[22 July 2008]
My colleague Thomas Roessler writes:
[The monotonic semantics of RDF] guarantee that you won’t run into a world of inconsistency when you discover additional information, and they also guarantee that you can learn things about the world piece by piece.
My evil twin Enrique responds: So let us start with the information that “The individual denoted by http://www.w3.org/People/cmsmcq/2008/ns1#joe is identical to the individual named http://www.w3.org/People/cmsmcq/2008/ns2#Josephus”, which I assume I can express using some predicate like the OWL sameAs.
And now let us discover additional information in another triple store, which contains the information that “The individual denoted by http://www.w3.org/People/cmsmcq/2008/ns1#joe is distinct from the individual named http://www.w3.org/People/cmsmcq/2008/ns2#Josephus”, which it expresses using some predicate like the OWL differentFrom.
I’m having trouble understanding (concludes Enrique) how we can do this without either running into a world of inconsistency (a small world, perhaps, bounded in a nutshell, but still a world big enough for joe and Josephus to be both the same and different), or else running into a world in which we find that “inconsistency” has been defined to have a highly technical meaning under which the two triples just described are not actually inconsistent in the technical sense (why do I expect someone to start lecturing me about Herbrand models any moment now?), even though any application relying on the usual notions of identity and difference may find itself at a loss as to what to make of seeing them both in the same graph.
I reminded Enrique of the American pioneer Daniel Boone, who proudly claimed that he had never been lost in his life. Never? Never. [Pause.] “But I was a mite bewildered once for three days.” [Rimshot.]
“Writing a really general parser is a major but different undertaking, by far the hardest points being sensitivity to context and resolution of ambiguity.” — Graham Nelson
Well, technically it’s only OWL that has the concept of contradictory predicates. At the pure RDF level, there is no reason to suppose that sameAs and differentFrom can’t both hold of the same subject-object pair. The most we can say is that we don’t find any such instances in our particular corpus of triples.
A more significant issue around RDF predicates is “Should we believe them?”, which extends “Who says so?”.
The point that Enrique hits here is, to first approximation, – as John says – a point about a specific interpretation of a vocabulary. If the example was asserting that the cat is at the same time dead and alive, then Enrique would utter nonsense in the interpretation that he probably intended, but the graph could still be true (as would both of the “contradictory” triples) in, say, an interpretation in which “dead” means “sleeping under the couch” and “alive” means “well-fed”. So, in the intended interpretation, the initial statement is false; but in all interpretations in which the initial statement holds, we can derive the constituent statements. And in that sense, RDF entailment is indeed monotonic. (Enrique, meet Humpty Dumpty.)
http://www.w3.org/TR/rdf-syntax-grammar/
http://www.w3.org/TR/rdf-mt/
The specific example, though, is more interesting than my feeble attempt at putting Schroedinger’s cat on the semantic web, since OWL isn’t just any vocabulary on top of RDF that might be modelled (or interpreted) poorly: There is indeed a definition of OWL semantics that’s supposed to be compatible with RDF semantics (for some interpretation of “comaptible”). It’s somewhat scary stuff (and no, I don’t think I fully appreciate it). Ultimately, though, what things boil down to (again) is that once you fix an interpretation, you can utter falsehoods in RDF.
http://www.w3.org/TR/owl-semantics/rdfs.html
(At least that’s my admittedly imperfect understanding of these things. Scary? You bet.)
If the protection offered by monotonic semantics boils down to guaranteeing that there is some possible interpretation of the sentences, then it sounds as if what it’s managing is just to ensure that an RDF graph G does not correspond to a logical inconsistency, i.e. M(G), or not L(not G), if we use M for ‘logically possible’ and L for ‘logically necessary’ and allow G to mean now a graph, now the conjunction of the sentences in the graph.
That is a good guarantee to have, I suppose; I have enough appreciation of formal methods to like the sense of security it gives.
But it sounds a lot weaker than the claim from which we started (“[The monotonic semantics of RDF] guarantee that you won’t run into a world of inconsistency”), and somewhat less compelling as a story, either about the safety of merging RDF graphs or about RDF’s semantic richness compared to XML. Does anyone seriously believe that no similar consistency guarantee could be constructed for XML? It would, of course, guarantee only that there is some interpretation of the XML document which is consistent, not that the interpretation you happen to be interested in is consistent, but if I am understanding this discussion properly that is pretty much what RDF’s guarantee says, too.
That is, we are in the second of the two possible worlds Enrique mentioned, one in which someone is telling us that there is a technical sense in which there is no contradiction here, and the fact that we have not seen a road or a location we recognize for three days does not, really, mean that we’re lost. Just a mite bewildered.
Yup, icky details of the Herbrand voodoo are in http://www.w3.org/TR/rdf-mt/
As soon as we introduce layers over dumb RDF triples where contradictions can be explicitly stated, we’re in territory where new data can contain troublesome suprises. But that mess was already there even if only visible at a human level even without OWL.
When it comes right down to it, there is no way to compel RDF to actually be about anything: as the Löwenheim-Skolem theorem tells us, any formal theory about anything whatsoever can by a mere reinterpretation of terms become a theory about the natural numbers. So RDF authors are half-poets: they affirm, but they deny nothing. (Philip Sidney said, “The poet never affirmeth, and therefore never lieth.”)