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>

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 UTC

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