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Re: RDF Semantics - Intuitive summary needs to be scoped to interpretations (ISSUE-149)

From: Pat Hayes <phayes@ihmc.us>
Date: Sat, 9 Nov 2013 21:54:39 -0600
Cc: Antoine Zimmermann <antoine.zimmermann@emse.fr>, www-archive <www-archive@w3.org>, "Peter F. Patel-Schneider" <pfpschneider@gmail.com>, Ivan Herman <ivan@w3.org>, Sandro Hawke <sandro@w3.org>
Message-Id: <71710066-4483-4C17-8F61-902D574849BE@ihmc.us>
To: David Booth <david@dbooth.org>

On Nov 8, 2013, at 11:00 PM, David Booth <david@dbooth.org> wrote:

> Hi Pat,
> 
> Thanks very much for your excellent summary.  Detailed responses below.  (And sorry it takes me so long to write these things up.)
> 
> On 10/31/2013 02:32 AM, Pat Hayes wrote:
>> Hi David
>> 
>> Rather than respond point-by-point, I will again try to summarize.
>> However, there are a few responses that are needed first:
>> 
>>> ... at least in principle, anything that can be described in, say,
>>> English prose could instead be described in RDF.
>> 
>> Most emphatically, no. Even if you substitute the most expressive
>> formal logic available (say, full higher-order modal tense logic) ,
>> this would not be even remotely correct. RDF is so inexpressive that
>> it cannot manage something as simple as "Fathers are not mothers."
>> OWL cannot define the idea of an uncle, and full first-order logic
>> cannot define the idea of a natural number.
> 
> I was assuming suitably expressive standard semantic extensions.

But that begs the question. How do these miraculous extensions acquire *their* full meaning? Eventually you will ground out in NL prose (perhaps plus some mathematics, etc.) in documents read by human readers. 

>  But regardless, are you claiming that this would fundamentally change the argument?

Im still not sure what the argument is, but I think this is a basic point about how meanings are communicated on the Web. (Or indeed anywhere else.) 

> The context and purpose of this assumption is to bar any "then a miracle occurs" steps from the process of IRI definition. Again, a primary assumption is that for scalability, in the vast majority of cases IRI definitions must be provided using description -- not ostension.

Well, if http-range-14 is accepted, then the vast majority - down to several places of decimals after 99% - are going to be determined by what we might call http-ostention. 

>  In principle, descriptions could either be formal (in RDF) or informal (in English prose, say).  Are you claiming that English prose definitions can somehow avoid underdeterminism that leads to divergence) where RDF descriptions cannot?

Yes, that is exactly what is meant by saying that English is more expressive than RDF.

>  For example, are you claiming that there exists a resource property whose value can be specified in English prose but whose value cannot be specified in RDF (assuming suitably expressive generic semantic extensions)?

Yes

>  If so, how?

?? By writing the English prose. (? I must be missing your point.) For example, take how xsd:decimal is defined in http://www.w3.org/TR/xmlschema11-2/#decimal : "decimal represents a subset of the real numbers, which can be represented by decimal numerals." 

>>> But AFAICT, the trend is inevitably *toward* mismatch as more
>>> statements are published, assuming that: (a) parties publish data
>>> independently (without knowledge of each other)
>> 
>> But why would you assume this? Usually, if B is publishing data using
>> A's IRIs (the only case that is of interest here) then B will have
>> access to *some* published information which will help determine what
>> A's intentions were regarding A's intended meaning. For example, if
>> you use DBpedia IRIs then there are large pages of information
>> available, in multiple languages. The entire Semantic Web/linked-data
>> enterprise is predicated on the idea that IRIs both denote entities
>> and also provide links to sources of more information about those
>> entities (or, if you like, more information about what the IRI is
>> intended to denote.) So the idea of two RDF authors using the same
>> IRI but without any knowledge of what the owner of the IRI intended
>> it to refer to, is SW/LD-pathological.
> 
> It sounds like you misunderstood what I meant.  I was not talking about RDF authors not having access to IRI definitions.  I am assuming that A's IRIs have published definitions that are provided in RDF.  I am talking about the situation where A publishes IRI definitions, and B1, B2 etc. are other parties who independently use A's IRIs in publishing other data.  The assumption is that B1, B2 etc. are all aware of A's IRI definitions, but are not aware of each other (or each other's data).
> 
> When this occurs, AFAICT the trend will inevitably be toward mismatch as B1, B2, etc. publish more data, even if each of those datasets is *individually* fully compatible with A's IRI definitions.

I am much less convinced of this than you are. Of course this does happen in practice, but there is also a lot of awareness of the dangers and people trying to take care to avoid it. I acknowledge the possibility but push back on the "inevitability".

>>> If the problem is disagreement then yes, you would have to choose
>>> between the source graphs.  But if the problem is divergence then
>>> you have to do some more work -- resource identity splitting -- but
>>> can still use both source graphs after splitting.
>> 
>> Changing the IRIs in a graph gives you a different graph. So you
>> would not be using both source graphs, but some modification of the
>> source graphs. And you would be obliged to *not* use - that is, to
>> reject - at least one of the source graphs, when they are mutually
>> inconsistent.
> 
> Are you just dotting i's and crossing t's here?  The point is that the original graphs are *not* rejected, but you're right that they are not merged **as is** either. 

If they are "changed", then that is a form of rejection. Let us be precise (the WG spent months getting this right): graphs cannot change. What you can do is reject one graph and use a modified one, but that modification is a *different* graph. 

