[SCL] Re: Observation

[pat hayes]

>On Sun, Jun 01, 2003 at 09:14:51PM -0500, Pat Hayes wrote:
>>  >On Fri, May 30, 2003 at 10:54:32AM -0500, Pat Hayes wrote:
>>  >> [Menzel wrote:]
>>  >> >The point Ian seemed to be making against SCL was that there are
>>  >> >sentences in HIS familiar first-order language that suddenly change
>>  >> >their logical properties when you interpret THAT VERY LANGUAGE via
>>  >> >SCL model theory. Not so on our latest SCL model theory in which the
>>  >> >relation between R and I is unspecified.  The logical properties of
>>  >> >a traditional first-order SCL language are identical whether
>>  >> >interpreted a la Tarski or a la SCL.  What you *can't* do is start
>>  >> >with a gonzo type-free SCL language and expect that a superficially
>>  >> >traditional first-order sentence ripped from that context will have
>>  >> >the same logical properties when interpreted a la Tarski.
>>  >>
>>  >> Ah, but that is what Ian WOULD expect. The point is that all
>>  >> sentences one comes across are 'ripped' from their context in this
>>  >> way: there are no contexts on the Web. There are only sentences. Or,
>>  >> to put the same point differently, if it looks like first-order, then
>>  >> it IS first-order. Or to put the point differently again: it must be
>>  >> possible to recognize the intended context simply by parsing the
>>  >> sentences. A file of sentences must be interpretable in one definite
>>  >> way without any other information being made available that is not
>>  >> somehow encoded in the form of the expressions in that file.?
>>  >
>>  >Well, how about then that parsers default to the least amount of
>>  >type-freedom?
>>
>>  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. That property will now hold. 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.  And we arent saying that its meaning 
*changes* in a non-FOL context, after all.

>
>>  > 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. 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. In any case, its not accurate: there isn't any assumption 
being made, and the parser doesn't need to default to anything.  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 }

>
>>  >One could then only get a Horrocks sentence in an inconsistent set (I
>>  >think).
>>
>>  Right, but the same applies without making the special assumption.
>>
>>  > Thus, if you receive, say,
>>  >"(x)(Px <-> ~Qx) & (x,y)x=y", then you assume "P" and "Q" aren't also
>  > >individual constants and hence don't denote individuals.
>  >
>>  No, you just assume that you don't know that they do.
>
>Point taken, but this invocation of what you don't know has no real
>place in the definition of a language.
>
>>  They might: the next thing you see might be R(P), and you don't want
>>  that to contradict anything.
>
>Then you would simply retract the assumption in questin.

The are no assumptions that need to be retracted. One just has a 
model theory which applies to sets of sentences.

>
>>  > No Horrocks
>>  >problem.  The only way to force them to denote would be add sentences
>>  >to the file in which "P" and "Q" occur as arguments.
>>
>>  The only way to FORCE them, yes.
>>
>>  > But then the
>>  >resulting file would be provably inconsistent for pretty much
>>  >standard first-order reasons:  you'd be able to prove "P=Q" from
>>  >"(x,y)x=y" and "~P=Q" from "(x)(Px <-> ~Qx)" and the usual rules of
>>  >identity.  (Suppose "P=Q".  then by the substitutivity of identicals
>>  >we have "(x)(Px <-> ~Px)", which is easily shown to be contradictory
>>  >in FOL.)
>>  >
>>  >I think you were saying something like this on the phone the other
>>  >day, but I didn't grok it at the time.
>>
>>  I thought this was what YOU had been saying for the past week :-)
>
>Almost, but not quite.
>
>>  >> If this requires that the sublanguage in fact be a small
>>  >> subontology, as with the suggested use of Rel, then I reckon we
>>  >> ought to as far as possible absorb these little extensions in to
>>  >> the core language, so this ought to be scl:Rel and be given its
>>  >> semantic burden as part of the core model theory. That way, SCL has
>>  >> all the tools it requires to present its sublangauges to the world.
>>  >> I am pretty sure that we can do all the cases we are interested in
>>  >> with just a few of these: we decided a while back that Rel or
>>  >> something like it was going to be needed for GOFOL expressivity in
>>  >> any case, right?
>>  >>
>>  >> 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. 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.

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?

Pat
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