SUO: Re: Intension & Extension
Jon and others,
----- Original Message -----
From: "Jon Awbrey" <jawbrey@oakland.edu>
To: "Robert E Kent" <rekent@ontologos.org>
Cc: "Ontology" <ontology@ieee.org>
Sent: Wednesday, January 23, 2002 12:16 AM
Subject: Re: Intension & Extension
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> RK = Robert Kent
>
> RK: 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:
>
> RK:
http://suo.ieee.org/IFF/versions/20020102/IFFClassificationOntology.pdf
>
> RK: As an illustrious example, see the 19 formal concepts (with
> explicit intent and extent) in the concept lattice of the
> "Living Classification" on pages 74-76 of the same IFF
> Classification Ontology document.
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> Robert,
>
> I have been looking at the Living Classification Lattice that
> you mentioned, and I have a couple of questions just to start:
>
>
o-------o-------o-------o-------o--------o-----o------o--------o--------o---
------o
> | | needs | lives | lives | needs | di | mono | motile | limbed |
suckles |
> | | water | in | on | chloro | cot | cot | | |
young |
> | | | water | land | phyll | | | | |
|
>
o-------o-------o-------o-------o--------o-----o------o--------o--------o---
------o
> | leech | 1 | 1 | | | | | 1 | |
|
> | bream | 1 | 1 | | | | | 1 | 1 |
|
> | frog | 1 | 1 | 1 | | | | 1 | 1 |
|
> | dog | 1 | | 1 | | | | 1 | 1 |
1 |
> | hemiz | 1 | 1 | | 1 | | 1 | | |
|
> | reed | 1 | 1 | 1 | 1 | | 1 | | |
|
> | bean | 1 | | 1 | 1 | 1 | | | |
|
> | maize | 1 | | 1 | 1 | | 1 | | |
|
>
o-------o-------o-------o-------o--------o-----o------o--------o--------o---
------o
>
> FE = Formal Extent
> FI = Formal Intent
>
> 1. Are you using "object" and "instance" as synonyms?
Yes. "Object" is the term used in Formal Concept Analysis, "token" is the
termed used in Information Flow, and "instance" is the term I have been
using in the IFF. All are equivalent in this mathematical context.
> 2. Can you explain to me in what sense it makes sense to you to say
> that {bream, frog} is the FE of the fo.co. "bream", or similarly,
> that {maize, reed} is the FE of the fo.co. "maize", as generators?
_Within the context of the living formal context (living classification)_
"bream", "frog", "maize" and "reed" are four objects (using FCA
terminology). The object "bream" is a living being that needs water, lives
in water, is motile and has limbs. So it generates the formal concept whose
intent (comprehension) is a "limbed motile living being that lives in
water". In the (possibly limited) world of the living formal context, the
extent of this concept is just "frog" and "bream". One can always choose a
different model by expanding and/or contracting the formal context (adding
or subtracting either objects or attributes), and of course you will have
maps (called infomorphisms) that link these operations.
Of course, it gets more interesting when you apply these techniques to more
interesting classifications (formal contexts). In fact, the classification
of greatest interest here is the *truth* classification where the instances
(FCA formal objects) are model-theoretic structures, the types (FCA
attributes) are sentences of a 1st order language, and the incidence
relation is satisfaction. In the infinite matrix for this classification,
put a "1" to the right of a structure M and directly under a sentence S when
the structure is a model for (satisfies) the sentence M |= S. Then, the
formal concept generated by a structure M has as its intent (comprehension)
the theory generated by the structure th(M). And a sentence S1 labels
(generates) a concept below another concept labeled by sentence S2 when S1
entails S2 (S1 has more models than S2).
Robert E. Kent
rekent@ontologos.org