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Re: Unique denotation assumption

From: Gregg Reynolds <dev@mobileink.com>
Date: Thu, 12 Sep 2013 14:18:53 -0500
Message-Id: <204E787B-9C06-4D11-95C5-6941E37CBF7B@mobileink.com>
Cc: david@dbooth.org
To: www-archive@w3.org
Hi David,

Since you made a point of posting to the public list I hope you won't mind a comment from the peanut gallery.  (I was not subscribed at the time so this message was generated by the "Respond" link on the archive website, which didn't copy the original body.  So I'll cut and paste.)

DB: [
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, ]

I think you are correct - in my view the passage reflects a fundamental misunderstanding - but for the wrong reasons.  

Background: I think the problem, which seems to be pervasive in the RDF community, is confusion of the empirical and logical on the one hand, and misreading/misuse of model theory on the other.  Model theory is not about interpretation or denotation, it's about logical consequence.  It uses a mathematical device - a function - that it calls an "interpretation",  but only as a means of getting at logical consequence.  It is not /about/ interpretation.  The result (and maybe the goal in the first place) is to arrive at a concept of logical consequence that swings free of any particular interpretation.  (For "interpretation", we can substitute "denotation" and various other terms with no change in meaning.)

It should be stressed that the purely mathematical notion of "interpretation" in model theory is entirely distinct from the non-mathematical interpretations we routinely assign to symbols - call that "hermeneutic interpretation" as opposed to "model-theoretic interpretation".  MT is obviously indifferent to hermeneutic interpretation, but it is also indifferent to particular MT-interpretations.

Note btw that the MT-interpretation function is total - it maps every symbol in the language.  So it is not possible to have more than one assignment to a given symbol for a given interpretation.  This point is relevant to your g1/g2 example, more below.

So I think the main problem with the passage is the mention of denotation.  It's a fundamental mistake to think of MT as having something to say about what the extra-logical symbols of a language "really" mean (i.e. denote).  In fact, it's a mistake to think that MT has anything at all to say about meaning that extends beyond logical consequence.  But it's no less a mistake to think that model theoretic semantics has anything to say about "the" MT-interpretation of any symbol.  MT semantics does not determine the denotations of extra-logical symbols, so it is just wrong to say that "Two different appearances of an IRI denote the same resource."  At least, it would be wrong to claim that MT supports such a statement.  Furthermore it cannot be taken as a stipulation, since it says nothing about how to determine denotations, let alone decide when they're equal.  So it would be pointless to declare by fiat that an IRI must have the same denotation everywhere, since nobody could act on it.  What *is* the same everywhere is the rules of logical consequence.  Distinct "appearances" of the same RDF statement in distinct "appearances" of the same context (i.e. graph) have the same logical consequences, regardless of denotation.

Among other things, this means that while MT has a notion of logical truth, but is indifferent to ordinary truth, and in particular it has absolutely nothing to say about the truth values of empirical statements.

To get a true conclusion you need true premises and valid inference.  MT semantics addresses inference/consequence only.  It has nothing to say about the truth of, for example, observational statements of Physics or Chemistry, but quite a lot to say about drawing consequences from them if they are true.

The lesson for RDF is that MT has nothing to say about the empirical truth of any RDF statement, nor about the empirical referent of any IRI.  And it is indifferent to particular MT-interpretations (models) of extra-logical symbols.  It does not determine the denotations, formal or informal, of any such symbols.  (Just as formalized group theory doesn't care whether you interpret it in terms of this or that concrete group.) The only thing a model-theoretic semantics can do is show that the sorts of inferences one can make in RDF (e.g. from :a a :b, :b rdfs:subClassOf :c to :a a :c) are (or correspond to) logical consequences.  Specific extra-logical meanings of :a :b and :c are irrelevant, so there is no point in bringing them up.

Another way to put it:  there is no fact of the matter as to what IRIs denote, model theory or not.

DB [
This is wrong, because an IRI can and often does denote different 
resources in different RDF interpretations.  And this, in practice, 
means that an IRI often denotes different resources in different 
*graphs*, because any graph has a set of satisfying interpretations, and 
different graphs may have different sets of satisfying interpretations. ]

No, the interpretation function is a total function.  One symbol, one denotation.  Furthermore, an interpretation that satisfies some particular graph but fails to satisify the RDF axioms would not be a model for RDF, so it would be irrelevant - no consequences would follow.

The reason the passage you quote above is wrong is not that, as you argue, interpretations may vary.  It's wrong (or at least misguided) because particular interpretations of particular IRIs are irrelevant to the MT semantics of the language.

DB [
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).]

Actually you only need one graph to make your point here; since you have sets of interpretations you can just pick two distinct interpretations (models) for the one graph.  But again, that is irrelevant to the MT-semantics, which only cares about whatever models make for logical consequence.  If the interpretation is such a model, various other statements follow as logical consequences; if not, then not.  As to whether or not any particular interpretation is true in the sense of describing a fact, RDF has nothing to say about it, indeed cannot say anything about it.

A further point: the sort of models of interest to MT semantics - the ones where, if the axioms come out true, so do the theorems - are /global/ models.  They cover the entire language, including every sentence that can be constructed in it.  So the sort of localized interpretations you describe - distinct graphs having distinct interpretations - really boil down to a matter of distinct models, each covering all graphs expressible in the language.  Or to put it a little less verbosely: local denotational idiosyncracies of the sort you describe are ruled out by the definition of an interpretation function in MT.

To sum up, I think the focus on denotation is misguided so I would probably just drop the passage.  But if I had to fix it I would suggest something like:  

Original:
  "IRIs have global scope: Two different appearances of an IRI
   denote the same resource."

Suggested:
  "IRIs have global scope.  This follows from the fact that the interpretation function is a total function: it maps every symbol to a denotation. Two different appearances of an IRI, under a given interpretation, have the same denotation; this follows from the fact that the interpretation is a function.  Only interpretations that are models of the RDF axioms are relevant to RDF semantics.  However, no one model is privileged; different uses can and do place different interpretations on occurences of the same IRI.  The establishment of authoritative, shared interpretations (models) is a matter of social convention and therefore beyond the scope of the definition of RDF."

But I would also add language to try clear up some of the confusion, something like:

   Particular interpretations - whether informal (hermeneutic) for formal (model-theoretic) - of IRIs are irrelevant to the formal semantics of RDF.  The formal semantics of RDF guarantees that certain statements follow as logical consequences from any set of RDF statements regardless of what specific interpretations are placed on them by people or systems, and whether or not they state true facts, so long as those interpretations count as models of the RDF axioms.  There may be many - even infinitely many - interpretations that fit this description.  RDF semantics says nothing about the empirical truth of RDF statements, nor about real-world referents of IRIs; nor does it determine the denotations of extra-logical symbols (IRIs), formal or informal.  For RDF semantics, the only constraint on an interpretation is that it be a model of the RDF axioms, but it can be any such model.  Users and systems can assign whatever (informal) meanings they please to IRIs; in particular, different people may interpret the same or different instances of an IRI in different ways.  So long as their interpretations are models of the RDF axioms, they are equivalent with respect to RDF semantics, and no one of them is primitive.
  
My $0.02,

Gregg
Received on Thursday, 12 September 2013 19:19:24 UTC

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