### Ontolog Forum

# Upper Ontology Summit Discussion

- In support of the UpperOntologySummit

## (1) Proposed Methodology Outline

- A suggested methodology for finding the maximum commonality among ontologies:

- By Michael Uschold

- I have long believed that the best solution for reaching agreement on

ontologies at any level is:

- Agree on everything/anything that you can that is uncontentious [or

contentious only at a superficial level] in terms of the 'things of interest'.

- If you can also agree on the same terms for the things of interest,

then great. If not, then use different terms and map/record them as synonyms.

- Agree to disagree on other things, when there are good reasons for

different stakeholders that have different needs (e.g. 3d/4d)

- where possible, map between the diff 'things of interest' so that a

user can to the maximal extent possible, enjoy the experience of a virtual CEO, even though it is more messy under the bonnet/hood. If a lattice of theories works for this, then great.

- Patrick Cassidy Commented:
- Mike's four suggested stages might well serve as an outline of the process that will succeed. I would only add that the custodians of the existing upper ontologies may conclude that small modifications of their own ontologies in the interest of increasing the level of commonality would create significant benefits with minimal cost.

## (2) Theoretical Relations among ontologies

- By Michael Gruninger

The notes for the Upper Ontology Summit contain two key ideas: -develop methods to relate the existing upper ontologies to each other. - create a common subset ontology that is compatible with all of the linked upper ontologies.

- There are two fundamental relationships between ontologies that we need

to consider:

- 1. extension
- 2. definable interpretation

- First, a few definitions:
- A theory is a set of sentences in a language conformant with Common Logic.
- An ontology is a set of theories.
- Suppose that the nonlogical lexicon of a theory T2 is a subset of the nonlogical lexicon of the a theory T1. T1 is an extension of T2 if the axioms in T1 entail the axioms in T2.
- T1 is definably interpretable in T2 iff for each symbol in the nonlogical lexicon of T1 the relation/function/constant denoted by the symbol is definable by a sentence S in the language of T2.
- Theory T1 generalizes theory T2 iff T1 is definably interpretable in a theory T3 and T2 is a consistent extension of T3.
- (The intuition is that the more general a theory, the weaker it is, so that theories are extensions of the theories in the Common Subset Ontology (CSO), with definable interpretation being used in cases where different nonlogical lexicons are used.)
- The idea is that we can design a Common Subset Ontology (CSO) by solving the following problem for the theories contained in the set of existing upper, mid-level, and domain-specific ontologies:

- Given two theories T1 and T2, determine whether there exists a theory that generalizes both.
- Theories that do not have any generalizations are candidates for inclusion in the CSO.

- Additional Comments
- 1. This is a well-posed problem with a definite solution; it is not a matter of philosophical differences.
- 2. Evaluation of the relationships between ontologies is made using their axioms alone; it cannot rely on intended models of concepts that are not axiomatized. If the axioms of an ontology are insufficient to capture their users' intended semantics, then there is little progress that can be made towards integration; we risk descent into logomachy, as far too many previous efforts have done.
- 3. Theories may be generalizations of each other. For example, Hilbert's geometry and Tarski's geometry are definably interpretable in each other, even though they

have different primitives and different relations. In such cases, either or both theories could be included in the CSO.

- michael gruninger