we present the logics of typical and atypical (LTA) in order to give directly a logical approach of typicality / atypicality associated to a concept by a more common way than in LDO, in using only classes and not determination operators. It is introduced a distinction between predicative property and concept defined with its intension and its essence, a part of intension. A typical instance of a concept inherits all properties of intension; a typical instance inherits only properties of essence but it is a full member of the category associated to a concept and not a member with a weak degree of membership. In natural categorization, there are often instances (the exceptions) which do not inherit some properties of the essence; they cannot be considered as atypical instance and belong to the boundary of the category.