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

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Mon, 22 Feb 2010 16:53:50 -0600
Message-id: <1266879230.11366.40.camel@xxxxxxxxxxxxxx>
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.

> 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.

> 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.

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