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Re: N3 contexts vs RDF reification

From: pat hayes <phayes@ai.uwf.edu>
Date: Thu, 3 May 2001 00:29:11 -0500
Message-Id: <v04210104b71692f0b19d@[205.160.76.182]>
To: "Seth Russell" <seth@robustai.net>
Cc: 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?

I already did this, in a message sent to you on 4/26/01 which, 
ironically, you appended to this very message.

>  Maybe, at least, specify what language this 'meaning' is to be
>expressed in and provide an example.

It is expressed in model theory.  That is the only way one can make 
the first-order/second-order contrast properly, in fact. (You can 
specify this using common or garden mathematics; if you want to get 
very formal and kosher, you could do it in set theory, but that would 
be overkill. )

>  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?

No; the point is more that you havnt said what your notation means, 
and so adding more of the notation can't possibly provide the meaning.

>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 cannot see this at all. The meanings might be stateable in DAML, 
since DAML does have a model theory. But (1) you havnt said how to 
translate your diagrams into DAML, and (2) as a matter of fact, DAML 
can't express second-order meanings: it is strictly first-order. But 
more to the point, no matter how much you add to the diagram, all you 
are doing is adding to the diagram. No amount of such adding can 
specify what all this diagramming actually means. You need to provide 
a semantics for your diagrams.

Look, you call your diagrams "assertions". OK, then what exactly are 
they asserting? What would the world have to be like in order to make 
one of them true in it? You need to say how your diagrams are to be 
interpreted: what the parts of them denote, and how the denotations 
of a larger diagram is specified in terms of the denotations of its 
parts.

Simple example from logic: proposition letters denote truthvalues; 
truth-values of expressions like (P and Q) and (P or Q) are computed 
from the truthvalues of P and of Q using truth-tables; an 
interpretation is an assignment of truthvalues to proposition 
letters; a sentence is satisfied in an interpretation if that 
assignment of truthvalues to its letters makes it true; what a 
sentence *means* is that the world satisfies it. That works fine for 
purely propositional logic and is all very easy. For the quantifiers 
you need a fancier notion of a world (it has a universe of things 
that the quantifiers range over and the relation names denote 
relatons on, and so forth) and it gets more interesting, but the 
basic idea is the same. For a functional language like LISP you would 
need another kind of universe with lots of functions in it, and the 
math gets hairier; for a modal langauge you need sets of possible 
worlds; for a language with reification, you need to have expressions 
in the universe as well as things, which makes the notion of 
'satisfiable' much trickier to state properly; and so on. For a 
second-order language you need a universe which has things and also 
has relations in it (not just defined on it, but actually contained 
in it.) Get the idea?

Pat Hayes

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Received on Thursday, 3 May 2001 01:29:25 GMT

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