Logical Connections of Statements at the Ontological Level

In the classical formal logics, the negation can only be applied to formulas, not to terms and predicates. In (frame-based) knowledge representation, an ontology contains descriptions of individuals, concepts and slots, that is statements about individuals, concepts and slots. The negation can be applied to slots, concepts and statements, so that the logical implication should be considered for all possible combinations of individuals, concepts, slots and statements. In this regard, the logical implication at the ontological level is different from that at the logical level. This paper attempts to give such logical implications between individuals, concepts, slots, statements and their negations. InTroduCTIon In the first-order logic, there are three logical connectives Ú , Ù and negation Ø, where Ø can only be applied on formulas to form new formulas, and for any formula Øj and any model M, Øj is true in M if and only if j is not true in M . In natural languages, the connectives and negations have many forms. For example, the exclusive disjunction (exclusive or) and inclusive disjunction (inclusive or). For the negation, the forms are varying. The negation can be applied to a statement (He is not happy), a concept (not a happy man), an individual (Not he is happy) and a value of an attribute (unhappy). DOI: 10.4018/jcini.2010070105 60 International Journal of Cognitive Informatics and Natural Intelligence, 4(3), 59-85, July-September 2010 Copyright © 2010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. To formalize the different forms of the negation in natural languages, we consider the negation at the ontological level, where the levels are a classification of the various primitives used by knowledge representation systems, firstly defined by Brachman (1979), based on which Guarino (1994) added the ontological level to the levels: • The logical level; • The epistemological level; • The ontological level; • The conceptual level, and • The linguistic level. We believe that every level has its own negation. The negation at the logical level is the logical negation Ø on formulas. In the first order logic, the negation Ø is applied only to formulas, i.e., if j is a formula then so is Øf ; and j is false if and only if Øj is true. Hence, j and Øj are contradictory. The negation at the epistemological level is the negation on formulas and on modalities. For example, let B be the epistemological modal believe, and j be a first-order formula. Then, we have the following formulas: B B B B j j j j ,( ) , ( ), ( ); Ø Ø Ø where the negation in ( ) ØB j is at the epistemological level; the negations in B( ) Øj and Ø( ) Bj are at the logical level. For example, the following sentences Lois believes that Clark Kent is strong. Lois does not believe that Clark Kent is strong. Lois believes that Clark Kent is not strong. It is not true that Lois believes that Clark Kent is strong.