On Sep 26, 2008, at 4:20 PM, Berg-Cross, Gary Ctr SAF/XCPAB wrote:
Yup, thanks. Don't know why the other one didn't display right, sorry.
These are important discussions regarding things like formal semantics and real world semantics. So I'd like a bit more help here. Concerning what the "Tarskian universe" is about I've seen write ups like the one below from "Semantics for the WWW" Dieter Fensel1, Jim Hendler, Henry Lieberman, and Wolfgang Wahlster. " The most common classic formalism, also the most amenable to the use of ontologies, is so-called
declarative or Tarski semantics as may be found in various places in the database and AI
literature, as in Reiter’s seminal paper [Reiter, 1988] or in the book [Genesereth & Nilsson,
1987]. Essentially it replaces “reality” (the domain) by a conceptualization, a mathematical
object that typically consists of very elementary constructs such as a set of objects and of
(mathematical) relations. Semantics is then formally defined simply as an interpretation
mapping from the system (or rather from the language describing a system instance in some
syntax) to this conceptualization." Do you find this consistent with your interpreation?
No. This is the same misunderstanding that John has exhibited. (I would note before proceeding that Tarskian semantics did not originate with Reiter, Genesereth or Nilsson, none of whom even wrote on the topic, and that - while they are all eminent in their fields - none of the authors of this report are philosophers or logicians.) In fact, this quote above makes, or hints at, several fundamental mistakes.
1. A 'set of objects' is indeed a 'very elementary' mathematical notion, but it is not a "construct". A set is simply a collection, in a very general and generic sense. It is very important not to make the mistake of thinking that because the notion of 'set' itself is in some reasonable sense abstract, that the things that the set is a set of, its members, must therefore be "abstract" or "formal" or in some other way ontologically attenuated. Set language is just a very basic mathematical language for talking about stuff. A set can be a set of anything, including concrete physical particulars.
2. The term "(mathematical) relations" suggests a restriction to a certain kind of relation, "mathematical" ones, presumably as opposed to other, non-mathematical relations. This is nonsense. There are no 'mathematical' relations: there are only relations, described mathematically. Mathematics is simply a language - actually, an entire family of languages - for talking about things precisely.
Here is a Tarskian structure: the universe set comprises you and me, and the relation denoted by the symbol 'R' is that of being further south at 10:00 am CT on the 29 September, 2008. All that we need to know about this relation in order to make the semantics well-defined is the set of pairs of things it holds true of, which in this case is the set {<me, you>}, which is enough to tell us that in this interpretation, (R Pat Gary) and (exists (x)(not (R x Pat))) are true and (R Gary Gary) is false. But note, the entities in this universe are real, and the relationship mentioned is a geographical/temporal relation, described mathematically, not a "mathematical relation".
3. The interpretation mapping is from a syntax to an interpretation structure. It is not from the language of a meta-description to an interpretation structure, even if that meta-language describes a syntax. Computing folk seem to be particularly susceptible to his kind of use/mention confusion, perhaps because they are used to thinking of syntax as something evanescent, and specified dynamically by a parser.
4. The most egregious mistake is the use of the word "replace". I have already tried to explain why any semantic theory must describe reality in some way, and that this description is not a replacement or a change of topic, but simply a(nother) description in a meta-language. If these authors think that Tarskian semantics is 'replacing' the world by a mathematical construct, how would they propose to begin to state any kind of semantic theory which does not so 'replace' it? One has to describe reality somehow. One cannot simply wave at it and hope that a theory will emerge.
But let me say: none of these issues in some sense 'really matter' to the actual practice of using semantics to analyze ontological formalisms. They are all essentially philosophical points, the ultimate in pedantry. So I wouldn't want to be quoted as critiquing Fensel et. al.., who probably weren't intending to write a philosophical paper. If we weren't having a rather pedantic discussion, none of this would matter a great deal. The OWL and RDFS semantic specifications would be unchanged, whichever side one takes a firm position on in this debate. The debate matters only when people want to argue for the need to also be doing something else to connect the formal semantics to the real world, because 'formal' semantics is somehow incomplete, and needs to be supplemented by 'real', 'non-mathematical', 'practical' semantics.
On Tarski's main example that snow is white
To be fair to Tarski, he may not have intended the example to bear this much weight of analysis. But let us proceed. , and he was talking about snow, not a mathematical simalcrum of snow ("snow") there seem to be people who argue that one needs something interpretive between the real and formal concept, as in the paragraph below from Chapter 4 pg 80 (Truth Values) of Joseph Margolis' A Conceptual Primer for the Turn of the Millennium/ disquotational theory of truth, I mean that theory that accepts as its paradigm (disputatiously attributed to Tarski the formula: 'Snow is white' is true if and only if snow is white. To ascribe truth, it is said (notably by Quine, who accepts the formulation), one merely "cancels the quotation marks." My own view is that the disquotational theory is either utterly vacuous (in the form just given), privileged (if taken as correct as far as it goes, as on Davidson's reading of Tarski , or else flatly false (since, admitting intransparency, some interpretive tertium will be needed). The objective of these strenuous (but ineffectual) moves against the legitimation of truth (a fortiori, the legitimation of truth-value assignments)—on which Quine and Davidson agree—is simply to endorse what has come to be called "naturalizing."
What Margolis seems to be arguing against here is a theory that says that sentences of the form " 'foo' means foo." amount to, or constitute, a theory of truth, all by themselves, and that no other formulation is ultimately possible. That is the disquotational theory, as I understand it. Quine has argued this position, though in a somewhat more refined formulation. If I have this right, then I would agree with Margolis that this theory is inadequate and in fact close to vacuous. It assumes for the metatheory what it sets out to establish for the representational language. As I tried to make clear, I wasn't arguing for this position, only that the metatheory must indeed have a semantics. But its role is not to simply be a kind of dequoted mirror of the representational language, but to play a structural meta-theoretic role of describing the represented world (in its own - in our case, set-theoretic - terms) and the relationships which establish truth in it.
Seems to me that what we might call the disquotation phenomenon, that " 'foo' means foo ", should be seen not as a theory of truth in itself, but a as kind of quick test for a theory of truth. If the theory predicts things of the form " 'foo' means baz. " then that ought to be at the least a good reason to re-examine the theory more critically.
So, to respond to your point, I don't see Margolis here as arguing for the need for interpretation between real and formal concepts, so much as arguing for the need for concepts at all, as opposed to just more language.
Pat
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