Re: SUO: Montague's Type System
On Tue, Feb 17, 2004 at 04:11:13PM -0800, Richard Cooper 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}.
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...
regards
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