Re: SUO: Re: ONT Intension & Extension
John,
Thanks for your comments.
(my responses below).
----- Original Message -----
From: "John F. Sowa" <sowa@bestweb.net>
To: "Robert E. Kent" <rekent@ontologos.org>
Cc: "Jon Awbrey" <jawbrey@oakland.edu>; "SUO"
<standard-upper-ontology@ieee.org>; "Ontology" <ontology@ieee.org>
Sent: Sunday, January 20, 2002 12:41 PM
Subject: Re: SUO: Re: ONT Intension & Extension
> Robert,
>
> I did the comparison you suggested, and the definition in the IFF
> document is wrong.
No, it is not wrong! The definitions, terminology and axiomatics in all of
the IFF follow the definitions, standard terminology and axiomatics of
well-known mathematical ideas.
> REK> Compare Alonzo Church's discussion (below) of the intension and
> extension of
> > a concept with the mathematical (Formal Concept Analysis) (Rudolf Wille
and
> > Garrett Birkhoff) definitions of the intent and extent of a formal
concept
> > on the middle of page 30 in the IFF Classification Ontology document
> >
[http://suo.ieee.org/IFF/versions/20020102/IFFClassificationOntology.pdf].
>
> The error is clearly seen on page 31 of the IFF document where it
> says that a concept c1 is a subconcept of c2 when the extent of c1
> is a subset of the extent of c2 "or equivalently" when the
> intent of c1 includes (I don't know how you intend to read
> the superset symbol when comparing intensions, so I used the
> word "includes") the intent of c2.
The definition in the IFF is precisely the definition of a formal concept in
FCA (definition 20, page 18, "Formal Concept Analysis: Mathematical
Foundations" by Bernhard Ganter and Rudolf Wille). The extent consists of
all objects (IFF instances) that satisfy the attributes of the formal
concept and the intent consists of all attibutes (IFF types) satisfied by
the objects of the formal concept. A formal concept is a closed element of
the Galois connection of FCA derivation. And more specifically, the
equivalence noted in the IFF document is precisely stated in definition 21,
page 19, of the Ganter-Wille text.
Robert E. Kent
rekent@ontologos.org