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Re: N3 contexts vs RDF reification

From: Seth Russell <seth@robustai.net>
Date: Wed, 2 May 2001 20:58:43 -0700
Message-ID: <00b501c0d385$5a7063c0$b17ba8c0@c1457248a.sttls1.wa.home.com>
To: "Peter F. Patel-Schneider" <pfps@research.bell-labs.com>
Cc: <phayes@ai.uwf.edu>, <www-rdf-interest@w3.org>
From: "Peter F. Patel-Schneider" <pfps@research.bell-labs.com>

Re: http://robustai.net/mentography/TransitiveProperties.gif

> Pretty pictures might be useful for some things, but they certainly are
not
> sufficient to show that your can represent second-order sentences in RDF.
> Sure you may have a syntactic encoding of second-order sentences in
> RDF, but you can also have an encoding of second-order sentences in
> XML or even HTML.  To have a second-order logic, you have to provide a
> second-order meaning for these encodings, either derived from the meaning
> of the encoding language or independent of that meaning.  You have done
> neither.

Ok, apparently I have made some mistakes in diagramming the quantifications,
I'll correct those and resubmit  .... but ...

Could you perhaps sketch for me what a "second-order meaning" would look
like?  Maybe, at least, specify what language this 'meaning' is to be
expressed in and provide an example.   Now, obviously, in my simple diagram
of Pat's description of transivity ( in KIF, i presume) I did not elaborate
the rest of the ontology and logical constructs that would complete a
functional model.  Is your criticism of my diagram that I have not make that
elaboration?  Wouldn't such a criticism be kind of like criticizing a quick
sketch of a cartoon character because it didn't animate itself?  Can you not
see that if the extra assertions necessary to elaborate the modes are in
fact in the DAML schemas, that they could be easily added to the diagram?

... i am very confused as to what you and Pat are requiring?

