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Re: N3 contexts vs RDF reification

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Wed, 02 May 2001 18:33:59 -0400
To: seth@robustai.net
Cc: phayes@ai.uwf.edu, www-rdf-interest@w3.org
Message-Id: <20010502183359R.pfps@research.bell-labs.com>
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 independant of that meaning.  You have done
neither.

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 Wednesday, 2 May 2001 18:34:47 GMT

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