> The graphs are used by systematically transforming one or both of them (by properly substituting IRIs -- a process very similar to skolemization) and then merging the result, so as to join and use the relevant information content of *both* graphs. This is fundamentally different from disagreement, in which relevant information content must be *discarded*. By "relevant information content" I'm talking about the information content of the graph other than the particular choice of IRIs that were assigned -- a kind of graph isomorphism (though more general than the notion of isomorphism that is defined in the RDF Semantics, which is only isomorphism over bnode relabeling rather than IRI relabeling).

>>> ... RDF data does not generally describe the real world, it
>>> describes a particular *conceptualization* of the real world
>> 
>> It describes the world *using* a conceptualization (is there any
>> other way to describe anything?). It does not (usually) describe the
>> conceptualization.
> 
> Maybe we're just quibbling over words here, but I don't see how RDF data that models the world as flat could possibly be describing the real world, since the real world is *not* flat.

It is locally, approximately flat. If your interpretation of "flat" allows for this locality and approximation, then it can be true to describe the world as flat in that sense. 

>   In essence, unless RDF is 100% accurate then it isn't describing the real world, is it?

Of course descriptions may be loose or approximate or incomplete, and yet still be true. I have worked as part of a framing crew, where someone on a ladder will call down to someone else operating a chop saw, and say: "94 and three quarters, fat!" and then complain when the cut 2x4 he is passed does not fit the frame. These utterances are about the real world and have truth values, even if they are not accurate to thousands of an inch or fail to mention the species of the tree from which the timber was cut. 

>  It could just as well be claimed to be describing my left earlobe.
> 
>> 
>>> false graphs aren't very useful, because they entail everything
>> 
>> Just as a technical point: *logically false* graphs  contradictions,
>> false in *every* interpretation  entail everything. Mere falsity
>> does not get you quodlibet.
> 
> Interesting point.  Do you think that has any impact on this analysis? If so, how?  The context of that statement was just pointing out that false graphs aren't very useful.

Well, it means that the case for inutility has to be re-stated. 

> 
>> 
>> ---------
>> 
>> There are two substantive points of disagreement between us, and one
>> complete mismatch (divergence?) where I fail to understand what you
>> are saying. Let me deal with the two points first.
>> 
>> 1.  The reality thesis: that the real world is one of the satisfying
>> interpretations, and data is (usually) about the real world. I find
>> this obvious, so obvious indeed that it should not even need to be
>> said. You apparently find it either mistaken or meaningless, and in
>> any case think it is misleading as a guide to intuition. I am not
>> sure how to persuade you to my way of thinking, but let me ask you:
>> if all this linked data is not about reality, what do you think it
>> *is* about?
> 
> Informally (and imprecisely) we can say that published RDF data is about the real world.

It might be informal, but it is true, and fundamental.

>  But in general I think it would be more precise to say that it is *directly* about a particular conceptualization of the world

No, It is not ABOUT a conceptualization. (Well, not usually. Arguably, the SKOS defining documents are about a conceptualization, for example, but then they *use* a different conceptualization.) This is a very basic point in discussing semiotics at all: we have to keep a clear distinction between what a description is *about*  - what it describes - and *how* it describes it, the conceptualization it uses.

> , and only *indirectly* about the real world, through an informal and unspecified correspondence between that conceptualization and the real world.

To use a conceptualization to describe X is not describing something other than X. It is describing X *as being* the thing so described. For example, I can think of you as a dynamic process involving a host of chemical events inside cells, rather than thinking of you as a person. But it would still be *you* I am thinking about, whichever way of thinking *of you* I choose to adopt.  

>  For example, some RDF data may describe a flat-earth conceptualization of the world, and that flat-earth conceptualization corresponds in some way to the real world.  RDF data does not *directly* describe the real world, because the real world is *not* flat.

It describes it through an imprecise or approximate conceptualization. But approximation does not change the object of the description. 

> Specifically, such RDF would ascribe properties to the earth that the earth does not possess.
> 
>> And why do we find it useful, if it does not provide us
>> with information about the actual world?
> 
> It is useful because, when we feed that data into our applications, our applications produce the desired outputs.  It doesn't matter what the data is about as long as it produces the desired outputs.  I'm not saying that to be flippant, but to emphasize that the whole notion of the real world is completely irrelevant to machine processing.  Machines just push the bits around, and if they come out with the answers that we want, we're happy.  What really matter is usefulness of the data.  But I agree that, all other things being equal (which they aren't), conceptualizations are *much* more useful when they more closely resemble the real world.

We kind of agree and kind of don't agree here, but it would take too long to sort out the threads. (Basically, I think that utility almost always only arises from truth, in this kind of a context. With exceptions such as deliberate lying or confabulation to achieve a political or military end, but this goes outside 'normal' RDF data usage.) Just in passing, conceptualizations do not *resemble* the world. They are ways of thinking about the world. Arguably, we cannot think about or describe the world at all except through some conceptualization or other. Even quantum physics has its conceptualizations. 

