[SCL] Re: Observation

[Chris Menzel]

On Mon, Jun 02, 2003 at 12:28:40PM -0500, Pat Hayes wrote:
> >> >> [Menzel wrote:]
> >> But we also want the language to be monotonic, so Im suspicious of
> >> defaults. But why do we need to do this? My understanding of your
> >> semantic proposal was always that the MT is simply agnostic; it does
> >> not need to make assumptions one way or the other. Given a set of
> >> sentences, there are many models; the satisfaction conditions require
> > > that relation symbols (which are in relation position) must denote
> > > relations in R and individual symbols (which are in the argument tuple
> > > positions) must denote elements of I, and we quantify over I.
> >> Textbook stuff, right?  The ONLY thing that is nonstandard is that we
> >> forgot to specify that the syntactic domains are disjoint, and if you
> >> read it quickly you probably didn't even notice that :-)  To the
> >> extent that the axioms get the syntactic categories overlapped, the
> >> semantic domains also must overlap at least to that extent in any
> >> satisfying interpretation: that follows from the satisfaction
> >> conditions. That is all we need to say, right?
> >
> >You seem to be ignoring a point you made forcefully in a previous
> >message, viz., that the Horrocksian hordes would expect the meaning of
> >an FOL-looking sentence to have the same logical properties in every
> >context.  A Horrocks sentence won't, relative to SCL's semantics.  What
> >happened to that point?
> 
> They will expect it to mean in SCL exactly what it means in FOL, if 
> it is read in a FOL context. 

But sounds very much like the point I made a couple msgs ago, adding
only that it would be illicit to expect a first-order sentence ripped
from a fuller SCL context to have the same meaning that it had in that
context -- to which I *though* you replied that Ian *would* expect to
have the same meaning in an FOL context (which strikes me as entirely
unreasonable).  Now you seem to be backing away from that claim.

> That property will now hold. 

Exactly my earlier point.

> The Horrocks rhetorical point was always that SCL isnt even a genuine
> extension of FOL because it distorts the meaning of FOL sentences even
> in a FOL context.

Which (as you are saying) is no longer so in the current semantics.

> 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.  That's exactly what a Horrocks sentence is.  Thing is, a
Horrocks sentence can only have purchase in a context where it would not
be a reasonable thing to assert.

> >> > That is, reasoners/parsers/etc receiving a file should
> >> >assume that a predicate is NOT an individual constant unless it is
> >> >used explicitly as such in that file.
> >>
> >> Well, that is a cackhanded way to say it. Rather, they should NOT
> >> assume that it IS in the domain of quantification;
> >
> >These amount to exactly the same thing in my book (ignoring the fact
> >that you've given the point a semantic rather than syntactic spin).  To
> >make the assumption in question does not mean it can't later be
> >overridden.
> 
> OK, but your way of saying it makes it sound like a default which 
> later needs to be overridden. 

Well, it isn't.

> Since there are (other) hordes out 
> there who want to use nonmonotonic logics which really do support 
> genuine assumption/retraction reasoning, I would prefer to avoid 
> language which they will read as referring to nonmonotonic rule 
> systems. 

Point taken, but I was talking to our little group, who know better.

> In any case, its not accurate: 

It's fine from the perspective of language definition.

> there isn't any assumption being made, and the parser doesn't need to
> default to anything.  

This is another of those stylistic difference thingies.  You are talking
about parsers processing stuff snagged higgledy-piggeldy off the web,
while I'm talking about the definition of a language.  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.  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.  This isn't to say that these relations will be
crystal clear on the web for any given lexicon.

> Its just that
> p(a) & not (exist (?y) ?y(a) )
> has some models which are ruled out when you add, say,
> R(p)
> The fact that a contradiction may arise when sentences are added 
> shouldn't be remarkable: sets of sentences are just like that, eg 
> consider adding (not p) to the set { p }

Well, I thought I made that point as well.

Sometimes I feel like this debate is proceeding a bit like this:

Menzel: "Smith is an utter moron."
Hayes:  "Nonsense.  Smith is a blithering idiot."
Menzel: "I beg to differ with you -- he's a moron."
Hayes:  "You're confused: Smith is a paradigm of idiocy."

:-)

> >> >> BTW, yet another tweak occurs to me. Having Rel in GOFOL as a
> >> >> predicate is illegal of course since there isnt anything there for
> >> >> it to be predicated of.?
> >> >
> >> >Oh but there is -- it is true of exactly those individuals that are
> >> >relations.
> >>
> >> In GOFOL there aren't any relations which are individuals, by
> >> definition.
> >
> >That is false.  The domain of quantification in an first-order
> >interpretation is any nonempty set of things.  Those things might be
> >relations, either extensional or intensional.
> 
> BUt the conventional MT for FOL assumes a segregated vocabulary, maps 
> relation symbols to relations which are *identified* as sets of 
> tuples of members of I, and quantifies over I; and is usually 
> interpreted relative to Z_F set theory which has the axiom of 
> foundation. 

Well, of course.  But that is neither here nor there.  Your claim was 
that the extension of "Rel" would be empty.  That is false.  It's
extension would be I intersect R.  The extension of "Ind" (better,
"Obj" or "Particular" or the like) would be I - R.

> So the things in the domain cannot be the relations named by the
> relation symbols, in such a picture. It is this fact plus the (false,
> but widely believed) idea that the axiom of foundation is a safety
> bulwark against paradox, which I strongly suspect is the reason for
> the passionate Horrocks/Patel-Schneider defense of conventional FOL
> semantics.

I suspect something like that might be true -- in addition to the fact 
that they don't want meanings of TFOL sentences to change in TFOL 
contexts.

> BTW, all this makes me think that it might be worth stating the MT in
> 'conventional' terms ( ie without an explicit extension mapping) but
> being explicit that we are using Aczel's set theory rather than Z_F:
> what do you think?

Surely wouldn't hurt to provide it as an alternative formalization.

-chris