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

[pat hayes]

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).  My version of the  issue arises from 
your style of writing MT, which depends on the textbook ideas of 
signatures being part of the syntax, so that lexical items have a 
lexical category to which the MT recursions refer.  I don't think 
that way, and so we have been slightly at cross purposes.  Rather 
than answer your email point-by-point, let me try to give an overview 
of my way of talking, which I think avoids the difficulties.

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

>
>>  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.

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. 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.

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.

-----

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: it is discovered - or more 
properly, constraints on it are discovered - as a side-effect of the 
parsing process. But in fact, one doesn't even need to go into this 
'language' (=categorized lexicon) way of talking at all, or even 
speak of syntactic roles: the lexical categories are only there in 
the service of the interpretations in any case, and the only role of 
the syntactic categories is to provide the ability to make the 
distinctions noted in the truth-recursion for atoms; but in fact 
there is no need to make those distinctions, since they are implicit 
in the 'else false' parts of the clauses. In this way of talking, 
they are merely the domains of two projections of a partial mapping 
from V, and it is this map which constitutes the actual 
interpretation. So the very *idea* of a symbol having a pre-ordained 
syntactic 'category' isn't needed in SCL (still less so in CL), and 
in fact it gets in the way. Since we allow the categories to overlap, 
there is no way to impose a unique lexical identity on any name 
independently of the axiomatic context; to do so in the metatheory 
just makes all sentences linguistically ambiguous, creating 
artificial problems for no useful reason (that I can see.)

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. 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.

>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.)

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, not by the language. All that 
expressions can do is to constrain these categories in any satisfying 
interpretation.

>This isn't to say that these relations will be
>crystal clear on the web for any given lexicon.

If there is some information about SCL syntax that is centrally 
relevant to interpreting sentences but is not crystal clear  on the 
Web, then we havn't done our job right.

Pat

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'.
-- 
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