> 
> > Are the records of the
>> transactions in your bank account about your actual wealth? Would
>> that change if the bank started using RDF?
>> 
>> Your objections to the idea include the observation cited above about
>> conceptualizations. Yes, of course data is stated *using* a
>> conceptualization, just like all assertions in every language or
>> formalism. But that does not make it any less about reality. It
>> really is a fact about the real world that Hilary and Tensing climbed
>> Everest in 1953; that we conceptualize the world here in terms of
>> people and mountains does not make this any less true. I am not sure
>> what the point of your "toucan" example is, but apparently the real
>> world can satisfy both the bird assertions and the website
>> assertions, by appropriate choice of an interpretation mapping. (If
>> the complete set of assertions is inconsistent then of course nothing
>> can satisfy it.) Your third point concerned approximations and
>> idealizations, such as the flat-earth geography of road maps. But
>> examples like this do not argue against the reality thesis. An
>> approximate or idealized description of X is still a description of
>> X. Bear in mind that if some RDF can be satisfied by an approximation
>> or simplification of the real world, then it can also be satisfied by
>> the more complicated real world, since one can add (an infinite
>> amount of) structure to an interpretation freely without making any
>> RDF triples false. (This is a consequence of RDF being a positive
>> logic without negation.) The map example is quite instructive, as
>> quite a lot of geolocation information (eg lat/long coordinates) is
>> in fact describing spherical space rather than flat space, even
>> though we project it onto flat surfaces.
> 
> First of all, let me distinguish between two cases, to be sure that we are talking about the same case.  One case is where the conceptualization is carefully crafted to acknowledge the fact that it is an approximation, so it might say, for example, something to the effect that "pi is greater than 3.1 and less than 3.2".  I'm not talking about this case.
> 
> The other case is where the conceptualization in effect says "pi equals 3.1".  In other words, it asserts something that may be good enough for many applications, but in fact it turns out to be false when scrutinized in detail in the real world.  

Well, is it false? That depends on what exactly "pi" and  "equals" are interpreted to mean. If you mean the actual mathematical terms then yes, it is false. But if all quantities are understood here to be rounded to one place of decimals then this is true.

> This is the case that I'm talking about, and I used a flat earth conceptualization as an example, because it is useful for some applications but clearly wrong in terms of (directly) describing the real world.

I disagree about its being clearly wrong, and I disagree about most of the data being based on it. In fact, geolocation data uses one or another virtual spheres projected from BPS satellites, and yet we use it together with 'flat earth' data such as road mileages between cities without this concpetualiztion mismatch being an issue. 

> 
> Furthermore, AFAICT this second case is where divergence inevitably leads (barring splitting, which mints new URIs).  The reason is that divergence leads to a situation where one graph A in essence asserts "X P V" and another graph B asserts "X P NotV"

? Do you mean X NotP V ?

> , which means that from some perspective, one of those graphs is *wrong*.  And yet, within each of those graph's intended interpretations, the graph is *not* wrong, it is true.  I.e., within each graph's *conceptualization* of the world, the graph's statements are consistent with that conceptualization.

What is a *graph's* conceptualization? Graphs use IRIs to refer, and IRI reference transcends graphs. It makes sense to speak of a particular IRI vocabulary as comprising a conceptualization, but not a graph as having one in isolation. 

Thinking about this later, maybe what you mean here is the interpretations of the graph intended *by the author of the graph*, ie something like what the author *had in mind* when they composed and published the RDF, what is was they were *trying to say*. In which case, yes indeed two graphs may have divergent interpretations in this sense. But that is not what RDF Semantics is trying to capture: it is not about what an RDF author is *trying* to say, but what *actually gets said* by the RDF they publish. This might be a good deal less than what they were trying to say (and so have a lot more interpretations than are in the set of "intended interpretations", which is something that nobody but the author, and maybe not even her, can possibly have any access to.)

>> To say that some assertion is about the real world, or that it is
>> factual, is not to claim that it is in some metaphysical sense the
>> final truth or the definitive description, or that it is the last
>> word, or that its truth has to have ended science. It is just saying
>> that it is true.
> 
> I think what you mean here is that there may be *additional* facts that are also true.  Right?  If you are telling me that there are various different notions of truth, then you'll have to explain more.

No, I want a single definition of truth. But I don't want to have that notion lumbered with extra baggage of being completely accurate or completely complete in some impossible way. 

>> You say:
>>> ... The "real world" interpretation is largely irrelevant -- both
>>> to the formal semantics and to understanding how the Semantic Web
>>> *actually* works.
>> 
>> I strongly disagree. Many IRIs have fixed interpretations in the
>> actual world,
> 
> I was wondering when you would bring that up.  :)   I partially agree, but I think the degree is far different.  I would say that a *few* IRIs have fixed interpretations -- meanings that are fixed by ostension.  But the vast majority are defined by description, and those by necessity are far less fixed.
> 
>> determined by all kinds of social, technical and
>> linguistic conventions and meanings entirely outside RDF.
> 
> We need to be careful about the assumptions that we make when stepping outside of RDF, to avoid "then a miracle occurs" steps whenever possible. My assumptions are:
> 
> - A very few IRIs are defined by ostension.  (This is one "then a miracle occurs" assumption.)
> 
> - All other IRIs are defined by description.  For simplicity, let's assume either English prose or RDF.
> 
> - For the most part, English prose definitions do not fix the interpretations significantly more than RDF definitions, because: (1) machines cannot understand English prose, and the Semantic Web is intended for machine processing; and (2) an English prose definition could in principle be written instead as RDF (assuming suitable standard semantic extensions).

OK, I definitely do not accept this last assumption. Your point (2) is just flat wrong, unless the extensions are themselves defined using natural language (which makes the point vacuous), by Goedel's theorem. No computable axiom system can even define the integers.

> 
> I know you disagree with point #2 of that last assumption, and you may well be correct about that in a theoretical sense, but the question is: is there really a big difference in our ability to avoid unintended interpretations via English prose definitions versus RDF definitions?

Yes.

> And if so, is that difference big enough to override the effect of point #1?

Yes, I would say. (Although of course English *can* also fail to describe or define meanings adequately.) English prose (all natural languages) relies on a huge mass of terms and ideas which are connected to the real world through our senses, perhaps by ostention but more likely through reading other NL prose. No axiomatic formal system can reproduce this kind of connection, but it can use it and extend it. 

