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**Tuesday, December 20, 2005**

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Lattice of ontologies

Function/structure descriptions

I changed the name of this thread from

Theories, Models, Reasoning, Language, and Truth

because the key concern/excitement I have is that the pure mathematical concept of a Boolean lattice might be revealed in a simple fashion.

Your note opens onto such a simple explanation.

http://colab.cim3.net/forum/ontac-forum/2005-12/msg00127.htm l

Lattice theory in physics, particularly in thermodynamics, does not have the same properties as does the Boolean lattice (abstracted from the ( U, subset, smallest element, largest element ) ) . Lattice theory (as in thermodynamics) is something I know a little about since I published in this area as early as 1987.

The ordered quadruple designation, you use, for the Boolean algebra is incomplete, but this is not important technical issue.

The ordering relationship and the definition of U is the critical issue, in terms of ontology construction and entailments. But we need to look closely at what a Boolean lattice is and what is proposed as a lattice of theories. What is the binary relationship in this lattice of theories?

It seems to some of us that one needs to be able the talk separately about the definition of the elements of an ontology, and that these elements should be in all cases "concepts".

The entailment of a set of concepts is then in two parts:

**inferential entailment:** how the concepts are used by some type of
machine or formal inference

**structural entailment:** how the concept
referent sits inside of specific situations.

So if one starts with a "universal set" as a set of concept representations, or taxa in structural bioinformatics, we can then impose an ordering relationship to organize this universal set into a tree like structure (taxonomy). This does not produce a Boolean lattice, it just produces a tree (a specific type of graph).

So how are lattice of theories developed?