SUO: JA v/s JD on Peirce, Hillman, FCA, etc...
Hi there,
There has been quite a lot of arguments going back and forth between
Jon Awbrey and me, only part of which appeared on the Ontology & Gdsemiocom lists.
So, it's probably not too clear what point I am trying to make.
Actually the original subject even got completely lost in comments
of comments about details and hair splitting.
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.
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:
> Objet:
> SUO: *Date 27 Feb 2002 -- Extension x Comprehension = Information
> Date:
> Wed, 27 Feb 2002 16:16:02 -0500
> De:
> Jon Awbrey <jawbrey@oakland.edu>
> A:
> Stand Up Ontology <standard-upper-ontology@ieee.org>
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> SUO WG Members,
>
> If you are looking for a new perspective on that
> fearful symmetry between extension and intension,
> and want to see the bearing of a precursory idea
> of information on doing so then you may find the
> following series of excerpts from Peirce's early
> lectures of some interest:
>
> Extension x Comprehension = Information
> [snip]
What I regret in JA's approach is that it is not that much enlightening
and, rather than piling up quotations upon quotations of famous authors,
it is better to *cut* thru them and try to synthesize.
Definitions
-----------
Reminding the definitions from Hillman (also from Wille), we can state
that the barebones of an ontology is just a relation R, i.e. a subset
of the product XxY, where:
- The elements of X are all the "things" in the universe actually known
to the ontology "owner". In CG parlance these would be individual markers.
Some subsets of X are the 'extents' of the known 'concepts' (Hillman)
as well as the 'extensions' of the known 'words' (Peirce) or of what
Peirce call 'true symbols', that is concepts with compound names.
But not all subsets are extensions of proper concepts or true symbols.
- The elements of Y are all the 'attributes' (common parlance) or 'predicates'
(Hillman) or 'characters' (Peirce) actually known to the ontology "owner".
Some subsets of Y are the 'intents' of the known 'concepts' (Hillman)
as well as the 'comprehensions' of the known 'symbols' (Peirce).
But not all subsets are comprehensions of proper concepts or true symbols.
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
I am trying to present the smallest model upon which formal arguments
about the ontology structure can still be made.
No "true one and only" ontology
-------------------------------
Although concepts must be "proper", i.e. closed under <||> and |><|,
so we can go unambiguously from extension to comprehension and vice-versa,
I choose to name the concepts from their extensions rather than from
their comprehensions for the following reasons:
Whenever we are acquiring new information from the outside world and
updating our knowledge by means of changes to our ontology, we certainly
don't want to disturb most of our previous knowledge and not recast
known individuals outside the categories under which we have filed them.
When we learn that penguins don't fly, we still have a strong preference
to cast them as birds and we are more willing to alter the comprehension
of the name "birds" than keep the name "birds" it's previous definition
and create a new awkward concept "birds OR penguins" which, while still
a perfectly proper concept and a superset of the known "birds" concept,
will make it a real drag to extend to penguins most of what we know
about birds.
We prefer to keep with penguins as "birds" because it is (likely) much
less work to deal with the exceptions due to the fact that penguins don't
fly, yet are still called "birds", than to update the complementary set
of knowledge statements that, except where flying is involved, all which
applies to "birds" also applies to penguins and for that matter to the
new "birds OR penguins" proper concept for which we have no name yet.
The previous "birds" concept, naming both it's extension and comprehension,
could now be renamed "flying birds" to the purpose of attaching statements
which apply only to this specific category, but this is of no cost until
we need it and of very little cost when we do.
*************************************************************************
* But, depending on the amount of knowledge we already accumulated when *
* we meet such kind of an exception, it may happen that we choose to do *
* the other way around: there is no "true" way to shape our ontology it *
* all depends on where we stand with respect to our current knowledge. *
*************************************************************************
Disjunctive terms rarely "useless"
----------------------------------
I object to Peirce considerations on "useless" disjunctive terms
such as 'man and horse and kangaroo and whale' as stated in:
http://suo.ieee.org/ontology/msg03752.html
If for some reason we are talking about such an extension
(subset of X) and it happens that there is actually a 'character',
'attribute', 'predicate' (a member of Y, whatever you like to call it)
which is in fact shared only by men, horses, kangaroos, and whales among
mammals, in this case there *is* a possible new "comprehension which is
adequate to the limitation of the extension".
Just add this new 'attribute' to the comprehension of mammals, this will
be a proper concept, closed under |><| and which extension will be exactly
'man and horse and kangaroo and whale' individuals and be closed under <||>.
Then give a name to this new concept, say "weirdo-mammals".
If you don't do that, and as Peirce suggest you speak only about mammals,
you would abusively make statements about mammals which should apply only
to "weirdo-mammals".
Of course, Peirce is right if you cannot find any such specific attribute.
But he is wrong again when he say:
"the person who has found out this, knows more about this thing than that
it is a mammal; he therefore knows which of these four it is"
What a fancy assumption about the working conditions of the ontology
builder! He may be working from some indirect report of someone not
willing to provide full information and he *may* indeed *not* know
which one of the four is actually involved.
********************************************************************
* So sometimes he may not even be able to enter this new knowledge *
* into the ontology, it has to be stored outside the realm of the *
* ontology structure for lack of a proper place to put it. *
********************************************************************
Beware of oranges Horatio
-------------------------
I also object to Peirce's considerations on conjunctive terms
as stated in:
http://suo.ieee.org/ontology/msg03757.html
Assuming for a moment with Peirce that "the only spherical bright
fragrant juicy tropical fruit we know is the orange", the closure
under |><| of 'spherical, bright, fragrant, juicy, tropical fruit'
brings us back to a more specific comprehension which happens to be
the set of all "essential" attributes of an orange.
