Stating the model theory free from lexical categories (was: Re: [SCL] Re: Observation)

[Chris Menzel]

On Tue, Jun 03, 2003 at 02:50:49PM -0500, Pat Hayes wrote:
> Chris, after the talk to day I now see what you have been talking 
> about and why you have been saying these odd-to-me-sounding things 
> (and, no doubt, vice versa). ... Rather than answer your email
> point-by-point, let me try to give an overview of my way of talking, 

It is definitely just a way of talking -- a very reasonable one, I
agree.

> which I think avoids the difficulties.

If you mean communication difficulties, then I agree.

> This is the key point:
> 
> > Not to assume,
> >i.e., stipulate, in the definition of a lexicon that a predicate is
> >an individual constant is to assume, i.e., stipulate, that it isn't.
> >One explicitly defines that members of each lexical category.
> 
> Well, this One doesn't do that, 

Well, in a sense you do -- it's just that you like to situate yourself
"after the fact".  I'm just abstracting away from that.

> for very good pragmatic reasons: there is NO WAY to do that on the
> Web; and in practice, no way to do it in almost all deployed
> machine-oriented logical systems, very few of which have an explicit
> 'language declaration' syntax for the very good reason that such
> syntax would provide no useful functionality.  

I agree.  My approach was never intended to be taken as a guide to
writing web ontologies.

> This idea of a stipulated lexical categorization is a theorist's dream
> of reason, but its still a dream. 

I.e., an abstraction; yes.

> In brief: the only lexical categories that are permissible are those
> that can be detected by a parser. In TFOL one can have such categories
> by a notational convention. In SCL we can't do that, and there is no
> need to do it in any case.

Right.
 
> >> And we arent saying that its meaning *changes* in a non-FOL context,
> >> after all.
> >
> >But we are certainly saying that a sentence *could* change in an SCL
> >context.
> 
> No, I am NOT saying that. It is the SAME sentence both syntactically 
> and semantically, and it is part of the same language (SCL): it is 
> parsed the same way and the model theory applies to it in the same 
> way. There is no need to even talk of 'context' here: there are no 
> contexts, only sets of sentences.

I think I agree with this way of putting it now.

> Let me sketch how this works.
> 
> Define a vocabulary V to be a set of names. SCL syntax is defined in 
> the usual CL way over a vocabulary. We can actually take V to be a 
> globally fixed set of all possible names; the only purpose of V is to 
> provide a single domain for the interpretation mapping, it plays no 
> role in the actual machinery of the semantics.
> 
> An interpretation M (of V, if you like) is defined by:
> 1.  Nonempty sets I and R (no disjointness or subset conditions imposed)
> 2. A partial mapping (also denoted by M) from V to (I union R)
> 3. A mapping extM from R to I*
> 
> M satisfies S when standard truth-recursions, but note:
> ...
> M([atomic-sentence rel arg1 ... argn]) = true iff extM(M(rel)) (is 
> defined and) contains <M(arg1),...,M(argn)> (which is defined), 
> otherwise false
> 
> M([Uquant var body]) = true iff M[var/x](body) = true for all x in I, 
> otherwise false
> 
> Now, given a set O of axioms over V, say that RO is the subset of V 
> whose members occur in a relation position in O and IO the subset of 
> V whose members occur in an argument tuple in O. 

Right, fine; you are just describing a lexicon "backwards".  Instead of
starting with a predefined lexicon with its determinate lexical
categories, you want to look at a set of axioms and extract the lexical
catetories.  This is essentially what I was talking about with my
awkwardly-named principle of "least type-freeness".  

> Any interpretation M in which a name in RO isn't mapped to something
> in R, or in which something in IO isn't mapped to I, is going to make
> O false, so the mere act of asserting O rules out those
> interpretations. And since M is one single mapping, if any symbol is
> in both RO and IO then any satisfying interpretation has to have I and
> R overlapping at least to that extent.

Right.

> So in our example:
> 
> P(a)
> 
> is satisfied by a model M in which M(P) in R, extM(M(P)) = {<a>} and 
> I ={M(a)}, and also in which I={M(a), M(P)} - the extra thing in the 
> universe is harmless - but only the first of these interpretations 
> would also satisfy
> 
> not exists (?x) ?x(a)
> 
> and we can rule that one out by also adding
> 
> R(P)
> 
> since M(P) must then be in I in order for this to be true, in any 
> interpretation.

Check.

> -----
> 
> In this way, the ontology contains its own implicit 'declarations' of 
> the syntactic role of the terms in its vocabulary and also of whether 
> or not its language can be understood as a TFOL language. The 
> syntactic role of a name is not fixed by prior (invisible) fiat when 
> defining a 'language' in your sense: 

Or, equally, it was as fixed before as after, we just didn't happen to
know which language was relevant til we saw the ontology.

> So the very *idea* of a symbol having a pre-ordained syntactic
> 'category' isn't needed in SCL (still less so in CL), 

It is, as you have already said, a convenient and harmless abstraction.

> and in fact it gets in the way. 

Not for the definition of the model theory -- though it might be the
case that people are misled into thinking we are talking about
implementation.

> This is the major reason why Ive always disliked your use of the
> textbook "language" terminology, by the way: the fact that each (S)CL
> ontology wears its syntax on its sleeve, as it were, is a major
> feature of this thing we have been building, and the 'logical language
> signature' way of talking obscures that feature. 

I don't believe anything we've done to this point contradicts that.  But
it might be a good thing to reflect this point more strongly in our
approach.  But I'm just not sure how it would work.  Practically
speaking, one starts with an ontology, a set of formulas.  But a formula
is typically something defined in terms of an antecedent lexicon.

> As you know, I havnt liked your 'standard' way of defining languages
> from day one, but Ive shut up about it because its just us guys
> talking and you are the one writing the documents. But I think that we
> have got to a point where we have to get this thrashed out, because
> this notion of a logical language is now causing problems. SCL isn't a
> language in this sense: its an ontology notation with a model theory.
> The 'languages' are just side-effects of ontologies.

I really don't know how to write things up much differently.  Where does
the notion of a formula come from?

> >Granted, we don't stipulate overlap relations between lexical
> >categories in SCL, because SCL permits any kind of overlap in its
> >instances; but a *given* lexicon will have to stipulate those
> >relations explicitly -- theoretically at least.
> 
> I disagree, see above. In fact I don't think we even need the idea of
> a lexicon (other than some kind of global lexicon to enable a parser
> to distinguish names from eg whitespace.)

Well, *that* might be true, but the point is that you DO need a lexicon
of basic syntactic elements to bring any semblance of order to the
language and its semantics.

> It would be OK to have a notion of the lexicon of an interpretation,
> actually; but it has to be understood to be a semantic notion, not a
> syntactic one.  The lexical category of an interpreted symbol is
> defined by the interpretation mapping, 

I'll try to work the details of the idea out.

> PS. BTW, the reason we weren't communicating, I think, is that I have
> been consistently understanding 'relation symbol' to mean 'symbol
> which occurs in a relation position in some sentence', whereas you
> have been consistently thinking of it as meaning 'symbol which is
> categorized as a relation symbol in the language's lexicon'.  --

That's part of the story, but doesn't explain everything.  I think a
more accurate explanation is this difference in standpoint.

I'm very tired and fuzzy-headed as I write this, so I'm not at all
confident of its coherence...

-chris