> If we assume that English prose descriptions fundamentally fix the interpretations substantially more than RDF descriptions, then it represents a second "then a miracle occurs" step in the process -- a step that surely is not scalable for the Semantic Web in the same way that RDF descriptions are scalable.

The point is not that the semantic web machinery can take part in the miracle of human communication, but that it can as it were ride upon it. The machinery will only 'know' that part of the meaning of IRIs that can be reflected in the formal model theory, but the conclusions it draws will be expressed using IRIs which have a richer meaning than the machinery itself can make use of, but which will be available to human readers of the mechanically drawn conclusions. This works when the mechanical inference does not destroy or modify any of that richer meaning, of course; but if the human notions of truth can be expressed in model-theoretic terms (even if they use constraints that would be miraculous if we were limited to RDF) then this will always be the case. Exceptions to this happy picture might for example include social constraints on the appropriateness or otherwise of certain descriptive vocabularies. The Turkish government gets very upset when the anyone refers to "genocide" of Armenians, even if this is factually correct. RDF semantics does not deal with such social and diplomatic delicacies in language use, however. 

>> We still
>> want to be able to use RDF to describe these referents. For example,
>> I am a consultant on a project (http://www.imagesnippets.com/) to add
>> RDF markup to images. These RDF descriptions use IRIs which identify
>> (and in the RDF refer to) images, regions in the images, people and
>> places and colors and objects described in DBpedia and many other
>> real (no scare quotes) things in the real world. None of these
>> denotation mappings are specified by RDF descriptions, and most of
>> them could not be. Most  I would claim, virtually all  RDF linked
>> data uses IRIs like this to refer to real things. It is centrally
>> important that the formal semantics works with such identifying
>> IRIs.
>> 
>> 'Edmund Hilary climbed Everest in 1953' says something true about the
>> actual, real, world. It expresses a fact. Just a mundane, simple bit
>> of data. So, how is this factuality of this fact related to
>> model-theory semantics? By the actual, real, world being one of the
>> satisfying interpretations of it. Because if the real world was not a
>> satisfying interpretation of this sentence, then it *couldn't
>> possibly* be true (in the real world.)
>> 
>> But we can, if you like, simply agree to disagree about this, as it
>> has no direct bearing on the basic point we have been arguing about,
>> which is...
>> 
>> 2.  The idea of an IRI denoting something "in a graph".  Your gloss
>> on this phase, as I now understand it from your email (the first time
>> you have explained your intended meaning) is as follows: you take all
>> the interpretations which satisfy the graph (and there will be
>> different such sets for different graphs, of course) and then you
>> ask, what does the IRI denote in those interpretations? And that is
>> what the IRI denotes "in the graph". (Do I have that right?)
> 
> Yes, exactly right.
> 
>> 
>> But that does not define anything, because for any consistent graph
>> G, and any IRI U in that graph, there are interpretations which
>> satisfy G and in which U denotes things different from what it
>> denotes in other interpretations satisfying G. There is no graph
>> which 'pins  down' the interpretations of the URIs which occur in it
>> in the way that your definition requires.  (Here is a simple proof.
>> Let x be something which is not an IRI. The interpretation I with
>> universe {x} and IEXT(x)={<x,x>} and I(u)=x for every URI u,
>> satisfies G. The Herbrand interpretation H of G also satisfies G. But
>> H(U) = U =/= x = I(U), by construction. QED.) In fact, one can make a
>> stronger statement: truth in an interpretation does not depend on the
>> identity of the referents of IRIs *at all*, because one can take
>> *any* satisfying interpretation and produce another isomorphic one
>> with the identities permuted in any way one likes, as long as the
>> IEXT mappings are permuted to match. (In fact, this applies to *any*
>> axiomatizable, complete formal logic, no matter how expressive.) In a
>> nutshell: model theory does not determine reference.
>> 
>> This should not be too surprising, actually, if you think about how
>> model theory is defined. The very definition of interpretation
>> presumes complete referential freedom: any IRI can denote anything.
>> And truth is determined solely by how those things stand in relations
>> to one another. The entire apparatus of model theory makes no
>> reference to the *actual identity* of the things in the universe
>> being described. So creating real constraints on reference -
>> attaching, as it were, a name to a thing - has to be done by other
>> means. In practice, we rely on notions of naming and reference
>> already in use in the larger world (as I did when using "Everest" to
>> refer to the highest mountain, and how ImageSnippets does when using
>> 'http://schema.org/Person' to refer to the class of human beings) and
>> sometimes on predefined mappings (as we do when fixing the referents
>> of literals using datatypes) and perhaps even by ostention (arguably,
>> http-range-14 can be seen as declaring HTTP GET/200 to be a form of
>> ostention.) And this all works quite nicely (a lot of the time)
>> because we can all (more or less) agree on what these referring names
>> actually refer to, at least well enough to transfer meanings
>> successfully by using them as referring names in sentences.
>> 
>> So, as I believe I have said several times, phrases such as
>> "interpretation of an IRI in a graph" are not meaningful. It is not
>> that this is a different perspective on model theory, or an
>> alternative viewpoint. It is that it, quite literally, does not mean
>> anything.
> 
> Yes and no.  I fully agree with your proof, and the fact a Hebrand interpretation will always satisfy the graph, so let's not get hung up on that.

Well I agree lets not get hung up on it, but let us also not simply ignore it, as it refutes your claim.

> But you have left out two key things.  One is the existence of *some* IRIs that -- as you pointed out above -- *do* have their interpretations fixed, presumably by ostension.  Those will (by the rules of entailment) cause the possible interpretations of *other* IRIs to be reduced also -- usually still not uniquely, but nonetheless reduced.  A simple example is if <http://example/aa> has a fixed interpretation, and <http://example/aa> owl:sameAs <http://example/AA>, then by entailment <http://example/AA> also has a fixed interpretation.

