Logical Connections of Statements at the Ontological Level

Logical Connections of Statements at the Ontological Level

Cungen Cao (Chinese Academy of Sciences, China), Yuefei Sui (Chinese Academy of Sciences, China) and Yu Sun (Chinese Academy of Sciences, China)
DOI: 10.4018/978-1-4666-1743-8.ch004
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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.
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In the first-order logic, there are three logical connectives 978-1-4666-1743-8.ch004.m01, 978-1-4666-1743-8.ch004.m02 and negation 978-1-4666-1743-8.ch004.m03 where 978-1-4666-1743-8.ch004.m04 can only be applied on formulas to form new formulas, and for any formula 978-1-4666-1743-8.ch004.m05 and any model 978-1-4666-1743-8.ch004.m06 is true in 978-1-4666-1743-8.ch004.m07 if and only if 978-1-4666-1743-8.ch004.m08 is not true in 978-1-4666-1743-8.ch004.m09

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).

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:

  • 978-1-4666-1743-8.ch004.m10 The logical level;

  • 978-1-4666-1743-8.ch004.m11 The epistemological level;

  • 978-1-4666-1743-8.ch004.m12 The ontological level;

  • 978-1-4666-1743-8.ch004.m13 The conceptual level, and

  • 978-1-4666-1743-8.ch004.m14 The linguistic level.

We believe that every level has its own negation.

The negation at the logical level is the logical negation 978-1-4666-1743-8.ch004.m15 on formulas. In the first order logic, the negation 978-1-4666-1743-8.ch004.m16 is applied only to formulas, i.e., if 978-1-4666-1743-8.ch004.m17 is a formula then so is 978-1-4666-1743-8.ch004.m18; and 978-1-4666-1743-8.ch004.m19 is false if and only if 978-1-4666-1743-8.ch004.m20 is true. Hence, 978-1-4666-1743-8.ch004.m21 and 978-1-4666-1743-8.ch004.m22 are contradictory.

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