From: Ian Horrocks <horrocks@cs.man.ac.uk>

Date: Tue, 5 Jul 2005 14:08:55 +0100

Message-Id: <fded42c420c5cd7f4bb757a435b8d45e@cs.man.ac.uk>

Cc: www-rdf-rules@w3.org

To: drew.mcdermott@yale.edu

Date: Tue, 5 Jul 2005 14:08:55 +0100

Message-Id: <fded42c420c5cd7f4bb757a435b8d45e@cs.man.ac.uk>

Cc: www-rdf-rules@w3.org

To: drew.mcdermott@yale.edu

On 2 Jul 2005, at 03:39, Drew McDermott wrote: > > >>> [me] >>> I am somewhat baffled. If two systems use the same syntax, and >>> employ >>> the same vocabulary with the same (Tarskian) semantics ... [then >>> they can interoperate by exchanging messages] > >> [Ian Horrocks] >> [...] >> They use the same model structure, but LP semantics admits many fewer >> models than FO semantics, and fewer models means more entailments. > > But entailments just _cannot_ be part of the meaning of a set of > assertions. If that were the case, then a nonmonotonic system > couldn't interoperate with itself! I hope we are not talking about "meaning" in any higher sense - I am but a humble computer scientist, and know little of this meaning of which you speak! All I am talking about is a formal "language" that, given a theory (set of statements), specifies the set of models that are admitted by the theory. The specification typically also defines entailment in terms of models, i.e., a statement is entailed by a theory iff it is true in every model admitted by the theory. > > Suppose Cn(X,Y) is a nonmonotonic consequence relation, giving the > entailments from X when combined with the disjoint set Y. (Cn can be > defined using stable models, well-founded models, or whatever.) > If we focus just on the entailments of X itself, we can define E(X,Y) > to be Cn(X,Y)\Cn(Y,{}), the statements entailed by X over and above > the statements entailed by Y. But then E(X,Y) can differ from E(X,Z) > in many cases. Must we say that X means something different when > conjoined with Y than what it means when conjoined with Z? Two > nonmonotonic systems can interoperate, according to this view, only if > they happen to have reached the same conclusions. This seems to be a > reductio ad absurdum of the whole idea. I don't see the problem. What you describe above is simply one of the properties of a non-monotonic language, i.e., for two theories X and Y, the set of models admitted by X+Y may not be a subset of the set of models admitted by X. Two languages can interoperate if, given the same theory, they admit the same models (and perhaps under other circumstances, e.g. if, as has been suggested w.r.t. DLP, we restrict our means of examining models so that we are unable to distinguish some differences in models). This is the case, e.g., for an OWL-Lite system and a SWRL system (assuming that the theory is syntactically within the OWL-Lite subset of SWRL). I am not sure how to express a notion such as "what X means when conjoined with Z" in the context of this framework. > > The whole "minimal model," "stable model," etc. family of mechanisms > are devices for specifying precisely --- and hopefully efficiently --- > what follows from a set of premises. But the use of the word "model" > all over this landscape doesn't mean we're talking about the > _semantics_ of the statements involved. A theorem of the form > > The entailments from statements S > = the set of statements true in > all models of S with property P > > should not be read as if it were changing the set of models of S. According to my simple view of formal languages, if the semantic specification of the formal language restricts all models to have property P, then the set of models of S is the same as the set of models of S with property P, so "the entailments from statement S" is, by definition, equal to "the set of statements true in all models of S with property P". If the semantic specification doesn't so restrict models, and the language allows me to express a statement of the form "all models must have property P", then I can ask about the entailments from S \cup {all models must have property P}, which is, by definition, equal to "the set of statements true in all models of S with property P". In neither case does the query change the models of S. Ian > > -- Drew > > > -- > > -- Drew McDermott > Yale University > Computer Science Department > >Received on Tuesday, 5 July 2005 13:09:03 UTC

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