RE: SUO: Montague's Type System
Pierluigi Miraglia wrote:
> > Anyone,
> >
> > I'm having trouble understanding a small passage
> > in the CoreLex thesis. It goes like this:
> >
> > ----------------------------
> > "(17) John and every (other) student went to her party.
> >
> > In Montague's type system (e for individual; t for proposition)
> > [every (other) student]NP is of type <<e,t>,t>, whereas [John]PN
> > is of type e. Although of different type they are coordinated
> > on the same level by the conjunction operator (functor) 'and'.
> > One can soleve this problem by shifting [John]PN from type e
> > to type <<e,t>,t> [Partee, 1987]. The semantical motivation
> > behind this is that 'John' can be interpreted as the set of
> > properties of 'John', which is exactly expressed by the
> > type <<e,t>,t>: a function from sets of properties (<e,t>) to
> > truth values (t). "
> > ---------------------------
> >
> > I understand that (and) takes two truth values, not an individual
> > and a proposition. What throws me is the meaning of "<<e,t>,t>"
> > as a type notation. Can anyone explain what words to use when
> > speaking "<<e,t>,t>" out loud?
> >
> > Thanks,
> > Rich
> >
>
>
> There is more than one way, but here is one that usually works for me:
> for any type S, you identify a set of objects of type S with its
> characteristic function -- a function from S to {0, 1}.
In programming languages like Delphi, the type name
is also used as a characteristic function. So
----------------------
type TStudent = Class(...);
var John : TObject; {or some descendent class thereof}
...
if TStudent(John)
then Enroll(John);
---------------------
So I interpret the characteristic function as sort of built
into the language. I guess the example above would be:
<John, t> is an individual (the var Alpha) of type TStudent.
<<John, t>, t> is the class TStudent, including all its
individual instances.
Is that a correct interpretation?
But wouldn't the class TStudent include John a second time?
So <<John, t> t> includes John twice, unless there is
some key that can indicate there is really only one John.
But if so, why use such a round about way of saying that
"every (other) student" is a class of the same class as John?
It seems like a difficult notation compared to modern
programming languages - maybe it historical paths are the
confusing part here.
> Since 't' as you note is the primitive type Truth-Value (or Boolean),
> you can think of anything of the form <S, t> as the type of all
> functions from S to t, and therefore the type (class) of all sets of
> objects of type S.
>
> So, given the primitives e and t:
>
> <e, t> = type of sets of objects of type e
> = type of sets of individuals
> = type of (extensions of) unary predicates (the
> denotations of N = "Noun")
>
> <<e, t>, t> = type of sets of objects of type <e, t>
> = type of sets of properties (= unary predicates)
> ([[John]] has this type because it is the set of all
> properties John has; [[every dog]] is of this type, too)
>
> <<e, t>, <<e, t>, t>>
> = type of functions from properties to sets of properties
> = type of quantifiers (the denotation of 'every', e.g., is
> \lambda P \lambda Q [\forall x Px \rightarrow Qx])
>
> etc.
>
> The 'and' of NP coordination denotes an operator different from the
> truth-functional, sentential 'and'.
>
> (The best explanation I know of all this is in Dowty-Wall-Peters.)
> Hope this serves, maybe somebody will be kind enough to correct my own
> misconceptions...
Thanks, I'll google up some Dowty-Wall-Peters references.
Rich
> regards
>
> --
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> Pierluigi Miraglia Cycorp, Inc.
> Ontologist 3721 Executive Center Dr.
> (512) 514-2988 Austin, TX 78731
>
> sr
>