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Re: [ontolog-forum] Ontologies vs Theories / Axioms vs Rules

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Mon, 24 Oct 2011 08:52:13 -0400
Message-id: <4EA55F7D.2020103@xxxxxxxxxxx>
I would like to return to the original topic of this thread, which Ali
started while I was traveling:    (01)

> Michael and Todd were talking about differences between Ontologies & Theories,
> and Rules & Axioms. I've actually encountered a number of people who make
> this distinction -- that an ontology doesn't really express "rules", and
> that these rules are somehow different from "axioms". The line is still
> blurry to me, but it seems to be that in this terminology, a "rule" refers
> to anything that OWL doesn't support, whereas in the CL (and FOL world)
> there is no distinction between these "rules" and "axioms".    (02)

The terminology is hopelessly confused because of the mixture of
many independently developed tools, methodologies, and disciplines.    (03)

 From the online Merriam-Webster, an ontology is    (04)

    "a particular theory about the nature of being or the kinds
    of things that have existence"    (05)

That implies that every ontology is a theory.  That is a comfortable way
of talking for people who have taken a course in logic.  Anyone with
that background would feel comfortable in talking about the "axioms"
of a theory.  By extension, they talk about the axioms of an ontology.    (06)

On the other hand, there are large numbers of "rule-based" languages,
which are usually Horn-clause (or quasi-Horn-clause) notations.
Their most obvious characteristic is a statement that uses the
word "if" or some variant, such as an arrow.    (07)

As for OWL, there is a distinction that the medieval Scholastics
had clarified a thousand years ago.  There were two logic-based
traditions that had evolved since antiquity:    (08)

  1. Categorical syllogisms, which are based on Aristotle's original
     notation for reasoning about the categories in his ontology.    (09)

  2. Hypothetical syllogisms, which are if-then "rules" that were
     developed a couple of centuries later.    (010)

In categorical syllogisms, the most common type is named Barbara:    (011)

    All A is B.
    All B is C.
    Therefore, all A is C.    (012)

But the medieval Scholastics pointed out that categorial syllogisms
such as Barbara can be converted to hypothetical syllogisms:    (013)

    If X is A, then X is B.
    If X is B, then X is C.
    Therefore, if X is A, then X is C.    (014)

Since OWL is an extension of Aristotle's original syllogisms,
it uses a categorial style of notation and reasoning.    (015)

Every university graduate from the 13th century to the early
20th century would have known these points.  But Bertrand Russell
considered syllogisms to be a competitor to symbolic logic.  So
he wanted universities to stop teaching Aristotelian logic and
switch to symbolic logic.    (016)

Good old Bertie won half the battle.  He got them to stop
teaching syllogisms.  Now most university graduates don't know
any kind of logic.    (017)

John    (018)

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