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

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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