Yes, of course. (In an old publication, I used the analogy of tying down a fabric to a surface by attaching it at enough points and relying on the strength of the fabric to hold the rest of it.) This is exactly how meanings can be transferred across entailments. But this relies upon the interpretations of occurrences of IRIs being *independent* of the particular circumstances of their occurrence. If (two instances of) the same IRI in two places can mean different things, then these fixed interpretations cannot be preserved through inferences. 

> The second is the fact that every application that consumes RDF has a preconceived set of *assumed* interpretations.

I simply do not agree with this. In fact I do not think it is coherent. How would this preconceived set yield any difference in behavior, for a conforming RDF engine? 

>  That set of assumed interpretations represents the particular conceptualization of the world in which that application operates.

No. Conceptualizations are not graphs and they are not sets of interpretations. 

>   For example, in the toucan example, one application uses a conceptualization of the world involving web pages, and in that conceptualization, <http://example/toucan> becomes fixed to a web page -- not the string "http://example/toucan" (as in a Hebrand interpretation), and not the bird either, but the web page.   Similarly, in a different application whose conceptualization of the world involves birds (but not web pages), the interpretation of <http://example/toucan> becomes fixed to a bird.

As far as I understand the point of this example (and I am not sure that I do), both of these hypothetical applications are simply drawing invalid conclusions about what the IRI means. (The web page app is assuming that something that might be a bird is in fact a web page, and oppositely for the other case.) Which is fine: they are free to do so. But the semantics has nothing constructive to say about that, other then that they are indeed drawing invalid conclusions. 

> Finally, if we back up one step from the idea that each consuming application has a set of assumed interpretations, we are back to the idea that each graph has a set of intended interpretations, which represent the set of assumed interpretations that the graph author intended to support.The net effect is that it *is* meaningful to talk about the idea of an interpretation of an IRI in a graph.

No, we are not back at that idea. In fact, I fail to understand how you even see any connection between these two observations and this central claim of yours. Suppose for the moment that some IRI meanings are determined by ostention and that some applications draw logically invalid conclusions from RDF. How does all of this imply that it is meaningful to speak of an interpretation of an IRI in a graph? By the way, the point made in my earlier reponse, that any referent can be simply replaced by another without changing truth values as long as the extension mappings are redefined to match, applies to *any* referent not itself fixed by ostention or in some other rigid way. 

>   In fact, I would conjecture that that is how many people in the Semantic Web think of IRI meaning, whether they are conscious of it or not.   It's really a very sensible and intuitive way to think of it, IMO

I believe that I have shown quite rigorously that it is incoherent and meaningless. But even if I am wrong about that, it is certainly in violation of the RDF Semantics specifications as published, which define an interpretation to be a mapping from IRIs to referents, not from IRI occurrences in graphs to referents. So those who do think this way are misunderstanding the RDF specs and may be in danger of writing non-conformant apps. 

> , though as we know, people's intuitions differ.
> 
>> 
>> ---------
>> 
>> Now the place where I fail to understand what it is you are saying.
>> 
>> At the end of your email you list all the advantages of an "other
>> way" of approaching model theory. But as far as I can tell, this
>> "alternative" is simply standard model theory.
> 
> I agree!  That's what I've been saying all along.  But AFAICT, based on your other comments, it does seem to involve a slightly different way of *thinking* about the standard model theory.
> 
>> For example:
>> 
>>> The other way to think of the RDF Semantics is in terms of
>>> *multiple* interpretations
>> 
>> This is the only correct way. As I have said to you before, *of
>> course* we think in terms of multiple interpretations. That is the
>> entire point of defining the notion of interpretation. The very
>> definition of entailment refers to multiple interpretations.
> 
> Well yes, but see your next statement.
> 
>> 
>>> , instead of attempting to assume or impose a single "real world"
>>> interpretation.
>> 
>> Well, it is fine to assume that the real world is *one*
>> interpretation, but nobody has ever suggested "imposing" a single
>> interpretation. Certainly, nothing in the RDF Semantics document
>> speaks of anything like this.
> 
> Uh . . . but that is *exactly* what this offending "intuitive introduction" did (before it was removed) and why I insisted that it needed to be scoped to an interpretation:
> [[
> An RDF graph is true exactly when:
>  1.  . . . etc.

No, this was saying that the graph is TRUE when, etc.. That does not appeal to the model-theoretic notion, but to a pre-theoretic notion of what it is for a sentence to be true. It was intended to convey the intuition that the model theoretic acount of truth was, when stripped of its mathematical jargon, a very close fit to this pre-theoretic notion. Which was the point of the (now deleted) section in the semantics. 

> ]]
> 
> And you argued:
> http://lists.w3.org/Archives/Public/www-archive/2013Oct/0021.html
> [[
> To say that a graph (or any other assertion or sentence) is true, is to say that when it is interpreted *in the actual world*, its truth-value is true.
> ]]
> 
> That is *exactly* the kind of implicit assumption

What "implicit assumption" do you think is being made here? That there is only one interpretation of the graph? No, that is not being assumed. That there is only one world that we all inhabit? Well, yes, I do assume that, but I will not be drawn into a debate about that in this forum. 

> that I think is misleading, and why I have been trying hard to explain how a "multiple interpretations" view of the semantics is a slightly different view from thinking of the semantics in terms of an "in the actual world" interpretation.

