John, thank you for the comment. I am familiar with lattices from,
among others, your KR book, which was one of the sources of inspiration
for my previously mentioned complaints. (01)
I hope for some more, *serious* discussion from the side of the OBO
John F. Sowa wrote:
> I'm happy that you were making a point that anybody who
> knew anything about logic would consider obvious:
> > I was repeatedly complaining on the OBO- and BFO-related
> > lists about the insistence, within that framework, on
> > single inheritance.
> Those people were probably confusing some of the practical
> problems in programming languages (where few, if any,
> programs are formally defined) with issues in logic,
> where everything is formally defined.
> > I mention this because the answer to my complaints, if any,
> > was invariably that single inheritance a) increases efficiency
> > of reasoning, b) is more natural and easier to use, and c) is
> > good for interoperability.
> When it comes to logic, those three statements are hopelessly
> false, confused, and misleading:
> a) There are *always* multiple ways to derive a proof of
> any theorem. Failing to permit them in a hierarchy does
> *nothing* to speed up theorem proving, and it can in many
> cases block important short cuts.
> b) Every animal, vegetable, and mineral on planet earth
> can be classified in an open-ended number of ways.
> Arbitrarily picking one and prohibiting the others is
> unnatural, confusing, and horribly difficult to use.
> c) It does *nothing* to promote interoperability. On the
> contrary, it can *block* interoperability when one
> system arbitrarily blocks paths that another uses.
> d) You can add the problems created by single inheritance
> in trying to merge ontologies. The result invariably
> has a superset of the inheritance paths of both. If
> multiple inheritance is prohibited, merging becomes
> In previous notes, I mentioned the large group of people who
> use Formal Concept Analysis to derive lattices, which support
> multiple inheritance in very clear, efficient, and easily
> visualizable ways.
> For a very brief summary, see Section 7 on lattices in
> my tutorial on math and logic:
> For more detail, including lots of software, see the FCA home page:
> Formal Concept Analysis Homepage
> For some pretty pictures of lattices generated by FCA, see
> Formal Concept Analysis Examples
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