#### Quantitative Information

In MOTEL we use the cardinality-based approach proposed by Owsnicki-Klewe [12] for dealing with number restrictions. Unfortunately, this approach is incomplete for languages in which concept disjointness is expressible.

The approach of Baader and Hollunder [1], by contrast, provides a complete tableau method for , but has some disadvantages:

1. The approach is not adequate for dealing with large numbers. Consider the following example: Suppose the universe consists of at most thirty objects. If there are at least twenty objects in C and there are at least twenty objects in D, then there are at least ten objects in the intersection of C and D.

The human ability to draw this conclusion is completely independent of the numbers we are using. Multiplying all numbers occurring in the example by a factor of 10 wouldn't make it any harder for us come up with the correct answer. Quite the opposite is true for the tableau method.

2. The basic inference mechanism provided by tableau theorem provers is consistency checking for knowledge bases. This is adequate for answering queries that can be solved by checking the consistency of a suitably extended knowledge base, for example, for problems like subsumption, instantiation, and classification.

But the most suggestive class of queries for knowledge bases in , e.g. the question `How many objects are in C and D?' in the example above, cannot even be formulated.

A promising approach to quantitative reasoning with numerical quantifiers seems to be that of Hustadt, Ohlbach, and Schmidt [9], who investigate a translation technique which translates modal logics with graded modalities into a fragment of many-sorted first-order logic. For, expressions can be associated directly with modal expressions.

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