I wish I did not have to keep saying this, but you really do need to get a better grasp of the basic foundational ideas of model theory. It is intended to be a formal theory of a natural phenomenon in the real world, which is that sentences really can express truths. The theory is inherently about "multiple interpretations", especially when it is used to characterize valid entailment relationships. To say that a sentence is true when what it says about the actual world is correct, is not to deny the inherent multiplicity of interpretations. The first is a statement about nature; the second is a statement about the formal apparatus of model theory.

>  The intuition seems to be different.
> 
>> 
>>> By this I mean, for example, that:
>>> 
>>> - Two different graph authors may have different sets of intended
>>> interpretations in mind when they publish their RDF graphs, and the
>>> same URI may indeed denote different resources in those
>>> interpretations.
>> 
>> Different sets of interpretations in mind, yes, of course (standard).
>> URIs denoting different things in different interpretations, yes of
>> course (standard). URIs denoting different things in different *sets*
>> of interpretations, yes, if we are talking about sets of
>> interpetations an author *has in mind*. But URIs denoting things in a
>> set of all interpretations which satisfy a given graph? No, for the
>> reasons described above. That idea is incoherent.
> 
> I've addressed that above.

Actually you have not. And later in the message, you claim to be talking about set of interpretations the author *has in mind*, not sets which satisfy a given graph. This shifts the ground of the discussion in a crucial way.

>>> - The most accurate way to understand a graph is to interpret it in
>>> the way that the author intended it to be interpreted. Since we
>>> have no other reliable way of knowing what that might be, we can
>>> assume that the author's intended interpretations for a graph are a
>>> subset of the graph's **satisfying interpretations**.  I.e., we
>>> take the graph's meaning at face value, rather than attempting to
>>> interpret it according to some hidden, assumed "real world"
>>> interpretation.
>> 
>> Yes, this is exactly what the RDF model theoretic semantics presumes.
>> Asserting a graph effectively claims that interpretations must be
>> such as to make it true, i.e. to satisfy it. Each graph makes some
>> claims about how the world is structured,
> 
> Yes.  :)
> 
>> and the claims made by
>> multiple graphs are connected by their common use of global IRIs.
> 
> *If* the graphs are merged.

No. This has nothing at all to do with whether the graphs are merged or not. Merging (actually, because of the new treatment of shared blank node scopes, unioning, but let us ignore this) two graphs does not change the meaning of the triples in those graphs in ANY WAY. 

>  If the graphs are not merged, then the RDF semantics says nothing about them using the same interpretations.

WRONG. Utterly and disastrously wrong. The RDF semantics speaks of interpretations, and each interpretation fixes the referent of every IRI (and the IEXT mappings, ie what the relational extension is when this thing is understood to be a relation), and hence the truthvalue of every triple. Every *possible* triple, even ones that have not yet ever been written down. The truth conditions defined by the semantics have got nothing whatever to do with what graphs have been published or created or merged on the Web. 
 
>  Of course, any two graphs *may* be merged.  But the point is that it is sometimes useful to look at the semantics of each graph *individually*

That phrase has no meaning. The semantics does not give individual interpretations to a single graph, It interprets the entire vocabulary of all IRIs, and that in turn determines the truth values of triples, and that of graphs. There is no such thing as a semantics of "graphs individually" versus those of two graphs merged. Let G1 and G2 be graphs and H be their merge. Then every interpretation assigns a truthvalue to G1 and to G2 and to H  all three of them  and H is true in that interpretation exactly when both G1 and G2 are true in it. That is exactly what the RDF semantics says about merging. By the way, even when H is constructed, (graphs isomorphic to) G1 and G2 are still subsets of it. Merging two graphs creates a bigger graph but it does not destroy the pre-merged graphs. 

> (if the graphs are being used separately, by separate applications) just as it is also sometimes useful to look at the semantics of the merge (such as if those graphs are being used together).

The "semantics of the merge" is identical to the semantics of the two graphs considered separately. The RDF semantics assigns truth values to *all* graphs, including graphs that only have a Platonic existence. Consider the set of all RDF graphs that have ever been published on the Web at the moment I hit the period at the end of this sentence. Define a 'survey' graph consisting of one triple from each of the graphs in that set. That RDF graph, which by construction has never been constructed in any RDF document or datastructure, is one of those which has a determined truthvalue in every RDF interpretation. 

BTW, each triple in that graph is both in its published graph and in this 'survey' graph, as well as in many many others (maybe infinitely many, depending on how one defines "all graphs"). So what exactly determines how it is to be interpreted, in your view? 

>>> Some benefits of looking at the formal semantics this way
>> 
>> What "way" are you talking about? Look, *of course* each graph has a
>> set of satisfying interpretations, and asserting the graph is saying
>> that the world being described by the graph is one of those
>> satisfying interpretations. (Or if we want to give authors the
>> ability to be vague about exactly what they are talking about, then
>> the interpretations of whatever the author had in mind are a subset
>> of the satisfying interpretations.) And of course we should take a
>> graph at face value, as you put it, as saying exactly this. All this
>> is *exactly* what the current semantics itself says (or presumes). As
>> far as I can see, you are simply agreeing with standard
>> model-theoretic intuitions here.
> 
> I am *entirely* agreeing with the model-theoretic semantics.  But apparently, as has been evidenced in this discussion, our intuitions sometimes differ.

It not a matter of intuitions. If you agree with the RDF model theory, then you agree that IRI reference is global (because that is what the model theory says.) But you do not agree with the latter, so you apparently do not agree with the former. BTW, your comment ot the RDF WG illustrates this perfectly. The RDF specs say that IRIs have a global meaning. You objected to this. So apparently you do *not* agree with the current RDF model. Or do you? You can't have it both ways. 

