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
