From: pat hayes <phayes@ai.uwf.edu>

Date: Thu, 28 Jun 2001 00:17:43 -0500

Message-Id: <v04210115b75d66689b6a@[205.160.76.188]>

To: Dan Connolly <connolly@w3.org>

Cc: w3c-rdfcore-wg@w3.org

Date: Thu, 28 Jun 2001 00:17:43 -0500

Message-Id: <v04210115b75d66689b6a@[205.160.76.188]>

To: Dan Connolly <connolly@w3.org>

Cc: w3c-rdfcore-wg@w3.org

>pat hayes wrote: >[...] > > The RDF distinction between anonymous nodes and named nodes is > > logically invisible, as far as I can see. > >I have been convinced otherwise. I'm still not certain, >but let's see if I can reproduce the argument >and its ability to convince... > > > All nodes have an identity, > > so giving one of them a URI doesnt really change anything about it or > > what it means. All nodes could be non-anonymous and nothing would > > change in the meanings of any RDF. Anonymity seems to be a kind of > > syntactic sweetener which works by removing some bitterness rather > > than by adding sugar. > >I think anonymous nodes have non-trivial expressive power. > >In sum: skolemization is *not* a two-way street: >while a proof of a skolemized formula is generally >accepted as a proof of the formula with existential >quantifiers, it doesn't work the other way around. Yes and no. Its true that you can't get from a proof of the existential to a proof with any particular skolem constant, ie if you fix on one skolem constant and then go back to the existential proof and ask for a proof containing *that* skolem constant, you can't get it from the existential proof, because to get a skolemised proof you have to re-skolemise, and every time you do that you will get a *new* skolem constant. But this new one and the old one are what one might call functionally equivalent, in that anything that you can prove about one of them you can also prove about the other (except of course being equal to itself). Skolemising an existential is like saying 'OK, at least one thing exists ..... Choose one of them and call it FOO.' Now, once you have done that, you can again say 'choose one of them and call it BAZ', but you can't legitimately use 'FOO' again, since you have no way to know if you are choosing the same one. >You lose information when you replace an >existentially quanitified variable/term with >a URI. You don't gain or lose information by skolemising. What you do do, is provide a name for something that was previously nameless; but you don't gain any new information about it (unless you count that new name as new information; but since you generated it, that seems rather generous.) >In detail: let's suppose you're buying a book. >You transmit, to the bookseller, a description >of the book you want to buy; roughly, >"it's by Barnsley and it's called >Fractals Everywhere. I want it in >hardback, tomorrow." Ingenious argument. But you have done something slightly odd in your logic here (maybe there is no way to avoid doing this in RDF, which is another issue, I'll return to that) by writing this as a simple existential. If you (just) say that "there exists a book by Barnsley and its called "Fractals Everywhere" and ..." then, indeed, the bookseller really has no way to know what you want. He has a book that satisfies this description, but there's nothing in a simple existential to guarantee that there might not be something else that satisfies it as well, so he can have no logical confidence that the thing he has that satisfies the description is the *same* as the thing that you have in mind that satisfies the description. If all that he has to go on is a simple existential, that is. In order to know that the thing that you are saying exists is the *same* as the thing he knows exists, he has to know something else as well, which is that there is only one such thing (we are talking about books here as editions rather than copies, I assume). That can be said explicitly in KIF (FOL) by using universal quantification and identity, ie instead of just saying (exists (?x) (and (by ?x Barnsley)...)) you include a statement of identity: (exists (?x) (and (and (by ?x Barnsley)...) (forall (?y)(implies (and (by ?y Barnsley)...) (= ?x ?y)))) which is what Russell gave as the logical gloss for a definite description, ie this is what you get when you say something about 'THE x which is by Barnsley and ......'. The 'THE' is what pins down the meaning to a single entity. Now, if you apply this to your example, there can be two URI's, but it would follow (either from what you would have said, ie that you wanted THE book...., or from general knowledge that the bookseller has about uniqueness of author/title information, or something) that the two names (URIs) were equivalent, ie had the same denotation; so now there would be no loss of information at all. This analysis requires logical apparatus which goes well beyond what RDF1.0 can do, but another way to think about how to convey the uniqueness assumption in your order, is to say that the URIs in the following example are not simple logical constants, but something more like public names. So the idea would be, here, that instead of the existential being in your Kbase, then getting sent to the bookseller, then processed by him in his Kbase, that something more like a process of setting up a 'common ground' happens, where some of your assertions and some of the booksellers assertions are brought into a common area of mutually accepted beliefs, like what seems to happen in a conversation. If that were possible, then it would be more like the two sets of beliefs being made to intersect, rather than thinking of them as communicating with