>>> Is this making any more sense to you?
>> 
>> No. I don't know what the "it", that is supposed to provide all these
>> advantages, actually is. If it is the idea that asserting a graph
>> amounts to saying that the intended interpretation is one of those
>> satisfying the graph, then this is what model theory says already. If
>> it is the idea that an IRI can refer to one thing in one graph and a
>> different thing in a different graph, then that is false (by
>> definition)
> 
> Wrong.  Stop right there.  Please re-read that last sentence, and notice that it did *not* stipulate a particular interpretation.  I.e., it was *not* scoped to a single interpretation.

It does not matter. In fact, it is so scoped, implicitly, simply by its referring to what an IRI refers to. (It did not say, what an occurrence of an IRI in a graph refers to.) An interpretation, you may recall, is simply a mapping which identifies one way to interpret the universe of IRIs. But all those 'ways to interpret'  ALL of them  treat all occurrences of an IRI in the same way. So none of them treat different occurrences of a single IRI differently when they occur in different graphs. This is true if we are talking about one interpretation or about several interpretations or about all interpretations. There is simply no construct that is defined, or AFAIK can possibly be defined, within the framework of the current RDF semantics, which allows two occurrences of a given IRI to be assigned distinct referents, under any circumstances whatever. (Just for the record, you have not yet defined your notion of "refer in a graph" with enough precision to analyse it. I do not believe that you will ever be able to do so, but would be most interested to be proved wrong.)

>   If you had instead said: "the idea that, **within a particular interpretation**, an IRI can refer to one thing in one graph and a different thing in a different graph, then that is false", then that would be perfectly true.

An interpretation is simply a formal way of expressing what it is that IRIs refer to. IRIs refer, RDF assumes, globally: that is, the IRI itself does the referring, and all tokens of that IRI refer in the same way. I agree this may be an idealization, but it is a widely assumed presumption, and (which is the point here) the RDF specifications require, normatively, that IRIs be interpreted in this way when considering RDF entailment.

Even if one wants to argue (as many have done, including myself on occasion) that this idealization is unrealistic, and that we need to have a way for different occurrences of a single IRI to denote differently in context-sensitive ways, the idea of graphs being the required contexts does not make sense. A given *occurrence* of an IRI in a single triple is simultaneously "in" an infinitude of different graphs, so the idea of *the* containing graph is meaningless. Something else has to provide the needed context. And the truth conditions have to be rewritten so as to include the context (whatever that context is presumed to be) into the semantic equations. Of course, this can all be done (in a variety of ways) but the result is a different semantics from that described in the RDF specs.

>   But the whole point of my long explanation above about a graph's intended interpretations is that is is perfectly normal and natural to think of applying different interpretations to different graphs.

And I have already explained, many times, why this is neither normal nor correct, as far as the RDF specs are concerned. In fact it is explicitly prohibited when considering validity of RDF inference. 

>  Thus, when a single interpretation has not stipulated, it is *wrong* to assume that a single interpretation is being applied.

The text to which you are objecting does not assume this. The point at issue between us applies in *every* interpretation. 

> 
> I accept the fact that your intuition may be different than mine -- and perhaps everyone else's also.  But you *cannot* make unqualified statements like that, containing free variables, and expect everyone else to have the same intuition about what they mean as you do.  If you want to make a statement that is only true in one interpretation, then you *must* stipulate that single interpretation.

David, let me try to respond to you in the manner which your tone here seems to invite. You may go to hell. You do not have the right nor the intellectual authority to tell me what I must or must not do or say. (And your point here about 'free variables" is weird enough to be something out of Monty Python, by the way.)  You are just WRONG. I have explained to you that you are wrong, how you are wrong, and given you actual mathematical proofs that you are wrong. Apparently none of this has made the slightest impact on your misapprehensions. So I fail to see that prolonging this conversation is likely to produce any more enlightenment or to be any use to either of us. 

It is not a matter of rival intuitions. I understand your intuition, but you are using the wrong formal way to describe it. Your 'refer in a graph' idea simply does not make sense, and if you were to fix it so that it does make sense, you would have a different RDF semantics. None of this is remotely a matter for debate or opinion. 

You think that you have a rival intuition which makes a different kind of sense of conventional model theory. In which case, let me urge you to write up your account of this rival intuition in enough detail and with enough precision that it can be published in one of the many venues for such reviewed publications, and publish it there so that everyone can understand it and refer to it. And until you do this, I suggest that you might consider refraining from commenting further on the matter in public, as all that you are succeeding in doing by such comments is convincing a wider and wider set of readers that you do not know what you are talking about. 

> So, returning to what "it" is, a central theme here is the idea that different graphs have different sets of intended interpretations.

Of course they might. Nobody has ever disputed this. But the *intended* interpretations refer only the mental state of the author of the RDF, and that is a matter that will always be inaccessible to objective description. The RDF semantics is concerned only with what the published graphs actually say, not what they were intended to say.

> Therefore, it is entirely normal and sensible to think and talk about the same IRI denoting different things in different graphs

I will not argue about normality or sensibleness, but I will continue to assert that the actual published normative RDF specs are based on the assumption that IRIs denote independently of the graphs they occur in: put another way, that it is IRIs that denote, not occurrences of IRIs in graphs. 

> : an IRI may denote one thing in graph A's intended interpretations, and a different thing in graph B's intended interpretations.
> 
> Notice that if I were to commit the same sin that you committed above (in making a statement with unbound variables) I would shorten that last sentence into: An IRI may denote one thing in graph A and a different thing in graph B.  And the problem seems to be that when you read "An IRI may denote one thing in graph A and a different thing in graph B" *your* intuition causes you to read that sentence as "**Within a given interpretation**, an IRI may denote one thing in graph A and a different thing in graph B", and the sentence is obviously false by that reading.  But *my* intuition causes me to read it as "An IRI may denote one thing in graph A's **intended interpretations** and a different thing in graph B's *intended interpretations**", and the sentence is obviously true by that reading.

