truth in a structure

I have some questions about the 'truth-in-a-structure' notion of 
model-theoretic semantics, in particular with what seem to be certain inevitable
issues that arise once people consider the jump from model-theoretic structures
to their real-world counterparts.

These questions are related to earlier comments by Ian [1] and Peter [2].

Ian says in [1], "... Jane is the mother of John just in case that, in all 
models of our ontology, the pair of objects that Jane and John map to is a 
member of the set that isMotherOf maps to."

Where does this set live, the set that isMotherOf maps to? 

In other words, and supposing that this set is never given a single extensional 
definition that all users would agree accurately defines 'isMotherOf', 
where does a user go to find this set and see if Jane is the mother of John?

Further, considering 'isMotherOf' as it applies to a woman and a child, 
how does a model-theoretic semantics deal with the following situation?

For one community, 'IsMotherOf' deals only with the biological bearer of 
the child.  For another community, 'IsMotherOf' deals only with the woman whose
egg was used to conceive the child.  For yet another community, 'IsMotherOf'
deals with the woman who is the legal guardian (in some jurisdiction at some
time, ...) of the child, etc.

What I am trying to get at is this: When model-theory is used in mathematics
to deal with satisfaction conditions of language elements (i.e., 'to say
what the symbols mean'), it seems that one never steps outside
the model-theoretic structure (a set, with certain constants, and functions
and relations defined on it) itself to determine whether the truth and 
satisfaction conditions actually obtain.

But as soon as we apply the model-theoretic approach to a real-world 
situation like 'isMotherOf' it seems we inevitably do/must step outside
the model theoretic structure and concern ourselves with at least these two 
isues: 

(1) how people find, and assess membership in, for instance, the set that 
IsMotherOf maps to, and

(2) how to account for, make manifest, and possibly bring into harmony
the different understandings of 'IsMotherOf' that different users will have
and will want to see reflected in the semantics before they agree employ
that semantics in some application?

The first issue, I suspect, is dealt with in a straightforward way, though
I'm wondering what provisions are typically made in the implementation of
the semantics to let users determine whether the set that IsMotherOf maps to 
makes sense for their purposes.

The second issue relates to Peter's comments below from the last part of [2],
and it seems pretty important to the efforts of the Semantic Web.

I think it's a fair statement that there are often significant difficulties 
among user communities when it comes to assessing whether or not a set like 
'the set that IsMotherOf maps to' captures the understanding of 'IsMotherOf' 
that a given community has. Does the Semantic Web community take any position
on these difficulties?

On the second issue above, it seems that some writers on model-theoretic 
semantics (e.g., Hodges in [3]) make the appeal that model-theoretic truth is 
ordinary truth (the argument that "Snow is white" is true iff, well, 
snow is really white) and model-theoretic satisfaction is ordinary satisfaction 
("Jane is the mother of John" iff, well, Jane really is the mother of John). 
But it seems that these appeals also skirt the difficulties that
people have in determining whether a given statement obtains in the real world.
Where in the Semantic Web community's approach to semantics are such
real-world modeling issues treated? Or are these issues considered to be
out of scope of what the rdf-logic and webont folks intend to provide?

I'm not aware of much discussion from the Semantic Web community that
deals with this second issue. If I've missed it, please send me pointers.
This issue seems to be one that users will inevitably have to confront
as they try to evaluate whether or not they can profitably use a
model-theoretic semantics on the Web. If some of you think it is not
important or it is best left treated somewhere else, can you please say why?

Thanks very much,

Jim

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[1] http://lists.w3.org/Archives/Public/www-rdf-logic/2002Aug/0029.html
[2] http://lists.w3.org/Archives/Public/www-rdf-logic/2002Aug/0026.html
[3] http://www.maths.qmw.ac.uk/~wilfrid/joburg.pdf

Received on Monday, 19 August 2002 10:50:33 UTC