# [DL] incoherence in DL semantics?

Enrico Franconi franconi at cs.man.ac.uk
Sat Jun 29 15:09:30 CEST 2002

```On June 28, Rod Moten writes:
> Naturally, we expect Meters and Centimeters should be related,
> i.e. minimally Meters should subsume Centimeters, which I write as
> Centimeters < Meters. If this is true then for every interpretation
> I, I(Centimeters) is a subset of I(Meters), written I(Centimeters)
> subseteq I(Meters), This is where I am having a problem.
> I(Centimeters) subseteq I(Meters) means that every element of
> I(Centimeters) is an element of I(Meters).  However, each element in
> I(Centimeters) has a different meaning in I(Meters). For example, 200 cm
> is not the same as 200 m, but 200 cm is 2 m. How do you reconcile this
> problem of incoherence in Description Logics? Is there something about DL I
> missed?

First of all, you have to clearly state what kind of extension the two
classes Centimeter and Meter do have. From your example, it seems that
in your interpretation the instances of these two classes are pairs of
type <integer,length-unit>. In this case, it is wrong to assume that
Centimeters < Meters, since no instance of Centimeter is also instance
of Meters. So, your "natural" and "minimal" assumption is definitely
wrong in this context. You are confusing the subsumption relation in
Description Logics with a generic ordering relation between terms
(like the one that holds between Centimeters and Meters).

So, there is no incoherence in Description Logics, but just in the use
you make of DLs.

Hope this helps
cheers
-- e.

Enrico Franconi                     - franconi at cs.man.ac.uk
University of Manchester            - http://www.cs.man.ac.uk/~franconi/
Department of Computer Science      - Phone: +44 (161) 275 6170
Manchester M13 9PL, UK              - Fax:   +44 (161) 275 6204

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