It may be true with that reading (which has never surfaced in these conversations before this email, by the way) but that is irrelevant to how RDF is used on the Web and to RDF semantics, as I have explained above. For the record, by the way, I do not think it always makes sense even with that reading (because our very thoughts might have multiple interpretations which allow the IRI reference to vary.)

> 
>> but in any case would not provide all these claimed
>> advantages that 'it' is supposed to have, even if it could be made
>> somehow true.
> 
> Please tell me exactly which ones you disagree with, and why.  Every one of them is supported by an unbroken chain of reasoning, so if you're not reaching the same conclusions as me, then we need to identify what differing assumptions we're making along the way and why.
> 
>> 
>>> Have I explained myself in sufficient detail, or do you still think
>>> that "David . . . does not properly understand the intuitive
>>> foundations of semantics" and my points are mere "inanity", as you
>>> previously concluded?
>> 
>> I regret if my usage here seemed impolite, but I do (still) find your
>> posts, including this one, to be a strange mixture of basic ideas
>> about model theory (re)stated as though they were somehow a new
>> insight or an alternative to the standard view (which I referred to
>> as "inanity") and strangely stubborn basic mistakes which do, I am
>> afraid, strongly suggest that you have not grokked the basic ideas of
>> model theory.
> 
> I am kind of amazed that you *still* seem to think that I am making some kind of mistake in my understanding of model theory, since it seems to me that we have repeatedly confirmed that my understanding corresponds *exactly* with standard model-theoretic semantics.  Precisely what basic ideas of model theory do you think I have not grokked?

I believe I have already answered this question in my previous response. 

The pattern of our conversation is like this: A entails B. You repeatedly assert that you agree with and accept A, but you refuse to agree with B. I conclude that, in spite of your protestations, you do not really understand and agree with A, since if you did, you would see why B follows immediately from it. 

>>> And do you *still* think I merely need to go read a book on model
>>> theory, or have we now (I hope) got past that?  If not, what
>>> aspects of model theory do you still think I misunderstand?
>> 
>> Well, I guess, the basic idea of an interpretation. An RDF
>> interpretation, by definition, is a mapping from IRIs to referents.
>> It is not a mapping from IRIs-in-graphs or from IRI-occurrences or
>> from IRIs-in-a-context. Ergo, every interpretation treats all
>> occurrences of an IRI in the same way, as referring to the same
>> thing, regardless of which graphs the IRI happens to occur in.
> 
> Yes.  I don't know what I have to do to convince you that I understand that, but believe me, I do.

You apparently don't understand the consequences of it. But there is no point in continuing this correspondence. You may do and say whatever you like, and the RDF specs will say something else. I will not undertake to try further explanations. 

>> Therefore, the notion of what an IRI denotes "in a graph" is
>> meaningless.
> 
> No, as already explained in some detail above, that does not follow. The idea that an IRI denotes something "in a graph" certainly *is* meaningful if you think of it in terms of the graph's **intended interpretations**.  If you think of it that way, it is a very sensible notion indeed.

Perhaps, although I doubt it. But that is completely irrelevant to almost everything that has been said until now in all these various threads, which have not mentioned "intended" interpretations, but interpretations of graphs. The "intended" language is also completely absent from the RDF Semantics specs, of course. 

>> This basic fact  and it is a very basic and
>> foundational point  still seems to elude you.
> 
> I hope the above explanations have helped you to see why I keep pointing out that that conclusion is wrong.
> 
>> To emphasize, this is
>> not a "perspective" which admits alternatives, it is simply a fact
>> about how interpretations are defined.
> 
> The fact that an interpretation is (by definition) a mapping from IRIs to referents has not eluded me at all.  I am fully aware of that.  The perspective part comes in after that, when talking about a graph's intended interpretations.  One perspective is to assume a single (real-world) interpretation to by which all graphs are evaluated. Another perspective is to evaluate each graph by its intended interpretations.  Both views are fully consistent with standard model theory, but AFAICT they seem to appeal to different intuitions.

The only useful way to proceed is to consider all graphs that come your way, and draw valid conclusions from them. This amounts to giving RDF free rein to interpret its graphs in any way that the semantics permits. Neither of your "perspectives" is relevant to the description of  RDF entailment. 

>>> The bottom line here is that some of the statements -- and
>>> intuition -- in the existing RDF drafts are just plain *wrong* and
>>> need to be corrected.  In particular, the statement in RDF Concepts
>>> that says "IRIs have global scope: Two different appearances of an
>>> IRI denote the same resource" is just factually *wrong*.
>> 
>> It is a presumption of the RDF data model.  The semantics, in
>> particular, is based on it. I don't quite see how it can be factually
>> wrong, since RDF *defines* the notion of denotation. (If it had said
>> "identiifies" then it might be factually wrong, but it doesn't.)
> 
> Hopefully my explanations above have by now clarified why this is wrong as stated

For the record, they have not. If anything, your argument is now even more confused than it appeared to be before this email. 

> , so I won't go over it again here.  But if you disagree please let me know where and why, so that I can address it.

I already have let you know, in painful detail. I will do you the honor of not repeating myself yet again. As I say, if you think you have a case, write it up and publish it in a refereed journal. 

Good luck.

Pat 

> 
> Best,
> David

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Received on Sunday, 10 November 2013 03:55:18 UTC

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