Re: [ontolog-forum] Semantic interoperability, DL's, Expressive Logics

[Ali Hashemi <ali.hashemi+ontolog@xxxxxxxxxxx>]

Hi Chris,

Thanks for the feedback. Comments below.

On Mon, Feb 22, 2010 at 5:53 PM, Christopher Menzel <cmenzel@xxxxxxxx> wrote:

On Mon, 2010-02-22 at 01:43 -0500, Ali Hashemi wrote:
> ...

On the whole this is a very useful post.  A couple of niggling remarks.

> I'll begin with a very high level recap of what it means to have a
> formal ontology. If I am describing a domain in a formalism, I will
> have statements (axioms) written in some logic. These axioms are
> interpreted and essentially allow a bunch of "models." We say that the
> axioms are satisfied by a model iff every statement of the theory
> holds true for a given model.

What is the connection between "the theory" and "the axioms" here?
Presumably, by the theory you mean the deductive closure of the set of
axioms.  But, assuming your system is sound, the axioms are true in (=
satisfied by) a given model if and only if all of their deductive
consequences are.  So your biconditional here looks trivial -- assuming
(as I think you intend) that "a (given) model" on each side refers to
the *same* model.  Are you wanting to define satisfiability?  Let S be a
set of statements in some logical language L.  Then we say S is
*satisfiable* if and only if there is a model M of L such that every
statement in S is true in M.

Hmm, my phrasing was awkward. All I meant to say here is that associated with any set of axioms are a set of permissible models... I wasn't aiming to define satisfiabilty. Just pointing out something that is perhaps trivial, mainly because I don't know what the background of the general reader on this forum is.

> I speak here of course, about models in
> the sense of Tarksi:
> http://en.wikipedia.org/wiki/Model_theory#First-order_logic

I myself wouldn't recommend that article; it's a bit of a mess.  A much
better introduction is Hodges' "First-order Model Theory" in the Stanford
Encyclopedia of Philosophy.

Yes, the wiki article is a mess, thanks for the link.

> If a theory is a non-conservative extension of another,
> all of its models will be a subset of that more general theory.

I think you mean the set of its models will be a subset of *the set of
models of* that more general theory.  But that's not true in general, as
the extension might contain vocabulary not included in the original
theory.  So you'd need to state that a bit more carefully.

Also true. One way of addressing this concern re the introduction of new "orthogonal" vocabulary (i.e. conservative extensions) is to organize theories / modules in a repository according to something we term "core-hierarchies." 

A core hierarchy consists of theories which are non-conservative extensions of a particular intuition. Note you can still introduce new terms, but they will be extending previous ones. Slide 5 of Michael Gruninger's presentation at the Feb 19 conference call ( http://ontolog.cim3.net/file/work/OpenOntologyRepository/2010-02-19_OOR-Developers-Panel/COLORE--MichaelGruninger_20100219.pdf ) includes the definition for core hierarchies:

===========================
Definition:
A core hierarchy is a set of core theories T1, ..., Tn such that:

L(Ti) = L(Tj) for all i,j
===========================

(Heh, it just occurred to me that organizing theories in such a way could inadvertently lead to and is possibly very similar to the quest for primitives.)

Alternatively, if you lift this restriction, you must accommodate what I awkwardly and tentatively phrased as the n-dimensional interactions between sets of models with different vocabularies. Instead of simply slicing out models in a single plane, you have an n-dimensional space corresponding to the different vocabularies... However, there is no way I am going to start trying to explicate what this entails or how it would work via email :D.
 
Best,
Ali

 
-chris

_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
 

-- 
(•`'·.¸(`'·.¸(•)¸.·'´)¸.·'´•) .,., 
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/  
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/  
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/ 
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (01)