After prototypo's comment I thought that I should try to be clearer in my criticism of
A system will always be in two of four states. It can be valid or invalid, and it can be consistent or inconsistent. A valid system is true under every possible interpretation, while an invalid system has at least one interpretation where it is not true. An inconsistent system is false under every possible interpretation, while a consistent system has at least one interpretation where it is true.
With the open world assumption there are always going to be a lot of interpretations open to us. We cannot tell a user that the data has a problem until we know that the data cannot possibly be right. This means that we want our system to be consistent, so we have to report an inconsistent system. Invalid systems are fine, as they could well evaluate to true, depending on future data. Of course, valid systems are also OK, as they always evaluate to true.
If we can prove inconsistency, then we can report the problem, and Kowari will be done.
The problem is that
owl:minCardinality statements will always be consistent in an open world. On occasion they will even be valid (when all the objects are
owl:differentFrom each other). So what is the point of statements with the
owl:minCardinality predicate? They can't ever be false, so they don't tell you anything! It's kind of like a belt and braces when your belt is unbreakable.
owl:minCardinality is not useful in the open world, then what is the point? I have to infer that it takes on the more intuitive meaning, and gets applied in a closed world sense. In that case it becomes useful to a database like Kowari. It is for this reason that I think that the contributor of this statement wasn't thinking clearly about the open world assumption. Otherwise, why include a statement of no value?
OK, I should concede that a system based on an
owl:minCardinality statement can be false... but only when the range of the predicate is a class with a restricted set of instances, where the size of that set is less than the cardinality. However, the Description Logic Handbook actually defines consistency as ABox data which meets the criteria of TBox data. This example demonstrates inconsistent TBox data, which isn't quite the same thing. In other words, it's a bad ontology, not bad data being described by the ontology. It's still inconsistent, but it doesn't meet the inconsistency definition put forward by the DL Handbook.
Thursday, March 31, 2005