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Re: [ontolog-forum] Ontology-based database integration

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Sun, 11 Oct 2009 13:34:13 -0500
Message-id: <4B6C735D-1816-4CD0-B918-779CAC96E511@xxxxxxxx>
On Oct 9, 2009, at 3:29 PM, rick wrote:
Christopher Menzel wrote:
And model theoretic semantics is entirely silent on those questions.

Sounds like model theoretic semantics is overdue for an update because the rich discussion of meaning is still ongoing in philosophy.

I believe this betrays a misunderstanding of what model theoretic semantics is about.  Your comment suggests that you think that, because it contains the word "semantics", model theoretic semantics purports to be all there is to semantics.  To the contrary, model theoretic semantics one rather small, though vital, niche in the body of research that constitutes semantics generally.  The point of my remark above was precisely to highlight the limitations of model theoretic semantics vis-a-vis the broader semantical enterprise.  Let me give it another shot.

The typical purpose of a model theoretic semantics is threefold:

1. To provide precise meanings to those lexical items in a given class of formal languages that are considered logical and hence whose meanings are held constant across all interpretations. 

In the languages of classical first-order logic, for example, these include the boolean operators and the quantifiers.  Extensions of classical logic might include temporal operators, modal operators, epistemic operators, etc.

2. To provide formal representations of the kinds of meanings to be assigned to the basic non-logical lexical categories of a given language.

In classical first-order model theory, for example, individual constants are assigned objects within a given domain of discourse and n-place predicates are assigned sets of n-tuples of members of the domain — the latter constituting the classical "extensional" representation of properties and relations.  The development of possible world semantics by Kripke led to very creative (and controversial) representations of properties and relations.  These ideas were extended to other semantic objects and applied to more and more kinds of syntactic constructs by, most notably, Montague, whose work in turn has generated a huge body of research in linguistics on the semantics of natural language.  Well known non-Montagovian developments in model theoretic semantics include Kamp's Discourse Representation Theory and the information theoretic semantics of Barwise and Seligman that arose out of situation semantics and which draws not only the the heritage of classical model theory but also category theory to model basic semantic categories.  (I should note that Barwise and Seligman's work is also concerned with a lot more than simply providing a semantics for formal languages.)

There is an important qualification here: One might apply a model theoretic semantics to define a specific intended interpretation for a given language — e.g., one might specify that the intended interpretation for the language of arithmetic is the usual natural number structure, or that the intended interpretation of a language designed to represent employee information is a specific company and its actual employees and their relevant properties and relations.  The task of identifying and defining intended interpretations is what I called applied semantics, wherein a given model theoretic semantics is being used to provide a specific interpretation for a specific language, and that of course is a very important activity; but it is not itself part of model theoretic semantics proper.  That is the point I was making by saying that "model theoretic semantics is entirely silent on those questions", i.e., on the question of the precise meaning of a specific name or predicate.  If you design a language for employee information, model theoretic semantics cannot tell you what the intended interpretation for that language is; it cannot tell you what the meaning of the name "CEO" or the predicate "WorksInDept" are.  It can only tell you that "CEO" denotes a member of the domain of the interpretation and that "WorksInDept" picks out a set of pairs of members of the domain.

3. To provide a rigorous account of how the meanings assigned to a complex _expression_ are determined systematically by the meanings assigned to their immediate, semantically significant syntactic parts and, ultimately, by the meanings assigned to its logical connectives and the items in its basic non-logical lexical categories.

This last feature of a semantics is known as compositionality.  Compositionality is critical, as it provides a finite definition of meaning for languages with infinitely many expressions.  Our own semantical knowledge, it seems, must take some sort of compositional form.  For there seems no other adequate explanation for how it is that we, with our finite intellects, have the ability to interpret an unbounded number of sentences that we have never heard before.  Pretty clearly, our knowledge must consist in some finite set of semantic principles that enable us to assign meanings to complex sentences in terms of the meanings of their semantically significant parts and, ultimately, in terms of our knowledge of individual words.

Take for example Scott Soames' recent "The Unity of the Proposition."


This is a very interesting paper (typically so for Soames, who is an excellent philosopher), but I'm afraid it does not have the relevance to the issue at hand that you think it does.  The paper is in fact, for the most part, a discussion of a well-known historical question concerning the metaphysics of propositions in the work of Frege and Russell.  Only in the last few pages does he offer a "sketch" of a general notion of proposition that might be useful in a modern theory of meaning — an issue that has been at the heart of much discussion in linguistics and the philosophy of language for many years.  In any case, the general philosophical issues Soames broaches are entirely orthogonal to the nature of model theoretic semantics. 

Given that model theory implies satisfying an interpretation based on structures which preserve truth

Sentences are simply true or false in a given structure (relative to a given interpretation); structures do not in any clear sense "preserve truth".  Truth preservation is typically a property applied to inference rules in a logic.  Specifically, a rule: "From sentences of the form S1, ..., Sn, infer a sentence of the form S*" is said to be truth preserving if, under any interpretation in which sentences of the forms S1, ..., Sn are true, the relevant sentence of the form S* is true as well.  Truth is preserved from the premises of a rule to the sentence inferred by means of the rule.

implied by the assumption of material adequacy

I guess you are alluding to Tarski here.  Material adequacy is a condition on a theory of truth, and a rather simple one, roughly, that all and only those sentences declared by the theory to be true are, in fact, true; a little more rigorously (but still roughly), a materially adequate theory should be able to prove all instances of what Tarski called "Convention T", viz,:

  "S" is true (under interpretation M) if and only if (under M) S.

Tarski's famous example is:

  "Snow is white" is true if and only if snow is white.

Looking at the paper you pointed us to, I see that you are under a rather serious misimpression about what material adequacy actually is in Tarski's theory.

and signatures which lack enough symbols to represent meaning, 

A signature is simply a declaration of the non-logical lexicon in a language.  I don't know what you mean by a signature "which lacks enough symbols to represent meaning". 

I'm surprised that someone hasn't already proposed revising the definition of interpretation to include structures where *fully interpreted* requires meaning,

I don't know what you mean by "requiring meaning".  As noted, a model theoretic semantics assigns robust meanings to its logical connectives and gives only general semantical directions with regard to the assignment of meanings non-logical lexical items and complex expressions.

not truth (or satisfaction) and complex enough signatures to include symbols that are interpretants, signs and objects.

You lost me.

Here's a piece I've been working on for a while that speaks to this issue and has a useful diagram that extends the Triangle of Meaning.

It's called Linked Data: Interpretants and Interpretation.

Of course there's more than a life time of work to be done to properly develop what I've proposed, but silence on these important questions won't last forever !

I appreciate your efforts, but 150 years of intensive work — in philosophy, linguistics, psychology, sociology, artificial intelligence, and cognitive science  — on the many aspects of the problem of meaning has been anything but silent on the questions you raise.  That those questions might not be answered in model theoretic semantics — a rather small aspect of an enormous body of research — is a reflection only of the fact that it was not designed to answer them.

Chris Menzel

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