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Re: Unique denotation assumption

From: David Booth <david@dbooth.org>
Date: Fri, 13 Sep 2013 12:07:39 -0400
Message-ID: <5233384B.4000300@dbooth.org>
To: Pat Hayes <phayes@ihmc.us>
CC: www-archive <www-archive@w3.org>
Hi Pat,

I think the big disconnect between us is that I am looking at the 
semantics from the perspective of multiple interpretations, whereas you 
seem to keep assuming a single-interpretation viewpoint.  I'll try to 
explain . . .

On 09/11/2013 11:15 PM, Pat Hayes wrote:
>
> On Sep 11, 2013, at 6:14 PM, David Booth wrote:
>
>> hi Pat,
>>
>> Since the RDF WG does not have a public discussion list to which
>> non-members can post, I'm copying the public www-archive list.  I
>> want to propose a couple of changes to the RDF specs, and thought
>> it might save the WG and others time if I first discuss them with
>> you (and anyone else you think would be interested).  One of them
>> is the following.
>>
>> In
>> https://dvcs.w3.org/hg/rdf/raw-file/default/rdf-concepts/index.html
>>
>>I see this statement:
>>
>> "IRIs have global scope: Two different appearances of an IRI denote
>> the same resource."
>>
>> This is wrong
>
> It is a foundational assumption of RDF. Without this assumption,
> graph merging would never be a valid operation, and no RDF engines
> could combine informationfrom different sources to draw conclusions.
>
>> , because an IRI can and often does denote different resources in
>> different RDF interpretations.
>
> That is irrelevant to the point being made in the sentence you cite
> above. No interpretation allows two occurrences of a single IRI to
> denote different things.

The problem is that that sentence is not constrained to a single 
interpretation.  If it said "**For each RDF interpretation** IRIs have 
global scope . . . . " then it would be fine.

As written, that sentence ("IRIs have global scope: Two different 
appearances of an IRI denote the same resource") is just patently false, 
as your example of G1 and G2 illustrated:

> G1: ex:a ex:p "23"^^xsd:number .
>
> G2: ex:p rdfs:range xsd:string .

Assuming XSD semantics, ex:p clearly denotes something different in G1 
than it denotes in G2.

>
>> And this, in practice, means that an IRI often denotes different
>> resources in different *graphs*
>
> No, it does not. What is true, of course, is that the sets of
> interpretations which satisfy the two graphs will be different, and
> each will likely be different from the set of interpretations
> satisfying both of them (their union.) But that does not imply that
> IRIs in different graphs denote differently.

When you say "But that does not imply that IRIs in different graphs 
denote differently", notice that you are implicitly assuming that the 
*same* interpretation(s) will be applied to both graphs.  (Because 
obviously if different interpretations are applied, the same URI may 
denote different things.)  But my point was that if you look at it from 
the perspective of "What are the sets of satisfying interpretations for
each of these graphs (separately)?", then we *are* talking about 
different sets of interpretations, and the same IRI *may* denote 
different things in them.

>
>> , because any graph has a set of satisfying interpretations, and
>> different graphs may have different sets of satisfying
>> interpretations.
>
> Yes, but that does not imply what you seem to think it implies.

I doubt that what I think is what you think I think.  What do you think 
I think it implies?

To be clear, I think it implies the following.  For graphs g1 and g2, 
URI u, let s1 and s2 be the sets of satisfying interpretations for g1 
and g2 (respectively).  Let rs1 be the set of resources {i(u) such that 
i is in s1}, and similarly let rs2 be the set of resources {j(u) such 
that j is in s2}.   Claim: Every member of s1 is "what u denotes in g1" 
and every member of s2 is "what u denotes in g2".  (But note that they 
came from different sets of interpretations: s1 and s2, respectively.) 
Claim: "what u denotes in g1" may be *different* from "what u denotes in 
g2".  The truth of this should be obvious.

Notice that nowhere am I making the single-interpretation assumption. 
AFAICT, the disconnect between us seems to be that you are assuming that 
one must only be talking about the *same* interpretation(s) when talking 
about g1 and g2.  That would be necessary if we were talking about the 
union of g1 and g2, but it is *not* necessary if we are talking about g1 
and g2 as separate graphs.  The RDF Semantics can perfectly well be 
applied to each graph separately.

>
>> For example, suppose graphs g1 and g2 have sets of satisfying
>> interpretations s1 and s2, respectively, and those sets may be
>> disjoint.  Then colloquially (and technically) we can say that an
>> IRI may map to one resource in g1 (i.e., in some interpretation in
>> s1) and a different resource in g2 (i.e., in some interpretation in
>> s2).
>
> No, that would not be a correct way to express this condition, even
> colloquially. What you are describing is two graphs which contradict
> each other in some way.