Seth

>
> Peter Patel-Schneider
> Bell Labs Research
>
>
>
> From: "Seth Russell" <seth@robustai.net>
> Subject: Re: N3 contexts vs RDF reification
> Date: Tue, 1 May 2001 12:51:34 -0700
>
> > Well, sorry,  I'm still not getting this.  Could I impose on you again
to
> > refer to a new mentograph:
> >
> > [1] http://robustai.net/mentography/TransitiveProperties.gif
> >
> > See  you say ...
> >
> >     "What you are talking about is meta-language statements
> >     (statements about other statements),
> >     not higher-order statements."
> >
> > Yet I have been able to transform the two examples of  "higher-order
> > statements" that you gave below into RDF statements merely about other
> > statements - which I take to mean that statements can be objects of
other
> > statements.   So I am at a loss to make the distinction you require.
> >
> > A couple of notes on my diagram.
> >
> > * I use a short-hand notation for RDF reification - explained at
> > [2] http://robustai.net/mentography/reification.gif
> >
> > * I had to change your example slightly away from unary relations so as
to
> > correspond with the RDF way of doing things.  But I was able to
duplicate
> > the problem to which you referred and then resolve it with a arc labeled
> > "not" between the variable class and the designated class.
> >
> > So, what am I missing ??
> >
> > Thanks for your patience with this troublesome student ...
> >
> > Seth
> >
> > .... in response to your examples below ...
> > >
> > > >.. i have these concepts all smushed together in
> > > >my mind ... and am playing catch up with my education.  But I still
don't
> > > >get what makes logic higher order.  I have tried to depict my
> > understanding
> > > >of your description in the graph at
> > > >http://robustai.net/mentography/higherOrder.gif  which I have also
put on
> > > >the Public CMap server under the SemanticWeb Project.  If you do find
the
> > > >time to answer me, maybe you could show where I have gone wrong by
> > mutating
> > > >my graph.
> > >
> > > Hmm, not sure I follow the graph , I'm afraid. Sorry, I tend to work
> > > with words better.
> > > The higher-order/firstorder distinction is rather a subtle one to get
> > > exactly right.  Let me sketch it first and then correct the sketch
> > > later, OK?
> > >
> > > Sketch
> > > First-order logic asserts relations between things, so you can say
things
> > like
> > > (IsBiggerThan Bill Fred)
> > > ie relation IsBiggerThan holds between things Bill and Fred,  and it
> > > quantifies over the things, so you can say
> > > (forall (?x) (exists (?y)(IsBiggerThan ?y ?x)))
> > > ie for any thing x there is something y which is bigger than it, ie
> > > everything has something bigger than it. (I didnt say it was true,
> > > only that you can say it.)
> > >
> > > OK. In second-order logic, you can also quantify over the
> > > (first-order) relations and have (second-order) relations on
> > > relations, so for example you could say that IsBiggerThan is
> > > transitive:
> > > (Transitive IsBiggerThan)
> > > and define Transitive:
> > > (forall (?R)
> > >     (iff (Transitive ?R)
> > >            (forall (?x ?y ?z)(implies (and (?R ?x ?y)(?R ?y ?z))
> > >                                                   (?r ?x ?z)))
> > > ))
> > > Notice that the ?R ranges over (first-order) relations, not just
things.
> > > In third-order, you can have relations on second-order relations, and
so
> > on...
> > > Higher-order means you can go as high up the ladder of relations of
> > > relations of... as you want. In practice nobody much wants to go
> > > beyond second-order, most of the time, but you never know.
> > >
> > > Real Story
> > >
> > > What *really* makes a logic higher-order is that when you quantify
> > > over 'all relations', that really does mean ALL relations, not just
> > > the ones you happen to mention. There are a hell of a lot of
> > > relations; more than you probably ever want to think about. For
> > > example, consider the property (unary relation, ie relation with one
> > > argument) of being further north than the oldest plumber born in
> > > Philadelphia. Hey, its a perfectly good property; but when you said
> > > (forall (?p)...) did you really have that in mind as a possibility?
> > > Answer: if you are a mathematical logician, yes, you did. The moral
> > > of which is that real higher-order logic is probably more use to
> > > mathematicians than anyone else. For another example, suppose you
> > > wanted to say that two people had something in common, and thought of
> > > using a second-order sentence like
> > > (exists (?P) (and (?P Bill)(?P Joe)))
> > > to say it (ie there is some property true of Bill and of Joe), and
> > > you were thinking of ?P's like 'eye-color' or 'watches baseball'. It
> > > wouldnt do the job for you, since in real higher-order logic, this is
> > > trivially true of any two things, since the property of 'being either
> > > Bill or Joe' satisfies it. Written using lambda this would be
> > > (lambda (?x) (or (= ?x Bill)(= ?x Joe))). Obviously this is true of
> > > Bill (who is equal to Bill) and also of Joe, so it works for ?P. No
> > > good saying "that's not a real property": in real second-order logic
> > > it is, tough luck.
> > > The connection with lambda-calculus is that any lambda-expression
> > > with a sentence body defines a relation. ANY lambda-expression. So
> > > higher-order logic has an inference rule (called 'comprehension',
> > > sometimes its phrased as an axiom) which allows you to make any
> > > sentence into a lambda-expression. If you can say it, its can be used
> > > to define a relation, is the idea.
> > >
> > > ------
> > >
> > > As you can see, none of this has got anything to do with sentences
> > > about sentences: its all to do with sentences about relations.
> > >
> > > Hope this helps.
> > >
> > > Pat Hayes
> > >
> > > ---------------------------------------------------------------------
> > > IHMC (850)434 8903   home
> > > 40 South Alcaniz St. (850)202 4416   office
> > > Pensacola,  FL 32501 (850)202 4440   fax
> > > phayes@ai.uwf.edu
> > > http://www.coginst.uwf.edu/~phayes
> > >
> > >
> >
>
Received on Thursday, 3 May 2001 00:04:47 GMT

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