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Re: Unique denotation assumption

From: David Booth <david@dbooth.org>
Date: Wed, 25 Sep 2013 23:56:05 -0400
Message-ID: <5243B055.2090705@dbooth.org>
To: Gregg Reynolds <dev@mobileink.com>
CC: www-archive@w3.org
Hi Gregg,

On 09/12/2013 03:18 PM, Gregg Reynolds wrote:
> 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.

Sure, no problem.  It is intended to be a public discussion.  I just 
didn't want to clutter the RDF comments list with it.

> (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.

*Which* MT-interpretation function?  My point is that the semantics 
permits *many* interpretation functions.  People often seem to assume 
that there is only one, but the semantics are very clear that there may 
be many interpretations.

> 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,

Okay, but . . .

> 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.

. . . I don't see how that follows.  Even if MT semantics cannot 
determine the denotations of extra-logical symbols -- and by that I 
assume you mean the resources denoted by IRIs -- and does not know or 
care what they are, it seems to me that an interpretation function can 
still be required to be, well, *functional*, such that any appearances 
of an IRI map to the *same* resource.

> 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.

Okay, I *think* I'm following what you mean, but I also think that that 
is the reason for allowing multiple interpretations, i.e., the allowance 
of multiple interpretations is the mathematical way of punting on the 
"real" meaning of an expression.

>
> 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.

No, that isn't *quite* the only thing model-theoretic semantics can do. 
  It can also *constrain* the set of valid interpretations.

> Specific extra-logical meanings of :a :b and :c are
> irrelevant, so there is no point in bringing them up.

I don't entirely agree that that means there is no point in bringing 
them up.  I think we can still discuss constraints on them even if we 
don't know what they are.

>
> 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.

That is true of *each* interpretation function, but my point is that the 
semantics permits *many* interpretation functions.  It is perfectly 
reasonable to ask: "What interpretations would make graph G1 true?" and 
"What interpretations would make G2 true?".  And, given those to sets of 
interpretations, it is reasonable to ask: "What does URI u denote, in 
each of those sets of interpretations?"  And it may well be a different 
set of resources relative to G1 than to G2.

> 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.

A function is a function.  Unless you are telling me that there is some 
additional restriction or magic going on, I see no reason why only one 
such function can exist.

David


>
> 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, 26 September 2013 03:56:33 UTC

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