Yes: the intersection of their sets of satisfying interpretations is 
empty.  I.e., the graphs contradict each other.

> Which is impossible  in RDF but much easier
> to arrange in RDFS or OWL or other more expressive languages, but in
> any case it is often only possible to detect the contradiction when
> we presume that IRIs denote uniformly across graphs. Take this
> example, using some rdfs:
>
> G1: ex:a ex:p "23"^^xsd:number .
>
> G2: ex:p rdfs:range xsd:string .
>
> Each is consistent (has satisfying interpretations) but together they
> are inconsistent (recognizing XSD). Why? Because if ex:p has range
> xsd:string, then the first triple in G1 implies that a nuber must be
> an XSD string, and XSD says these are disjoint. But notice, this line
> of reasoning assumes that the ex:p in the G1 is the same ex:p as in
> G2.

By "inconsistent", presumably you mean either; (a) that the graph that 
is the union of G1 and G2 has no satisfying interpretations; or (b) that 
the intersection of the sets of satisfying interpretations for G1 and G2 
is empty.  In case (a), we would be talking about a different graph that 
has G1 and G2 as subgraphs, so yes, ex:p in G1 must be the same as ex:p 
in G2, and there are no interpretations that satisfy this requirement. 
In case (b), ex:p in G1 clearly denotes a different resource than ex:p 
in G2, and that is why the intersection of their satisfying 
interpretations is empty.

> If it wasn't  if the same IRI in two different graphs could
> denote different things  then there would be no inconsistency here,
> because G2 might be talking about a different thing than the ex:p
> that G1 is talking about.

Indeed, it is quite clear that the ex:p that G1 is talking about *is* 
different than the ex:p that G2 is talking about, and this is why these 
graphs are incompatible.  This is evident because in every satisfying 
interpretation of G1, the value of ex:p is an xsd:number, whereas in 
every satisfying interpretation of G2, the value of ex:p its value is an 
xsd:number, whereas in G2 its value is required to be an xsd:string, and 
an xsd:string cannot be an xsd:number.  I.e., the intersection of their 
sets of satisfying interpretations is empty, so the graphs are 
necessarily inconsistent.

>
>>
>> This requires thinking about graphs in terms of sets of satisfying
>> interpretations -- an important and valid perspective -- rather
>> than assuming that one looks at them only through the lens of a
>> single interpretation.
>
> Of course we think of graphs in terms of sets of satisfying
> interpretations. That does not imply that different occurrences of an
> IRI can denote differently.

Sure it does.  ex:p denotes *different* things in the set of satisfying 
interpretations of G1 than it does in the set of satisfying 
interpretations of G2, as we just established above.

>
>>
>> In short, I think the above statement needs to be qualified
>> somehow, such as:
>>
>> "IRIs are intended to have global scope: Two different appearances
>> of an IRI are intended to denote the same resource. (But see RDF
>> Semantics for caveats regarding different RDF interpretations.)"
>
> The semantics document has no such "caveats".
>>
>> And then we would have to add something to the RDF Semantics
>> document to explain how an IRI can denote different things in
>> different interpretations.
>
> Of course IRIs denote different things in different interpretations.
> That is what interpretations are all about: each interpretation is
> one way to construe what IRIs mean, one mapping from IRIs to denoted
> things. But each one of them construes each IRI in a particular way.
> It does not construe one occurrence of a given IRI in one way, and a
> different occurrence of the same IRI in a different way.

Correct: a single interpretation does not construe one occurrence of a 
given IRI in one way and a different occurrence of the same IRI in a 
different way.  But that is *exactly* what multiple interpretations do.

> That
> condition  we might call it a 'split reference'  is not possible in
> any interpretation. In fact, it is explciitly ruled out by the very
> construction of interpretations, which are defined as mappings from
> IRIs to referents. Not from *occurrences* of IRIs to referents,
> notice.

Agreed.  I'm not talking about a single interpretation.  I'm talking 
about multiple interpretations, such as we may be considering when we 
ask: "What are the sets of satisfying interpretations for g1, g2 and g3 
(respectively)?"

>
>>
>> Does this make sense to you?
>
> Clearly not; but you knew this, surely, from our earlier extended
> discussions on this topic.

Hey, I'm trying to make headway toward a common understanding here!  :) 
  Have I made any progress?

Best wishes,
David
Received on Friday, 13 September 2013 16:08:10 UTC

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