You can pretty much always get what you want (if you geach sometime…)
One of the reasons why I abandoned the idea to have the domain restriction as part of the meaning of the quantifier/determiner was because I couldn’t see a way to restrict the x in [[the]]: λP[ιx[P(x)]]. Polly suggested that I try working with the raised version of [[the]], Montague’s et,ett type [[the]]. And I knew that I had thought about this problem before, tried to crack, came up against something tricky and decided to stick with have the domain restrictions being nominal restrictions.
It turns out that all we need is a (generalized) conjunction operator and our good friend geach. My N/RC shift is just this: λP[λQ[P ∏ Q]], where in an ordinary NP like the dog, P would be the set of dogs and Q some other restriction on individuals in that domain. I used ∏ and not ∩ in my rule because it was the easiest way to generalize that rule so that you could get functional domain restrictions as well (as in the woman who he loves who every man invited…is his mother, where the restriction is type ee,t). With this rule, you don’t even have to concern yourself with working with a higher-typed [[the]] (though it turns out that if you geach the type-lift operator and then apply it to e-typed [[the]], you get the higher-typed meaning for free). You just need to geach [[the]] twice and then apply it to the conjunction operator and you’ll get λP[λQ[ιx[P(x) & Q(x)]]]. It turns out that this is the same trick that will get you the domain restriction into quantifiers.
emma :: Aug.17.2007 :: syntax, semantics, logic, linguistics :: 6 Comments »
i have a couple thoughts on this:
i’m not sure deploying conjunction as a combinator, as you’ve done here, isn’t going to have too much generative power. for instance:
(1) x is a vole and scurries
(2) ?x is a vole scurries
so you might just say the is mapped to domain-restricting the, without committing yourself to a GC combinator.
but there’s something a little strange about domain-restricting the, namely that it has two mechanisms for picking up contextually salient individuals: ? and the domain restriction. perhaps then ?’s function is reduced to picking out the singleton member of the intersection of sets P and Q.
i have a couple thoughts on this:
i’m not sure deploying conjunction as a combinator, as you’ve done here, isn’t going to have too much generative power. for instance:
(1) x is a vole and scurries
(2) ?x is a vole scurries
so you might just say the is mapped to domain-restricting the, without committing yourself to a GC combinator.
but there’s something a little strange about domain-restricting the, namely that it has two mechanisms for picking up contextually salient individuals: ? and the domain restriction. perhaps then ?’s function is reduced to picking out the singleton member of the intersection of sets P and Q.
i have a couple thoughts on this:
i’m not sure deploying conjunction as a combinator, as you’ve done here, isn’t going to have too much generative power. for instance:
(1) x is a vole and scurries
(2) ?x is a vole scurries
so you might just say the is mapped to domain-restricting the, without committing yourself to a GC combinator.
but there’s something a little strange about domain-restricting the, namely that it has two mechanisms for picking up contextually salient individuals: iota and the domain restriction. perhaps then iota’s function is reduced to picking out the singleton member of the intersection of sets P and Q.
(you cant post iotas in your comments for some reason)
oh weird it gave me db errors and posted my comment anyway.. ah well, delete away
oh come on delete those
lol x 3