But this is only true until the ontology builder meet *another* kind
of fruit, which despite being a 'spherical, bright, fragrant, juicy,
tropical fruit' is *not* an orange, at which point all previous
replacements of 'spherical, bright, fragrant, juicy, tropical fruit'
by 'an orange' will be screwed up!
I made a similar point a while ago in the CG list about unicity of
individual markers, in that case the screwed info was on the "things" side.
Peirce also makes the same kind of fancy assumption about the source of
informations he made when talking of 'men, horses, kangaroos, and whales':
"On the other hand, if the conjunctive term is subject and we know that
every spherical bright fragrant juicy tropical fruit necessarily has
certain properties, it must be that we know more than that and can
simplify the subject".
No, we cannot.
He obviously speaks as if talking of a "finished" ontology that
will not need any revision due to new knowledge acquisition.
This is not the "real" situation we have to work with.
******************************************************************
* Again, sometimes, the new knowledge cannot be encoded into the *
* ontology for lack of a proper place to put it. *
******************************************************************
Do not mess with induction
--------------------------
I also object to Peirce's views on induction as stated in:
http://suo.ieee.org/ontology/msg03756.html
The union of extensions for "Neat, swine, sheep, and deer" will map thru |>
to 'cloven-hoofed herbivorous animals' and it is uncontroversial that this
is a proper concept (closed under |><|) and that it *includes* Neat, swine,
sheep, and deer but this is all that we can say.
Whether there exist 'cloven-hoofed herbivorous animals' other than the ones
mentioned or whether we want to assume that all 'cloven-hoofed animals' are
herbivorous are entirely separate questions and should not interfere with
concept building considerations.
Do not let the koans bug you
----------------------------
This is a remark about Jon's musings on zen philosophy as stated in:
http://suo.ieee.org/ontology/msg03768.html
Defining a human being as an 'apterous biped' is of course asking for trouble.
Choosing a poor subset of the comprehension of a concept brings the risk that
the |><| closure will not map back to the whole comprehension but to a smaller
one which larger extension will then contain most strange things.
Many subsets of the comprehension will be selective enough such as to map back
to the full comprehension under |><| but not *all* subsets will do.
Where does the information hides?
---------------------------------
To clarify a point so dear to Jon Awbrey about getting new information
whenever either the Extension or the Comprehension grows without the
counterpart shrinking, I will restate Peirce blind man example from:
http://suo.ieee.org/ontology/msg03780.html
A blind man having been told that 'red' is a 'color' and which objects he
know are 'red' is now told that *all* 'red' things are also 'non-blue'.
First I will say that the way Peirce use the word "information" is quite
loosy an only qualitative. We do feel that he gained some knowledge but
what is it?
This new information will make the blind man to change *his* knowledge
database R = subset of XxY (if he is using Hillman's data representation
for such purpose...) in such a way that all instances of 'red' things will
have, not only a pair (x , 'red') in R but also a pair (x, 'non-blue').
Which means that doing the |><| closure on the {'red'} set in Y will now
map back to {'red', 'non-blue'}, an "enriched" comprehension.
The ambiguous point is what do we now mean by "red" when we speak of
'Content' along Peirce's terminlogy? He gives no hint about how his
new "information" is stored.
Does "red" still means only {'red'}, in which case I cannot fathom
where we got any new information at all.
This is the reason to choose the concept names mapping to be from
the set of names to the power set of "things", pow(X).
Then, the extension of all 'red' things will be unchanged under the
<||> closure but the comprehension will now be {'red', 'non-blue'},
in which case I, for myself, can clearly see where some new information
has been introduced and can even try to measure it according to the
cardinality of comprehension before and after the change.
Conversely, when we acknowledge new "things" of exactly the same kind
(same comprehension) of existing ones the changes to the cardinality
of the extension can be used to measure the amount of information.
Anyway, figuring how to measure the information gain during ontology
building does not seem to me to be the most critical problem.
Those two last cases are unproblematic in that the new knowledge
is fully accounted for and retrievable from the updated ontology.
This may not always be the case as we have seen above if we
have no "proper" concept to "hang on" the new information.
Summary
-------
It should be kept in perspective that "real world" ontology building
relies on gathering information from new instances of "things" and
checking for their attributes.
This is an always evolving process for which *no* finalised ontology
can be contemplated and during which the very structure of the lattice
of concepts can undergo drastic changes for which no "best way" or
canonical form can be found.
Futhermore, and I partly agree with Peirce here, some information
possibly cannot be folded into the ontology structure.
********************************************************************
* DO NOT EXPECT A "META" ONTOLOGY COULD BE BUILT, NO ULTIMATE *
* FINALIZED ONTOLOGY CAN BE CONTEMPLATED BECAUSE THE DIFFICULTIES *
* COME FROM THE BUILDING PROCESS, NOT THE LEVEL OF MEANING NOR THE *
* WORLD OBJECT INSTANCES. *
* THEREFORE THE MOST IMPORTANT PROBLEM IS HOW TO MATCH AND MERGE *
* ONTOLOGIES FROM DIFFERENT SOURCES. *
********************************************************************
And I didn't even care about how to name or define new attributes...
Cheers.
-- Jean-Luc Delatre
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