[Top] [All Lists]

Re: [ontolog-forum] Thing and Class

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Tue, 16 Sep 2008 16:16:06 -0400
Message-id: <48D01406.3010701@xxxxxxxxxxx>
Pat and Chris,    (01)

CM> Pat is careful to talk about Kripke *structures* here, as
 > distinct from Kripke *models*, which is what I thought you
 > were talking about.    (02)

In my first note in this thread, I explicitly mentioned the Kripke
triple (K,R,Phi) of the set of worlds K, the accessibility relation R,
and the evaluation function Phi.  When I spoke about a mapping from
Kripke semantics to Dunn semantics, I meant that for every choice of
(K,R,Phi) and for every world w in K, there is a unique pair (M,L) of
facts and laws in Dunn's sense.  I apologize if any of my statements
weren't sufficiently qualified to be clear.    (03)

JFS>> But they do not have an existence in the world that is independent
 >> of the ontology that we use to characterize them and identify
 >> particular instances.    (04)

CM> Well, yeah, I mean, you do have to have people, tables, and cabbages
 > in your ontology to include them in the domain of a model.  You seem
 > to be wanting to say something deeper, but I'm not seeing it.    (05)

I'm glad that you gave some examples, which on the surface seem to be
relatively straightforward:  "people, tables, and cabbages", but as we
have seen there are hotly debated questions about when an egg becomes
a person and where is the borderline between the species Homo sapiens
and the genus Homo or the broader categories of hominid vs. hominin.
Many other issues arise with birth defects, Siamese twins, people
with no discernible brain function, etc.    (06)

There are equally thorny questions about the borderline between tables
and other flat surfaces.  And the single species Brassica oleracea has
been bred into cabbages, cauliflower, broccoli, and other varieties
that a nonspecialist wouldn't even recognize as a member of the genus
Brassica.  When you get to relations that are commonly expressed as
prepositions, verbs, adverbs, and abstract nouns, all bets are off.    (07)

PH> Actually there seems to be a fairly robust and widely accepted
 > consensus that semantic structures can be parts of reality.    (08)

CM> By definition a model is a mathematical entity of a certain
 > sort, usually an ordered n-tuple of some ilk.  The real world,
 > whatever it is, isn't an n-tuple.    (09)

The professor of statistics and industrial engineering, George Box,
made the very widely quoted statement "All models are wrong, but
some are useful."  I can imagine armchair logicians and philosophers
who might say "Semantic structures can be parts of reality", but I
can't imagine Box or anybody who quotes him accepting that statement
with a straight face.    (010)

PH> As you point out above, a Kripke/Tarski (might as well just say
 > model-theoretic) structure consists of entities and relationships,
 > understood mathematically.  It does NOT determine the signature
 > of a language.  A given structure can be used to interpret a wide
 > range of languages, and even the same language in a wide range of
 > ways. So to simply speak of "the set of sentences true in" such
 > a structure does not specify anything.  That could be just about
 > any set.    (011)

If we consider the triple (K,R,Phi), the evaluation function Phi
is defined in terms of the syntax (or at least the abstract syntax)
of some specific language L.  That language L would be the one to
use in order to define the mapping from a possible world w in K to
a Dunn-style pair of laws and facts.    (012)

PH> ... your construction is under-defined. You have to specify how
 > to determine the set of sentences.    (013)

Given a Kripke triple (K,R,Phi), for each world w in K, the set of
all sentences of L that are true in w constitute the facts of w:    (014)

    Facts of w  =df  {s in L | Phi(s,w)}    (015)

The set of all sentences of L that are necessarily true in w
constitute the laws of w.  According to Kripke, those are the
sentences that are true in every world w' accessible from w:    (016)

    Laws of w  =df  {s in L | forall w', R(w,w') implies Phi(s,w')}    (017)

PH> ... there is a meaningful notion of relational structure
 > independent of any ontology.  Kripke's own definitions are
 > independent of ontology.    (018)

I agree that the notion of 'Kripke structure' is independent of
any ontology.  But any specific set of Kripke worlds K uniquely
determines everything that exists or can exist in any world and
all the possible relationships presupposed by K.    (019)

For any world w in K, the relational structure (D,R) of w determines
the set of true ground-level sentences about w in any given syntax.
The collection of all those ground-level statements for every w in K
determines all the entities that exist or can exist in any world
of K and all the possible relationships among them.    (020)

What would you call that other than an ontology?    (021)

John    (022)

Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/  
Subscribe/Config: 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 Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (023)

<Prev in Thread] Current Thread [Next in Thread>