- From: pat hayes <phayes@ai.uwf.edu>
- Date: Thu, 3 May 2001 00:29:11 -0500
- 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 --------------------------------------------------------------------- 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 01:29:25 UTC