Dear Ravi and John,
====================
Ravi:
I still don't fully comprehend your question, but will attempt an answer. In my email, I was using theory in a technical sense, specifically a set of axioms expressed in some formalism. Hence, a formal ontology captures a fragment of our natural language understanding of something, which is itself a fragment of what actually is. Be that as it may, our computers will be using the formal ontology as a software artifact to aid in whatever problem we are undertaking. I am definitely not suggesting that the formal ontology is all there actually is, just that it has certain properties that we can exploit.
Lastly, I used the word *model* strictly in the Tarski sense, and am not sure what you meant by its use in your first email.
====================
John:
I agree with pretty much your entire post, some comments:
AH> Lastly, the idea of logical "primitives" isn't really new,
> though it hasn't quite been articulated or implemented in the
> way I sketched out below....
[JFS] I agree. This is another issue that will have to be clarified
and explained in the guidelines. In fact, we might not have
a clear idea of what kind of primitives we should have until
somebody develops a clearly specified set of them and
demonstrates how they can be used to define other terms.
I don't think it is necessary to *a priori* specify a set of primitives, at least in this sense. Pretty much anyone using a formal ontology will be reusing concepts culled from various other disciplines. I refer you again to the forthcoming paper in FOIS 2010 entitled "Ontology Verification with Repositories" by Gruninger, Hahmann, Hashemi and Ong.
Briefly, in your conceptualization of some domain when you utilize certain axioms, chances are, many fragments of your conceptualization map into various mathematical theories. The process of generating these mappings will begin to identify if there are in fact a set of primitives. Note, right now, COLORE has some 80 ontologies represented in CLIF, so one could add a new ontology and see if it (or parts of it) maps into one of the pre-existing ones.
Moreover, we've also developed a procedure that would semi-automatically identify such mappings between theories / ontologies / modules in the repository. All one would need to provide is a mapping question such as "is the before relation ranging over timepoints equivalent to partial orderings?" or more formally:
(forall (t1 t2) (if (before t1 t2)
(leq t1 t2)))
This simple statement can be provided as input to our procedure and can determine if the *before* relation over timepoints is equivalent or at least relatively interpretable to some notion of ordering.
AH> Allowing and capturing all different (but logically equivalent)
> axiomatizations of an intuition is laudable goal, but it introduces
> serious problems in terms of _possibly_ exponential growth for the
> repository...
[JFS] The lattice of all theories is infinite. It doesn't "grow" because
it is always as big as it can be. But the guidelines can specify
various rules, regulations, and restrictions on what is stored,
what is highlighted, what is deprecated, what has been contributed
but not vetted, what is archived, etc.
True, but we are talking here about a software artifact, which _will_ grow. Moreover, if the idea of generating an FO is to enable interoperability, and we maintain parallel hierarchies of equivalent theories, extending one theory in one of these parallel hierachies would put them out of sync. Just an issue that needs to be addressed.
Cheers,
Ali
On Fri, Feb 19, 2010 at 11:39 AM, ravi sharma <drravisharma@xxxxxxxxx> wrote:
Ali
Ontology is relationship among objects and is in our minds as theories. What other aspects of theories come to your mind?
Theories are not fully expressible therefore we model them by math, logic, experience and language as well as visualization tools in some cases.
Thanks.
Ravi
On Fri, Feb 19, 2010 at 11:33 AM, (•`'·.¸(`'·.¸(•)¸.·'´)¸.·'´•) .,., <asaegyn@xxxxxxxxx> wrote:
I'm sorry Ravi, I do not understand your question. Might you rephrase?
Ali
On Fri, Feb 19, 2010 at 11:23 AM, ravi sharma <drravisharma@xxxxxxxxx> wrote:
Ali
Is it not the modules where our Foundation Ontologies reside and the languages and models (visualization for example) describe how close we can get to describe or visualize Ontologies with reality in the depicted model space? Are there any aspects other than logic and experience (including observations) that help us understand the theories (top layer) better?
--
Thanks.
Ravi
(Dr. Ravi Sharma)
313 204 1740 Mobile
Dear Ontolog,
The discussion on growing a family of interlingua ontologies in a repository is very stimulating.
I would suggest the following things need to be considered:
==================================
1) What is the granularity of a module that one would wish to allow? A single statement in logic is ostensibly a theory, and can be considered a module. However, this leads to an exponential proliferation of modules. Any repository should have some guidelines as to what constitutes a module.