each other, so that this existential about this book could then be mutually accepted. Then if it were skolemised *when in the common ground*, then only a single skolem name would be generated, but it - the new name - would be part of both your beliefs, and you could both use it to refer to the book you have agreed to talk about. This is a bit like a conversation between two mathematicians going like this: A: 'I want to talk about a finite group with ten elements.' B: 'OK.' A: 'Call this group G' B: 'OK, now what else do I need to know about G?' .... where they can agree on a mutually acceptable name even when one of them doesnt yet know exactly what the other is talking about. Once A has said enough about G to enable B to figure out that there is only one thing that satisfies the description, B might say: 'now I know what you mean'. (Another way to think about it is that you and the bookseller set up a temporary existential quantifier whose scope extends across sentences in both your Kbase and his, a kind of spanning quantifier; and then when you skolemise that, you both get beliefs involving the same name. I like this way of thinking, since it seems to suggest a way to analyse public names in general as implicit linking quantifiers whose scope extends across an entire community. ) >Formally, in n-triples: > >---8<--- >_:g0 <http://purl.org/dc/elements/1.1/title> "Fractals Everywhere" . >_:g0 <http://purl.org/dc/elements/1.1/creator> "Barnsley" . >_:g0 <http://booksellers.example/vocab#binding> ><http://booksellers.example/vocab#hardback> . >_:g0 <http://booksellers.example/vocab#shipping> ><http://booksellers.example/vocab#nextDay> . >---8<--- > >In KIF, by my understanding of anonymous nodes: > >---8<--- >(exists (?x0 ) > (and > (PropertyValue http\:\/\/purl\.org\/dc\/elements\/1\.1\/title > ?x0 > "Fractals Everywhere") > (PropertyValue http\:\/\/purl\.org\/dc\/elements\/1\.1\/creator > ?x0 > "Barnsley") > (PropertyValue http\:\/\/booksellers\.example\/vocab\#binding > ?x0 > http\:\/\/booksellers\.example\/vocab\#hardback) > (PropertyValue http\:\/\/booksellers\.example\/vocab\#shipping > ?x0 > http\:\/\/booksellers\.example\/vocab\#nextDay) >) ) >---8<--- > >Now if I use something like Sergey's algorithm[1,2] >to compute a URI for this anonymous term, I get >something like: > >---8<--- ><http://skolem.example#432oj34oij2o3ijo23j> ><http://purl.org/dc/elements/1.1/title> "Fractals Everywhere" . ><http://skolem.example#432oj34oij2o3ijo23j> ><http://purl.org/dc/elements/1.1/creator> "Barnsley" . ><http://skolem.example#432oj34oij2o3ijo23j> ><http://booksellers.example/vocab#binding> ><http://booksellers.example/vocab#hardback> . ><http://skolem.example#432oj34oij2o3ijo23j> ><http://booksellers.example/vocab#shipping> ><http://booksellers.example/vocab#nextDay> . >---8<--- > >i.e. in KIF: > >---8<--- > (and > (PropertyValue http\:\/\/purl\.org\/dc\/elements\/1\.1\/title > http\:\/\/skolem\.example\#432oj34oij2o3ijo23j > "Fractals Everywhere") > (PropertyValue http\:\/\/purl\.org\/dc\/elements\/1\.1\/creator > http\:\/\/skolem\.example\#432oj34oij2o3ijo23j > "Barnsley") > (PropertyValue http\:\/\/booksellers\.example\/vocab\#binding > http\:\/\/skolem\.example\#432oj34oij2o3ijo23j > http\:\/\/booksellers\.example\/vocab\#hardback) > (PropertyValue http\:\/\/booksellers\.example\/vocab\#shipping > http\:\/\/skolem\.example\#432oj34oij2o3ijo23j > http\:\/\/booksellers\.example\/vocab\#nextDay) >) >---8<--- > >Now suppose the bookseller's inventory database shows: > >---8<--- ><http://booksRus.example/inv2001-06-25#item342323> ><http://purl.org/dc/elements/1.1/title> "Fractals Everywhere" . ><http://booksRus.example/inv2001-06-25#item342323> ><http://purl.org/dc/elements/1.1/creator> "Barnsley" . ><http://booksRus.example/inv2001-06-25#item342323> ><http://booksellers.example/vocab#binding> ><http://booksellers.example/vocab#hardback> . ><http://booksRus.example/inv2001-06-25#item342323> ><http://booksellers.example/vocab#shipping> ><http://booksellers.example/vocab#nextDay> . >---8<--- > >i.e. > >---8<--- >(and > (PropertyValue http\:\/\/purl\.org\/dc\/elements\/1\.1\/title > http\:\/\/booksRus\.example\/inv2001\-06\-25\#item342323 > "Fractals Everywhere") > (PropertyValue http\:\/\/purl\.org\/dc\/elements\/1\.1\/creator > http\:\/\/booksRus\.example\/inv2001\-06\-25\#item342323 > "Barnsley") > (PropertyValue http\:\/\/booksellers\.example\/vocab\#binding > http\:\/\/booksRus\.example\/inv2001\-06\-25\#item342323 > http\:\/\/booksellers\.example\/vocab\#hardback) > (PropertyValue http\:\/\/booksellers\.example\/vocab\#shipping > http\:\/\/booksRus\.example\/inv2001\-06\-25\#item342323 > http\:\/\/booksellers\.example\/vocab\#nextDay) >) >---8<--- > > >This information from the inventory database clearly >provides a match (i.e. a proof, or the interesting >bits of one, anyway) for the >description of the book stated as a formula involving >an existentially quantified variable, ?x0. No, not a description; it just proves that a book exists which satisfies the existential. But since there might be several things that satisfy an existential, nothing follows about whether the bookseller's item#342323 actually IS the particular thing that you had in mind when writing your order. (Not without some uniqueness assumption from somewhere.) So the bookseller is in the same position in either case. If he knows the thing is unique, he can figure out what you want; if he doesn't, he can't. Of course, if you had phrased your order not as an existential but as a universal , so that you say: "if it is a book and has author...and