Re: Unique denotation assumption

Howdy,

On Wed, Sep 25, 2013 at 10:56 PM, David Booth <david@dbooth.org> wrote:

> 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.
>
>
> 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.


Agreed.  I guess my question then is why you think it significant (if you
do) to pick out specific interps.  Interps that are not models are
uninteresting; interps that are models are equivalent "up to isomorphism".

>
>
>  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.


If I recall the point I was trying to make is that absolute statements
about denotation are out-of-bounds.  So any claim like "two different
occurences of an IRI denote the same thing" must be relativized to an
interpretation.

>
>
>  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.


Well, at least on the non-logical meaning.  On the other hand, Etchemendy
claims that going from the ordinary truth of a universal closure (i.e. all
instances/interpretations of a formula are true) to the logical truth of
each instance is invalid.  So I guess it depends on whom you ask.

>
>
>
>> 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.


Whoa nelly!  That's exactly what you *cannot* do in MT.  If you constrain
the set of allowable interpretations you end up in a circle: sentences that
turn out to be true under the allowed interpretations are declared
logically valid only because "logically valid" is defined by exclusion of
falsifying cases.  So why bother with model theory if we're going to cheat
anyway?  Etchemendy identifies this as another problem with standard MT; I
happen to think it's a rather glaring problem with RDF-MT, but that's
another discussion.


>
>
>  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.


True enough, but then we move beyond MT semantics and into denotational
semantics.  We can do that if we have a language with an more-or-less
clear, antecedently available intended interpretation, such as the language
of chemistry or group theory.  Then we can ask whether our formal vocab
symbols adequately capture the (non-logical) semantics that we already
understand or just stipulate e.g. that '2' SHALL denote two.  But again,
that's not relevant to logic and model theory; you can have an intended
interpretation, but you cannot use it to delimit the class of acceptable
interpretations.  Once you start constraining interpretation, you surrender
logical validity.

>
>
>> 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.


Replace "interpretations" with "models of RDF".  You can have distinct
models and ask what objects each assigns to a particular IRI - but why
bother?  By definition such assignments are inessential, replaceable - all
that matters is the set of inferences that are valid under all models, no
matter how the extra-logical symbols are interpreted.


>  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.


I think we actually agree, mostly.  Of course any number of interp
functions can be defined.  More importantly, no one such function can be
priviledged as *The One True Interpretation*.  Interpretatons that are
models are
"all the same" in just that sense, that there is no way pick out any one of
them as "better" than the others.

My recollection is that your original post was motivated by the observation
that the official RDF docs seem to say or at least give the (false)
impression that there is an "Official RDF Interpretation".  On that I would
whole-heartedly agree, but it's largely a literary/editorial matter and it
looks to me like the chance of that changing is about zero.

Thanks,

Gregg