SUO: Re: JA v/s JD on Peirce, Hillman, FCA, etc...
Jean-Luc Delatre and others,
(comments below -- mostly a FYI)
----- Original Message -----
From: "Jean-Luc Delatre" <jld@club-internet.fr>
To: "Jon Awbrey" <jawbrey@oakland.edu>
Cc: "Stand Up Ontology" <standard-upper-ontology@ieee.org>; "Gdsemiocom"
<gdsemiocom@univ-perp.fr>
Sent: Friday, March 08, 2002 2:25 PM
Subject: SUO: JA v/s JD on Peirce, Hillman, FCA, etc...
[snip]
> I will here try to highlight my position.
> I am posting to SUO because I feel the conclusions are relevant to
> the goal of SUO, which I think is misdirected.
The SUO starter document called the IFF (Informationm Flow Framework)
represents many basic notions of Set Theory, Order Theory and Formal
Concept Analysis (FCA) in both the (set-theoretically) large and small. The
current version of the IFF represents the large notions, principally in
the IFF Core Ontology
<http://suo.ieee.org/IFF/versions/20020102/IFFCoreOntology.pdf> and the IFF
Classification Ontology
<http://suo.ieee.org/IFF/versions/20020102/IFFClassificationOntology.pdf>.
These are located in the upper metalevel. Eventually, there will also be a
representation for the small naotions, and these will be located in the
lower metalevel. Let us use the abbrevations: IFF-CORE for the IFF Core
Ontology and IFF-CLS for the IFF Classification Ontology. In particular, the
notions that you discuss below are all represented and axiomatized in the
IFF.
> I found what I deem a remarkably concise and usefull set of definitions
> related to FCA in pages 3 and 4 of "What is a concept?",
> from Chris Hillman: http://www.math.washington.edu/~hillman/papers.html
> and this prompted me to make some critical comments to JA's message:
[snip]
> According to Hillman what makes a subset a proper concept is the fact that
it
> is closed under <||> if taken from X and closed under |><| if taken from
Y.
> (duality allows to halve the amount of demonstrations, see Hillman's)
>
> Where: |> A = {y : (x ,y) in R for all x in A}
> <| B = {x : (x ,y) in R for all y in B}
> <||> A = {x : (x ,y) in R for all y in |> A}
> |><| B = {y : (x ,y) in R for all x in <| B}
>
> So we can beef up our ontology structure with a set of 'concepts' which
> will be names for subsets of X which happen to be extensions of proper
> concepts. That is, from now on, the ontology will be build upon
> X, Y, R c XxY, N, pow(X), C c N x pow(X) where:
> - N is a set of Names for the concepts
> - pow(X) is the power set of the "things" set X
> - C is a mapping from names to "proper concepts" subsets of X
[snip]
The power set operator pow(-) is represented by the term 'SET$power' and is
axiomatized on page 8 of the IFF-CORE.
The Cartesian product operator (-x-) is represented by the term
'SET.LIM.PRD2$binary-product' and is axiomatized on page 19 of the IFF-CORE.
__________
See Diagram 2 on page 28 of the IFF-CLS which illustrates the main FCA
operators represented and axiomatized in the IFF.
The operator |> is called 'CLS.CL$left-derivation' in the IFF and the
operator <| is called 'CLS.CL$right-derivation' in the IFF. These are
axiomatized on pages 28-29 of the IFF-CLS.
The operator <||> is called 'CLS.CL$instance-closure' in the IFF and the
operator |><| is called 'CLS.CL$type-closure' in the IFF. These are
axiomatized on page 29 of the IFF-CLS.
The *set* of formal concepts is represented by the term 'CLS.CL$concept',
which is axiomatized on page 30 of the IFF-CLS. The *partial order* of
formal concepts is represented by the term 'CLS.CL$concept-order', which is
axiomatized on page 31 of the IFF-CLS. The *complete lattice* of formal
concepts is represented by the term 'CLS.CL$complete-lattice', which is
axiomatized on page 32 of the IFF-CLS. The *concept lattice* of formal
concepts is represented by the term 'CLS.CL$concept-lattice', which is
axiomatized on page 36 of the IFF-CLS.
The set N of Names for formal concepts corresponds to the set of indexes in
a collective concept. This is represented by the term 'CLS.CONC$index' and
axiomatized on page 41 of the IFF-CLS.
__________
The IFF Model Theory Ontology (coming soon to the SUO-WG) is based heavily
on the notion of classification (called formal context in FCA termnology)
(and also on a suitable notion of hypergaph).
Robert E. Kent
rekent@ontologos.org