title ...., then I want it", then the bookseller can figure out what you want. However, you might now get several books, unless you give enough of a description to pin down one uniquely. >But it doesn't match/prove the formula where _:g0 >is replaced by the skolem constant ><http://skolem.example#432oj34oij2o3ijo23j> . Right. And it might not be it. On the other hand it is possible that those names corefer, ie (= http\:\/\/booksRus\.example\/inv2001\-06\-25\#item342323 <http://skolem.example#432oj34oij2o3ijo23j>) and if the bookseller could somehow conclude this (eg by knowing that only one thing satisfied the description) then he could get you the book you want from your order, skolemised or not. >So an RDF document with anonymous nodes is *not* >interchangeable with a skolemized version of that document. >Q.E.D. It is if everyone is using valid logical reasoning. > >I think this is the important part of Pat's message >that I disagree with is: > > > RDF is supposed to > > consist of assertions, not equations, and they are pretty simple > > assertions of the logical form (V s o) . Those s's and o's are > > logical names = logical constants, ie they denote things, one thing > > each in a given interpretation. Now, logical names are *just* like > > existentially quantified variables where the quantifers are at the > > top level (the only level there is in RDF), ie not inside any > > universal quantifiers. > >The s's and o's would be just logical constants if we >left out anonymous/existentially-quantified terms. >But RDF does have such terms, as demonstrated above. >And the semantics of such terms is non-trivially >different from the semantics of constants: >an existentially quantified formula is satisfied >by an interpretation if there's *some* thing >you can bind the existential term to that makes >it work out. Yes, and that is exactly the truth conditions for a logical name, as well; it has to have a denotation that makes the sentences containing the name work out true. >Skolemizing the formula reduces >the number of interpretations that satisify the >formula. No, it doesnt. Technically, if E is an existential sentence and E' is a skolemisation of it, then E and E' are actually in different logical languages, because the signature of E' has a new symbol in it, so it takes a little care to compare interpretations. If I is an interpretation of the smaller langauge (of E, without the new skolem name) then I can be easily extended to an interpretation I' of the larger language (of E') which is identical to I on the E-language and makes E' true (just make the skolem name denote one of the things that satisfies the existential.) And in the language of E' (with the skolem name), any interpretation which satisfies E' will also satisfy E, ie E' entails E in the E'-language. So either way round, skolemising the formula doesnt *reduce* the number of interpretations. Theres quite a lot of depth to that notion of a 'new' name. In natural deduction systems where introducing new names is a legal inference step, the rules for scoping new names can get very complicated. >It's one of those things that usually >works "without loss of generality," No, it always works. You can see why if you transcribe it into second-order logic, since then the skolemisation process is basically just a quantifier exchange, and its actually valid in both directions. The first-order case seems clunkier only because in first-order order, any change to the nonlogical vocabulary goes outside the usual inference process and so isn't technically valid; but thats only a technicality. >but doesn't, >in the case of RDF; i.e. in the case when constant >symbols that are shared between documents. Ah, but wait: that 'shared' begs the question. A skolem constant generated inside one document cannot possibly be shared with another document; it has to be a new name, not one used anywhere else. (Possible thanks to the Power of the URI, right?). Now, once generated, it might later get transferred to another document, but that transfer process doesnt add any new information to it. If a name is shared between documents in such a way that each of them says something nontrivial about it (ie each one constrains the possible interpretations), then it can't be a skolem constant. (Unless maybe it was somehow introduced to them both simultaneously by some kind of 'common ground' maneuver, as sketched above.) BTW, an overall comment: all this discussion has been in the context of classical logic. If we are using a nonmonotonic logic of some kind, eg by assuming a unique-name condition, then skolemisation may well not be correct at all, and we would need to re-think the entire game. It may (MAY) be that Euler is using some such assumptions to do its LCU 'unification'; I honestly cannot follow what Euler is up to. >p.s. This message is really about an existing issue >http://www.w3.org/2000/03/rdf-tracking/#rdfms-identity-anon-resources > >The fact that there are graphs involving anonymous nodes >that are not expressible in RDF/XML is, I agree, another >issue; one that, I'm afraid, belongs in the >postpone-to-next-version pile. > >[1] Re: RDF API 1.0 Draft / algorithm for anonymous URIs >Sergey Melnik (Wed, 08 Dec 1999) >http://lists.w3.org/Archives/Public/www-rdf-interest/1999Dec/0046.html > >[2] what I actually used was random scratching at the keyboard ;-) > >-- >Dan Connolly, W3C http://www.w3.org/People/Connolly/ 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/~phayesReceived on Thursday, 28 June 2001 09:49:01 UTC

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