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Re: truth in a structure

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Mon, 19 Aug 2002 11:38:22 -0400
To: jim@spatial.maine.edu
Cc: www-rdf-logic@w3.org
Message-Id: <20020819113822M.pfps@research.bell-labs.com>

From: Jim Farrugia <jim@spatial.maine.edu>
Subject: truth in a structure
Date: Mon, 19 Aug 2002 10:42:56 -0400 (EDT)

> 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? 

This set is in an interpretation.  Each interpretation has a (potentially)
different set for isMotherOf, generally called the extension of isMotherOf.

> 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?

Now we are going from logic to philosophy.

In general, such sets are not constructed, not even by people.  Instead,
there is some notion (or notions, see below) of the intension of isMotherOf
used by a person, along with, perhaps, a number of isMotherOf
relationships.  This intension is applied to a particular state of affairs
to come up with an extension that includes the specific relationships.

In logics, the situation is somewhat similar.  One provides a number of
axioms (or definitions, or statements) that constrain the extension of
isMotherOf in any interpretation, these axioms may include axioms that play
the part of the intension above, such as stating that the spouse of the
father at the time of birth is the mother (actually this would be the other
way around), or axoims that play the part of the extension above, such as
stating that Mary 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.

Most logics just ignore this and assume that isMotherOf refers to one of
these meanings.  

> 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.

Correct.  Semantics is just a formal system.  Any connection between it and
the real world, assuming that there is a real world of course, has to be
made outside the semantics.  

That said, the whole idea behind model theories is that they do correspond
somehow to the real world, namely that an interpretation is an abstraction
of the way the real world might be, and that a model for a KB is an
abstraction of the way the real world could be, assuming that every axiom
in the KB is true.

> 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.

See above.

I don't think that the SW makes any difference here.  There are logics that
worry about different knowers (or believers) and how their knowledge (or
belief) can be combined in an environment where not everyone knows (or believes)
everything everybody knows (or believes).

> 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?

Well, the SW does have a slightly interesting viewpoint on this, namely
that names, like isMotherOf, should be the property of believers (i.e.,
owners of web pages), and are part of namespaces.  This means that version
of isMotherOf would be foo:isMotherOf and another would be bar:isMotherOf,
potentially a totally different property.  To make connections between the
different properties either requires explicit use of someone else's names
or a way of relating different names.

> 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?

Well, I, for one, find some of the ideas on this in SW literature
completely repugnant.  For example, there are statements to the effect that
the meaning of isMotherOf for the SW *is* ordinary truth, i.e., that there
is some innate relationship between the logical formalisms underlying the
SW and the real world.  My view, is that the logical formalisms underlying
the SW are just symbol systems, and that any connection to the real world
has to be made outside the logical formalisms, possibly by programs that
use information in the SW and also access the real world.

> 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?

The problem with the discussion, such as it is, on these issues is that it
is fragmented and that the viewpoints of many players are implicit and
hidden.  I have tried on a number of occasions to get elucidation on
related points and have made very little headway.  Some of the work of the
W3C RDF Core WG has turned up some deeply-held beliefs by major SW players
in this field.

> Thanks very much,
> Jim
> -----------------------------------------------------------------------------
> [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

Peter F. Patel-Schneider
Received on Monday, 19 August 2002 11:39:47 UTC

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