Type-shift
I had this revelation on Monday about sets and type-shifting, and even though it’s sort of something that I should have understood already, I was quite tickled and pleased with this mini epiphany.
It just dawned upon me that type-shifting is much more than a mechanical backdoor trick around some type-mismatch problems. Type-shifting takes a set and creates a new set describing the original set. So in the standard example of lifting an NP of type e to the generalized quantifier type et, t, not only is this a neat trick for getting around things like conjunction reduction, but it is actually shedding light onto what assumptions are made upon accepting the notion of a set; taking the singleton set and making a set of properties which hold of the member of the singleton set shows that whenever we make a set of objects, we are cutting across certain properties which may vary among individual members of a set and saying that for our purposes, these differences are irrelevant. What matters are the specific properties we wish to highlight, which all the members of the set share. Considering this, once you establish a certain set, you can of course make a set whose central property concerns its interaction with the initial set.
I’m not sure if I’m really expressing myself with the clarity and precision that I’d like, but as I get more comfortable with this idea, I’ll post a more coherent description of what it is that I’d finally realized this week.
It’s clear to me now that the job of the semanticist is not just to develop clever formal tools to describe the way grammar composes meaning, but also to recognize what the philosophical implications of adopting certain tools are. Previously, I’d just thought of these tools as wrenches and hammers; now I realize that they are also nails and screws. I’d taken for granted these logical operations as purely mechanical devices, but I’m beginning to see how they also have very deep philosophical roots.
emma :: Jan.31.2007 :: misc, semantics, philosophy :: 3 Comments »
Here’s something I was thinking about in semantics class today. I haven’t asked Elena about it, and it might just be stupid. But I’m going to take the comments form of this blog post as an opportunity to write it down and thus maybe sort it out in my head a bit.
We’re using the Heim & Kratzer textbook and the system described therein, so we say that the denotation of a verb like ’smokes’ is a function from individuals to truth values (returning 1 in this specific example iff the person smokes). We write something like the following:
[[smokes]] = lambda(x) . x smokes
While that may look illuminating, it’s really just a clever label for the function itself, which is e.g. the following set:
{(Mary, 1), (John, 0), (Lisa, 1)}
Now I understand that one of the prime assumptions of the whole theory is that we as native speakers don’t have to know the actual truth value of an utterance, but rather only the truth conditions — what would be required to make it true. So the fact that the function is that specific set in this specific world is immaterial; what matters is that we know how to judge whether any particular set would work out for the given utterance and state of affairs.
So then isn’t it sort of a contradiction to say that the denotation of ’smokes’ is the function described in lambda notation above? Because if that’s what we are saying, then we are also saying that the denotation of ’smokes’ is that set:
[[smokes]] = {(Mary, 1), (John, 0), (Lisa, 1)}
But we said that we don’t have to know (and in many cases *can’t* know) the actual composition of sets like this. So how can the set be the denotation of the word we’re using? If we’re to say that there is any sort of psychological reality to our compositional theory of function application, shouldn’t we actually be able to *know* (in some epistemological sense I apparently can’t quite articulate) the objects we are talking about?
Perhaps I’m misunderstanding the meaning of the term ‘denotation’.
Aaron, good question. I’ll post a reply attempting to clarify that issue soon :-D
[…] A comment on my earlier post on type-shift: Here’s something I was thinking about in semantics class today. I haven’t asked Elena about it, and it might just be stupid. But I’m going to take the comments form of this blog post as an opportunity to write it down and thus maybe sort it out in my head a bit. […]