An Intuitionisticaly based Description Logic

Classical Description Logic has been widely used as a basis for ontology creation and reasoning in many knowledge specific domains, including Legal AI. As in any other domain, consistency is an important issue for legal ontologies. However, due to its inherently normative feature, coherence (consistency) in legal ontologies is more subtle than in other domains. Consistency, or absence of logical contradictions, seems more difficult to maintain when more than one law system can judge a case. This is called a conflict of laws. There are some legal mechanisms to solve these conflicts, some of them stating privileged fori, other ruling jurisdiction, etc. In most of the cases, the conflict is solved by admiting a law hierarchy or a law precedence. Even using these mechanisms, coherence is still a major issue in legal systems since each layer in this legal hierarchy has to be consistent. As consistency is a direct consequence of how one deals with logical negation and subsumption. Negation and subsumption play a central role in ontology coherence. An adequate intuitionistic semantics for negation in a legal domain comes to the fore when we take legally valid individual statements as the inhabitants of our legal ontology. This allows us to elegantly deal with particular situations of legal coherence, such as conflict of laws, as those solved by Private International Law analysis. In [7, 5, 8] we present an Intuitionistic Description Logic, called iALC for Intuitionistic ALC (for Attributive Language with Complements, the canonical classical description logic system). A labeled sequent calculus for iALC based on a labeled sequent calculus for ALC [10], was also presented. In these previous articles, we discussed the jurisprudence foundation of our system, and show how we can perform a coherence analysis of “Conflict of Laws in Space” by means of iALC. This conflict happens when several laws can be applied, with different outcomes, to a case depending on the place where the case occurs. Typical examples are those ruling the rights of a citizen abroad. In [6], we presented the semantics of iALC on the framework for constructive modal logics presented by Simpson [12] and adapted to description languages by Paiva [4]. We apply this logic to the problem of formalizing legal knowledge. Description Logics are an important knowledge representation formalism, unifying and giving a logical basis to the well known AI frame-based systems of the eighties. Description logics are very popular right now. Given the existent