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 ::
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