==================================
2) Allowing and capturing all different (but logically equivalent) axiomatizations of an intuition is laudable goal, but it introduces serious problems in terms of _possibly_ exponential growth for the repository (i.e. maintain several parallel modules and hence lattice of theories). But more importantly, it is also nightmare in terms of updating, maintaining and growing the repository. It can be done, but it is not a trivial task, and would need very clear guidelines for how the repository would handle updates / changes to any particular module. Moreover, how might one handle the extension of one module? There are several possible solutions, but each have their drawbacks.
I realize i'm a pretty abstract person, so i'll ground this in an example.
There are at least two ways of characterizing the notion of a lattice. One can do it with the "less than or equal" relation, or one can use the functions of meet and join. Both result in a set of isomorphic models. Let us assume that the repository contains both characterizations of this intuition. Now say I decide to extend the module for lattice based on meet-join. Should I also create a parallel entry for this extension based on leq? Perhaps one might respond -- well, whichever axiomatization people use is the one we need to worry about, if someone wants to extend the alternate description, they can create a parallel entry.... This is certainly one approach, but I believe it introduces serious concerns for maintaining and updating the repository.
==================================
3) I humbly suggest there are two ways one can construct "upper ontologies" -- one is to focus on things as they are in the real world (the traditional upper ontology approach), and the other is to focus on the language one is using to describe said things. Both approaches provide significant advantages for reuse, but ignoring one introduces significant hurdles. I seem to have been unsuccessful thus far in communicating this idea, but will try once more:
The infinite lattice of theories can take on various guises. The most straightforward application involves an ontological stance, where one describes things such as "spatio temporal thing", and specializes from there by adding axioms. Refer the diagram below for how adding axioms / extending theories results in a corresponding tightening of what I term "model spaces." In the diagram below, arrows pointing right constitute "non-conservative extensions" of a module (think of the lattice of theories turned sideways, where down is now right). The "model space" below is the set of permissible models (in the Tarski sense) that are allowed by the axioms. The more general modules (i.e. less axioms) permit more models, and each non-conservative extension "slices up" the model space of the more general theory.
A complementary lattice of theories which doesn't fit neatly into the above is a fundamentally more abstract, but just as useful exposition of descriptions in the language. Instead of focusing on the pragmatics, it exploits the semantics of the theory, specifically the relationship between axioms and the set of allowable models (Tarski sense).
As an example, let us consider a common characterization of time as branching points. For the sake of brevity, let's assume time is discrete moments, (it really doesn't matter for our purposes here). While the pragmatics of our axioms might connect the symbols
(forall (t1 t2 t3)
(if (and (before t1 t2) (before t2 t3))
(before t1 t3))
*t1, t2, t3* are time points or moments, and *before* captures an intuition for how time points relate to one another, a common analogy to help conceptualize this is of course, literally branching time...
The above is also quite clearly a classic application of the axioms for transitivity applied to the notion of ordered time points. Quite aside from the above axiom being the specialization of some concept of time, it is also a specialization of the logical structures that arise from describing time in a language such as first order logic. In a way, there are two sorts of "primitives' (i use this term very loosely) here. One is the "primitive" notion of timepoint, or perhaps more generally, time. The other is the "primitive" notion of ordering, or specifically for branching time, partial ordering.
i.e. one can remain agnostic on ontological commitments, and simply flesh out the logical implications of the statements. Obviously, this approach on it's own is useless, since in the end we need to connect our theories to the real world, but it would be a serious oversight to not consider and incorporate this perspective in the construction of any "foundation ontologies."
More specifically, if in the construction of any such repository, we keep track of both extensions (ontological and logical), we would be able to greatly ameliorate the task of generating semantic mappings and hence interoperability. (If you would like more details on how this might work, a bit outdated but perhaps still useful exposition on this idea can be found in my Master's thesis.)
==================================
I'll end here for now. I have much more to add, but this email is already quite long and I imagine some of these issues might be touched upon in this morning's conference call.
I would also suggest that those interested in the types of mappings available when one uses Common Logic to check out the forthcoming FOIS 2010 proceedings. There will be a paper there that will quite rigorously spell out many (though not all) of the possible connections / mappings between theories (both ontologically and logically). It would probably save a lot of effort in terms of reinventing wheels and should help clarify many of the issues that undertaking the creation of such a repository will entail.
Best,
Ali
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